differential geometry, theory and applications (contemporary applied mathematics) (philippe g...

0 iffe re n t i a G e o met r y D hTheory a n d A p p ications

Philippt G C rlet Ta-Tsien L i

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0 if fe re n t i a I G e o 1 e t r y : Theory a n d A p p l i c a t i o n s

Series in Contemporary Applied Mathematics CAM

Honorary Editor: Chao-Hao Gu (Fudun University) Editors: P. G. Ciarlet (City University ofHong Kong),

Ta-Tsien Li (Fudan University)

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Mathematical Finance - Theory and Practice

New Advances in Computational Fluid Dynamics - Theory, Methods and Applications

Actuarial Science - Theory and Practice

Mathematical Problems in Environmental Science and Engineering

Ginzburg-Landau Vortices

Frontiers and Prospects of Contemporary Applied Mathmetics

Mathematical Methods for Surface and Subsurface H yd rosystems

(Eds. Deguan Wang, Christian Duquennoi, Alexandre Ern) Some Topics in Industrial and Applied Mathematics

(Eds. Rolf Jeltsch, Ta-Tsien Li, Ian Hugh Sloan) Differential Geometry: Theory and Applications

(Eds. Philippe G. Ciarlet, Ta-Tsien Li)

(Eds. Yong Jiongmin, Rama Cont)

(Eds. F. Dubois, Wu Huamo)

(Eds. Hanji Shang, Alain Tosseti)

(Eds. Alexandre Ern, Liu Weiping)

(Eds. HaYmBrezis, Ta-Tsien Li)

(Eds. Ta-Tsien Li, Pingwen Zhang)

Series in Contemporary Applied Mathematics CAM 9

0 if fe I e n t i a I G e o m e t ry :

editors

Philippe G Ciarlet City University of Hong Kong, China

Ta-Tsien Li Fudan University, China

Higher Education Press

xe World Scientific NEW J E R S E Y * LONDON SINGAPORE B E l J l N G * SHANGHAI * HONG KONG - TAIPEI CHENNAI

Theory and Applications

Philippe G. Ciarlet Ta-Tsien Li Department of Mathematics School of Mathematical Sciences City University of Hong Kong Fudan University 83 Tat Chee Avenue 220, Handan Road Kowloon, Hong Kong Shanghai, 200433 China China

Editorial Assistants: Zhou Chun-Lian

Copyright @ 2008 by Higher Education Press 4 Dewai Dajie, Beijing 100011, P. R. China, and World Scientific Publishing Co Pte Ltd 5 Toh Tuch Link, Singapore 596224

All rights reserved. No part of this book may be reproduced or transmittkd in any form or by any means, electronic or mechanical, including photocopying, recording or by

any information storage and retrieval system, without permission in writing from the

Publisher

ISBN 978-7-04-022283-8

Printed in P. R. China

Preface

The ISFMA-CIMPA School on “Differential Geometry: Theory and Applications” was held on 07 August ~ 18 August 2006, in the building of the Chinese-French Institute for Applied Mathematics (ISFMA), Fudan University, Shanghai, China. This school was jointly organized by the ISFMA and the CIMPA (International Centre for Pure and Applied Mathematics), Nice, France. About sixty participants from China, Hong Kong, France, Cambodia, India, Iran, Pakistan, Philippines, Romania, Russia, Sri-Lanka, Thailand, Turkey, Uzbekistan and Vietnam attended this highly successful event.

The first objective of this school was to lay down in a self-contained and accessible manner the basic notions of differential geometry, such as the metric tensor, the Riemann curvature tensor, the fundamental forms of a surface, covariant derivatives, and the fundamental theorem of surface theory etc. Although this field is with good reasons often considered as a “classical” one, it has been recently “rejuvenated”, thanks to the manifold applications where it plays an essential role.

The second objective of this school was to present some of these applications, such as the theory of linearly and nonlinearly elastic shells, the implementation of numerical methods for shells, and mesh generation in finite element methods.

To fulfill these objectives, four series of lectures, each series comprising ten 50min-lectures, were delivered under the following titles: “In- troduction to differential geometry” , “Introduction to shell theory” , “A differential geometry approach to mesh generation”, and “Numerical methods for shells”. This volume gathers the materials covered in these lectures. As such, this volume should be very useful to graduate students and researchers in pure and applied mathematics.

The organizers take pleasure in thanking the various organizations for their generous support: The ISFMA, the CIMPA, the French Embassy in Beijing, the Consulate General of France in Shanghai, the National Natural Science Foundation of China, Fudan University, Higher Educa- tion Press and World Scientific. Finally, our special thanks are due to Mrs. Zhou Chun-Lian for her patient and effective work in editing this book.

Philippe G. Ciarlet and Ta-Tsien Li February 2007

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Contents

Preface

Philippe G. Ciarlet: An Introduction to Differential Geometry in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Philzppe G. Ciarlet, Cristinel Mardare: An Introduction to Shell Theory ............................................... 94

Dominique Chapelle: Some New Results and Current Challenges in the Finite Element Analysis of Shells . . . . . . . . . . . 185

Pascal Frey: A Differential Geometry Approach to Mesh Generation ............................................. 222

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1

An Introduction to Differential Geometry in R3

Philippe G. Ciarlet Department of Mathematics, City University of Hong Kong

83 Tat Chee Avenue, Kowloon, Hong Kong, China E-mail: [email protected]

Introduction

These notes’ are intended to give a thorough introduction to the basic theorems of differential geometry in R3, with a special emphasis on those used in applications.

The treatment is essentially self-contained and proofs are complete. The prerequisites essentially consist in a working knowledge of basic notions of analysis and functional analysis, such as differential calculus, integration theory and Sobolev spaces, and some familiarity with ordinary and partial differential equations.

In Part 1, we review the basic notions, such as the metric tensor and covariant derivatives, arising when a three-dimensional open set is equipped with curvilinear coordinates. We then prove that the vanishing of the Riemann curvature tensor is sufficient for the existence of isometric immersions from a simply-connected open subset of E3 equipped with a Riemannian metric into a three-dimensional Euclidean space. We also prove the corresponding uniqueness theorem, also called rigidity theorem.

In Part 2, we study basic notions about surfaces, such as their two fundamental forms, the Gaussian curvature, and covariant derivatives. We then prove the fundamental theorem of surface theory, which asserts that the Gauss and Codazzi-Mainardi equations constitute sufficient conditions for two matrix fields defined in a simply-connected open subset of R2 to be the two fundamental forms of a surface in a three-dimensional Euclidean space. We also prove the corresponding rigidity theorem.

‘With the kind permission of Springer-Verlag, these notes are extracted and adapted from my book “An Introduction to Differential Geometry with Applica- tions to Elasticity”, Springer, Dordrecht, 2005, the writing of which was substantially supported by two grants from the Research Grants Council of Hong Kong Special Administrative Region, China [Project No. 9040869, CityU 100803 and Project No. 9040966, CityU 100604].

2 Philippe G. Ciarlet

1 Three-dimensional differential geometry

Outline

Let R be an open subset of R3, let E3 denote a three-dimensional Eu- clidean space, and let 0 : R 4 E3 be a smooth injective immersion. We begin by reviewing (Sections 1.1 to 1.3) basic definitions and properties arising when the three-dimensional open subset O(R) of E3 is equipped with the coordinates of the points of R as its curvilinear coordinates.

Of fundamental importance is the metric tensor of the set O(R), whose covariant and contravariant components gij = gji : R + R and 9'3 = gja : R + R are given by (Latin indices or exponents take their values in {I, 2,3}):

. .

j 9 " ' 23 - - gi . g j and gt3 = g' . g 3 , where gi = 8iO and gj . gi = hi.

The vector fields gi : R + R3 and gj : R + R3 respectively form the covariant, and contravariant, bases in the set O(R).

It is shown in particular how volumes, areas, and lengths, in the set O(R) are computed in terms of its curvilinear coordinates, by means of the functions gij and gij (Theorem 1.3-1).

We next introduce in Section 1.4 the fundamental notion of covariant derivatives vilij of a vector field wigi : R 4 R3 defined by means of its covariant components wi over the contravariant bases gi. Covari- ant derivatives constitute a generalization of the usual partial derivatives of vector fields defined by means of their Cartesian components. In particular, covariant derivatives naturally appear when a system of partial differential equations with a vector field as the unknown, e.g., the displacement field in elasticity, is expressed in terms of curvilinear coordinates.

It is a basic fact that the symmetric and positive-definite matrix field ( g i j ) defined on R in this fashion cannot be arbitrary. More specifically (Theorem 1.5-1), its components and some of their partial derivatives must satisfy necessary conditions that take the form of the following relations (meant to hold for all i , j , k , q E {1,2,3}): Let the functions rijq and be defined by

rijq = z (a jg iq+dig jq-dqgi j ) and l7$ = g p q r i j q , where (gpq) = (g i j ) - ' .

. . . .

1

Then, necessarily,

An Introduction to Differential Geometry in R3 3

The functions rij, and are the Christoffel symbols of the first, and second, kind and the functions

are the covariant components of the Riemann curvature tensor of the set 0 ( R ) .

We then focus our attention on the reciprocal questions: Given an open subset R of R3 and a smooth enough symmetric and

positive-definite matrix field (g i j ) defined on R, when is it the metric tensor field of an open set @(R) c E3, i.e., when does there exist an immersion 0 : R 4 E3 such that gi j = &O ' a j 0 in R?

If such an immersion exists, to what extent is it unique? As shown in Theorems 1.6-1 and 1.7-1, the answers turn out to be

remarkably simple to state (but not so simple to prove, especially the first one!): Under the assumption that R i s simply-connected, the necessary conditions

R q i j k = 0 in R

are also su f ic ien t for the existence of such an immersion 0. Besides, i f R is connected, this immersion i s unique up t o isometries

of E3. This means that, if 0 : R + E3 is any other smooth immersion

g . . $3 - - a i ~ . aj0 in R, there then exist a vector c E E3 and an orthogonal matrix Q of order three such that

satisfying -

~ ( x ) = c + Q ~ ( X ) for all 5 E R.

Together, the above existence and uniqueness theorems constitute an important special case of the fundamental theorem of Riemannian geometry and as such, constitute the core of Part 1.

We conclude this chapter by indicating in Section 1.8 that the equivalence class of 0, defined in this fashion modulo isometries of E3, depends continuously o n the matrix field ( g i j ) with respect to various topologies.

1.1 Curvilinear coordinates

To begin with, we list some notations and conventions that will be con- sistently used throughout.

All spaces, matrices, etc., considered here are real. Latin indices and exponents range in the set {1,2,3} , save when

otherwise indicated, e.g., when they are used for indexing sequences, and the summation convention with respect to repeated indices or exponents


is systematically used in conjunction with this rule. For instance, the relation

Si(X) = gij ( X ) d (XI

gi(x) = Cgij(x)gi(z) for i = 1 , 2 , 3 .

means that 3

j=1

Kronecker’s symbols are designated by dl, dij, or 6aj according to the context.

Let E3 denote a three-dimensional Euclidean space, let a. b and a A b denote the Euclidean inner product and exterior product of a, b E E3, and let la1 = f i denote the Euclidean norm of a E E3. The space E3 is endowed with an orthonormal basis consisting of three vectors il? = h Zi. Let 2i denote the Cartesian coordinates of a point 2 E E3 and let ai := a/a2ii.

In addition, let there be given a three-dimensional vector space in which three vectors ea = ei form a basis. This space will be identified with R3. Let xi denote the coordinates of a point x E EX3 and let 13i := a /dx i , aij := d2/axidxj, and aijk := a 3 / d ~ i d ~ j a ~ k .

Let there be given an open subset 6 of E3 and assume that there exist an open subset R of R3 and an injective mapping 0 : R -+ E3 such

Figure 1.1-1: Curvilinear coordinates and covariant bases in an open set 8 c E3. The thr? coordinates z1,x2,z3 of x E R are the curvilinear coordinates of P = O ( x ) E R. If the three vectors gi(z) = &O(z) are linearly independent, they form the covariant basis at 9 = O(z) and they are tangent to the coordinate lines passing through 9.

An Introduction to Differential Geometry in lit3 5

that O(R) = 6. Then each point 2 E fi can be unambiguously written as

and the three coordinates zi of 2 are called the curvilinear coordinates of 2 (Figure 1.1-1). Naturally, there are infinitely many ways of defining curvilinear coordinates in a given open set R, depending on how the open set R and the mapping 0 are chosen!

Examples of curvilinear coordinates include the well-known cylindrical and spherical coordinates (Figure 1.1-2).

h

2 = O(2), 2 E R,

h

b z

E3

Figure 1.1-2: Two familiar examples of curvilinear coordinates. Let the mapping 0 be defined by

0 : (cp, p, z ) E R 4 (p cos 'p, p sin 9, z) E E3. Then (p, p, z ) are the cylindrical coordinates of P = 0(cp, p, z) . Note that (p + 2kn, p, z) or (p + T + 2 k ~ , -p, z ) , k E Z, are also cylindrical coordinates of the same point 2 and that cp is not defined if Z is the origin of E3.

Let the mapping 0 be defined by 0 : ( c p , $ , ~ ) E C l -i (rcos$cosp,rcos$sinp,rsin$) E E3.

Then (9, $, T ) are the spherical coordinates of P = 0(p, $, T ) . Note that ( ( p + Z k ~ , $+ 2!7r, r ) or ('p + 2 h , $ + K + 2!~, - T ) are also spherical coordinates of the same point P and that cp and $ are not defined if 2 is the origin of E3.

In a different, but equally important, approach, an open subset R of B3 together with a mapping 0 : R 4 E3 are instead_ a priori given.

If 0 E Co(R;E3) and 0 is injective, the set R := O(R) is open by the invariance of domain theorem (for a proof, see, e.g., Nirenberg [1974, Corollary 2, p. 171 or Zeidler [1986, Section 16.4]), and curvilinear coordinates inside R are unambiguously defined in this case.

If 0 E C1(R; E3) and the three vectors &O(z) are linearly independent at all 2 E R, the set fi is again open (for a proof, see, e.g., Schwartz [1992] or Zeidler [1986, Section 16.4]), but curvilinear coordinates may be defined only locally in this case: Given z E R, all that can be asserted (by the local inversion theorem) is the existence of an open neighborhood

h


V of x in R such that the restriction of 0 to V is a C1-diffeomorphism, hence an injection, of V onto O(V) .

1.2 Metric tensor

Let R be an open subset of R3 and let

be a mapping that is diflerentiable at a point x E R. If SX is such that (x + Sx) E R, then

O(z + Sx) = O(x) + VO(x)bx + o(Sx), where the 3 x 3 matrix V O ( x ) and the column vector Sx are defined by

&@I &@I 8301

8 1 0 2 8 2 0 2 8 3 0 2 ) (x) and Sx = f::) . 8103 8 2 0 3 8 3 0 3 6x3

Let the three vectors gi(x) E R3 be defined by

i.e., gi(x) i s the i-th column vector of the matrix VO(x). Then the expansion of 0 about x may be also written as

O(x + Sx) = O(x) + 6xigi(.) + o(6x) If in particular Sx is of the form SX = Gtei, where bt E R and ei is

one of the basis vectors in R3, this relation reduces to

O(x + 6tei) = O ( x ) + 6tgi(x) + o(6t).

A mapping 0 : R -+ E3 is an immersion at x E R if it is differentiable at x and the matrix VO(x) is invertible or, equivalently, if the three vectors gi(x) = &O(x) are linearly independent.

Assume that the mapping 0 is an immersion at x. Then the three vectors gi(x) constitute the covariant basis at the point 2 = O(x) .

In this case, the last relation thus shows that each vector gi(x) i s tangent to the a-th coordinate line passing through 2 = O ( x ) , defined as the image by 0 of the points of R that lie on the line parallel to ei passing through x (there exist to and t l with to < 0 < t l such that the i-th coordinate line is given by t E ]to,tl[ -+ fi( t) := O ( x + tei) in a


neighborhood of 2; hence fl(0) = &O(x) = gi(x)); see Figures 1.1-1 and 1.1-2.

Returning to a general increment 6x = 6xiei, we also infer from the expansion of 0 about x that (recall that we use the summation convention) :

p ( x + 62) - @ ( % ) I 2 = 6xTVO(x)TVO(x)6x + o(16z12) = 6xig&) 'gj(x)6xj + o(16s12).

Note that, here and subsequentIy, we use standard notations from matrix algebra. For instance, 6xT stands for the transpose of the column vector 6x and V O ( Z ) ~ designates the transpose of the matrix VO(x), the element at the i-th row and j - th column of a matrix A is noted (A)ij, etc.

In other words, the principal part with respect to 6x of the length between the points O(x+6x) and O ( x ) is {Gxigi(x).gj(x)Sxj}1/2. This observation suggests to define a matrix (gij (x)) of order three, by letting

gij(x) := gz(x) . gj(x) = (vo(x)Tvo(x))ij.

The elements gij (x) of this symmetric matrix are called the covariant components of the metric tensor at 2 = O(x) .

Note that the matrix VO(x) is invertible and that the matrix (gij(z)) is positive definite, since the vectors gi(x) are assumed to be linearly independent.

The three vectors gi(x) being linearly independent, the nine relations

g2(x) . g&) = 6;

unambiguously define three linearly independent vectors gi(x). To see this, let a priori gi(x) = Xik(z)gk(x) in the relations gi(z) .gj(x) = 6;. This gives Xik(x)gkj(x) = 6;; consequently, Xik(x) = gik(x) , where

(gij(x)) := (gij(2))y.

Hence gi(x) = gik(x)gk(x). These relations in turn imply that

g i ( 4 . s w = (gik(z)g&)) . (gje(4ge(4) = gz"x)gje(x)gke(x) = gZk(2)6% = g"(x),

and thus the vectors gi(x) are linearly independent since the matrix (gij(z)) is positive definite. We would likewise establish that gi(x) =

The three vectors gi(x) form the contravariant basis at the point f = O ( x ) and the elements gij(x) of the symmetric positive definite

gij (4Sj (x).


matrix (g i j ( x ) ) are the contravariant components of the metric tensor at 2 = O ( x ) .

Let us record for convenience the fundamental relations that exist between the vectors of the covariant and contravariant bases and the covariant and contravariant components of the metric tensor at a point x E R where the mapping 0 is an immersion:

gi j (x ) = g i (x ) . g j ( x ) and gZj(x) = g i (x ) . g j ( x ) , g i ( x ) = gi j (x )g j (x ) and gi (x) = g i j ( x )g j ( x ) .

A mapping 0 : R -+ E3 is an immersion if it is an immersion at each point in R, i.e., if 0 is differentiable in R and the three vectors g i (x ) = & O ( x ) are linearly independent at each x E R.

If 0 : R -+ E3 is an immersion, the vector fields gi : R + R3 and gi : R -+ R3 respectively form the covariant, and contravariant bases.

To conclude this section, we briefly explain in what sense the components of the “metric tensor” may be “covariant” or “contrawariant”. - Let R and 6 be two domains in R3 and let 0 : R + E3 and 6 : R + E3 be two C’-diffeomorphisms such that O(R) = 6(6) and such that the vectors g i (x ) := & O ( x ) and Gi(Z) = of the covariant bases at the same point O(z) = 6(Z) E E3 are linearly independent. Let g i (x ) and $(Z) be the vectors of the corresponding contravariant bases at the same point 2. A simple computation then shows that

1 -- where x = ( x j ) := 0 o 0 E C1(R;6) (hence Z = ~ ( z ) ) and =

Let gij ( x ) and Tij (2) be the covariant components, and let gzj ( x ) and ?j(Z) be the contravariant components, of the metric tensor at the same point O(z) = 6(Z) E E3. Then a simple computation shows that

(T) := x-1 E C l ( 6 ; 0).

These formulas explain why the components gij(x) and $j (z ) are respectively called “covariant” and ‘kontravariant” : Each index in g i j ( x ) “varies like” that of the corresponding vector of the covariant basis under a change of curmilinear coordinates, while each exponent in gij (x) (‘varies like” that of the corresponding vector of the contravariant basis.

Remark. What is exactly the “second-order tensor” hidden behind its covariant components gij ( x ) or its contravariant exponents g i j ( z )

An Introduction to Differential Geometry in W3 9

is beautifully explained in the gentle introduction to tensors given by Antman [1995, Chapter 11, Sections 1 to 31; it is also shown in ibid. that the same “tensor” also has “mixed” components gj(x), which turn out

0

In fact, analogous justifications apply as well to the components of all the other “tensors” that will be introduced later on. Thus, for instance, the covariant components .{(.) and Gi(x), and the contravariant components d ( x ) and Gi(x) (both with self-explanatory notations), of a vector at the same point O ( x ) = 6 ( E ) satisfy (cf. Section 1.4)

to be simply the Kronecker symbols 6;.

?Ji(x)gZ(z) = Gz(.)g(.) = ?J~(x)g i ( . ) = 3(.)iji(.).

It is then easily verified that

In other words, the components w{(x) “vary like” the vectors gi(z) of the covariant basis under a change of curvilinear coordinates, while the components d ( x ) of a vector “vary like” the vectors g i (x ) of the contravariant basis. This is why they are respectively called %ovariant” and rrcontrauariant”. A vector is an example of a “first-order” tensor.

Note, however, that we shall no longer provide such commentaries in the sequel. We leave it instead to the reader to verify in each instance that any index or exponent appearing in a component of a “tensor” indeed behaves according to its nature.

The reader interested by such questions will find exhaustive treatments of tensor analysis, particularly as regards its relevance to elasticity, in Boothby [1975], Marsden & Hughes [1983, Chapter 11, or Simmonds [1994].

1.3 Volumes, areas, and lengths in curvilinear coordinates

We now review fundamental formulas showing hzw volume, area, and length elements at a point 2 = O ( x ) in the set R = O(R) can be expressed either in terms of the matrix VO(x), or in terms of the matrix

These formulas thus highlight the crucial r61e played by the matrix ( g i j ( x ) ) for computing “metric” notions at the point 2 = O(z). Indeed, the “metric tensor” well deserves its name!

A domain in Wd, d 2 2, is a bounded, open, and connected subset D of Wd with a Lipschitz-continuous boundary, the set D being locally on

( S i j ( X ) ) .


one side of its boundary. All relevant details needed here about domains are found in NeEas [1967] or Adams [1975].

Given a domain D c R3 with boundary r, we let dx denote the volume element in D , d r denote the area element along r, and n = nii? denote the unit (In1 = 1) outer normal vector along I? ( d r is well defined and n is defined dr-almost everywhere since r is assumed to be Lipschitz-continuous) .

Note also that the assumptions made on the mapping 0 in the next theorem guarantee that, if D is a domain in R3 such that D C R, then {6}- c R, {O(D)}- = O(D), and the boundaries 6’6 of 5 and dD of D are related by 86 = O(aD) (see, e.g., Ciarlet [1988, Theorem 1.2-8 and Example 1.71).

If A is a square matrix, C o f A denotes the cofactor matria: of A. Thus C o f A = (det A)A-T if A is invertible.

Theorem 1.3-1. Let R be an open subset of R3,-let 0 : R 4 E3 be an injective and smooth enough immersion, and let R = @(a).

(a) The volume element d f at f = O(x) E R is given in terms of the volume element dx at x E R by

-

df = I det VO(x) ldz = m d x , where g(x) := det(gij(x)).

(b) Let D be a domain in R3 such that c R. The area element df(f) at 3 = O(x) E 8 6 is given in terms of the area element dF(x) at x E d D by

d f ( f ) = I CofVO(x)n(x) ldr (x) = m d n i ( x ) g i j ( x ) n j ( a : ) d I ’ ( x ) ,

where n(x) := ni(x)ei denotes the unit outer normal vector at x E dD. (c) The length element d?(f) at f = O(x) E fi is given by

. 112 dT(f) = { S X ~ V O ( ~ ) ~ V O ( X ) S ~ } ~ ” = {6xigij(x)6xJ} ,

where Sx = 6xiei.

Proof. The relation d2 = )de tVO(x) l dx between the volume elements is well known. The second relation in (a) follows from the relation g(x) = I detVO(x)I2, which itself follows from the relation

Indications about the proof of the relation between the area elements d?(f) and dr(x) given in (b) are found in Ciarlet [1988, Theorem 1.7-11 (in this formula, n(x) = n,(x)e2 is identified with the column vector in R3 with n,(x) as its components). Using the relations Cof(AT) =

(gz3(x)) = v o ( x ) T v o ( z ) .

An Introduction to Differential Geometry in EX3 11

(CofA)T and Cof(AB) = (CofA)(CofB), we next have:

I CofVo(x)n(x)/2 = n(x )T Cof (vo (x )Tvo(x ) )n (x ) = g (x) ni (x)gij ( x )n j (x) .

Either expression of the length element given in (c) recalls that dF(5) is by definition the principal part with respect to 6x = Gxaei of the length l@(x + 62) - O ( x ) / , whose expression precisely led to the introduction

0

The relations found in Theorem 1.3-1 F e used in particular for computing volumes, areas, and lengths inside R by means of integrals insid: R , i.e., in terms of the curvilinear coordinates used in the open set R (Figure 1.3-1):

Let D be a domain in R3 such that 25 c 0, let 5 := O ( D ) , and let f ^ ~ L1(6) be given. Then

of the matrix ( g i j ( x ) ) in Section 1.2.

0

\ I R t

Figure 1.3-1: Volume, area, and length elements an curvilinear coordinates. The

elements dZ,dF(Z), and d@Z) at P = O(z) E fi are expressed in terms of dz, d f (z ) , and 6x at z E R by means of the covariant and contravariant components of the metric tensor; cf. Theorem 1.3-1. Assume that R is a domain and that 0 is a C1- diffeomorphism of R onto {a}-. Then, given a domain V such that V C R and a dr-measurable-subset A of 20, the corresponding relatio_ns are used for computing the volume of V = O(V) c R, the area of A = @ ( A ) c 8 2 , and the length of a curve (? = O(C) c {fi}-, where C = f ( I ) and I is a compact interval of R.


h

In particular, the volume of D is given by

h

volD := d 2 = m d x .

Next, lct r := do_, let CJe a dr-measurable subset of r, let 5 := O ( C ) c d D , and let h E L1(C) be given. Then

s,- h(2)df;(2) = L ( x o O)(x)md-dl?(x).

h

In particular, the area of C is given by

h

areaC := s, dF(2) = m d w d l ? ( x ) .

Finally, consider a curve C = f ( I ) in R, where I is a compact interval of R and f = f iei : I + R is a smooth-enough injective mapping. Then the length of the curve 6 := O(C) c R is given by

This relation shows in particular that the lengths of curves inside the open set O(R) are precisely those induced by the Euclidean metric of the space E3. For this reason, the set O(R) is said to be isometrically immersed in E3.

1.4 Covariant derivatives of a vector field

Suppose that a vector field is defined in an open subset 6 of E3 by means of its Cartesian component_s Ci : 6 + R, i.e., this field is defined by its values Gi(2)2? at each 2 E R, where the vectors Zi constitute the orthonormal basis of E3; see Figure 1.4-1.

Suppose now that the open set R is equipped with curvilinear coordinates from an open subset R of EX3, by means of an injective mapping 0 : R + E3 satisfying O(R) = 6.

How does one define appropriate components of the same vector field, but this time in terms of these curvilinear coordinates? It turns out that the proper way to do so consists in defining three functions ui : R -+ R by requiring that (Figure 1.4-2)

h

-


Figure 1.41: A vector field in Cartesian coordinates. At each point 3 E 6, the vector Gi(?@' is defined by its Cartesian components Gi(. ) over an orthonormal basis of E3 formed by three vectors Zi.

Figure 1.4-2: A vector field in curvilinear coordinates. Let there be given a vector field in Cartesian coordinates defined at each 3 E 6 by its Cartesian components &(Z) over the vectors 8" (Figure 1.4-1). In curvilinear coordinates, the same vector field is defined at each z E R by its covariant components vi(z) over the contravariant basis vectors gi(z) in such a way that vi(z)gi(z) = Gi(3)ei, 3 = O(z).

where the three vectors gi(x) form the contravariant basis at 2 = O(x) (Section 1.2). Using the relations gi(x) . g j ( z ) = 6; and 2 .2j = $, we immediately find how the old and new components are related, viz.,

The three components vi (x) are called the covariant components


of the vector v i ( x )g i ( x ) at 2, and the three functions vi : R + R defined in this fashion are called the covariant components of the vector field vigi : R + E3.

Suppose next that we wish to compute a partial derivative ajGi(2) at a point 2 = O ( x ) E R in terms of the partial derivatives &vk(z) and of the values vq(x) (which are also expected to appear by virtue of the chain rule). Such a task is required for example if we wish to write a system of partial differential equations whose unknown is a vector field (such as the equations of nonlinear or linearized elasticity) in terms of ad hoc curvilinear coordinates.

As we now show, carrying out such a transformation naturally leads to a fundamental notion, that of covariant derivatives of a vector field.

Theorem 1.4-1. Let R be an open subset of R3 and let 0 : R + E3 be a n injective immersion that is also a C2-diffeomorphism of R onto 6 := O(R). Given a vector field Gigi : fi + R3 in Cartesian coordinates with components Gi E C'(fi), let vigi : R + R3 be the same field in curvilinear coordinates, i.e., that defined by

h

h

h h -2 vi(x)e = v i ( x )gz (x ) for all 2 = O ( x ) , x E R.

Then vi E C'(R) and for all x E R, h

ajGi(2) = (vklle[gkli[gelj)(x), 2 = O ( x ) ,

where 21. . := 8 . v . - r ? .v and rp. := gp . aig j ,

2113 3 2 23 P 23

and [Si(2)]k := g y x ) . Zk

denotes the i - t h component of g i ( x ) over the basis { Z I , Z ~ , Z ~ } .

Proof. The following convention holds throughout this proof The simultaneous appearance of 2 and x in an equality means that they are related by 2 = O ( x ) and that the equality in question holds for all x E R.

(i) Another expression of [gi(x)]k := g i (x ) . Z k .

Let O ( x ) = Ok(x)Zk and 6(2) = @(2)ei, where 6 : 6 + E3 denotes the inverse mapping of 0 : R + E3. Since 6 ( O ( x ) ) = x for all x E R, the chain rule shows that the matrices VO(x) := (a jOk(x ) ) (the row index is k ) and ?6(2) := ( a k W ( 2 ) ) (the row index is i) satisfy

- h .

V6(2)VO(x) = I,

An Introduction to Differential Geometry in Iw3 15

or equivalently,

The components of the above column vector being precisely those of the vector g, (x), the components of the above row vector must be those of the vector gi(x) since gi(x) is uniquely defined for each exponent i by the three relations gi(x) .gj(x) = 6 j , j = 1,2 ,3 . Hence the Ic-th component of gi(x) over the basis {21,22,23} can be also expressed in terms of the inverse mapping 0, as:

h

h A .

[gi(x)]k = ako"2).

(ii) The functions := gq . &g, E Co(R).

We next compute the derivatives &gq(x) (the fields gq = gq'g' are of class C1 on R since 0 is assumed to be of class C2). These derivatives will be needed in (iii) for expressing the derivatives &Ci(2) as functions of LC (recall that Ci(2) = uk(z)[g'(x)]i). Recalling that the vectors gk(x) form a basis, we may write a priori

aesq(x) = -r&(4gk(4,

thereby unambiguously defining functions expressions in terms of the mappings 0 and 0, we observe that

: R + Iw. To find their h

r;k(x) = r;m(x)6r = r jm(z)gm(x) . gk(x) = -aegq(x) . gk(x). h h

Hence, noting that &(gq(x) . gk(x)) = 0 and [gq(x)lp = aP@(2), we obtain

r;k(x) = gq(x). aegk(x) = 5pSq(~)aekop(~) = rig(+ Since 0 E C2(R;E3) and 6% E C1(6;Iw3) by assumption, the last

relations show that r;k E Co(0). h

(iii) The partial derivatives of the Cartesian components of the vector field Giza E C'(6; EL3) are given at each 2 = 0 ( x ) E 6 by

5,Gi(2) = u k l l e ( 4 [&)li [ge(41j,


and [ g k ( z ) ] i and r&(z) are defined as in (i) and (ii). h

We compute the partial derivatives ajGi(2) as functions of z by means of the relation Zi(2) = v k ( z ) [ g k ( z ) ] i . To this end, we first note that a differentiable function w : R -+ R satisfies

& w ( 6 ( 2 ) ) = aew(z)@e(2) = a e w ( z ) [ g e ( z ) ] j ,

by the chain rule and by (i). In particular then,

5jGi (2) = 5j V k (6 (2)) [gk (z)] i + vq (z) 5j [gq (6 ($))I i

= aevk(z) [ge(z) ] j [Sk(") l i + v,(.)(ae[9q(z)li)[ge(z)lj = (aevk(z) - % ( Z ) % ( 4 ) [ g k ( ~ ) l i [ g e ( z ) l j ,

since &gq(z ) = -1 '&(z )gk(z ) by (ii). 0

The functions v. . = a.v. - rP.v 4 3 3 2 23 P

defined in Theorem 1.4-1 are called the first-order covariant derivatives of the vector field vigi : R -+ R3.

The functions = g p . a. zgj

are called the Christoffel symbols of the second kind (the Christoffel symbols of the first kind are introduced in the next section).

The following result summarizes properties of covariant derivatives and Christoffel symbols that are constantly used.

Theorem 1.4-2. Let the assumptions on the mapping 0 : R -+ E3 be as in Theorem 1.4-1, and let there be given a vector field vigi : fl -+ R3 with covariant components vi E C'(fl).

(a) The first-order cowariant derivatives villj E Co(R) of the vector field vigi : R -+ R3, which are defined by

villj := ajvi - I?$vp, where I 'P. .= gp . & g j , 2.7 .

can be also defined by the relations

a j (v ig i ) = vzl,jgi * VZllj = {aj(wkg"} ' g i .

(b) The Christoffel symbols I?;' := g p . d ig j = I?yi E Co(R) satisfy the relations

&gP = -rP.gj 23 and a j g , = I?jqgi.


Proof. It remains to verify that the covariant derivatives villj, defined in Theorem 1.4-1 by

may be equivalently defined by the relations

Villj = ajvi - qjVp,

a j (V ig2) = vzlljgi

These relations unambiguously define the functions vUillj = { a j ( V k g ' ) } ' g i since the vectors gz are linearly independent at all points of R by assumption. To this end, we simply note that, by definition, the Christoffel symbols satisfy &gp = - r : jg j (cf. part (ii) of the proof of Theorem 1.4-1); hence

a j (v ig i ) = ( a j V i ) g Z + Vidjg i = ( a j V i ) g i - V i r j k g ' = 21' ZIlJSi. '

To establish the other relations a jg , = r i q g i , we note that

o = ' 9 , ) = -rp.gi 3% . g P + g p . d j g , = -r;j + g p . a j g , .

Hence a jg , = m, . s P ) g p = r : jgp .

Remark. The Christoffel symbols can be also defined solely in terms of the components of the metric tensor; see the proof of Theorem 1.5-1.

If the affine space E3 is identified with R3 and O(z) := z for all z E R, the relation 8 j (v ig i ) ( z ) = (vii l jgi)(z) reduces to aj(Gi(2)z) =

(8jGi(2))Zi. In this sense, a covariant derivative of the first order constitutes a generalization of a partial derivative of the first order in Cartesian coordinates.

h

1.5 Necessary conditions satisfied by the metric tensor; the Riemann curvature tensor

It is remarkable that the components gij = gji : R 4 R of the metric tensor of a n open set O(R) c E3 (Section 1.2), defined by a smooth enough immersion 0 : R -+ E3, cannot be arbitrary functions.

As shown in the next theorem, they must satisfy relations that take the form:

d j r i k q - dkrijq + I'plCqp - r;krjqp = o in 0,


where the functions rijq and 17!j have simple expressions in terms of the functions gi j and of some of their partial derivatives (as shown in the next proof, it so happens that the functions I':j as defined in Theorem 1.5-1 coincide with the Christoffel symbols introduced in the previous section; this explains why they are denoted by the same symbol). Note that, according to the rule governing Latin indices and exponents, these relations are meant to hold for all i , j , k , q E {1,2,3}.

Theorem 1.5-1. Let R be an open set in EX3, let 0 E C3(R;E3) be a n immersion, and let

denote the covariant components of the metric tensor of the set @(R). Let the functions rijq E C1(R) and

g i j := aio . ajo

E C1(R) be defined by

1 rijq := -(a. . + 8. . - a . . )

rp. := g P T i j q where ( g p q ) := ( g i j ) - ' .

2 3% d 3 q 4Qz.9 7

Then, necessarily,

djrikq - akrijq + rzjrkqp - r;krjqp = o in 0.

Proof. Let gi = a,@. It is then immediately verified that the functions rij, are also given by

r . . - 8. t j q - d l j ' g q .

For each II: E R, let the three vectors gj(II:) be defined by the relations g j ( z ) . g i ( z ) = 6;. Since we also have g j = gz jg , , the last relations imply that I?:' = &gj . gp. Therefore,

aigi = r ; jgp

&rijq = dikgj - g q + aigj . dkgql

a ig j . dkgq = r : j g p . dkg , = r:'rkqP.

since aigj = (&gj . gP)g,. Differentiating the same relations yields

so that the above relations together give

Consequently, dikgj ' g q = a k r i j q - r : j r k q p .

Since a i k g j = &jglc, we also have

aikgj . g q = 8 j r i k q - r:krjqp,


and thus the required necessary conditions immediately follow. 0

Remark. The vectors g i and g j introduced above form the covariant and contravariant bases and the functions g i j are the contravariant

0

As shown in the above proof, the necessary conditions R q i j k = 0 thus simply constitute a re-writing of the relations a i k g j = a k i g j in the form of the equivalent relations d i k g j . g q = a k i g j . g q .

components of the metric tensor (Section 1.2).

The functions

and rP. = g p q r i j q = d i g j . g p = rP.

23 3%

are the Christoffel symbols of the first, and second, kinds. We saw in Section 1.4 that the Christoffel symbols of the second kind also naturally appear in a different context (that of covariant differentiation).

Finally, the functions

R w k . . :=a.r. 3 t k q -akr . . z3q f r : j r k q p - r f k r j q p

are the covariant components of the Riemann curvature tensor of the set 0 ( R ) . The relations R q i j k = 0 found in Theorem 1.4-1 thus express that the Riemann curvature tensor of the set 0 ( R ) (equipped with the metric tensor with covariant components g i j ) vanishes.

1.6 Existence of an immersion defined on an open set in R3 with a prescribed metric tensor

Let M3,S3, and S; denote the sets of all square matrices of order three, of all symmetric matrices of order three, and of all symmetric positive definite matrices of order three.

As in Section 1.2, the matrix representing the F’rkhet derivative at 2 E R of a differentiable mapping 0 = (el) : R --+ E3 is denoted

VO(Z) := (ajo((2)) E M3,

where l is the row index and j the column index (equivalently, VO(z) is the matrix of order three whose j-th column vector is a j 0 ) .

So far, we have considered that we are given an open set R c R3 and a smooth enough immersion 0 : R -+ E3, thus allowing us to define a matrix field

c = (gq ) = VOTVO : R + S3>,


where gz3 : R --t IK are the covariant components of the metrac tensor of the open set @(R) c E3.

We now turn to the recaprocal questaons: Given an open subset R of R3 and a smooth enough matrix field

C = (gt3) : R 4 S,3>, when is C the metric tensor field of an open set @(a) c E3? Equivalently, when does there exzst an zmmerszon 0 . R 4

E3 such that c = V O ~ V O in R,

or equavalently, such that

gz3 = a,@ . a,@ in R?

If such an immersion exists, to what extent is it unique? The answers are remarkably simple: If R as szmply-connected, the

necessary condataons

found in Theorem 1.7-1 are also suficient for the existence of such a n immersion. If R is connected, this immersion is unique up t o isometries an E3.

Whether the immersion found in this fashion is globally injective is a different issue, which accordingly should be resolved by different means.

This result comprises two essentially distinct parts, a global existence result (Theorem 1.6-1) and a uniqueness result (Theorem 1.7-1). Note that these two results are established under different assumptions, on both the set R and the smoothness of the field (g i j ) .

In order to put these results in a wider perspective, let us make a brief incursion into Riemannian Geometry. For detailed treatments, see classic texts such as Choquet-Bruhat, de Witt-Morette & Dillard-Bleick [1977], Marsden & Hughes [1983], Berger [2003], or Gallot, Hulin & Lafontaine [2004].

Considered as a three-dimensional manifold, an open set R c R3 equipped with an immersion 0 : R + E3 becomes an example of a Riemannian manifold (0; ( g i j ) ) , i.e., a manifold, viz., the set R, equipped with a Riemannian metric, viz., the symmetric positive-definite matrix field (g i j ) : R ---f St defined in this case by gij := &@ .d j@ in R. More generally, a Riemannian metric on a manifold is a twice covariant, symmetric, positive-definite tensor field acting on vectors in the tangent spaces to the manifold (these tangent spaces coincide with R3 in the present instance).

This particular Riemannian manifold (R; (gij ) ) possesses the remarkable property that i ts metric is the same as that of the surrounding space E3. More specifically, (a; ( g i j ) ) is isometrically immersed in the Eu- clidean space E3, in the sense that there exists an immersion 0 : R + E3


that satisfies the relations gij = &O . aj0. Equivalently, the length of any curve in the Riemannian manifold (R; ( g i j ) ) is the same as the length of its image by 0 in the Euclidean space E3 (see Theorem 1.3-1).

The first question above can thus be rephrased as follows: Given a n open subset R of R3 and a positive-definite matrix field (g i j ) : R -+ s,3>, when is the Riemannian manifold (R; (g i j ) ) flat, in the sense that it can be isometrically immersed in a Euclidean space of the same dimension (three)?

The answer to this question can then be rephrased as follows (compare with the statement of Theorem 1.6-1 below): Let R be a simply- connected open subset of R3. Then a Riemannian manifold (0; ( g i j ) ) with a Riemannian metric ( g i j ) of class C2 in R i s f la t i f and only if i ts Riemannian curvature tensor vanishes in R. Recast as such, this result becomes a special case of the fundamental theorem on flat Riemannian manifolds, which holds for a general finite-dimensional Riemannian manifold.

The answer to the second question, viz., the issue of uniqueness, can be rephrased as follows (compare with the statement of Theorem 1.7-1 in the next section): Let R be a connected open subset of R3. Then the isometric immersions of a f la t Riemannian manifold (R; ( g i j ) ) into a Euclidean space E3 are unique up to isometries of E3. Recast as such, this result likewise becomes a special case of the so-called rigidity theorem; cf. Section 1.7.

Recast as such, these two theorems together constitute a special case (that where the dimensions of the manifold and of the Euclidean space are both equal to three) of the fundamental theorem of Riemannian Geometry. This theorem addresses the same existence and uniqueness questions in the more general setting where R is replaced by a p-dimensional manifold and E3 is replaced by a (p + q)-dimensional Eu- clidean space (the “fundamental theorem of surface theory”, established in Sections 2.8 and 2.9, constitutes another important special case). When the p-dimensional manifold is an open subset of E%P+q, an outline of a self-contained proof is given in Szopos [2005].

Another fascinating question (which will not be addressed here) is the following: Given again an open subset R of EX3 equipped with a symmetric, positive-definite matrix field (g i j ) : R -+ S,3>, assume this time that the Riemannian manifold (0; ( g i j ) ) is n o longer flat, i.e., its Riemannian curvature tensor no longer vanishes in R. Can such a Riemannian manifold still be isometrically immersed, but this t ime in a higher-dimensional Euclidean space? Equivalently, does there exist a Euclidean space Ed with d > 3 and does there exist an immersion 0 : R + Ed such that gij = &O .ajO in R?

The answer is yes, according to the following beautiful Nash theorem, so named after Nash [1954]: A n y p-dimensional Riemannian man-


ifold equipped with a continuous metric can be isometrically immersed in a Euclidean space of dimension 2p with a n immersion of class C1; it can also be isometrically immersed in a Euclidean space of dimension (2p + 1) with a globally injective immersion of class C1.

Let us now humbly return to the question of existence raised at the beginning of this section, i.e., when the manifold is an open set in R3.

Theorem 1.6-1. Let R be a connected and simply-connected open set in R3 and let C = (g i j ) E C2(R; S;) be a matrix field that satisfies

R q i j k := djrikq - a k r i j q + r;’rkqp - r;krjqp = 0 in 0,

where 1

239 ’- 2 3Q2q + a&,, - a n % ) ,

rp. Y := p r i j q with ($9 ) := (g i j ) - l .

r . . .- -(a. .

Then there exists an immersion 0 E C3(R;E3) such that

c = V O ~ V O in R.

Pro05 The proof relies on a simple, yet crucial, observation. When a smooth enough immersion 0 = (Oe) : R + E3 is a priori given (as it was so far), its components Oe satisfy the relations &jOe = ryjapOe7 which are nothing but another way of writing the relations &gj = ryjgp (see the proof of Theorem 1.5-1). This observation thus suggests to begin by solving (see part (ii)) the system of partial differential equations

&Fej = rP.Fep in R,

whose solutions Fej : R -+ R then constitute natural candidates for the partial derivatives ajoe of the unknown immersion 0 = (Be) : R 4 E3 (see part (iii)).

To begin with, we establish in (i) relations that will in turn allow us to re-write the sufficient conditions

23

d j r i k q - a k r i j q + rfjrkqP - rykrjqp = o in

in a slightly different form, more appropriate for the existence result of part (ii). Note that the positive definiteness of the symmetric matrices (g i j ) is not needed for this purpose.

(i) Let R be a n open subset 0fR3 and let there be given a field ( g i j ) E C2(R; S3) of symmetric invertible matrices. The functions rijq, I?;’, and gpq being defined by

1 rijq := p j g i q + aigjq - aqgij), rTj := g p q r . 23q7 ( S P 4 ) := (gijl-l,

An Introduction to Differential Geometry in IR3 23

define the functions

R q i j k := ajrikq - akrijq + r:jrkqp - rTkrjqP, R& := a j q k - akr$ + rfkrye - rf jqe.

Then Rp.. '23 k = gPqRqijk and R p i j k = gpqR:jk.

Using the relations

rjqe t rejq = ajgqe and r i k q = gqtr fk ,

which themselves follow from the definitions of the functions rijq and

(gPqajgqe + gqeajgPq) = aj (gPqgqe) = 0 ,

and noting that

we obtain

gPq(ajrikq - rLkrjqT) = ajrYk - rikqajgpq - re tkg Pq(ajg,e - rejq)

= a j q k t F f k r y Q - rfk(gPqajgqe t gqeajgPq)

= a j q k + &rye. Likewise,

gpq(dkri jq - rLjrkqT) = akrYj - rfjrie, and thus the relations R:jk = gPqRqijk are established. The relations R . . q23k - - gpqRpijk are clearly equivalent to these ones.

We next establish the existence of solutions to the system

ai Fej = I?:' Fep in R.

(ii) Let R be a connected and simply-connected open subset of IR3 and let there be given functions = E C'(R) satisfying the relations

a j q k - akrfj + rfkr;e - rfjrge = o in R,

ajrikq - akrijq + r$rkqp - rTkrjqP = o in R,

which are equivalent t o the relations

by part (i).

exists one, and only one, field (Fej) E C2(R; M3) that satisfies Let a point xo E R and a matrix (F&) E M3 be given. Then there

& F Q ~ ( x ) = I'Fj(x)Fep(x), x E 0, Fej(x0) = Fej. 0


Let x1 be an arbitrary point in the set R , distinct f rom xo. Since R is connected, there exists a path y = (ri) E C1([O, l] ;R3) joining xo to x1 in R; this means that

y(0) = xo, $1) = xl, and y ( t ) E R for all 0 < t 6 1.

Assume that a matrix field (Fej) E C1(R;M3) satisfies &Fej(x) = ryj(x)Fep(x), x E 0. Then, f o r each integer C E {1,2,3}, the three functions c j E C1([O, 11) defined by (for simplicity, the dependence on C is dropped)

C j ( t ) := Fej(r(t)), 0 6 t 6 1,

satisfy the following Cauchy problem for a Linear system of three ordinary differential equations with respect to three unknowns:

em = c;, where the initial values are given by

co 3 := F&.

Note in passing that the three Cauchy problems obtained by letting

It is well known that a Cauchy problem of the form (with self- C = 1,2, or 3 only differ by their initial values <:.

explanatory notations)

has one and only one solution C E C1([0,1];R3) if A E Co([0,1];M3) (see, e.g., Schwartz [1992, Theorem 4.3.1, p. 3881). Hence each one of the three Cauchy problems has one and only one solution.

Incidentally, this result already shows that, i f it exists, the unknown field (Fgj) is unique.

In order that the three values cj (1) found by solving the above Cauchy problem for a given integer C E {1,2,3} be acceptable candidates for the three unknown values Fej (XI), they must be of course independent of the path chosen for joining xo t o xl.

So, let yo E C1([O, l];R3) and y1 E C1([O, 1];R3) be two paths joining xo to x1 in R. The open set R being simply-connected, there exists a homotopy G = (Gi) : [0,1] x [0,1] -+ R3 joining yo to y1 in R, i.e., such that

G(.,O) = yo, G(., 1) = yl, G(t, A) E R for all 0 6 t < 1, 0 < A 6 1, G(0, A) = xo and G ( l , A) = x1 for all 0 6 X < 1,


and smooth enough in the sense that

G E C1([O, 11 x [0, 1];R3) and

Let <(.,A) = (cj(.,X)) E C'([0,1];R3) denote for each 0 6 X < 1 the solution of the Cauchy problem corresponding to the path G(.,X) joining 5' to d. We thus have

853 dGi -(t , A) = I':j(G(t, A))-(& X)&(t, A) for all 0 < t < 1, 0 < X < 1, aL at

<j(O, A) = sj" for all o 6 X Q 1.

Our objective is to show that

as, -(I, A) = 0 for all 0 < X < 1, dX

as this relation will imply that [ j ( 1 , O ) = (j (1, l), as desired. For this purpose, a direct differentiation shows that, for all 0 ,< t 6 1, 0 ,< X 1,

where

on the one hand (in the relations above and below, for r$(G(. , .)),akr:j(G(., .)), etc.).

the functions aj shows that, for all 0 < t < 1,0 < X < 1,

dkr:j, etc., stand

On the other hand, a direct differentiation of the equation defining

ac. dGi But -2 = so that we also have

at at

Hence, subtracting the above relations and noting that -

(G) and d~ (t) =

d a[. 13 dGi d dGi (x) by assumption, we infer that


The assumed symmetries = r;i combined with the assumed relations djI’yk - d,$$ + &rye - rfjrg, = 0 in R show that

a p kj - + rg,r% - r;jr;q = 0,

on the one hand. On the other hand,

since $(O, A) = Cj” and G(0, A) = zo for all 0 < X 6 1. Therefore, for any fixed value of the parameter X E [0,1], each function oj (., A) satisfies a Cauchy problem for an ordinary differential equation, viz.,

doj dGi -(t,X) =r:j(G(t,X))-(t,X)nq(t,X), at 0 < t 6 1, dt

q ( 0 , A) = 0.

But the solution of such a Cauchy problem is unique; hence aj (t, A) = 0 for all 0 6 t 6 1. In particular then,

= 0 for all 0 6 X 6 1,

and thus %( l ,X ) = 0 for all 0 6 X 6 1, since G(1,X) = z1 for all dX o < x < 1 .

(Fej) : R -+ R3 by letting For each integer C, we may thus unambiguously define a vector field

F e j ( d ) := Cj(1) for any z1 E R,

where 7 E C1([O, 11; R3) is any path joining zo to z1 in R and the vector field ( C j ) E C1([O, 11) is the solution to the Cauchy problem

Cj(0) = Cj”, corresponding to such a path.

To establish that such a vector field is indeed the C-th row-vector field of the unknown matrix field we are seeking, we need to show that (Fej)&I E C1(R; R3) and that this field does satisfy the partial differential equations &Fej = TpjFep in R corresponding to the fixed integer C used in the above Cauchy problem.


Let x be an arbitrary point in R and let the integer i E {1,2,3} be fixed in what follows. Then there exist x1 E R, a path y E C1([O, l];R3) joining xo to xl, T E ]0,1[, and an open interval I c [ O , l ] containing T

such that y( t ) = x + (t - 7)ei for t E I ,

where ei is the i-th basis vector in R3. Since each function <j is contin-

uously differentiable in [0,1] and satisfies " ( t ) = I?$(y(t))-(t)&(t) for all 0 < t < 1, we have

d<. dyz at d t

dSj <j(t) = <j((.) + (t - T) - (T ) + O ( t - 'T)

dt = <j.(T) + (t - T)r:j ( Y ( T ) ) < p ( T ) f o(t - T )

for all t E I . Equivalently,

Fej(x + (t - .)ei) = Ft j (x ) + (t - T)I':j(x)Fep(x) + o(t - x).

This relation shows that each function Fej possesses partial derivatives in the set 0, given at each x E R by

d i ~ e p ( ~ ) = r: j (x)Fep(x) .

Consequently, the matrix field (Fej) is of class C1 in R (its partial derivatives are continuous in Q) and it satisfies the partial differential equations 8iFe.j = I'$Fep in R, as desired. Differentiating these equations shows that the matrix field (FQ) is in fact of class C2 in 0.

In order to conclude the proof of the theorem, it remains to ade- quately choose the initial values F& at xo in part (ii).

(iii) Let R be a connected and simply-connected open subset of R3 and let ( g i j ) E C2(Q; S;) be a matr ix field satisfying

djrikq - dkri jq + r;jrkqp - r:Jjqp = o in R,

the func t ions rijq, I?:', and gpq being defined by

1 2 rijq := -(ajg,, + aigj, - aqgi j ) , ryj := p r i j q , ( p ) := ( g i j ) - l .

Given a n arbitrary point xo E R, let ( F g ) E S; denote the square

Le t (Fej) E C2(R;M3) denote the solution t o the corresponding sys- root of the matr ix ( g $ ) := (gij (xo)) E S;.

t e m

& F e j ( ~ ) = r:j(x)Fep(x), 2 E R, Fej(xo) = F&,


which exists and is unique by parts (i) and (ii). Then there exists an immersion 0 = (el) E C3(R; E3) such that

8 j@e = Fej and gij = &0 . aj0 in R.

To begin with, we show that the three vector fields defined by

gj := (Fej);=l E C2(R;R3)

satisfy 9.. g . = g” in R. a 3 23

To this end, we note that, by construction, these fields satisfy

&gj = q j g P in R,

gj (xO 1 = gj”,

where g: is the j- th column vector of the matrix (F&) E S53>. Hence the matrix field (gi . g j ) E C2(R; M3) satisfies

&(gz . S j ) = r;(g, . gi) + rK(g, . Sj) in a, (Si ’ g j 1 (xO) = go. 23 ’

The definitions of the functions Fij, and r’$ imply that

d k g i j = r i k j + r j k i and rijq = gPqF$

Hence the matrix field ( g i j ) E C2(R; S?) satisfies

d k g i j = r$grni + r Z g r n j in R , 0 0

Sij(Z ) = 9 . . 23 .

Viewed as a system of partial differential equations, together with initial values at xo, with respect to the matrix field ( g i j ) : R -+ M3, the above system can have at most one solution in the space C2(R; M3).

To see this, let x1 E R be distinct from xo and let y E C1([O, 11; R3) be any path joining xo to x1 in R, as in part (ii). Then the nine functions gi j (T( t ) ) , 0 < t < 1, satisfy a Cauchy problem for a linear system of nine ordinary differential equations and this system has at most one solution.

An inspection of the two above systems therefore shows that their solutions are identical, i.e., that gi ’ gj = gi j .

It remains to show that there exists a n immersion 0 E C3(R;E3) such that

&0 = gi in R,

An Introduction to Differential Geometry in JR3 29

Since the functions r$ satisfy = I':i, any solution (Fej) E C2(O; MI3) of the system

&Fej(z) = ry j ( z )Fep(z ) , z E R, Fej(xo) = Fe"j

satisfies &Fej = aj Fei in R.

The open set R being simply-connected, Poincare"s lemma (for a proof, see, e.g., Flanders [1989], Schwartz [1992, Vol. 2, Theorem 59 and Corollary 1, p. 234-2351, or Spivak [1999]) shows that, for each integer l , there exists a function Oe E C3((n) such that

ai@e = Fei in R,

or, equivalently, such that the mapping 0 := (el) E C3(R; E3) satisfies

&O = gi in R

That 0 is an immersion follows from the assumed invertibility of the 0 matrices ( g i j ) . The proof is thus complete.

Remarks. (1) The assumptions

made in part (ii) on the functions I'rj = I'yi are thus suficient conditions for the equations &Fej = r$Fep in R to have solutions. Conversely, a simple computation shows that they are also necessary conditions, simply expressing that, if these equations have a solution, then necessarily dikFej = &Fej in a.

It is no surprise that these necessary conditions are of the same nature as those of Theorem 1.5-1, viz., &gj = &jgk in R.

( 2 ) The assumed positive definiteness of the matrices ( g i j ) is used

The definitions of the functions r$ and rij, imply that the functions

only in part (iii), for defining ad hoc initial vectors gp.

satisfy, for all i, j , k , p ,


These relations in turn imply that the eighty-one suficient conditions

R q i j k = O in R for all i , j , k , q E {1,2,3},

are satisfied if and only if the six relations

R 1 2 1 2 = R 1 2 1 3 = R 1 2 2 3 = R 1 3 1 3 = 121323 = R 2 3 2 3 = 0 in fl

are satisfied (as is immediately verified, there are other sets of six relations that will suffice as well, again owing to the relations satisfied by the functions R q i j k for all i, j , k , 4) .

To conclude, we briefly review various extensions of the fundamental existence result of Theorem 1.6-1. First, a quick look at its proof reveals that it holds verbatim in any dimension d 3 2, i.e., with R3 replaced by Rd and E3 by a d-dimensional Euclidean space Ed. This extension only demands that Latin indices and exponents now range in the set {1,2,. . . , d } and that the sets of matrices MI3, S3 , and S; be replaced by their d-dimensional counterparts Md, S d , and Sd>.

The regularity assumption on the components gi j of the symmetric positive definite matrix field C = (gij) made in Theorem 1.6-1, viz., that gi j E C2(R), can be significantly weakened. More specifically, C. Mardare [2003] has shown that the existence theorem still holds if gij E C’(fl), with a resulting mapping 0 in the space C2(R;Ed). Then S. Mardare [2004] has shown that the existence theorem still holds if gij E WLCm(R), with a resulting mapping 0 in the space W$‘T((R; Ed) . As expected, the sufficient conditions R q i j k = 0 in of Theorem 1.6-1 are then assumed to hold only in the sense of distributions, viz., as

for all p E D(R). The existence result has also been extended “up to the boundary of the

set R” by Ciarlet & C. Mardare [2004b]. More specifically, assume that the set R satisfies the “geodesic property” (in effect, a mild smoothness assumption on the boundary dR, satisfied in particular if dR is Lipschitz- continuous) and that the functions gij and their partial derivatives of order 6 2 can be extended by continuity to the closure a, the symmetric matrix field extended in this fashion remaining positive-definite over the set a. Then the immersion 0 and its partial derivatives of order 6 3 can be also extended by continuity to 2.

Ciarlet & C. Mardare [2004a] have also shown that, if in addition the geodesic distance is equivalent to the Euclidean distance on R (a property stronger than the “geodesic property”, but again satisfied if the boundary dR is Lipschitz-continuous), then a matrix field (gij) E C 2 ( a ; S y )


with a Riemann curvatu_re tensor vanishing in R can be extended to a matrix field (&) E C2(R;S,n>) defined on a connected open set 6 containing and whose Riemann curvature tensor still vanishes in 6. This result relies on the existence of continuous extensions to a of the immersion 0 and its partial derivatives of order 6 3 and on a deep extension theorem of Whitney [1934].

1.7 Uniqueness up to isometries of immersions with the same metric tensor

In Section 1.6, we have established the existence of an immersion 0 : 52 c W3 -+ E3 giving rise to a set @(a) with a prescribed metric tensor, provided the given metric tensor field satisfies ad hoc sufficient conditions. We now turn to the question of uniqueness of such immersions.

This uniqueness result is the object of the next theorem, aptly called a rigidity theorem in view of its geometrical interpretation: It asserts that, if two immersions 6 E C1(R;E3) and 0 E C1(R;E3) share the same metric tensor field, then the set O(52) is obtained by subjecting the set 6 ( R ) either to a rotation (represented by an orthogonal matrix Q with det Q = l), or to a symmetry with respect to a plane followed by a rotation (together represented by an orthogonal matrix Q with det Q = -l), then by subjecting the rotated set to a translation (represented by a vector c) .

The terminology “rigidity theorem” reflects that such a geometric transformation indeed corresponds to the idea of a “rigid transformation” of the set 0(52) (provided a symmetry is included in this definition).

Let O3 denote the set of all orthogonal matrices of order three.

Theorem 1.7-1. Let 52 be a connected open subset of R3 and let 0 E C1(R; E3) and 6 E C1(R; E3) be two immersions such that their associated metric tensors satisfy

VOTVO = ~ 6 ~ ~ 6 in R.

Then there exist a vector c E E3 and an orthogonal matrix Q 6 O3 such that

o(IL:) = c + Q ~ ( x ) for all IL: E R.

Proof. Fhe three-dimensional vector space R3 is identified throughout this proof with the Euclidean space E3. In particular then, R3 inherits the inner product and norm of E3. The spectral norm of a matrix A E M3 is denoted

(A( := sup{lAb(; b E R3, (b( = 1).


To begin with, we consider the special case where 6 : R -+ E3 = R3 is the ident i ty mapping. The issue of uniqueness reduces in this case to finding 0 E C1(R; E3) such that

V O ( ~ ) ~ V O ( ~ ) = I for all x E R.

Parts (i) to (iii) are devoted to solving these equations.

(i) We first establish that a mapping 0 E C1(R; E3) that satisfies

V O ( ~ ) ~ V O ( ~ ) = I for all x E R

i s locally a n isometry: Given a n y point xo E 0, there exists a n open neighborhood V of xo contained in R such that

IO(y) - O(x)I = Iy - xi for all z, y E V

Let B be an open ball centered at xo and contained in R. Since the set B is convex, the mean-value theorem (for a proof, see, e.g., Schwartz [1992]) can be applied. It shows that

I O ( y ) - O(x)I < sup IVO(z)l/y - z/ for all z,y E B. zElx>Yl

Since the spectral norm of an orthogonal matrix is one, we thus have

lO(y) - O(x)1 < ly - xi for all z,y E B.

Since the matrix VO(zo) is invertible, the local inversion theorem (for a proof, see, e.g., Schwartz [1992]) shows that there exist an open neighborhood V of xo contained in R and an open neighborhood p of O ( x o ) in E3 such that the restriction of 0 to V is a C1-diffeomorphism from V onto p. Besides, thereis no loss of generality in assuming that V is contained in B and that V is convex (to see this, apply the local inversion theorem first to the restriction of 0 to B , thus producing a "first" neighborhood V' of zo contained in B , then to the restriction of the inverse mapping obtained in this fashion to an open ball V centered at O ( x o ) and contained in O(V')) .

Let 0-l : p + V denote the inverse mapping of 0 : V + p. The chain rule applied to the relation O-'(O(z)) = x for all z E V then shows that

VO-'(Z) = V O ( ~ ) - ' for all z = x E V.

The matrix VO-'(Z) being thus orthogonal for all 2 E p, the mean- value theorem applied in the convex set shows that

h le-'(Q) - W'(Z)I < 1 ~ - 21 for all Z , Q E V,


or equivalently, that

( y - z( < ( O ( y ) - O(z)( for all z, y E V.

The restriction of the mapping 0 to the open neighborhood V of zo is thus an isometry.

(ii) We next establish that, if a mapping 0 E C1(R;E3) is locally a n isometry, in the sense that, given any xo E R, there exists an open neighborhood V of xo contained in R such that I O ( y ) - O(z)( = Iy - Z I for all x , y E V , then i ts derivative i s locally constant, in the sense that

VO(Z) = VO(Z') for all z E K

The set V being that found in (i), let the differentiable function F : V x V + R be defined for all x = ( z p ) E V and all y = ( y p ) E V by

F(X,Y) := ( W Y ) - @e(x) ) (@e(y ) - @e(z)) - (Ye - Q)(Y! - ze)

Then F ( z , y ) = 0 for all z, y E V by (i). Hence

for all z, y E V . For a fixed y E V, each function Gi(., y ) : V + JR is differentiable and its derivative vanishes. Consequently,

dGa d@e dOe -(z, y ) = --(y)-(z) + 6ij = 0 for all z, y E V, dXi dya dzj

or equivalently, in matrix form,

Letting y = xo in this relation shows that

VO(z) = VO(X') for all z E V.

(iii) By (ii), the mapping VO : R + M3 is differentiable and its derivative vanishes in R. Therefore the mapping 0 : R + E3 is twice differentiable and its second Fre'chet derivative vanishes an a. The open set R being connected, a classical result from differential calculus (see, e.g., Schwartz [1992, Theorem 3.7.101) shows that the mapping 0 is a f i n e in R, i.e., that there exists a vector c E E3 and a matrix Q E M3 such that (the notation oz designates the column vector with components zi)

O(z) = c + Qoz for all x E 0.


Since Q = VO(zo) and VO(zO)TVO(zO) = I by assumption, the matrix Q is orthogonal.

- (iv) We now consider the general equations gij = g23 in R, noting

that they also read

vo(z)Tvo(z) = v G ( ~ ) ~ v G ( ~ ) for all z E R.

Given any point zo E R, let ;he neighborhoods V of zo and c of O(zo) and the mapping 0-1 : V -+ V be defined as in part (i) (by assumption, the mapping 0 is an immersion; hence the matrix V@(zo)

'"is invertible). Consider the composite mapping

& := 6 o 0-l : 9 -+ E3.

Clearly, 6 E C1@; E3) and

vqq = VG(z)V@-yq = v G ( ~ ) v E I ( ~ ) - ' for all 2 = ~ ( z ) , z E V.

Hence the assumed relations

V O ( ~ ) ' V O ( ~ ) = v G ( ~ ) ~ v G ( ~ ) for all z E R

e65(2)T6&(2) = I for all z E V. imply that

By parts (i) to (iii), there thus exist a vector c E R3 and a matrix Q E O3 such that

6(2) = G(z) = c + QO(Z) for all 2 = ~ ( z ) , z E V,

hence such that

=(z) := v G ( ~ ) v o ( ~ ) - ~ = Q for all z E V.

The continuous mapping E : V -+ M3 defined in this fashion is thus locally constant in R. As in part (iii), we conclude from the assumed connectedness of C2 that the mapping E is constant in R. Thus the proof is complete. 13

An isometry of E3 is a mapping J : E3 --+ E3 of the form J(z) = c+Q OX for all z E E3, with c E E3 and Q E O3 (an analogous definition holds verbatim in any Euclidean space of dimension d 2 2). Clearly, an isometry preserves distances in the sense that

IJ(y) - J(z)I = Iy - $ 1 for all z, y E R.


Remarkably, the converse is also true, according to the classical Mazur-Ulam theorem, which asserts the following: Let R be a connected subset in Rd, and let 0 : R -+ Rd be a mapping that satisfies

I @ ( Y ) - @ ( % ) I = ly - X I for all x , y E R.

Then 0 is an isometry of Rd. Parts (ii) and (iii) of the above proof thus provide a proof of this

theorem under the additional assumption that the mapping 0 is of class C1 (the extension from R3 to Rd is trivial).

In Theorem 1.7-1, the special case where 0 is the identity mapping of R3 identified with E3 is the classical Liouville theorem. This theorem thus asserts that if a mapping 0 E C1(R; E3) is such that V O ( x ) E O3 for all x E R, where R is a n open connected subset of R3, then 0 is a n isometry.

Two mappings 0 E C1(R;E3) and 6 E C1(R;E3) are said to be isometrically I equivalent if there exist c E E3 and Q E O3 such that 0 = c f QO in R, i.e., such that 0 = J o 6 , where J is an isometry of E3. Theorem 1.7-1 thus asserts that two immersions 0 E C1(R; E3) and 6 E C1(R; E3) share the same metric tensor field over a n open connected subset R of R3 i f and only i f they are isometrically equivalent.

Remark. In terms of covariant components gij of metric tensors, parts (i) to (iii) of the above proof provide the solution to the equations gi j = &i in R, while part (iv) provides the solution to the equations 9 . . 23 - - &0 . aj6 in R , where 6 E C1(R; E3) is a given immersion. 0

While the immersions 0 found in Theorem 1.6-1 are thus only defined up to isometries in E3, they become uniquely determined if they are required to satisfy ad hoc additional conditions, according to the following corollary to Theorems 1.6-1 and 1.7-1.

Theorem 1.7-2. Let the assumptions o n the set R and on the matrix field C be as in Theorem 1.6-1, let a point xo E R be given, and let Fo E M3 be any matrix that satisfies

FTF,, = c ( x o ) .

Then there exists one and only one immersion 0 E C3(R;E3) that satisfies

v @ ( x ) ~ ~ @ ( x ) = C ( X ) for all x E R, O(xo) = 0 and VO(x0) = Fo.

Proof. Given any immersion a E C3(R; E3) that satisfies

Va(z)TVa(x) = C ( x )


for all x E R (such immersions exist by Theorem 1.6-1), let the mapping 0 : R 3 R3 be defined by

O(x) := FoV+(xO)-l(+(x) - +(zo)) for all x E R.

Then it is immediately verified that this mapping 0 satisfies the announced properties.

Besides, it is uniquely determined. To see this, let 0 E C3(R;E3) and + E C3(R; E3) be two immersions that satisfy

v@(~)~vo(x ) = V + ( S ) ~ V + ( ~ ) for all x E R.

Hence there exist (by Theorem 1.7-1) c E R3 and Q E O3 such that a(x) = c + QO(x) for all 5 E $2, so that V@(x) = QVO(x) for all x E R. The relation VO(x0) = V+(xo) then implies that Q = I and

0 the relation O(z0) = ~ ( Z O ) in turn implies that c = 0.

Remark. One possible choice for the matrix Fo is the square root of 0

Theorem 1.7-1 constitutes the “classical” rigidity theorem, in that both mappings 0 and 6 are assumed to be in the space C1(R; E3). The next theorem is an extension, due to Ciarlet & C. Mardare [2003], that covers the case where one of the mappings belongs to the Sobolev space H1(R; E3).

The way the result in part (i) of the next proof is derived is due to F’riesecke, James & Muller [2002]; the result of part (i) itself goes back to Reshetnyak [1967].

Let 0: denote the set of all proper orthogonal matrices of order three, i.e., of all orthogonal matrices Q E O3 with det Q = 1.

Theorem 1.7-3. Let R be a connected open subset of R3, let 0 E C1(R; E3) be a mapping that satisfies

the symmetric positive-definite matrix C(x0).

d e t V O > 0 in R,

and let 6 E H1(R; E3) be a mapping that satisfies

det 06 > 0 a.e. in R and VOTVO = 06’06 a.e. in R.

Then there exist a vector c E E3 and a matrix Q E 0; such that

6(z) = c + Q@(Z) for almost all x E R.

Proof. The Euclidean space E3 is identified with the space R3 throughout the proof.


(i) To begin with, we consider the special case where Q ( x ) = x for all x E 0. In other words, we arc given a mapping 6 E H1(R) that satisfies vO(z) E 0: for almost all x E R. Hence

C o f V G ( z ) = (det V 6 ( x ) ) V 6 ( ~ ) - ~ = V ~ ( X ) - ~

for almost all II: E R, on the one hand. Since, on the other hand,

div C o f V 6 = 0 in (D ' (B) )3

in any open ball B such that B c R (to see this, combine the density of C2(B) in H 1 ( B ) with the classical Piola identity in the space C2(B); for a proof of this identity, see, e.g., Ciarlet [1988, Theorem 1.7.1]), we conclude that

A 6 = d i v C o f V 6 = 0 in (D ' (B) )3 .

Hence 6 = (6j) E (C"(R))3. For such mappings, the identity

a(a&aiGj) = a a i 6 j a i ( ~ 6 j ) + 2t4k6jaik6j, - -

together with the relations AC3j = 0 and &Oj&6j = 3 in R, shows that aikC3j = 0 in R. The assumed connectedness of R then implies that there exist a vector c E E3 and a matrix Q E 0; (by assumption, V ~ ( Z ) E 0; for almost all II: E 0) such that

-

6 ( x ) = c + Q ox for almost all x E R.

(ii) Consider next the general case. Let xo E R be given. Since 0 is an immersion, the local inversion theorem can b_e applied; there thus exist bounded open neighborhoods U of zo and U of Q(Q) satisfying U c R and {6}- c Q(R), such that the restriction QU of 0 to U can be extended to a C1-diffeomorphism from u onto {6}-.

Let 0,' : 6 4 U denote the inverse mapping of O U , which therefore satisfies VQG'(2) = VO(x)-l for all 2 = Q ( x ) E 6 (the notation 6 indicates that differentiation is carried out with respect to the variable 2 E G). Define the composite mapping

-

Since 6 E H1(U) and 0;' can be extended to a C1-diffeornorphism from {6}- onto g , it follows that 6 E H1(6;R3) and that

VG(2) = V6(2)60,1(2) = v o ( x ) v o ( x ) - '


for almost all P = O(z) E 6 (see, e.g., Adams [1975, Chapter 31). Hence the assumptions det VO > 0 in R, det V 6 > 0 a.e. in R, and VOTVO = VGTV6 a.e. in R, together imply that 6 & ( P ) E 0: for almost all 2 E 6. By (i), there thus exist c E E3 and Q E 0: such that

&(2) = 6(z) = c + Q 02 for almost all P = ~ ( z ) E U , h

or equivalently, such that

~ ( z ) := v O ( z ) v ~ ( z ) - l = Q for almost all z E U.

Since the point zo E R is arbitrary, this relation shows that E E Ltoc(R). By a classical result from distribution theory (cf. Schwartz [1966, Section 2.6]), we conclude from the assumed connectedness of R that E(z) = Q for almost all z E R, and consequently that

6(z) = c + Q O ( ~ ) for almost all z E Q. 0

Remarks. (1) The existence of 6 E H1(R; E3) satisfying the assumptions of Theorem 1.7-3 thus implies that 0 € H1(R;E3) and 6 E C1(R; E3).

(2) If 6 E C1(R;E3), the assumptions d e t V O > 0 in R and d e t V 6 > 0 in R are no longer necessary; but then it can only be concluded that Q E 03: This is the classical rigidity theorem (Theorem 1.7-1), of which Liouville's theorem is the special case corresponding to O(z) = z for all z E R.

(3) The result established in part (i) of the above proof asserts that, given a connected open subset R of R3, if a mapping 0 E H1(R; E3) is such that VO(z) E 0% for almost all z E R, then there exist c E E3 and Q E 0: such that O(z) = c + &ox for almost all z E R. This result thus constitutes a generalizatio_n of Liouville 's theorem.

(4) By contrast, if the mapping 0 is assumed to be instead in the space H1(R;E3) (as in Theorem 1.7-3), an assumption about the sign of det 06 becomes necessary. To see this, let for instance R be an open ball centered at the origin in R3, let O(z) = z, and let 6(z) = z if z1 2 0 and O(z) = ( - z 1 , 2 2 , 2 3 ) if z1 < 0. Then 6 E H1(R;E3) and V 6 E O3 a.e. in R; yet there does not exist any orthogonal matrix such that 6(z) = Qo.: for all z E R, since 6 ( R ) c {z E R3; z1 3 0) (this counter-example was kindly communicated to the author by Sorin Mardare).

(5) Surprisingly, the assumption d e t V O > 0 in R cannot be replaced by the weaker assumption det VO > 0 a.e. in R. To see this, let for instance R be an open ball centered at the origin in R3, let

-

An Introduction to Differential Geometry in R3 39 -

O ( x ) = ( x 1 ~ ~ , ~ 2 , 2 3 ) and let O ( x ) = O(z) if 22 2 0 and 6 ( x ) = ( - q x i , - 2 2 , ~ ) if 22 < 0 (this counter-example was kindly communicated to the author by HervB Le Dret).

(6) If a mapping 0 E C1(R;E3) satisfies d e t V O > 0 in 0, then 0 is an immersion. Conversely, if R is a connected open set and O E C1(R; E3) is an immersion, then either det VO > 0 in R or det VO < 0 in R. The assumption that det VO > 0 in R made in Theorem 1.7-3 is simply intended to fix ideas (a similar result clearly holds under the other assumption).

(7) A little further ado shows that the conclusion of Theorem 1.7- 3 is still valid if 6 E H1(R;E3) is replaced by the weaker assumption

6 E Hk,(R;E3). 0

Like the existence results of Section 1.6, the uniqueness theorems of this section hold verbatim in any dimension d 2 2, with R3 replaced by Rd and Ed by a d-dimensional Euclidean space.

1.8 Continuity of an immersion as a function of its metric tensor

Let R be a connected and simply-connected open subset of R3. Together, Theorems 1.6-1 and 1.7-1 establish the existence of a mapping 3 that associates with any matrix field C = ( g i j ) E C2(R; S;) satisfying

R q i j k := ajrikq - + r;$'kqp - I'ykrjqp = 0 in 0,

where the functions rij, and l7$ are defined in terms of the functions gi j as in Theorem 1.6-1, a well-defined element 3 ( C ) in the quotient set C3(R; E3)/R, where ( O , 6 ) E R means that there exist a vector a E E3 and a matrix Q E CD3 such that O(z) = a + Q6(z) for all 2 E R.

A natural question thus arises as to whether the mapping 3 defined in this fashion is continuous. Equivalently, is a n immersion a continuous function of its metric tensor?

When both spaces C2(R; S3) and C3(R; E3) are equipped with their natural Frkchet topologies, a positive answer to this question has been provided by Ciarlet & Laurent [2003].

More specifically, let R be an open subset of R3. The notation K R means that K is a compact subset of R. If g E C'(R;W),t 0, and K cz R, we define the semi-norms

where aa stands for the standard multi-index notation for partial derivatives. If 0 E Ce(R; E3) or A E Ce(R; M3), 1 2 0, and K @ R, we likewise


set

where 1.1 denotes either the Euclidean vector norm or the matrix spectral norm.

Then, for any integers C 3 0 and d 3 1, the space Ce(L?;Rd) becomes a locally convex topological space when it is equipped with the Fre'chet topology defined by the family of semi-norms / l . / l e , K , K R, defined above. Then a sequence (On),20 converges to 0 with respect to this topology if and only if

lim /I@" - Olle ,~ = 0 for all K G R. n-cc

Furthermore, this topology is metrizable: Let (Ki)+o be any sequence of subsets of R that satisfy

00

Ki CE L? and Ki C int Ki+l for all i 3 0, and R = u Ki. i=O

Then

lim 110" - O l l e , ~ = 0 for all K G R + lim &(On, 0) = 0, ?L--tco n-+m

where

For details about F'rkchet topologies, see, e.g., Yosida [1966, Chapter 11. Let C3(R; E3) := C3(R; E3)/R denote the quotient set of C3(R; E3)

by the equivalence relation R, where (0,G) E R means that 0 and 6 are isometrically equivalent (Section 1.7), i.e., that there exist a vector c E E3 and a matrix Q E O3 such that 0 ( x ) = c+ QG(x) for all x E R. Then it is easily verified that the set C3(R; E3) becomes a metric space when it is equipped with the distance d 3 defined by

&(6,+) = inf. d g ( ) i , x) = inf &(GI c + Q+), { ;z7 { z::

where 6 denotes the equivalence class of 0 modulo R.


The continuity of an immersion as a function of its metric tensor has then been established by Ciarlet & Laurent [2003], according to the following result (if d is a metric defined on a set X , the associated metric space is denoted { X ; d } ) .

Theorem 1.8-1. Let R be a connected and simply-connected open subset ofR3. Let

Cz(R; S;) := {(gij) E C2(R; s”,; R q i j k = 0 in a},

and, given any matrix field C = (g i j ) E C,2(R; S:), let F(C) E C3(R; E3) denote the equivalence class modulo R of any 0 E C3(R2;E3) that satisfies VOTVO = C in R. Then the mapping

F : {C,”(R; S;); d2) - {C3(R; E3); d 3 )

defined in this fashion is continuous. 0

As shown by Ciarlet & C. Mardare [2004b], the above continuity result can be extended “up to the boundary of the set R”, as follows. If R is bounded and has a Lipschitz-continuous boundary, the mapping F can be extended to a mapping that is locally Lipschitz-continuous with respect to the topologies of the Banach spaces C2(a;S3) for the continuous extensions of the symmetric matrix fields C , and C3@; E3) for the continuous extensions of the immersions 0.

Another extension, essentially motivated by nonlinear three-dimensional elasticity, is the following: Let R be a bounded and connected subset of W3, and let B be an elastic body with R as its reference configuration. Thanks mostly to the landmark existence theory of Ball [1977], it is now customary in nonlinear three-dimensional elasticity to view any mapping 0 E H1(R;E3) that is almost-everywhere injective and satisfies det V 0 > 0 a.e. in R as a possible deformation of B when B is subjected to ad hoc applied forces and boundary conditions. The almost- everywhere injectivity of 0 (understood in the sense of Ciarlet & NeEas [1987]) and the restriction on the sign of det VO mathematically express (in an arguably weak way) the non-interpenetrability and orientation- preserving conditions that any physically realistic deformation should satisfy.

It then turns out that the metric tensor field VOTVO E L1(R; S3) , also known as the Cauchy-Green tensor field in elasticity, associated with a deformation 0 E H1(R;E3) pervades the mathematical modeling of three-dimensional nonlinear elasticity (extensive treatments of this subject are found in Marsden & Hughes [1983] and Ciarlet [1988]). Conceivably, an alternative approach to the existence theory in three- dimensional elasticity could thus regard the Cauchy-Green tensor as the primary unknown, instead of the deformation itself as is usually the case.


Clearly, the Cauchy- Green tensors depend continuously o n the deformations, since the Cauchy-Schwarz inequality immediately shows that the mapping

0 E H1(O; E3) ---f VOTVO E L1(O; S3)

is continuous (irrespectively of whether the mappings 0 are almost- everywhere injective and orientation-preserving).

Then Ciarlet & C. Mardare [2004c] have shown that, under appropriate smoothness and orientation-preserving assumptions, the converse holds, i. e., the deformations depend continuously on their Cauchy-Green tensors, the topologies being those of the same spaces H1(R;E3) and L1 (0; S3) (by contrast with the orientation-preserving condition, the issue of non-interpenetrability turns out to be irrelevant to this issue).

This continuity result is itself a simple consequence of the following nonlinear Korn inequality, which constitutes the main result of ibid.: Let R be a bounded and connected open subset of R3 with a Lipschitz- continuous boundary and let 0 E C1(n;E3) be a mapping satisfying det V 0 > 0 in n. Then there exists a constant C(0) with the following property: For each orientation-preserving mapping 9 E H1(Q E3), there exist a proper orthogonal matrix R = R(9, 0) of order three (i.e., an'orthogonal matrix of order three with a determinant equal to one) and a vector b = b ( 9 , 0) in E3 such that

119 - (b + R O ) I ( H ~ ( ~ ; E ~ ) 6 C(@)IIV@V@ - V@TV@/I$n;s3).

That a vector b and an orthogonal matrix R should appear in the left- hand side of such an inequality is of course reminiscent of the classical rigidity theorem (Theorem 1.7-1), which shows that, if two mappings 6 E C1(R; E3) and 0 E C1(R; E3) satisfying det 06 > 0 and det V 0 > 0 in an open connected subset R of R3 have the same metric tensor field, then the two mappings are isometrically equivalent, i.e., there exist a vector-b in E3 and a proper orthogonal matrix R of order three such that 0(z) = b + RO(z) for all z E 0.

More generally, we shall say that two orientation-preserving mappings 6 € H1(R; E3) and 0 € H1(R; E3) are isometrically equivalent if there exist a vector b in E3 and an orthogonal matrix R of order three (a proper one in this case) such that

6(z) = b + Re(%) for almost all z E R.

One application of the above key inequality is the following sequential continuity property: Let Ok E H1(R;E3), k 3 1, and 0 E C1(2;E3) be orientation-preserving mappings. Then there exist a constant C ( 0 )


and orientation-preserving mappings 0 E H1(R; E3), k 2 1, that are isometrically equivalent to Ok such that

- k

- k Hence the sequence (0 )El converges to 0 in H1(R; E3) as k + 00 if the sequence ( ( v o ~ ) ~ v o ~ ) ~ = . _ , converges to V O ~ V O in L ~ ( R ; ~ 3 ) as k + m .

Should the Cauchy-Green strain tensor be viewed as the primary unknown (as suggested above), such a sequential continuity could thus prove to be useful when considering injimizing sequences of the total energy, in particular for handling the part of the energy that takes into account the applied forces and the boundary conditions, which are both naturally expressed in terms of the deformation itself.

They key inequality is first established in the special case where 0 is the identity mapping of the set R, by making use in particular of a fundamental “geometric rigidity lemma” recently proved by F’riesecke, James & Muller [2002]. It is then extended to an arbitrary mapping 0 E C1@ EXn) satisfying det V 0 > 0 in 2, thanks in particular to a methodology that bears some similarity with that used in Ciarlet & Laurent [2003].

Such results are to be compared with the earlier, pioneering estimates of John [1961], John [1972] and Kohn [1982], which implied continuity at rigid body deformations, i.e., at a mapping 0 that is isometrically equivalent to the identity mapping of R. The recent and noteworthy continuity result of Reshetnyak [2003] for quasi-isometric mappings is in a sense complementary to the above one (it also deals with Sobolev type norms).

2 Differential geometry of surfaces

Out line

We saw in Part 1 that an open set 0 ( R ) in E3, where R is an open set in R3 and 0 : R + E3 is a smooth injective immersion, is unambiguously defined (up to isometries of E3) by a single tensor field, the metric tensor field, whose covariant components gij = gji : R -+ R are given by

Consider instead a surface G = O(w) in E3, where w is a two- dimensional open set in R2 and 8 : w + E3 is a smooth injective immersion. Then by contrast, such a “two-dimensional manifold” equipped with the coordinates of the points of w as its curvilinear coordinates,

9 ” ’ 23 - - ai0 . aj0.


requires two tensor fields for its definition (this time up to proper isometries of E3), the first and second fundamental forms of G. Their covariant components a,p = ap, : w -+ R and bag = bp, : w -+ R are respectively given by (Greek indices or exponents take their values in {1,2}):

a,p = a, . a g and b,p = a3 . & u p ,

The vector fields ai : w ’-+ R3 defined in this fashion constitute the covariant bases along the surface G I while the vector fields ai : w --f R3 defined by the relations ai . aj = 6; constitute the contravariant bases along 2.

These two fundamental forms are introduced and studied in Sections 2.1 to 2.5. In particular, it is shown how areas and lengths, i.e., “metric notions”, on the surface G are computed in terms of its curvilinear coordinates by means of the components a,p of the first fundamental form (Theorem 2.3-1). It is also shown how the curvature of a curve on G can be similarly computed, this time by means of the components of both fundamental forms (Theorem 2.4-1). Other classical notions about “curvature”, such as the principal curvatures and the Gaussian curvature, are introduced and briefly discussed in Section 2.5.

We next introduce in Section 2.6 the fundamental notion of covariant derivatives qil, of a vector field qiai : w -+ R3 on G I thus defined here by means of its covariant components qi over the contravariant bases az. We establish in this process the formulas of Gauss and Weingarten (Theo- rem 2.6-1). Covariant derivatives of vector fields on a surface (typically, the unknown displacement vector field of the middle surface of a shell) pervade the equations of shell theory; see, e.g., Ciarlet [2000], Ciarlet [2005], or Ciarlet & C. Mardare [2007] in this Volume.

It is a basic fact that the symmetric and positive definite matrix field (a,g) and the symmetric matrix field (hap) defined on w in this fashion cannot be arbitrary. More specifically, their components and some of their partial derivatives must satisfy necessary conditions taking the form of the following relations (meant to hold for all a, p, (T, T E { 1,2}), which respectively constitute the Gauss, and Codazzi-Mainardi, equations (Theorem 2.7-1): Let the functions r a p T and be defined

1 bY

rapT = a(dpa,, + &upT - &-a,,) and I ’Ep = a U T a p T ,

where (auT) := (a,p)-’. Then, necessarily,

8.d’,,, - W , p , + r&ruTp - r:,,rPTp = b,,bpT - bapbU, in W ,

apb,, - aUb,g + rgubfip - I ’ ~ o b u p = O in w.


The functions rap7 and rgp are the Christoffel symbols of the first, and second, kind.

We also establish in passing (Theorem 2.7-2) the celebrated Theorema Egregium of Gauss: At each point of a surface, the Gaussian curvature is a given function (the same for any surface) of the components of the first fundamental form and their partial derivatives of order < 2 at the same point.

We then turn to the reciprocal questions: Given an open subset w of R2 and a smooth enough symmetric and

positive definite matrix field (a,p) together with a smooth enough symmetric matrix field (b,p) defined over w, when are they the first and second fundamental forms of a surface 8 ( w ) C E3, i.e., when does there exist an immersion 8 ; w + E3 such that

If such an immersion exists, to what extent is it unique? As shown in Theorems 2.8-1 and 2.9-1 (like those of their “three-

dimensional counterparts” in Sections 1.6 and 1.7, their proofs are by no means easy, especially that of the existence), the answers turn out to be remarkably simple: Under the assumption that w is simply-connected, the necessary conditions expressed by the Gauss and Codazzi-Mainardi equations are also suficient for the existence of such an immersion 8.

Besides, i f w is connected, this irnmersion is unique up to proper isometries in E. This means that, if 8 : w 3 E3 is any other smooth immersion satisfying

there then exist a vector c E E3 and a proper orthogonal matrix Q of order three such that

qY) = c + Q@(,) for all E W .

Together, the above existence and uniqueness theorems constitute the fundamental theorem of surface theory, another important special case of the fundamental theorem of Riemannian geometry already alluded to in Part 1. As such, they constitute the core of Part 2.

We conclude Part 2 by indicating in Section 2.10 how the equivalence class of 8, defined in this fashion modulo proper isornetries of E3, depends continuously o n the matrix fields (a,p) and (baij) with respect to appropriate topologies.


2.1 Curvilinear coordinates on a surface

In addition to the rules governing Latin indices that we set in Section 1.1, we henceforth require that Greek indices and exponents vary in the set {1,2} and that the summation convention be systematically used in conjunction with these rules. For instance, the relation

da(’%aZ) = (qflla - baflr73)aP + (r/3/a + b!7p)a3

means that, for a = 1 , 2 ,

Kronecker’s symbols are designated by b t , Sap, or Sap according to the context.

Let there be given as in Section 1.1 a three-dimensional Euclidean space E3, equipped with an orthonormal basis consisting of three vectors e = Zi, and let a . b, lal, and a A b denote the Euclidean inner product, the Euclidean norm, and the vector product of vectors a, b in the space E3.

In addition, let there be given a two-dimensional vector space, in which two vectors ea = e, form a basis. This space will be identified with EX2. Let y, denote the coordinates of a point y E EXz and let 8, := dldy, and d,p := d2/dyadyp.

Finally, let there be given an open subset w of R2 and a smooth enough mapping 0 : w 4 E3 (specific smoothness assumptions on 8 will be made later, according to each context). The set

--i

G := qW) is called a surface in E3.

unambiguously written as If the mapping 0 : w + E3 is injective, each point E G can be

i j = 8 ( Y ) , YEW,

and the two coordinates y, of y are called the curvilinear coordinates of 6 (Figure 2.1-1). Well-known examples of surfaces and of curvilinear coordinates and their corresponding coordinate lines (defined in Section 2.2) are given in Figures 2.1-2 and 2.1-3.

Naturally, once a surface 2 is defined as 2 = 8(w) , there are infinitely many other ways of defining curvilinear coordinates on 2, depending on how the domain w and the mapping 8 are chosen. For instance, a portion 2 of a sphere may be represented by means of Cartesian coordinates,


Figure 2.1-1: Curvilinear coordinates on a surface and covariant and contravariant bases of the tangent plane. Let G = B(w) be a surface in E3. The two coordinates y1,yz of y E w are the curvilinear coordinates of $ = B(y) E G. If the two vectors a,(y) = &B(y) are linearly independent, they are tangent to the coordinate lines passing through y^ and they form the covariant basis of the tangent plane to 2 at $ = B(y). The two vectors a"(y) from this tangent plane defined by a*(y) . ao(y) = 6; form its contravariant basis.

spherical coordinates, or stereographic coordinates (Figure 2.1-3). Inci- dentally, this example illustrates the variety of restrictions that have to be imposed on i3 according to which kind of curvilinear coordinates it is equipped with!

2.2 First fundamental form

Let w be an open subset of R2 and let

be a mapping that is dzflerentiable at a point y E w. If 6y is such that (y + 6y) E w , then


X Dy

Figure 2.1-2: Several systems of curvilinear coordinates on a sphere. Let C C E3 be a sphere of radius R. A portion of C contained “in the northern hemisphere” can be represented by means of Cartesian coordinates, with a mapping 8 of the form:

8 : (e, y) E w --t (2, y, {R2 - (z2 + y2)}l/’) E E3. A portion of C that excludes a neighborhood of both “poles” and of a “meridian”

(to fix ideas) can be represented by means of spherical coordinates, with a mapping 8 of the form:

8 : ( c p , $ ) E w + (Rcos$coscp,Rcos$sincp,Rsin$) E E3. A portion of C that excludes a neighborhood of the “North pole” can be repre-

sented bv means of StereomaDhic coordinates. with a maDDina 8 of the form: _ _ - 2R2v u2 + v2 - R2 e : (u,w) E +

The corresponding explanatory graphical conventions.


cp I I

/ I w /

Figure 2.1-3: Two familiar examples of surfaces and curvilinear coordinates. A portion ij of a circular cylinder of radius R can be represented by a mapping 8 of the form

8 : (p, z ) E w + (Rcos p, Rsin p, z ) E E3.

8 : ( p , ~ ) Ew+ ( ( R + r c o s ~ ) c o s p , ( R + r c o s x ) s i n p , ~ s i n x ) E E 3 , A portion G of a torus can be represented by a mapping 8 of the form

with R > T .

explanatory graphical conventions. The corresponding coordinate lines are represented in each case, with self-


where the 3 x 2 matrix V8(y) and the column vector 6y are defined by

Let the two vectors a,(y) E R3 be defined by

:= a , w = w2 (Y), (:::) i.e., a a ( y ) is the a- th column vector of the matrix VB(y). Then the expansion of 8 about y may be also written as

8 ( Y + SY) = O(Y) + 6Y"aa(Y) + O(6Y). If in particular 6y is of the form 6y = Ste,, where 6t E R and e, is

one of the basis vectors in EX2, this relation reduces to

e(y + bte,) = B(y) + Sta,(y) + o(6t ) .

A mapping 8 : w + E3 is an immersion at y E w if it is differentiable at y and the 3 x 2 matrix V8(y) is of rank two, or equivalently if the two vectors a,(y) = 8,8(y) are linearly independent.

Assume from now on in this section that the mapping 8 is an immersion at y. In this case, the last relation shows that each vector a,(y) is tangent to the a- th coordinate line passing through @ = 8(y), defined as the image by 8 of the points of w that lie on a line parallel to e, passing through y (there exist t o and tl with t o < 0 < tl such that the a-th coordinate line is given by t E ]to,tl[ + f , ( t ) := O(y + t ea) in a neighborhood of & hence f&(0) = da8(y) = a,(y)); see Figures 2.1-1, 2.1-2, and 2.1-3.

The vectors a,(y), which thus span the tangent plane to the surface 2 at = 8(y), form the covariant basis of the tangent plane to ij at see Figure 2.1-1.

Returning to a general increment Sy = 6 y a e a , we also infer from the expansion of 8 about y that (6yT and V8(y)T respectively designate the transpose of the column vector 6y and the transpose of the matrix VO(Y))

1% + SY) - q Y ) l Z = 6YTvqY)TvqlJ) 6 Y + O(l6Yl2) = b " a a ( y ) . ap(Y)6Yp + O(l6YI").

In other words, the principal part with respect to Sy of the length between the points e(y++y) and 8(y) is {6yaa,(y) .ap(y)6yP}l12. This observation suggests to define a matrix (a,p(y)) of order two by letting

a d y ) := d Y ) . a p ( y ) = (ve(Y)Tve(Y))a,.


The elements aap(y) of this symmetric matrix are called the covariant components of the first fundamental form, also called the metric tensor, of the surface 2 at $= 8(y).

Note that the m a t r i z (a,p(y)) is positive definite since the vectors a,(y) are assumed to be linearly independent.

The two vectors a,(y) being thus defined, the f o u r relations

a"(T/) . d y ) = 6;

unambiguously dejine two linearly independent vectors a"(y) in the tangent plane. To see this, let a priori a"(y) = Y""(y)ao(y) in the relations a*(y) . a p ( ~ ) = d;. This gives Y""(y)a,p(y) = 6;; hence Y*"(y) = ~"" (y ) , where

(aap(Y)) := (aaa(y ) ) - l

aa(Y) . 4 Y ) = a a u ( Y ) a w a , ( Y ) ' UAY)

Hence a" (y ) = a""(y)a,(y). These relations in turn imply that

= aau(y)a"(y)am7(y) = aau(y )6 t = u"p(y ) ,

and thus the vectors a"(y) are linearly independent since the matrix (a "p(y ) ) is positive definite. We would likewise establish that a a ( y ) =

The two vectors a*(y) form the contravariant basis of the tangent plane to the surface 2 at g = 8(y) (Figure 2.1-1) and the elements a"p(y ) of the symmetric matrix (a "p(y ) ) are called the contravariant components of the first fundamental form, or metric tensor, of the surface 2 at g = 8(y).

Let us record for convenience the fundamental relations that exist between the vectors of the covariant and contravariant bases of the tangent plane and the covariant and contravariant components of the first fundamental tensor:

a a a ( Y ) a P ( d .

a*p(y ) = a,(y). ag(Y) and a"P(d = a"(Y) .aP(Y), %(Y) = aap(Y)aP(Y) and a"(y) = a""Y)adY).

A mapping 8 : w -+ E3 is an immersion if it is an immersion at each point in w , i.e., if 8 is differentiable in w and the two vectors d a 8 ( y ) are linearly independent at each y E w.

If 8 : w -+ E3 is an immersion, the vector fields a, : w -+ R3 and a" : w + R3 respectively form the covariant, and contravariant, bases of the tangent planes.

A word of caution. The presentation in this section closely follows that of Section 1.2, the mapping 8 : w c R2 -+ E3 "replacing" the mapping 0 : R c R3 + E3. There are indeed strong similarities between


the two presentations, such as the way the metric tensor is defined in both cases, but there are also sharp differences. In particular, the matrix V8(y) is not a square matrix, while the matrix VO(z) i s square! 0

2.3 Areas and lengths on a surface

We now review fundamental formulas expressing area and length elements at a point g = 8(y) of the surface w = O(w) in terms of the matrix (a,p(y)); see Figure 2.3-1.

These formulas highlight in particular the crucial r6le played by the matrix (a,p(y)) for computing “metric” notions at y^ = 8(y). Indeed, the first fundamental form well deserves “metric tensor” as its alias!

Figure 2.3-1: Area and length elements on a surface. The elements dZi(g) and d&) at y^ = 8(y) E is are related to dy and 6y by means of the covariant components of the metric tensor of the surface 2; cf. Theorem 2.3-1. The corresponding relations are used for computing the area of a surface A = @ ( A ) C is and the length of a curve c^ = 8(C) C 2, where C = f ( I ) and I is a compact interval of R.

Theorem 2.3-1. Let w be an open subset of R2, let 8 : w 4 E3 be a n injective and smooth enough immersion, and let ij = 8(w) .

(a) The area element dZ@) at g= 8(y) E 2 is given in terms of the area element dy at y E w by

dZ@) = m d y , where a(y) := det(a,p(y)).

An Introduction to Differential Geometry in R3

(b) The length element dZ(fj) at c = 8(y) E w^ is given by

dZ(fj) = {by"~,p(y)by~}~'~.

53

Proof. The relation (a) between the area elements is well known. It c%n also be deduced directly from the relation between the area elements dr (2) and dr(z) given in Theorem 1.3-1 (b) by means of an ad hoc "three-dimensional extension" of the mapping 8.

The expression of the length element in (b) recalls that dZ@) is by definition the principal part with respect to 6y = by"ea of the length l8(y f 6y) - 8(y)/, whose expression precisely led to the introduction of the matrix (a,p(y)). 0

The relations found in Theorem 2.3-1 are used as follows for computing surface integrals and lengths on the surface w by means of integrals inside w, i.e., in terms of the curvilinear coordinates used for defining the surface ij (see again Figure 2.3-1). We assume again that 8 : w t E3 is an injective and smooth enough immersion.

Let A be a domain in R2 such that ?I c w (a domain in pz is a bounded, open, and connected subset of R2 with a Lipschitz-continuous boundary; cf. Section 1.3), let A := @ ( A ) , and let f ^ E L1(A^) be given. Then

In particular, the area of A is given by h

areaA := dZ(fj) = m d y .

Consider next a curve C = f(1) in w , where I is a compact interval of R and f = f "e" : I 4 w is a smooth enough injective mapping. Then the Zength of the curve 21 := O(C) c w is given by

The last relation shows in particular that the lengths of curves inside the surface 8 ( w ) are precisely those induced by the Euclidean metric of the space E3. For this reason, the surface 8 ( w ) is said to be isometrically immersed in E3.

2.4 Second fundamental form; curvature on a surface

While the image O(R) c E3 of a three-dimensional open set R c R3 by a smooth enough immersion 0 : R c R3 4 E3 is well defined by its


“metric” uniquely up to isometrics in E3, provided ad hoc compatibility conditions arc satisfied by the covariant components gi j : R -+ R of its metric tensor (cf. Theorems 1.6-1 and 1.7-1)1 a surface given as the image O(w) c E3 of a two-dimensional open set w c R2 by a smooth enough immersion 8 : w c R2 + E3 cannot be defined by its metric alone.

As intuitively suggested by Figure 2.4-1 the missing information is provided by the “curvature” of a surface. A natural way to give sub- stance to this otherwise vague notion consists in specifying how the curvature of a curve on a surface can be computed. As shown in this section, solving this question relies on the knowledge of the second fundamental form of a surface, which naturally appears for this purpose through its covariant components (Theorem 2.4-1).

I

Figure 2.4-1: A metric alone does not define a surface an E3. A flat surface GO may be deformed into a portion Gi of a cylinder or a portion 2 2 of a cone without altering the length of any curve drawn on it (cylinders and cones are instances of “developable surfaces”; cf. Section 2.5). Yet it should be clear that in general 20 and GI, or 20 and G2, or 2 1 and G2, are not identical surfaces modulo an isometry of E3!

Consider as in Section 2.1 a surface ij = 8(w) in E3, where w is an open subset of R2 and 8 : w C R2 + E3 is a smooth enough immersion. For each y E w, the vector

An lntroduction to Differential Geometry in R3 55

is thus well defined, has Euclidean norm one, and is normal to the surface G at the point y ^ = 8(y).

Remark. It is easily seen that the denominator in the definition of a3(y) may be also written as

l a d y ) A az(?/)I = m, where a(y) := det(a,p(y)) and that this relation holds in fact even if a(y ) = 0.

0

Fix y E w and consider a plane P normal to G at y^ = 8(y) , i.e., a plane that contains the vector ay(y>. The intersection 21 = P n 2 is thus a planar curve on G.

As shown in Theorem 2.4-1, it is remarkable that the curvature of at y^ can be computed by means of the covariant components a,p(y) of the first fundamental form of the surface G = 8(u) introduced in Section 2.2, together with the covariant components b,p(y) of the “second” fundamental form of 2. The definition of the curvature of a planar curve is recalled in Figure 2.4-2.

If the algebraic curvature of 21 at y^ is # 0, it can be written as -, and

R is then called the algebraic radius of curvature of the_curve C at y . This means that the center of curvature of the curve C at y^ is the point (y^+Ras(y)); see Figure 2.4-3. While R is not intrinsically defined, as its sign changes in any system of curvilinear coordinates where the normal vector as(y) is replaced by its opposite, the center of curvature i s intrinsically defined.

If the curvature of C at Q is 0, the radius of curvature of the curve 6 at y^ is said to be infinite; for this reason, it is customary to still write

the curvature as - in this case. 1 R

1

R -

1 R

Note that the real number - i s always well defined by the formula given in the next theorem, since the symmetric matrix (a,p(y)) is positive definite. This implies in particular that the radius of curvature never vanishes along a curve o n a surface 8 ( w ) defined by a mapping 8 satisfying the assumptions of the next theorem, hence in particular of class C2 o n w.

It is intuitively clear that if R = 0, the mapping 8 “cannot be too smooth”. Think of a surface made of two portions of planes intersecting along a segment, which thus constitutes a fold on the surface. Or think of a surface 8(u) with 0 E w and B(y1, y2) = )ylJ1+“ for some 0 < Q < 1, so that 8 E C1(u;E3) but 8 $! C2(w;E3): The radius of curvature of a curve corresponding to a constant y2 vanishes at y1 = 0.


Figure 2.4-2: Curvature of a planar curve. Let y be a smooth enough planar curve, parametrized by its curvilinear abscissa s. Consider two points p ( s ) and p ( s + As) with curvilinear abscissae s and s + As and let Aq5(s) be the algebraic angle between the two normals u ( s ) and u ( s + As) (oriented in the usual way) to y at those points.

When As + 0, the ratio ~ has a limit, called the “curvature” of y at p ( s ) . If this limit is non-zero, its inverse R is called the “algebraic radius of curvature” of y at p ( s ) (the sign of R depends on the orientation chosen on y).

The point p ( s ) + Ru(s), which is intrinsically defined, is called the “center of curvature” of y at p ( s ) : It is the center of the “osculating circle” at p ( s ) , i.e., the limit as As + 0 of the circle tangent to y at p ( s ) that passes through the point p ( s + As). The center of curvature is also the limit as As + 0 of the intersection of the normals u ( s ) and u ( s + As). Consequently, the centers of curvature of y lie on a curve (dashed on the figure), called “la d6velopp6e” in French, that is tangent to the normals to y.

As

Theorem 2.4-1. Let w be an open subset of R2, let 8 E C2(w;E3) be an injective immersion, and let y E w be fixed.

Consider a plane P normal to 2 = 8 ( w ) at the point c = 8 ( y ) . The intersection P n 2 is a curve e on 2, which is the image e = 8(C) of a curve C in the set W. Assume that, in a suficiently small neighborhood of y, the restriction of C to this neighborhood is the image f ( I ) of an open interval I c R, where f = fQea : I + R is a smooth enough injective mapping that satisfies - df“ ( t )e , # 0, where t E I is such that

d t y = f ( t ) (Figure 2.4-3).

1 h

Then the curvature - of the planar curve C at c is given by the ratio R

where a,p ( y ) are the covariant components of the first fundamental f o r m


Figure 2.4-3: Curvature on a surface. Let P be a plane containing the vector A , which is normal to the surface w = O(w). The algebraic

bl(Y) A an(y)l a3(Y) =

1 h

R curvature - of the planar curve C = P n ij = e(C) at 9 = e(y) is given by the ratio

where a,p(y) and b,p(y) are the covariant components of the first and sec-

ond fundamental forms of the surface G at y and - (t) are the components of the

vector tangent to the curve C = f ( Z ) at y = f ( t ) = f"(t)e,. If - # 0, the center of

curvature of the curve c^ at Q is the point (y + Ra3(y)), which is intrinsically defined in the Euclidean space E3.

d f a

dt 1 R

o f G at y (Section 2.1) and

Proof. (i) We first establish a well-known formula giving the curva-

ture - of a planar curve. Using the notations of Figure 2.4-2, we note that

1 R

sinAq!(s) = v ( s ) . T ( S + As) = -{v(s + As) - v ( s ) } . T ( S + As),

so that


(ii) The curve ( 0 o f ) ( I ) , which is a przori parametrized by t E I , can be also parametrized by its curvilinear abscissa s in a neighborhood of the point g. There thus exist an interval f c I and a mapping p : J -+ P , where J c R is an interval, such that

(0 o f ) ( t ) = p(s) and (a3 o f ) ( t ) = v ( s ) for all t E f, s E J.

1 R By (i), the curvature - of is given by

- 1 = - - (s) du ' T ( S ) , R ds

where

dP ds dt

4 0 O f) (t) g ds

T ( S ) = - ( S ) = ____

1 R To obtain the announced expression for -, it suffices to note that

by definition of the functions b,p and that (Theorem 2.3-1 (b))

The knowledge of the curvatures of curves contained in planes normal t o i3 suffices for computing the cuFature of any curve on i3. More specifically, the radius of curvature R at e of any smooth enough curve

C (planar or not) on the surface 6 is given by = - where cp is

the angle between the "principal normal" to at Q and a3(y) and - is given in Theorem 2.4-1; see, e.g., Stoker [1969, Chapter 4, Section 121.

The elements b,p(y) of the symmetric matrix (b,p(y)) defined in Theorem 2.4-1 are called the covariant components of the second fundamental form of the surface i3 = 0(u) at y^ = O(y).

- coscp 1

1 R

R R'

An Introduction to Differential Geometry in B3 59

2.5 Principal curvatures; Gaussian curvature

The analysis of the previous section suggests that precise information about the shape of a surface G = O(w) in a neighborhood of one of its points $ = O(y) can be gathered by letting the plane P turn around the normal vector a3(~) and by following in this process the variations of the curvatures at Q of the corresponding planar curves P n G, as given in Theorem 2.4-1.

As a first step in this direction, we show that these curvatures span a compact interval of R. In particular then, they "stay away from infinity".

Note that this compact interval contains 0 if, and only if, the radius of curvature of the curve P n G is infinite for at least one such plane P.

Theorem 2.5-1. (a) Let the assumptions and notations be as in Theorem 2.4-1. For a fixed y E w, consider the set P of all planes P normal to the surface G = O(w) at Q = O(y). Then the set of curvatures of the associated planar curves P n G, P E P, is a compact interval of

R, denoted [ -, -1 . 1 1

Rl(Y) R2(Y) (b) Let the matrix (bt(y)), Q being the row index, be defined by

bp,(Y) := aP"(Y)bao(Y),

where (a"P(y)) = (a,p(y))-l (Section 2.2) and the matrix (b,p(y)) is defined as in Theorem 2.4-1. Then

1 1 (c) 0 - # - there is a unique pair of orthogonal

Pl E P and P2 E P such that the curvatures of the associated curves PI n G and P2 n G are precisely - and -

Rl(Y) Ra(Y)'

1 1

Rib) RdYY

planes

planar

Proof. (i) Let A ( P ) denote the intersection of P E P with the tangent plane T to the surface G at Q, and let c ( P ) denote the intersection of P with G. Hence A ( P ) is tangent to E(P) at Q E 2.

In a sufficiently small neighborhood2f Q the restriction of the curve 6 ( P ) to this neighborhood is given by C(P) = (8 o f ( P ) ) ( I ( P ) ) , where I ( P ) c R is an open interval and f ( P ) = f a ( P ) e , : I ( P ) -+ R2 is a

smooth enough injective mapping that satisfies ~ df"(P) (t)ea # 0, where dt


t E I(P) is such that y = f (P) ( t ) . Hence the line A(P) is given by

A ( P ) = { Q + X d ( e o ~ ( P ) ) ( t ) ; X ~ R } ={Q+X["U,(~);XEE%},

where E" := df"(P) - (t) and taea # 0 by assumption. Since the line {y + pLJae,;p E R} is tangent to the curve C ( P ) :=

K 1 ( e ( P ) ) at y E w (the mapping 8 : w -+ R3 is injective by assumption) for each such parametrizing function f ( P ) : I(P) + R2 and since the vectors a,(y) are linearly independent, there exists a bijection between the set of all lines A(P) c T , P E P , and the set of all lines supporting the nonzero tangent vectors to the curves C(P).

Hence Theorem 2.4-1 shoTs tha: when P varies in P , the curvature of the corresponding curves C = C(P) at takes the same values as

dt

(ii) Let the symmetric-matrices A and B of order two be defined by

A := (a,p(y)) and B := (b,p(y)).

Since A is positive definite, it has a (unique) square root C, i.e., a symmebric positive definite matrix C such that A = C2. Hence the ratio

is nothing but the Rayleigh quotient associated with the symmetric matrix C-lBC-'. When Q varies in R2 - { 0 } , this Rayleigh quotient thus spans the compact interval of R whose end-points are the smallest and

largest eigenvalue, respectively denoted - and - , of the matrix

C-lBC-' (for a proof, see, e.g., Ciarlet [1982, Theorem 1.3-11). This proves (a).

1 1 Rl(Y) R2(Y)

F'urt hermore, the relation

c(c-~Bc-~)c-~ = ~ c - 2 = B A - ~

shows that the eigenvalues of the symmetric matrix C-'BC-l coincide with those of the (in general non-symmetric) matrix BA-'. Note that BA-' = (b t (y) ) with bi(y) := aoU(y)b,,(y), QI being the row index, since A-l = (a@(y)).

Hence the relations in (b) simply express that the sum and the product of the eigenvalues of the matrix BA-I are respectively equal to its


trace and to its determinant, which may be also written as det(bC@ (9)) d e t ( a 4 (Y))

since BA-l = (bt(y)) . This proves (b).

(:) (iii) Let q1 = ($) = Ctl and q2 = (“a) = Cez, with = rll V 2

and E2 = (:a) , be two orthogonal (qTq2 = 0) eigenvectors of the

1 symmetric matrix C-lBC-l , corresponding to the eigenvalues -

and - respectively. Hence R1 (Y) 1

R2(Y)’

T T 0 = qTq2 = C Ct2 = (:At2 = 0,

since CT = C. By (i), the corresponding lines A(P1) and A(P2) of the tangent plane are parallel to the vectors Eya,(y) and &‘ap(y), which are orthogonal since

{ I F % m } . {t,Paa(d} = %a(Y)Ix,P = E T M 2

1 the directions of the vectors ql and q2 are

uniquely determined and the lines A(P1) and A(P2) are likewise uniquely

1 If -

Rl(Y) # ma7 determined. This proves (c) .

We are now in a position to state several fundamental definitions: The elements b t ( y ) of the (in general non-symmetric) matrix (b t (y) )

defined in Theorem 2.5-1 are called the mixed components of the second fundamental form of the surface i3 = O(w) at 9 = 8(y).

(one or both possibly equal to The real numbers - and -

0) found in Theorem 2.5-1 are called the principal curvatures of Lj at

1 1

Rib) R2(Y)

the curvatures of the planar curves P n 2 are the If---

- 0, the 1 1

same in all directions, i.e., for all P E ‘P. If - = - - 1

point = 8(y) is called a planar point. If __ - - - # 0,fjis 1

called an umbilical point. It is remarkable that, i f all the points of i3 are planar, then i3 is a

portion of a plane. Likewise, if all the points of i3 are umbilical, then i3 is a portion of a sphere. For proofs, see, e.g., Stoker [1969, p, 87 and p. 991.

1 - 1 Rib) RZ(Yl)

Rib) RAY)

Rl(Y) RZ(Y)


Let c = 8 ( y ) E w be a point that is neither planar nor umbilical; in other words, the principal curvatures at S are not equal. Then the two orthogonal lines tangent to the planar curves PI nw and P2 nw (Theorem 2.5-1 (c)) are called the principal directions at y .

A line of curvature is a curve on 2 that is tangent to a principal direction at each one of its points. It can be shown that a point that is neither planar nor umbilical possesses a neighborhood where two orthogonal families of lines of curvature can be chosen as coordinate lines. See, e.g., Klingenberg [1973, Lemma 3.6.61.

1 1 If - # 0 and - # 0, the real numbers R l ( y ) and & ( Y ) are

Rl(Y) RdY) 1

called the principal radii of curvature of 2 at c. If, e.g., -

the corresponding radius of curvature R l ( y ) is said to be infinite, according to the convention made in Section 2.4. While the principal radii of curvature may simultaneously change their signs in another system of curvilinear coordinates, the associated centers of curvature are intrinsically defined.

which are the

principal invariants of the matrix (bE(y ) ) (Theorem 2.5-l), are respectively called the mean curvature and the Gaussian, or total, curvature of the surface w at y .

A point on a surface is an elliptic, parabolic, or hyperbolic, point according as its Gaussian curvature is > 0,= 0 but it is not a planar point, or < 0; see Figure 2.5-1.

An asymptotic line is a curve on a surface that is everywhere tangent to a direction along which the radius of curvature is infinite; any point along an asymptotic line is thus either parabolic or hyperbolic. It can be shown that, i f all the points of a surface are hyperbolic, any point possesses a neighborhood where two intersecting families of asymptotic lines can be chosen as coordinate lines. See, e.g., Klingenberg [1973, Lemma 3.6.121.

As intuitively suggested by Figure 2.4-1, a surface in R3 cannot be defined by its metric alone, i.e., through its first fundamental form alone, since its curvature must be in addition specified through its second fundamental form. But quite surprisingly, the Gaussian curvature at a point can also be expressed solely in terms of the functions a,p and their derivatives! This is the celebrated Theorema Egregium (“astonishing theorem”) of Gauss [1828]; see Theorem 2.6.2 in the next section.

Another striking result involving the Gaussian curvature is the equally celebrated Gauss Bonnet theorem, so named after Gauss [1828] and Bonnet [1848] (for a “modern” proof, see, e.g., Klingenberg [1973, The- orem 6.3-51 or do Carmo [1994, Chapter 6, Theorem 11): Let S be a

R I ~ Y ) = O’

1 1 1 The numbers - ( - + -) and

2 Rl(Y) R 2 ( Y ) R1 ( Y M 2 ( Y ) ’


h

A

I I

Figure 2.5-1: Different kinds of points o n a surface. A point is elliptic if the Gaussian curvature is > 0 or equivalently, if the two principal radii of curvature are of the same sign; the surface is then locally on one side of its tangent plane. A point is parabolic if exactly one of the two principal radii of curvature is infinite; the surface is again locally on one side of its tangent plane. A point is hyperbolic if the Gaussian curvature is < 0 or equivalently, if the two principal radii of curvature are of different signs; the surface then intersects its tangent plane along two curves.

smooth enough, “closed”, “orientable”, and compact surface in R3 (a “closed” surface is one “without boundary”, such as a sphere or a torus;


“orientable” surfaces, which exclude for instance Klein bottles, are defined in, e.g., Klingenberg [1973, Section 5.51) and let K : S -+ R denote its Gaussian curvature. Then

K(i7)dW) = 2 ~ ( 2 - 2g(S)),

where the genus g(S) is the number of “holes” of S (for instance, a sphere has genus zero, while a torus has genus one). The integer x(s) defined by x(S) := (2 - 2g(S)) is the Euler characteristic of s.

According to the definition of Stoker [1969, Chapter 5, Section 21, a developable surface is one whose Gaussian curvature vanishes everywhere. Developable surfaces are otherwise often defined as “ruled” surfaces whose Gaussian curvature vanishes everywhere, as in, e.g., Klin- genberg [1973, Section 3.71). A portion of a plane provides a first example, the only one of a developable surface all points of which are planar. Any developable surface all points of which are parabolic can be likewise fully described: It is either a portion of a cylinder, or a portion of a cone, or a portion of a surface spanned by the tangents to a skewed curve. The description of a developable surface comprising both planar and parabolic points is more subtle (although the above examples are in a sense the only ones possible, at least locally; see Stoker [1969, Chapter 5, Sections 2 to 61).

The interest of developable surfaces is that they can be, a t least locally, continuously “rolled out”, or “developed” (hence their name), onto a plane, without changing the metric of the intermediary surfaces in the process.

More details about these various notions are found in classic texts such as Stoker [1969], Klingenberg [1973], do Carmo [1976], Berger & Gostiaux [1987], Spivak [1999], or Kiihnel [2002].

2.6 Covariant derivatives of a vector field defined on a surface; the Gauss and Weingarten formulas

As in Sections 2.2 and 2.4, consider a surface i3 = 8 ( w ) in E3, where 8 : w c R2 + E3 is a smooth enough injective immersion, and let

Then the vectors a,(y) (which form the covariant basis of the tangent plane to 2 at ?j = B(y); see Figure 2.1-1) together with the vector a 3 ( y ) (which is normal to i3 and has Euclidean norm one) form the covariant basis at y .

An Introduction to Differential Geometry in IK3 65

Let the vectors u a ( y ) of the tangent plane to G at be defined by the relations u " ( y ) . u p ( y ) = 6;. Then the vectors u a ( y ) (which form the contravariant basis of the tangent plane at see again Figure 2.1-1) together with the vector u3(y) form the contravariant basis at 5; see Figure 2.6-1. Note that the vectors of the covariant and contravariant bases at y satisfy

uZ(y). U j ( Y ) = 8;.

Suppose that a vector field is defined on the surface G. One way to define such a field in terms of the curvilinear coordinates used for defining the surface i3 consists in writing it as viai : w 4 B3, i.e., in specifying its covariant components qi : w + IK over the vector fields uz formed by the contravariant bases. This means that qi(y)ui(y) is the vector at each point c= 8(y) 6

Our objective in this section is to compute the partial derivatives &(qiui) of such a vector field. These are found in the next theorem,

(Figure 2.6-1).

Figure 2.6-1: Contravariant bases and vector fields along a surface. At each point y^ = 8(y) E G = O(w), the three vectors a2(y), where a " ( y ) form the contravariant

basis of the tangent plane to ij at y^ (Figure 2.1-1) and a3(y) =

the contravariant basis at 2. An arbitrary vector field defined on G may then be defined by its covariant components qi : w 4 R. This means that q2(y)aZ(y) is the vector at the point Q.

al(Y) A az(y) , form b l ( Y ) A az(y)l


as immediate consequences of two basic formulas, those of Gauss and Weingarten. The ChristofSel symbols “on a surface” and the covariant derivatives of a vector field defined on a surface are also naturally introduced in this process.

A word of caution. The Christoffel symbols “on a surface” introduced in this section and the next one, viz., and r a p T , are thus denoted by the same symbols as the “three-dimensional” Christoffel symbols introduced in Sections 1.4 and 1.5. No confusion should arise, however. 0

Theorem 2.6-1. Let w be an open subset of R2 and let 6 E C2(w;E3) be an immersion.

(a) The derivatives of the vectors of the covariant and contravariant bases are given by

8,ap = I’&a, + bapa3 and = -I’guaa + b i a 3 , &a3 = &a3 = -bapaP = -b;a,,

where the covariant and mixed components b,p and bc of the second fundamental form of 2 are defined in Theorems 2.4-1 and 2.5-1 and

FEp := a,. &up.

(b) Let there be given a vector field qiaa : w -+ R3 with covariant components qi E C1(w). Then Viai E C’(w) and the partial derivatives &(viai) E Co(w) are given by

8a(rliai) = (&qp - rzpqu - bap773)aP + (&q3 + btqp)a3

= (vpla: - bapr/3)ap + (v3/a + b t q p ) a 3 ,

vpla := aaqp - C p q U and r/3la := 8,773.

where

Proof. Since any vector c in the tangent plane can be expanded as c = (c . a p ) a p = (c . a‘)a,, since &a3 is in the tangent plane (&a3 . a3 = . a3) = 0), and since &a3 . u p = -b,p (Theorem 2.4-l), it follows that

&a3 = (&a3 . a p ) a P = -bapaP.

This formula, together with the definition of the functions b! (The- orem 2.5-l), implies in turn that

= (&a3 . a‘)a, = -b,p(aP . ag)a, = -b,paPua, = -b“a a cr’


Any vector c can be expanded as c = (c . ai)ai = (c . aj)aj. In particular,

a,ag = (d,ap . au)a, + (&a0 . a3)a3 = r&a, + b,pa3,

by definition of and bag. Finally,

&aP = (&ap. a,)a" + (&ap . a3)a 3 = -riuau + bEa3,

since &aP. a3 = -aB. &a3 = bEa, . a0 = 01'

That viai E C'(w) if vi E C'(w) is clear since ai E C ' ( w ) if 8 E C2(w; E3). The formulas established supra immediately lead to the an-

0 nounced expression of d, (viai).

The relations (found in Theorem 2.6-1)

a,ap = rgpa, + ba0a3 and &aP = --J?iUau + bta3

a,a3 = &a3 = -b,gaP = -b"a 01 0,

and

respectively constitute the formulas of Gauss and Weingarten. The functions (also found in Theorem 2.6-1)

vgIa = - r",% and q3la = 801773

are the first-order covariant derivatives of the vector field viai : w + R3, and the functions

I'Eg := a,. d,ag = -&a". a g

are the Christoffel symbols of the second kind (the Christoffel symbols of the first kind are introduced in the next section).

Remark. The Christoffel symbols rEg can be also defined solely in terms of the covariant components of the first fundamental form; see the proof of Theorem 2.7-1. 0

The definition of the covariant derivatives v,lg = 8077, - l?EPvu of a vector field defined on a surface 8 ( w ) given in Theorem 2.6-1 is highly reminiscent of the definition of the covariant derivatives wi i l j = djwi - rpjwp of a vector field defined on an open set O(R) given in Section 1.4. However, the former are more subtle to apprehend than the latter. To see this, recall that the covariant derivatives willj = djvi - I?~'wp may be also defined by the relations (Theorem 1.4-2)

vi,,jgJ = aj(wigz).


By contrast, even if only tangential vector fields qaaa on the surface e ( w ) are considered (i.e., vector fields viai : w + EX3 for which 773 = 0), their couariant derivatives qaip = doq, -r" apqu satisfy only the relations

Valpa" = p {a/3(rlaaa)) , where P denotes the projection operator on the tangent plane in the direction of the normal vector (i.e., P(ciu2) ':= c,aa), since

ap(~aa") = 77,1paa + b;77aa3

for such tangential fields by Theorem 2.6-1. The reason is that a surface has in general a nonzero curvature, manifesting itself here by the "extra term" b;qaa3. This term vanishes in w if D is a portion of a plane, since in this case b; = bap = 0. Note that, again in this case, the formula giving the partial derivatives in Theorem 2.9-1 (b) reduces to

aa(77iai) = (77i,,)Ui.

2.7 Necessary conditions satisfied by the first and second fundamental forms: the Gauss and Codazzi-Mainardi equations; Gauss' Theorema Ggregium

It is remarkable that the components a,p = upa : w + R and b,p = bp, : w + E% of the first and second fundamental f o rms of a surface e(u), defined by a smooth enough immersion 8 : w + E3, cannot be arbitrary functions.

As shown in the next theorem, they must satisfy relations that take the form:

- aura,, + r$rorp - r&J'pTp = b,,bo7 - b,pb,, in w; sob,, - a,b,p + I'gubpp - r$b,, = 0 in w,

where the functions rap, and I?& have simple expressions in terms of the functions a,p and of some of their partial derivatives (as shown in the next proof, it so happens that the functions as defined in Theorem 2.7-1 coincide with the Christoffel symbols introduced in the previous section; this explains why they are denoted by the same symbol).

These relations, which are meant to hold for all a, p, c, T E {I, 2}, respectively constitute the Gauss, and Codazzi-Mainardi, equations.

Theorem 2.7-1. Let w be an open subset of R2, let 0 E C3(w;E3) be an immersion, and let


denote the covariant components of the first and second fundamental f o rms of the surface O ( W ) . Let the functions rapT E C' (W) and rgP E C' (W) be defined by

1 rap , := Z(8paar + &upT - &aap),

:= uurl?,pT where (auT) := (u,p)-'.

Then, necessarily,

dpraUT - aura,, + r$rUTp - r:,rpTp = b d p , - b,pb,, in W ,

apb,, - &b,p + r;,bp, - rEsb,, = 0 in w .

Proof. Let a, = 8,O. It is then immediately verified that the functions rapr are also given by

r a g r = &up . a,.

Let a3 = A a2 and, for each y E W , let the three vectors aj(y) be

defined by the relations aj (y) .a i ( y ) = 6i . Since we also have ap = uQBa, and a3 = a3, the last relations imply that I'gp = &ap . a,, hence that

la1 A a21

a,ap = r z p a , + bapa3,

since 8,ap = (&a, . a')a, + (&up . a3)a3. Differentiating the same relations yields

%rap, = %,up. a, + & u p . &a,,

so that the above relations together give

& u p . &ar = r$a, . &a, + bapa3 . duar = 17~pI',Tp + bapbur.

Consequently,

&,ap . a, = W , p , - r$&,, - b,pb,,.

Since &,up = aapa,, we also have

dauap . a, = dpr,,, - ~:J'p,, - baubpr.

Hence the Gauss equations immediately follow.

-&a, . a3, we have Since &a3 = (&a3. a,)a' + (&a,. a3)a3 and &a,. a, = -bau =

&a3 = -bauau.


Differentiating the relations b,p = & u p a3, we obtain

8, b,p = a,, u p . a3 + 8, u p . &a,.

This relation and the relations &ap = rZpa, + b,pas and &a3 = -b,,aU together imply that

Lieup ' &a3 = -r$bup.

Consequently, &,ap . a3 = a,baP I- r$b,,.

Since a,,ap = &paa, we also have

a,,ap . a3 = apb,, + I'z,bpp.

Hence the Codazzi-Mainardi equations immediately follow.

Remark. The vectors a, and u p introduced above respectively form the covariant and contravariant bases of the tangent plane to the surface Q(u), the unit vector a3 = a3 is normal to the surface, and the functions affP are the contravariant components of the first fundamental

As shown in the above proof, the Gauss and Codazzi-Mainardi equations thus simply constitute a re-writing of the relations &,up = &pa, in the form of the equivalent relations &,up . aT = &pa, . aT and &,up. a3 = i3,pa, . a3.

form (Sections 2.2 and 2.3).

The functions

and = a U T a p r = d,ap . a' = r;a

are the Christoffel symbols of the first, and second, kind. We recall that the Christoffel symbols of the second kind also naturally appeared in a different context (that of covariant differentiation; cf. Section 2.6).

Finally, the functions

sTapu := aprauT - aUrap,. + r:pruTp - rzurpTP are the covariant components of the Riemann curvature tensor of the surface Q(u).

and r a p r imply that the sixteen Gauss equations are satisfied if and only if they are satisfied for a: =

The definitions of the functions


1, ,D = 2, 0 = 1, r = 2 and that the Codazzi-Mainardi equations are satisfied if and only if they are satisfied for a = 1, p = 2, u = 1 and a = 1, p = 2, u = 2 (other choices of indices with the same properties are clearly possible).

In other words, the Gauss equations and the Codazzi-Mainardi equations in fact respectively reduce to one and two equations.

Letting a = 2, p = 1, u = 2, r = 1 in the Gauss equations gives in particular

S1212 = det(b,p)

Consequently, the Gaussian curvature at each point O ( y ) of the surface O(w) can be written as

- det(bap(y) (Theorem 2.5-1). By inspection of the since

function ,91212, we thus reach the astonishing conclusion that, at each point of the surface, a notion involving the “curvature” of the surface, viz., the Gaussian curvature, is entirely determined by the knowledge of the “metric” of the surface at the same point, viz., the components of the first fundamental forms and their partial derivatives of order < 2 at the same point! This startling conclusion naturally deserves a theorem:

1

Rl (Y)R2(Y) - det(aap(y))

Theorem 2.7-2. Let w be a n open subset of R2, let 8 E C3(w; E3) be a n immersion, let uap = d,B .ape denote the covariant components of the first fundamental f o r m of the surface O(w), and let the functions rapT and Sl212 be defined by

1

1

rap7 := Z(apaaT + &upT - dTaap) ,

s~~~~ := p 1 2 a 1 2 - &la22 - dz2all) + ~ ~ ~ ( r ~ ~ , r ~ ~ ~ - r11ar22B).

Then, at each point O(y) of the surface O(w), the principal curvatures 1 - and Rz(y) satisfy R1 (Y)

1

Theorem 2.7-2 constitutes the famed Theorema Egregium of Gauss [1828], so named by Gauss who had been himself astounded by his discovery.


2.8 Existence of a surface with prescribed first and second fundamental forms

Let M2, S2, and S: denote the sets of all square matrices of order two, of all symmetric matrices of order two, and of all symmetric, positive definite matrices of order two.

So far, we have considered that we are given an open set w c R2 and a smooth enough immersion 0 : w -+ E3, thus allowing us to define the fields (a,p) : w -+ S: and (b,p) : w -+ S2, where a,p : w -+ R and b,p : w -+ R are the covariant components of the first and second fundamental forms of the surface 0(w) c E3.

Note that the immersion 0 need not be injective in order that these matrix fields be well defined.

We now turn to the reciprocal questions: Given an open subset w of R2 and two smooth enough matrix fields

(a,p) : w + S: and (b,p) : w + S2, when are they the first and second fundamental forms of a surface O(w) c E3, i.e., when does there exist an immersion 0 : w -+ E3 such that

If such an immersion exists, to what extent is it unique? The answers to these questions, which turn out to be remarkably

simple, constitute the fundamental theorem of surface theory: If w is simply-connected, the necessary conditions of Theorem 2.7-1, i.e., the Gauss and Codazzi-Mainardi equations, are also suficient for the existence of such an immersion. If w is connected, this immersion is unique up to isometries in E3.

Whether an immersion found in this fashion is injective is a different issue, which accordingly should be resolved by different means.

Following Ciarlet & Larsonneur [2001], we now give a self-contained, complete, and essentially elementary, proof of this well-known result. This proof amounts to showing that it can be established as a simple corollary to the fundamental theorem on flat Riemannian manifolds established in Theorems 1.6-1 and 1.7-1 when the manifold is an open set in R3.

This proof has also the merit to shed light on the analogies (which cannot remain unnoticed!) between the assumptions and conclusions of both existence results (compare Theorems 1.6-1 and 2.8-1) and both uniqueness results (compare Theorems 1.7-1 and 2.9-1).

A direct proof of the fundamental theorem of surface theory is given in Klingenberg [1973, Theorem 3.8.81, where the global existence of the mapping 0 is based on an existence theorem for ordinary differential equations, analogous to that used in part (ii) of the proof of Theorem


1.6-1. A proof of the "local" version of this theorem, which constitutes Bonnet's theorem, is found in, e.g., do Carmo [1976].

This result is another special case of the fundamental theorem of Rie- mannian geometry alluded to in Section 1.6. We recall that this theorem asserts that a simply-connected Riemannian manifold of dimension p can be isometrically immersed into a Euclidean space of dimension ( p + q ) if and only if there exist tensors satisfying together generalized Gauss, and Codazzi-Mainardi, equations and that the corresponding isometric immersions are unique up to isometries in the Euclidean space. A substan- tial literature has been devoted to this theorem and its various proofs, which usually rely on basic notions of Riemannian geometry, such as connections or normal bundles, and on the theory of differential forms. See in particular the earlier papers of Janet [1926] and Cartan [1927] and the more recent references of Szczarba [1970] , Tenenblat [1971], Jacobowitz [1982], and Szopos [ZOOS].

Like the fundamental theorem of three-dimensional differential geometry, this theorem comprises two essentially distinct parts, a global existence result (Theorem 2.8-1) and a uniqueness result (Theorem 2.9- I), the latter being also called rigidity theorem. Note that these two results are established under different assumptions on the set w and on the smoothness of the fields (a,p) and (b,p).

These existence and uniqueness results together constitute the fundamental theorem of surface theory.

Theorem 2.8-1. Let w be a connected and simply-connected open subset of JR2 and let (a,p) E C2(w;S;) and (b,p) E C 2 ( w ; S 2 ) be two matrix fields that satisfy the Gauss and Codazzi-Mainardi equations, viz.,

where

1 rapr := 5 (dpa,, + 8aagr - araap),

17zp := aurI',p, where (aur) := (a,p)-'.

Then there exists a n immersion 6 E C3(w; E3) such that

a,p = d,6 . dp6 and b,p = in w.

Proof. The proof of this theorem as a corollary to Theorem 1.6-1 relies on the following elementary observation: Given a smooth enough


immersion 8 : w + E3 and E > 0, let the mapping 0 : w x 1 - E , E [ + E3 be defined by

O ( y , x3) := 8 ( y ) + x s a s ( y ) for all ( y , x 3 ) E w x 1 - E , E [ ,

A and let 1818 A 8281 ’ where a3 :=

gi j := 820 ’ ajo.

Then an immediate computation shows that 2 gap = a,p - 2x3bap + x 3 c a ~ and gi3 = hi3 in w x I-&, E [ ,

where aap and bap are the covariant components of the first and second fundamental forms of the surface 8 ( w ) and cap := aU7b,,bp7.

Assume that the matrices ( g i j ) constructed in this fashion are invertible, hence positive definite, over the set w x ]-&, &[ (they need not be, of course; but the resulting difficulty is easily circumvented; see parts (i) and (viii) below). Then the field ( g i j ) : w x I - & , € [ + S,”> becomes a natural candidate for applying the “three-dimensional” existence result of Theorem 1.6-1, provided of course that the iithree-dimensional” su f ic ien t conditions of this theorem, viz.,

ajrikq - akrijq + rTjrkqp - r Q j q P = o in R,

can be shown to hold, as consequences of the “two-dimensional” Gauss and Codazzi-Mainardi equations. That this is indeed the case is the essence of the present proof (see parts (i) to (vii)).

By Theorem 1.6-1, there then exists an immersion 0 : w x I-&, E [ -i

E3 that satisfies gi j = ai0 . a j 0 in w x ]-el &[. It thus remains to check that 0 := O(., 0) indeed satisfies (see part (ix))

The actual implementation of this program essentially involves elementary, albeit sometimes lengthy, computations, which accordingly will be omitted for the most part; only the main intermediate results will be recorded. For clarity, the proof is broken into nine parts, numbered (i) to (ix).

To avoid confusion with the “three-dimensional” Christoffel symbols, those “on a surface” will be denoted C& and Cap, in this proof (and only in this proof).

(i) Given two matrix fields (a,p) E C2(w; S;) and (b,p) E C2(w; S2) , let the matrix field ( g i j ) E C2(w x R;S3) be defined by

gap := aao - 2x3b,p + x3cap 2 and gi3 := hi3 in w x R

An Introduction to Differential Geometry in EX3 75

(the variable y E w is omitted; x3 designates the variable in R), where

cap := bLbpr and b: := auTbau in w.

Let wo be an open subset of R2 such that WO is a compact subset of w . Then there exists EO = E O ( W O ) > 0 such that the symmetric matrices ( g i j ) are positive definite at all points in no, where

Ro := wo x ] - E o , E o [ .

Besides, the elements of the inverse matrix ( g p q ) are given in 20 by

gap = x(n + l)x;a""(B"): and gi3 = di3 , n > O

where (B)! := b! and (B")! := b: . . . bEnPl for n 2 2,

i.e., (B")! designates for any n 2 0 the element at the a-th row and p- th column of the matrix B". The above series are absolutely convergent in the space C2 (no).

Let a priori g"p = Cn20 x:hEp where h,"p are functions of y E GO only, so that the relations gapgp7 = 6; read

It is then easily verified that the functions h$ are given by

h$ = (n + l ) a a g ( B n ) ! , n 2 0 ,

so that

n ) O

It is clear that such a series is absolutely convergent in the space C2(W0 x [ -Eo, E O ] ) if EO > 0 is small enough.

(ii) The functions Cgp being defined by

C& : = aaT Cap,,

where


define the functions

Furthermore, the assumed Codazzi-Mainardi equations imply that

b7,lp = b';la and baulp = baplu.

The above relations follow from straightforward computations based on the definitions of the functions b7,lp and bapiu. They are recorded here because they play a pervading r61e in the subsequent computations.

part (i), define the functions rij, E C1(Qo) and (iii) The functions gij E C2(D0) and gi j E C2(Do) being defined as in

E C1(no) by

1 r i jq := z(8jgiq + &gj, - 8,gij) and rfj := gPqrij,.

Then the functions F i j q = rjiq and = have the following expressions:

rap, Capo - x3(b7,lpaTU + 2C&bT,) + Xi(b7,1pbTu + C&cTu),

raP3 = -ra3P = baP - x3cap,

ra33 = r3p3 = r33q = 0,

r ip = bap - x3cap,

C p = c:p - c,,, x;+'bZlIp(B"):,

rt3 = -C n > O xz (B"+')E,

r;, = rg3 = 0,

where the functions cap, (B"):, and bLlp are defined as in parts (i) and (ii) .

All computations are straightforward. We simply point out that the assumed Codazzi-Mainardi equations are needed to conclude that the factor of x3 in the function rapu is indeed that announced above. We also note that the computation of the factor of xg in rapu relies in particular on the easily established relations

&cpo = b';(,b0, + b g ( d p p + C&cup + CEucpp.


(iv) The functions rijq E Cl (n0) and rfj E C'(n0) being defined as in part (iii), define the functions R q i j k E co((sz,) by

R q i j k := d j r i k q - a,+rijq -k r y ? k q p - r Y k r j q p .

Then , in order that the relations

hold, it is su f ic ien t that

The above definition of the functions R q i j k and that of the functions rijq and I?$ (part (iii)) together imply that, for all i,j, k , q ,

Consequently, the relation R 1 2 1 2 = 0 implies that Rapg7 = 0, the relations Rcu2p3 = 0 imply that R q i j k = 0 if exactly one index is equal to 3, and finally, the relations R a 3 p 3 = 0 imply that R q i j k = 0 if exactly two indices are equal to 3.

(v) The functions

satisfy R a 3 p 3 = 0 in no.

These relations immediately follow from the expressions found in part (iii) for the functions rijq and rp.. Note that neither the Gauss equations nor the Codazzi-Mainardi equations are needed here. a?

(vi) The functions

R a 2 P 3 := apr2301 - a3rZpa + r;pr3cup - r;3rpap

satisfy - R a 2 p 3 = 0 in Ro.

The definitions of the functions gap (part (i)) and rijq (part (iii)) show that


Then the expressions found in part (iii) show that

r;pr3ap - rg3rpap = r;ar2pu - r;3rapu = C Z p b u - c;pbau

+ ~ 3 ( b g l p b a u - bzlpb2u + Cipcau - C&c20),

and the relations R a 3 p 3 = 0 follow by making use of the relations

dacpu = bilabur + b f4 labpp + CLpcup + Cgucpp

together with the relations

d2bap - dab20 + C&b2u - CZpbau = 0,

which are special cases of the assumed Codazzi-Mainardi equations.

(vii) The function

R~~~~ := - d2rzll + rglrZlp - rg2rllp satisfies

R l z l 2 = 0 in no. The computations leading to this relation are fairly lengthy and they

require some care. We simply record the main intermediary steps, which consist in evaluating separately the various terms occurring in the function R 1 2 1 2 rewritten as

R~~~~ = (w221 - d2rzll) + (r72r12u - r71r22u) + (r123r123 - ~ 1 1 3 r 2 2 3 ) .

First, the expressions found in part (iii) for the functions rap3 easily yield

r123r123 - rll3rZz3 = ( b f 2 - + ~ ( b i i ~ z 2 - 2 b 1 2 ~ 1 2 + b 2 2 ~ 1 i ) + xz(cf2 - ~ 1 1 ~ 2 2 ) .

Second, the expressions found in part (iii) for the functions rapu and yield, after some manipulations:


Third, after somewhat delicate computations, which in particularmake use of the relations established in part (ii) about the functions6 |/3 and bap\a, it is found that

where the functions

are precisely those appearing in the left-hand sides of the Gauss equa-tions.

It is then easily seen that the above equations together yield

-Rl212 = {S1212 ~

— S3{

-hxj{SaTl2blbT2 + (c12c12 -

Since

C12C12 - C11C22 = (biibi2 - hib22){blbl - b\b\),

it is finally found that the function -R1212 has the following remarkableexpression:

#1212 = {S1212 - (&11&22 " ^126l2)}{l - X3(b\ + 6|) + x\{b\b22 - b\b\)}.

By the assumed Gauss equations,

51212 = 6ll&22

Hence R1212 = 0 as announced.(viii) Let w be a connected and simply-connected open subset of R2.

Then there exist open subsets we,e 2 0, of R2 such that We is a compactsubset of w for each e0 and

w=


Furthermore, for each C 2 0 , there exists Ee = &e(we) > 0 such that the symmetric matrices ( g i j ) are positive definite at all points in Ge, where

Re := we x I - E ~ , E [ [ .

Finally, the open set R := U Re

e>o

is connected and simply-connected. Let we, t 2 0, be open subsets of w with compact closures We C w

such that w = Ue20~e, For each C, a set Re := we x I - E ~ , E ~ [ can then be constructed in the same way that the set Ro was constructed in part

It is clear that the set R := Ue20 Re is connected. To show that R is simply-connected, let y be a loop in R, i.e., a mapping y E Co([O, 1];R3) that satisfies

(9.

y(0) = y(1) and y( t ) E R for all 0 6 t 6 1.

Let the projection operator 7r : R + w be defined by ~ ( y , 2 3 ) = y for all (y, x3) E R, and let the mapping po : [0,1] x [0,1] --t R3 be defined by

po(t ,X) := (1 - X)y(t) + X7r(y(t)) for all 0 Q t 6 I, O Q X 6 1.

Then po is a continuous mapping such that p o ( [ O , 11 x [0,1]) c R, by definition of the set R. Furthermore, po(t,O) = y ( t ) and p o ( t , l ) = 7r(y(t)) for all t E [0, I].

The mapping y:=7roy ECO([O,l];R2)

is a loop in w since T ( 0 ) = rr(y(0)) = x(y(1)) = y(1) and y ( t ) E w for all 0 Q t 6 1. Since w is simply connected, there exist a mapping p1 E Co([O, 11 x [0,1]; R2) and a point yo E w such that

p l ( t , 1) = 7 and p l ( t , 2) = yo for all 0 6 t 6 1

and pl( t ,X) E w for all 0 6 t < 1, 1 Q X 6 2.

Then the mapping p E Co([O, 11 x [0, 2];R3) defined by

p( t ,X) = po(t,X) p ( t , A) = pl(t, A)

for all 0 6 t 6 1, 0 < X 6 1, for all 0 Q t 6 1, 1 6 X 6 2,

is a homotopy in R that reduces the loop y to the point ( y o l o ) E R. Hence the set R is simply-connected.


(ix) B y parts (iv) to (viii), the functions rij, E C' (R) and I';' E

C1(R) constructed as in part (iii) satisfy

in the connected and simply-connected open set R. B y Theorem 1.6-1, there thus exists a n immersion 0 E C3(R; E3) such that

gij = 8iO. 8 j 0 in R,

where the matrix field (g i j ) E C2(R;S3>) is defined by

2 gap = a,p - 2~3b,p + x3cap and gi3 = 6 i 3 in R.

Then the mapping 8 E C3(w; E3) defined by

8 ( y ) = 0 ( y , O ) for all y E w :

satisfies

Let gi := 8i0. Then 8 3 3 0 = 83g3 = I'g3gp = 0; cf. part (iii). Hence there exists a mapping 8l E C3(w; E3) such that

@ ( Y , 2 3 ) = O ( Y ) + X ~ @ ( Y ) for all ( y , 2 3 ) E 0,

and consequently, g, = 8,8 + ~ 3 8 ~ 8 ' and g 3 = 8 l . The relations gi3 = gi . g 3 = &3 (cf. part (i)) then show that

(da8 + ~ 3 8 ~ 8 ' ) = 0 and 8l . 8 l = 1.

These relations imply that 8,8 . 8l = 0. 8 = -a3 in w , where

Hence either 8l = a3 or 1

dl8 A 828 ld18 A 8281 '

a3 :=

But 8' = -a3 is ruled out since

(818 A 828). 8l = det(gij)lz3,0 > 0.

Noting that

8,8. a3 = 0 implies 8,B. 8 p a 3 = - 8 4 3 . a3,

we obtain, on the one hand,

gap = ( a d + x d a a 3 ) . ( 8 ~ 8 + x38pa3)

= 8,8. 8p8 - 2x3dap8. a3 + xidaa3 . 8pa3 in R.


Since, on the other hand,

by part (i), we conclude that

a,p = d,8. ape and b,p = dap8. a 3 in w,

as desired. This completes the proof. 0

Remarks. (1) The functions cap = bzbp., = d,a3 . spa, introduced in part (i) are the covariant components of the third fundamental form of the surface O(w).

(2) The series expansion gap = c,,o(n + l ) x ~ a a f f ( B " ) ~ found in part (i) is known; cf., e.g., Naghdi [1972c

(3) The functions b:lp and b,plg introduced in part (ii) are the covariant derivatives of the second fundamental form of the surface 8(w); for details, see, e.g., [Ciarlet, 2005, Section 4.21.

(4) The Gauss equations are used only once in the above proof, for 0 showing that R l 2 1 2 = 0 in part (vii).

The regularity assumptions made in Theorem 2.8-1 on the matrix fields (a,p) and (b,p) can be significantly relaxed in several ways. First, Cristinel Mardare has shown by means of an ad hoe, but not trivial, modification of the proof given here, that the existence of an immersion 8 E C3(w;E3) still holds under the weaker (but certainly more natural, in view of the regularity of the resulting immersion 8) assumption that (b,p) E C1(w; S2), all other assumptions of Theorem 2.8-1 holding verbatim.

In fact, Hartman & Wintner [1950] had already shown the stronger result that the existence theorem still holds if (a,p) E C'(w;S;) and (baa) E Co(w;S2) , with a resulting mapping 8 in the space C2(u;E3). Their result has been itself superseded by that of S. Mardare [2003b], who established that if (asp) E Wll,',"(w; S;) and (hap) E Lgc(w; S2) are two matrix fields that satisfy the Gauss and Codazzi-Mainardi equations in the sense of distributions, then there exists a mapping 8 E W;"F(u; E3) such that (a,p) and (b,p) are the fundamental forms of the surface 8(w) . The last word in this direction seems to belong to S. Mardare [2005], who was able to further reduce these regularities, to those of the spaces W:d,p(w;S2>) and LFoc(w;S2) for any p > 2, with a resulting mapping 8 in the space W;":(w; E3).

An Introduction to Differential Geometry in JR3 83

2.9 Uniqueness up to proper isometries of surfaces with the same fundamental forms

In Section 2.8, we have established the existence of an immersion 8 : w c IR2 -+ E3 giving rise to a surface 8 ( w ) with prescribed first and second fundamental forms, provided these forms satisfy ad hoc sufficient conditions. We now turn to the question of uniqueness of such immersions.

This is the object of the next theorem, which constitutes another rigidity theorem, called the rigidity theorem for surfaces. It asserts that, if two immersions 2 E C 2 ( q E 3 ) and 8 E C2(u;E3) share the same fundamental forms, then the surface 8(w) is obtained by subjecting the surface g(w) to a rotation (represented by an orthogonal matrix Q with det Q = l), then by subjecting the rotated surface to a translation (represented by a vector c) .

Such a geometric transformation of the surface G(w) is sometimes called a “rigid transformation” , to remind that it corresponds to the idea of a “rigid” one in E3. This observation motivates the terminology “rigidity theorem” .

As shown by Ciarlet & Larsonneur [2001] (whose proof is adapted here), the issue of uniqueness can be resolved as a corollary to its “three- dimensional counterpart” , like the issue of existence. We recall that O3 denotes the set of all orthogonal matrices of order three. In addition, we let 0; := {Q E 03;de t Q = 1) denote the set of all proper orthogonal matrices of order three.

Theorem 2.9-1. Let w be a connected open subset of R2 and let 8 E C2(w; E3) and 2 E C2(w; E3) be two immersions such that their associated first and second fundamental forms satisfy (with self-explanatory

a,p = a,p and b,p = b,p in w.

Then there exist a vector c E E3 and a matrix Q E 0: such that

8(y) = c + QZ(y) for all y E w.

notations) - -

Proof. Arguments similar to those used in parts (i) and (viii) of the proof of Theorem 2.8-1 show that there exist open subsets we of w and real numbers ~g > 0, C .> 0, such that the symmetric matrices ( g i j ) defined by

gap := a,p - 2x3bap + x3c,p and gi3 = bi3, 2

where cap := a“‘b,,bp,, are positive definite in the set

Q := U W e x I - E ~ , E ~ [ .

e)o


The two immersions 0 E C1(R;E3) and 6 E C1(R;E3) defined by (with self-explanatory notations)

O(y, 23) := 8(y ) + 2303(y) and 6 ( y , 2 3 ) := 8(y) + 23&(y)

for all (y, q) E R therefore satisfy

g i j = & in 0.

By Theorem 1.7-1, there exist a vector c E E3 and an orthogonal matrix Q such that

~ ( y , x 3 ) = c + Q G ( ~ , x3) for all (y, x 3 ) E R.

Hence, on the one hand,

det VO(y, 23) = det Q det V6(y, 2 3 ) for all (y, 2 3 ) E R.

On the other hand, a simple computation shows that

for all ( y , ~ ) E 0, wliere

:= aP“(Y)b*o(Y), 3 E w ,

so that

det VO(y,z3) = det V 6 ( y , 2 3 ) for all (y,23) E 0.

Therefore det Q = 1, which shows that the orthogonal matrix Q is in fact proper. The conclusion then follows by letting 2 3 = 0 in the relation

@(y, 2 3 ) = c + Q&, 23) for all (y, 2 3 ) E 0.

A proper isometry of E3 is a mapping J+ : E3 + E3 of the form J+(z) = c + Qoz for all 2 E E3, with c E E3 and Q E 0;. Theorem 2.9-1 thus asserts that two immersions 8 E C1(w; E3) and is E C1 (w; E3) share the same fundamentaJ forms over an open connected subset w of R3 i f and only if 8 = J+ o 8, where J+ is a proper isometry of E3.

Remark. By contrast, the “three-dimensional” rigidity theorem (The- 0

Theorem 2.9-1 constitutes the “classical” rigjdity theorem for surfaces, in the sense that both immersions 8 and 8 are assumed to be in the space C2(w; E3).

orem 1.7-1) involves isometries of E3 that may not be proper.


As a preparation to our next result, we note that the second fundamental form of the surface 8(w) can still be defined under the weaker

E C1(w;E3) , by assumptions that 8 E C1(w;E3) and a3 =

means of the definition

a1 A a 2

la1 A a21

b,p := -acu. dpa3,

which evidently coincides with the usual one when 8 E C2(w; E3) . Following Ciarlet & C. Mardare [2004a], we now show that a sim-

ilar result holds under the assumptions that G E H 1 ( w ; E 3 ) and G3 := 61 A 6 2

E H1 (w; E3) (with self-explanatory notations). Naturally, our 161 A 621 first task will be to verify that the vector field 6 3 , which is not necessarily well defined a.e. in w for an arbitrary mapping 5 E H1(w; E 3 ) , is nevertheless well defined a.e. in w for those mappings G that satisfy the assumptions of the next theorem. This fact will in turn imply that the functions b,p := -a, . 8p63 are likewise well defined a.e. in w.

Theorem 2.9-2. Let w be a connected open subset of R2 and let 8 E C1(w; E 3 ) be an immersion that satisfies a 3 E C1(w; E3) . Assume that there exists a vector field 5 E H1(w; E3) that satisfies

I -

- - a,p = a,p a.e. in w, 5 3 E H 1 ( w ; E 3 ) ,

Then there exist a vector c E E3 and a matrix Q E 0: such that

and b,p = b,p a.e. in w.

- 8(y) = c + Q8(y) for almost all y E w.

Proof. The proof essentially relies on the extension to a Sobolev space setting of the “three-dimensional” rigidity theorem established in Theorem 1.7-3.

(i) To begin with, we record several technical preliminaries. First, we observe that the relations Zap = a,p a.e. in w and the

assumption that 8 E C1(w; E3) is an immersion together imply that

I

la1 A 6 2 1 = d m = Jdcto > o a.e. in w.

Consequently, the vector field 6 3 , and thus the functions b,p, are well defined a.e. an w.

,.,

Second, we establish that -

b,p = bp, in w and b,p = bp, a.e. in w ,


i.e., that a, . 8pa3 = ap . &a3 in w and Zi, . 8pZi3 = Zip ' daZ3 a.e. in w . To this end, we note that either the assumptions 8 E C1(w;E3) and a 3 E C 1 ( w ; E 3 ) together, or the assumptions 8 E H 1 ( w ; E 3 ) and a 3 E H 1 ( w ; E3) together, imply that a, . 8pa3 = 8,0 .8pa3 E LiOc(w), hence that 8,e. 8pa3 E D' (w) .

Given any cp E D ( w ) , let U denote an open subset of R2 such that supp cp c U and TI is a compact subset of w . Denoting by X' (., .)x the duality pairing between a topological vector space X and its dual X ' , we have

V ' ( W ) ' 8pa3i p ) v ( W ) = cpaCXO ' dy

Observing that a,8. a3 = 0 a.e. in w and that

= H - 1 ( u ; ~ 3 ) (8, (a, 0) 1 pa3 ) H A ( u ; ~3 ) 7

we reach the conclusion that the expression (8,8 . @a,, c p ) ~ ( ~ ) is symmetric with respect to Q and p since 8,pO = 8p,0 in ;D'(U). Hence 8,8 . dpa3 = ap0 . &a, in LtOc(w), and the announced symmetries are established.

Third, let - cap := d,Zi3, aPZ3 and cap := & a 3 . 8pa3.

Then we claim that .Zap = cap a.e. in w . To see this, we note that the matrix fields (Zap) := (Zap)-' and (amp) := (a,p)-l are well defined and equal a.e. in w since 8 is an immersion and Zap = a,p a.e. in w. The formula of Weingarten (Section 2.6) can thus be applied a.e. in w , showing that Cap = Zu7bu,b7p a.e. in w .

The assertion then follows from the assumptions b,p = b,p a.e. in w .

(ii) Starting from the set w and the mapping 0 (as given in the statement of Theorem 2.9-2), we next construct a set R and a mapping 0 that satisfy the assumptions of Theorem 1.7-2. More precisely, let

O(y, 2 3 ) := 8 ( g ) + 23a3(y) for all ( y , 2 3 ) E w x R.

Then the mapping 0 := w x IR + IE3 defined in this fashion is clearly continuously differentiable on w x R and

- - -


for all (y,x3) E w x R, where

Let wnr n 2 0 , be open subsets of R2 such that Zn is a compact subset of w and w = Un20wn. Then the continuity of the functions a,p, asp, b,p and the assumption that 8 is an immersion together imply that, for each n 3 0, there exists E, > 0 such that

det VO(y, 5 3 ) > 0 for all (y, 2 3 ) E zn x [ - E ~ , E ~ ] .

Besides, there is no loss of generality in assuming that E, < 1 (this property will be used in part (iii)).

Let then R := (J (wn x ]-En,&,[) .

n20

Then it is clear that R i s a connected open subset of R3 and that the mapping 0 E C1(R; E3) satisfies det VO > 0 in R.

Finally, note that the covariant components gij E Co(R) of the metric tensor field associated with the mapping 0 are given by (the symmetries b,p = bp, established in (i) are used here)

gap = a,p - 223b,p + x$c,p, ga3 = 0, g33 = 1.

(iii) Starting with the mapping 8 (as given in the statement of The- orem 2.9-2), we construct a mapping 0 that satisfies the assumptions of Theorem 1.7-2. To this end, we define a mapping 6 : R -+ IE3 by letting

O(y, 2 3 ) := 8(y) + 23?i3(y) for all (y,x3) E R,

where the set R is defined as in (ii). Hence 6 E H1(R;E3) , since R c w x 1-1, l[. Besides, det V6 = det VO a.e. in R since the functions bg := Zp"b,,, which are well defined a.e. in w , are equal, again a.e. in w , to the functions bg. Likewise, the components & E L1(R) of the metric tensor field associated with the mapping 0 satisfy & = gij a.e. in R since Zap = a,@ and b,p = b,p a.e. in w by assumption and E,p = cap a.e. in w by part (i).

-

- - -

-

(iv) By Theorem 1.7-2, there exist a vector c E E3 and a matrix Q E 0; such that

I

8(y) + x363(y) = c + Q(B(y) + 23a3(y)) for almost all (y, 23) E R.


Differentiating with respect to 2 3 in this equality between functions in H1(!22;E3) shows that &(y) = Qu3(y) for almost all y E w. Hence

0 8(y) = c + Q8(y) for almost all y E w as announced.

Remarks. (1) The existence of i? E H1(w; E3) satisfying the assumptions of Theorem 2.9-2 implies that i? E C1(w; E3) and G3 E C1(w; E3), and that 8 E H1(w; E3) and u3 E H1(w; E3).

(2) It is easily seen that the conclusion of Theorem 2.9-2 is still valid if the assumptions i? E H1(w; E3) and 6 3 E H1(w; E3) are replaced by the weaker assumptions i? E H:,,(W; E3) and Z3 E H;,,(w; E3).

2.10 Continuity of a surface as a function of its fundamental forms

Let w be a connected and simply-connected open subset of R2. Together, Theorems 2.8-1 and 2.9-1 establish the existence of a mapping F that associates with any pair of matrix fields (a,p) E C2(w; S;) and (b,p) E C2(w; S2) satisfying the Gauss and Codazzi-Mainardi equations in w a well-defined element F((a,p), (hap)) in the quotient set C3(w; E3)/R, where (0,s) E R means that there exists a vector c E E3 and a matrix Q E 0: such that 8(y) = c + Q&) for all y E w .

A natural question thus arises as to whether the mapping F defined in this fashion is continuous. Equivalently, is a surface a continuous function of its fundamental forms?

When both spaces C2(w; S2) and C3(w; E3) are equipped with their natural F'rkchet topologies, a positive answer to this question has been provided by Ciarlet [2003] , along the following lines.

To begin with, we introduce the following two-dimensional analogs to the notations used in Section 1.8. Let w be an open subset of W3. The notation K G w means that K is a compact subset of w . If f E Ce(w; R) or 8 E Ce(w; E3), C 2 0, and K G w , we let

\ \ f \ \ t ,K := SUP IW(Y)I I I ~ I I ~ , ~ := SUP i a y y ) i ,

{ r$<e {I::&.

where d" stands for the standard multi-index notation for partial derivatives and 1 . 1 denotes the Euclidean norm in the latter definition. If A E Ce(w; €MI3)), C 2 0, and K G w , we likewise let

llAlle,K = SUP la"A(Y)I, { I :::e

where 1 . 1 denotes the matrix spectral norm


We recall (see Section 1.8) that, for any integers k 2 0 and d 2 1, the space Ce(w; Rd) becomes a locally convex topological space when it is equipped with the &.e'chet topology defined by the family of semi-norms l l . l l e , K , IF, G w . Then a sequence (8".),,0 converges to 8 with respect to this topology if and only if

Furthermore, this topology is metrizable: Let ( I F , ~ ) + o be any sequence of subsets of w that satisfy

00

IF,^ c w and IF,^ c int IF,~+I for all i 2 0, and w = u IF,^

i=O

Then

lim I18n-8jle,K = 0 for all IF, G w e lim de(8", 8) = 0 , 71-00 n-+m

where

Let C3(w; E3) := C3(w; E 3 ) / R denote - the quotient set of C3(w; E3) by the equivalence relation R, where ( 8 , O ) E R means that there exist a vector c E E3 and a matrix Q E 0: such that 8 ( y ) = c + Q G ( y ) for all y E w. Then the set C 3 ( w ; E 3 ) becomes a metric space when it is equipped with the distance d 3 defined by

where 6 denotes the equivalence class of 8 modulo R. The continuity of a surface as a function of its fundamental forms

has then been established by Ciarlet [2003], according to the following result (if d is a metric defined on a set X , the associated metric space is denoted { X ; d } ) :

Theorem 2.10-1. Let w be connected and simply connected open subset ofR2. Let

C,"(w; S: x S2) := {((a,p) , (b,p)) E C2(w; 25:) x C2(w; S2); apcaur - aucapr + C&Curp - C&rCprp = baubpT - bapbur in W ,

apb,, - &b,p + C:,bp, - C$$,, = 0 in w }


Given any element ((a,p), (b,p)) E C;(u ;S; x S’), let F(((a,p), (hap))) E C3(w;E3) denote the equivalence class modulo R of any immersion 8 E C3(u ; E3) that satisfies

Then the mapping 3 ; d 3 F : { C ~ ( W ; S: x S2); dz} + {C3(w; E ) }

defined in this fashion is continuous. 0

The above continuity results have been extended “up to the boundary of the set w” by Ciarlet & C. Mardare [2005].

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I: Fixed-Point Theorems,

94

An Introduction to Shell Theory

Philippe G. Ciarlet Department of Mathematics, City University of Hong Kong

8 3 Tat Chee Avenue, Kowloon, Hong Kong, China E-mail: mapgcQcityu. edu.hlc

Cristinel Mardare Universite‘ Pierre et Marie Curie-Paris6

UMR 7598 Laboratoire Jacques-Louis Lions, Paris, F- 75005 France E-mail: [email protected]. f r

Introduction

These notes’ are intended to provide a thorough introduction to the mathematical theory of elastic shells.

The main objective of shell theory is to predict the stress and the displacement arising in an elastic shell in response to given forces. Such a prediction is made either by solving a system of partial differential equations or by minimizing a functional, which may be defined either over a three-dimensional set or over a two-dimensional set, depending on whether the shell is viewed in its reference configuration as a three- dimensional or as a two-dimensional body (the latter being an abstract idealization of the physical shell when its thickness is “small”).

The first part of this article is devoted to the three-dimensional theory of elastic bodies, from which the three-dimensional theory of shells is obtained simply by replacing the reference configuration of a general body with that of a shell. The particular shape of the reference configuration of the shell does not play any rBle in this theory.

The second part is devoted to the two-dimensional theory of elastic shells. In contrast to the three-dimensional theory, this theory is specific to shells, since it essentially depends on the geometry of the reference configuration of a shell.

‘With the kind permission of Springer-Verlag, some portions of these notes are extracted and adapted from the book by the first author “An introduction to Dif- ferential Geometry with Applications to Elasticity”, Springer, Dordrecht, 2005, the writing of which was substantially supported by two grants from the Research Grants Council of Hong Kong Special Administrative Region, China [Project No. 9040869, CityU 100803 and Project No. 9040966, CityU 1006041.

An Introduction to Shell Theory 95

For a more comprehensive exposition of the theory of elastic shells, we refer the reader to Ciarlet [18] and the references therein for the first part of the article, and to Ciarlet [20] and the references therein for the second part.

1 Three-dimensional theory

Outline

In this first part of the article, the displacement and the stress arising in an elastic shell, or for that matter in any three-dimensional elastic body, in response to given loads are predicted by means of a system of partial differential equations in three variables (the coordinates of the physical space). This system is formed either by the equations of nonlinear three- dimensional elasticity or by the equations of linearized three-dimensional elasticity.

Sections 1.2-1.4 are devoted to the derivation of the equations of three-dimensional elasticity in the form of two basic sets of equations, the equations of equilibrium and the constitutive equations. The equations of nonlinear three-dimensional elasticity are then obtained by adjoining appropriate boundary conditions to these equations. The equations of linearized three-dimensional elasticity are obtained from the nonlinear ones by linearization with respect to the displacement field.

Sections 1.5-1.6 study the existence and uniqueness of solutions to the equations of linearized three-dimensional elasticity.

Using a fundamental lemma, due to J.L. Lions, about distributions with derivatives in “negative” Sobolev spaces (Section 1.5), we establish in Section 1.6 the fundamental Korn inequality, which in turn implies that the equations of linearized three-dimensional elasticity have a unique solution.

In sections 1.7-1.8, we study the existence of solutions to the equations of nonlinear three-dimensional elasticity, which fall into two distinct categories:

If the data are regular, the applied forces are “small”, and the boundary condition does not change its nature along connected portions of the boundary, the equations of nonlinear three-dimensional elasticity have a solution by the implicit function theorem (Section 1.7).

If the constituting material is hyperelastic and the associated stored energy function satisfies certain conditions of polyconvexity, coerciveness, and growth, the minimization problem associated with the equations of nonlinear three-dimensional elasticity has a solution by a fundamental theorem of John Ball (Section 1.8).

96 Philippe G. Ciarlet, Cristinel Mardare

1.1

All spaces, matrices, etc., are real. The Kronecker symbol is denoted dl. The physical space is identified with the three-dimensional vector

space R3 by fixing an origin and a Cartesian basis (e1,e2,e3). In this way, a point x in space is defined by its Cartesian coordinates xi, x2,23 or by the vector x := xi xiei. The space R3 is equipped with the Euclidean inner product u . w and with the Euclidean norm 1u1, where u, w denote vectors in R3. The exterior product of two vectors u, w E R3 is denoted U A W .

For any integer n 2 2, we define the following spaces or sets of real square matrices of order n:

Notation, definitions, and some basic formulas

M": the space of all square matrices,

A": the space of all anti-symmetric matrices,

S": the space of all symmetric matrices,

My: the set of all matrices A E Mn with det A > 0, Sy : the set of all positive-definite symmetric matrices,

0": the set of all orthogonal matrices,

07: the set of all orthogonal matrices R E 0" with det R = 1.

The notation (ai j ) designates the matrix in M" with aij as its element at the i-th row and j-th column. The identity matrix in M" is denoted I := (6;). The space M", and its subspaces A" and S" are equipped with the inner product A : B and with the spectral norm (A1 defined by

i>j

1A) := sup{lAwl; w E R", I w I < I},

where A = (ai j ) and B = ( b i j ) denote matrices in M". The determinant and the trace of a matrix A = ( a i j ) are denoted det A and tr A. The cofactor matrix associated with an invertible matrix A E M" is defined by CofA := (det A)A-T.

Let R be an open subset of R3. Partial derivative operators of order m 2 1 acting on functions or distributions defined over R are denoted

where Ic = (ki) E N3 is a multi-index satisfying llcl := kl + k2 + k3 = m. Partial derivative operators of the first, second, and third order are also denoted di := d / d x i , dij := d2/dxidxj, and dijk := d 3 / 8 ~ l a ~ 2 a ~ 3 .


The gradient of a function f : R 4 IR is the vector field gradf := (aif), where i is the row index. The gradient of a vector field v = (vi) : R 4 R” is the matrix field Vv := (djwi), where i is the row index, and the divergence of the same vector field is the function div v := Ci &vi. Finally, the divergence of a matrix field T = ( t i j ) : R + M” is the vector field divT with components (Cj”=, 8jti j) i .

The space of all continuous functions from a topological space X into a normed space Y is denoted C o ( X ; Y ) , or simply C o ( X ) if Y = R.

For any integer m 2 1 and any open set R c It3, the space of all real- valued functions that are m times continuously differentiable over R is denoted Cm(R). The space em@), m 2 1, is defined as that consisting of all vector-valued functions f E P ( R ) that, together with all their partial derivatives of order 6 m, possess continuous extentions to the closure of R. If R is bounded, the space C” (2 ) equipped with the norm

is a Banach space. The space of all indefinitely derivable functions cp : R 4 R with

compact support contained in R is denoted D(R) and the space of all distributions over R is denoted D’(R). The duality bracket between a distribution T and a test function cp E D(R) is denoted (T, c p ) .

The usual Lebesgue and Sobolev spaces are respectively denoted Lp(R), and WmJ’(R) for any integer m 3 1 and any p 2 1. If p = 2, we use the notation Hm(R) = Wm>’(R). The space W,oT’p(R) is the space of all mesurable functions such that flu E WmJ’(U) for all U G R, where the notation flu designates the restriction to the set U of a function f and the notation U G R means that u is a compact set that satisfies u c R.

The space WT1p(R) is the closure of D(R) in WmJ’(R) and the dual of the space Wrlp(R) is denoted WPm+”(R), where p’ = 5. If the boundary of R is Lipschitz-continuous and if ro c 80 is a relatively open subset of the boundary of R, we let

-

w::(R) := {f E w’J’(R); f = o on ro}, w:;P(~) := {f E w~J’(R); f = 8,f = o on r,,},

where 8, denote the outer normal derivative operator along 8 R (since is Lipschitz-continuous, a unit outer normal vector (u i ) exists 80-almost everywhere along 80, and thus 8, = ui8i).

If Y is a finite dimensional vectorial space (such as R”, Mn, etc.), the notation Cm(R; Y ) , em@; Y ) , LP(R; Y ) and W”iP(R; Y ) designates the spaces of all mappings from R into Y whose components in Y are respectively in Cm(R), em@), LP(R) and Wm>P(n). If Y is equipped with


the norm I ' 1 , then the spaces LP(R2; Y ) and W">P(R; Y ) are respectively equipped with the norms

and

Throughout this article, a domain in R" is a bounded and connected open set with a Lipschitz-continuous boundary, the set R being locally on the same side of its boundary. See, e.g., Adams [2], Grisvard [54], or NeEas [73]. If R c R" is a domain, then the following formula of integration by parts is satisfied

div F . v d x = - .I F : Vv d x + l a ( F n ) . v da s, for all smooth enough matrix field F : R 4 Mk and vector field v : R -+

Rk, k 1 (smooth enough means that the regularity of the fields F and v is such that the above integrals are well defined; for such instances, see, e.g., Evans & Gariepy [47]). The notation da designates the area element induced on the surface dR by the volume element d x . We also record the Stokes formula:

1.2 Equations of equilibrium

In this section, we begin our study of the deformation arising in an elastic body in response to given forces. We consider that the body occupies the closure of a domain R c R3 in the absence of applied forces, henceforth called the reference configuration of the body. Any other configuration that the body might occupy when subjected to applied forces will be defined by means of a deformation, that is, a mapping

that is orientation preserving (i.e., det V + ( x ) > 0 for all z E 2) and injective on the open set R (i.e., no interpenetration of matter occurs). The image +(a) is called the deformed configuration of the body defined


by the deformation C P . The “difference” between a deformed configuration and the reference configuration is given by the displacement, which is the vector field defined by

u := CP - id,

where id : a + a is the identity map. It is sometimes more convenient to describe the deformed configuration of a body in terms of the displacement u instead of the deformation CP, notably when the body is expected to undergo small deformations (as typically occurs in linearized elasticity).

Our objective in this section is to determine, among all possible deformed configurations of the body, the ones that are in “static equilibrium” in the presence of applied forces. More specifically, let the applied forces acting on a specific deformed configuration fi := CP(Cl) be represented by the densities

f : fi + R3 and g : f’1 + R3,

where f’, c i3fi is a relatively open subset of the boundary of fi. If the body is subjected for instance to the gravity and to a uniform

pressure on f ‘ ~ , then the densities and g are given by f(5) = -g5(5)e3 and g ( 5 ) = - ~ f i ( 5 ) , where g is the gravitational constant, ,5 : fi + JR is the mass density in the deformed configuration, 5 denotes a generic point in {a}-, n(5) is the unit outer normal to afi, and T is a constant, called pressure.

These examples illustrate that an applied force density may, or may not, depend on the unknown deformation.

Our aim is thus to determine equations that a deformation CP corresponding to the static equilibrium of the loaded body should satisfy. To this end, we first derive the “equations of equilibrium” from a fundamental axiom due to Euler and Cauchy. The three-dimensional equations of elasticity will then be obtained by combining these equations with a “constitutive equation” (Section 1.3).

Let

s2 := {v E R3; 1v1 = l}

denote the set of all unit vectors in R3. Then, according to the stress principle of Euler and Cauchy, a body fi c W3 subjected to applied forces of densities f : fi --t R3 and g : f’1 -+ R3 is in equilibrium if there exists a vector field

z : {a}- x s2 -+ R3


such that, for all domains A C 6,

/A f d2 + / - i ( 2 , i i ( 2 ) ) dii = 0, LJA

2 A f d2 + LA 2 A i ( 2 , n ( 2 ) ) dii = 0 ,

where i i (2) denotes the exterior unit normal vector at Z to the surface aA (because is a domain, i i ( 2 ) exists for d2-almost all 2 E d A ) .

This axiom postulates in effect that the “equilibrium” of the body to the applied forces is reflected by the existence of a vector field i that depends only on the two variables 2 and i i (2) .

The following theorem, which is due to Cauchy, shows that the dependence of z on the second variable is necessarily linear:

Theorem 1.2-1. I fZ( . , i i ) : {6}- + R3 is of class C1 for all ii E S’J, i(Z, .) : Sz -+ R3 is continuous for all 2 E {6}-, and f : {6}- + R3 is continuous, then i : (62)- x S’J + R3 is linear with respect to the second variable.

Proof. The proof consists in applying the stress principle to particular subdomains in {a}-. For details, see, e.g., Ciarlet [18] or Gurtin &

In other words, there exists a matrix field ?’ : {6}- + M3 of class

Martins [55]. 0

C1 such that

i ( 2 , i i ) = T(2)ii for all 2 E (62)- and all ii E S’J.

Combining Cauchy’s theorem with the stress principle of Euler and Cauchy yields, by means of Stokes’ formula (see Section 1.1), the following equations of equilibrium in the deformed configuration:

Theorem 1.2-2. The matrix field T : (62)- + M3 satisfies

-divp(2) = f(2) for all 2 E 6, ? ‘ ( ~ ) i i ( ~ ) = g(2) for all 2 E Fl, (1.2-1)

?(z) E s3 for all 2 E 6.

The system (1.2-1) is defined over the deformed configuration a, which is unknown. Fortunately, it can be conveniently reformulated in terms of functions defined over the reference configurence R of the body, which is known. To this end, we use the change of variables


5 = @(x) defined by the unknown deformation @ : a + {fi}-, assumed to be injective, and the following formulas between the volume and area elements in {fi}- and a (with self-explanatory notations)

d5 = I det V@(x)/ dx, f i (5) dSi = CofV@(x)n(x) da.

We also define the vector fields f : R + R3 and g : Fl + R3 by

f(5) d5 = f (x) dx, g(5) dSi = g(x) da.

Note that, like the fields f and g, the fields f and g may, or may not, depend on the unknown deformation +.

First of all, assuming that + is smooth enough and using the change of variables 4j : 2 + (6)- in the first equation of (1.2-l), we deduce that, for all domains A c R,

f ( x ) dx + 1, T(@(x))CofV+(x)n(x) da = 0.

The matrix field T : a + M3 appearing in the second integral, viz., that defined by

~ ( x ) := T(+(x))CofV@(x) for all x E a, is called the first Piola-Kirchhoff stress tensor field. In terms of this tensor, the above relation read

which implies that the matrix field T satisfies the following partial differential equation:

-divT(x) = f (x) for all x E R.

Using the identity

V@ (x) ' T ( x) = V@ (x) [det V@ (x)?'( 9 (x) )] V+ (x) --T,

which follows from the definition of T ( x ) and from the expression of the inverse of a matrix in terms of its cofactor matrix, we furthermore deduce from the symmetry of the matrix T(5) that the matrix ( V @ ( x ) - ' T ( x ) ) is also symmetric.


It is then clear that the equations of equilibrium in the deformed configuration (see eqns. (1.2-1)) are equivalent with the following equations of equilibrium in the reference configuration:

-divT(z) = f(z) for all z E R, T(z )n ( z ) = g(z) for all z E rl, (1.2-2)

V + ( Z ) - ~ T ( Z ) E s3 for all z E R,

= +(ri). where the subset rl of a R is defined by

R -+ S3 be defined by Finally, let the second Piola-Kirchhoff stress tensor field E : -

~ ( z ) := V + ( Z ) - ~ T ( ~ ) for all z E R.

Then the equations of equilibrium defined in the reference configuration take the equivalent form

-div (V+(z)E(z)) = f(z) for all z E R, (V+(z)E(z))n(z) = g(z) for all z E r1,

in terms of the symmetric tensor field E. The unknowns in either system of equations of equilibrium are the

deformation of the body defined by the vector field + : 2 + R3, and the stress field inside the body defined by the fields T : -+ MI3 or X : n -+ S3. In order to determine these unknowns, either system (1.2-2) or (1.2-3) has to be supplemented with an equation relating these fields. This is the object of the next section.

(1.2-3)

1.3 Constitutive equations of elastic materials

It is clear that the stress tensor field should depend on the deformation induced by the applied forces. This dependence is reflected by the constitutive equation of the material, by means of a response function, specific to the material considered. In this article, we will consider one class of such materials, according to the following definition: A material is elastic if there exists a function Tn : n x M t 4 M3 such that

~ ( z ) = @(z, v+(z)) for all z E a. Equivalently, a material is elastic if there exists a function Xfl : a x M $ 4

S3 such that ~ ( z ) = @(z, v+(z)) for all z E n.

Either function Tn or Ed is called the response function of the material.


A response func t ion cannot be arbitrary, because a general axiom in physics asserts that any “observable quantity” must be independent of the particular orthogonal basis in which it is computed. For an elastic material, the “observable quantity” computed through a constitutive equation is the stress vector field i. Therefore this vector field must be independent of the particular orthogonal basis in which it is computed. This property, which must be satisfied by all elastic materials, is called the axiom of material frame-indifference. The following theorem translates this axiom in terms of the response function of the material.

Theorem 1.3-1. An elastic material satisfies the axiom of material frame-indifference i f and only if

Tn(z, Q F ) = QTn(z, F ) for all x E a and Q E 0: and F E Mi$,

or equivalently, if and only i f

Xn(x, Q F ) = Xn(x, F ) for all x E

The second equivalence implies that the response function En depend on F only via the symmetric positive definite matrix U := (FTF)’l2, the square root of the symmetric positive definite matrix (FTF) E S;. To see this, one uses the polar factorisation F = RU, where R := FU-’ E 0:, in the second equivalence of Theorem 1.3-1 to deduce that

En(,, F ) = Xfl(x, U ) for all z E n and F = RU E M:.

and Q E 0: and F E M:.

This implies that the second Piola-Kirchhoff stress tensor field E : -+

S3 depends on the deformation field @ : 32 + R3 only via the associated metric tensor field C := V@TV@, i.e.,

~ ( z ) = %(z, ~ ( z ) ) for all z E a, where the function % : fi x S; + S3 is defined by

%(x, C ) := Xn(x, for all z E a and C E Sj3>.

We just saw how the axiom of material frame-indifference restricts the form of the response function. We now examine how its form can be further restricted by other properties that a given material may possess.

An elastic material is isotropic at a point z of the reference configuration if the response of the material “is the same in all directions”, i.e., if the Cauchy stress tensor $(Z) is the same if the reference configuration is rotated by an arbitrary matrix of 0: around the point z. An elastic material occupying a reference configuration fi is isotropic if it is isotropic at all points of a. The following theorem gives a characteri- sation of the response function of an isotropic elastic material:


Theorem 1.3-2. An elastic material occupying a reference configuration R is isotropic i f and only if -

@(z, FQ) = Tfl(z, F ) Q for all z E 2 and Q E 0: and F E M:,

or equivalently, i f and only if

Efl(z, F Q ) = QTXfl(z, F ) Q for all z E and Q E 0: and F E M;.

Another property that an elastic material may satisfy is the property of homogeneity: An elastic material occupying a reference configuration a is homogeneous if its response function is independent of the particular point z E 2 considered. This means that the response function Tfl : 2 x M: 4 M3, or equivalently the response function Xfl : a x M : 4 S3, does not depend on the first variable. In other words, there exist mappings (still denoted) Tn : MLn"+ 4 M3 and Xfl : M: 4 S3 such that

T n ( z , F ) = T n ( F ) for all x E n and F E M:,

X n ( x l F ) = d ( F ) for all z E s2 and F E M:. The response function of an elastic material can be further restricted

if its reference configuration is a natural state, according to the following definition: A reference configuration a is called a natural state, or equivalently is said to be stress-free, if

and

T # ( ~ , I ) = o for all z E 2, or equivalently, if

~ f l ( z , I ) = o for all z E 2. We have seen that the second Piola-Kirchhoff stress tensor field X :

4 B3 as

X(z) = %(z, C(z)), where C(z) = VCP'(z)VCP(z) for all z E ;2.

If the elastic material is isotropic, then the dependence of X(z) in terms of C(z) can be further reduced in a remarkable way, according to the following Rivlin-Ericksen theorem:

Theorem 1.3-3. If a n elastic material is isotropic and satisfies the principle of material frame-indifference, then there exists functions -,! : R x B3 -+ EX, i E {1,2,3}, such that

X(z) = +yo(z)I + yi(z)C(z) + yZ(z)c2(z) for all x E 2,

- R 4 S3 is expressed in terms of the deformation field CP :

where ri(z) = $(z, tr C , tr(CofC), det C ) .


Proof. See Rivlin & Ericksen [74] or Ciarlet [18]. 0

Note that the numbers tr C(z), tr(CofC(z)), and det C(z) appearing in the above theorem constitute the three principal invariants of the matrix C(z).

Although the Rivlin-Ericksen theorem substantially reduces the range of possible response functions of elastic materials that are isotropic and satisfy the principle of frame-indifference, the expression of X is still far too general in view of an effective resolution of the equilibrium equations. To further simplify this expression, we now restrict ourselves to deformations that are “close to the identity”.

In terms of the displacement filed u : R + R3, which is related to the deformation @ : R + R3 by the formula

@(z) = z + ~ ( z ) for all z E 1, the metric tensor field C has the expression

C ( X ) = I + 2E(z),

E ( z ) := - (VuT(z ) + V u ( z ) + VuT(z)Vu(z) ) where

1 2

denotes the Green-St Venant strain tensor at z. Thanks to the above assumption on the deformation, the matrices

E ( z ) are “small” for all z E a, and therefore one can use Taylor expansions to further simplify the expression of the response function given by the Rivlin-Ericksen theorem. Specifically, using the Taylor expansions

tr C(z) = 3 + 2 tr E ( z ) , tr(CofC(z)) = 3 + 4 t rE(z ) + o((E(z)I),

det C(z) = 1 + 2 t rE (z ) + o(lE(z)I),

C2(z) = 1 + 4E(z) + o(IE(z)I),

tl and assuming that the functions yi are smooth enough, we deduce from the Rivlin-Ericksen theorem that

where the real-valued functions X(z) and p(z) are independent of the displacement field u. If in addition the material is homogeneous, then X and p are constants.

To sum up, the constitutive equation of a homogeneous and isotropic elastic material that satisfies the axiom of frame-indifference must be such that

X(X) = ~ t f ( z , I ) + X(tr E ( ~ ) ) I + 2 p ~ ( z ) + o z ( l ~ ( z ) ~ ) for all z E 1.


If in addition R is a natural state, a natural candidate for a constitutive equation is thus

~ ( z ) = X(tr E ( ~ ) ) I + 2 p ~ ( z ) for all z E a, and in this case X and p are then called the Lam6 constants of the material.

A material whose constitutive equation has the above expression is called a St Venant-Kirchhoff material. Note that the constitutive equation of a St Venant-KirchhoE material is invertible, in the sense that the field E can be also expressed as a function of the field E as

1 v ~ ( z ) = -E(x) - -(tr E(Z))I for all z E 2.

2P E

Remark. The Lami: constants are determined experimentally for each elastic material and experimental evidence shows that they are both strictly positive (for instance, X = 106kg/cm2 and p = 820000kg/cm2 for steel; X = 40000kg/cm2 and p = 1200kg/cm2 for rubber). Their explicit values do not play any r81e in our subsequent analysis; only their positivity will be used. The Lami: coefficients are sometimes expressed in terms of the Poisson coeficient v and Young modulus E through the expressions

X P(3X + 2cL) 2(X+p) and = X + P '

v =

1.4 The equations of nonlinear and linearized three-dimensional elasticity

It remains to combine the equations of equilibrium (equations (1.2-3) in Section 1.2) with the constitutive equation of the material considered (Section 1.3) and with boundary conditions on ro := dR\I ' l . Assuming that the constituting material has a known response function given by Tu or by Xu and that the body is held fked on ro, we conclude in this fashion that the deformation arising in the body in response to the applied forces of densities f and g satisfies the nonlinear boundary value problem:

-divT(s) = f(z), z E R, a(.)= 2, 2 E r,,, (1.4-1)

T(z)n(z) = g ( z ) , 5 r1,


where

T ( z ) = Tn(z, V@(z)) = V+(z)Xn(z, v@(z)) for all z E D. (1.4-2)

The equations (1.4-1) constitute the equations of nonlinear three- dimensional elasticity. We will give in Sections 1.7 and 1.8 various sets of assumptions guaranteeing that this problem has solutions.

Consider a body made of an isotropic and homogeneous elastic material such that its reference configuration is a natural state, so that its constitutive equation is (see Section 1.3):

where X > 0 and p > 0 are the Lam6 constants of the material. The equations of linearized three-dimensional elasticity are obtained from the above nonlinear equations under the assumption that the body will undergo a "small" displacement, in the sense that

+(z) = z + u(z ) with IVu(z)l << 1 for all z E 2.

Then, for all z E 2,

and

~ ( z ) = ~ + ( z ) ~ ( z ) = ( I + Vu(z))(X(tr E ( ~ ) ) I + 2 p ~ ( z ) )

X 2

= - tr(VuT(z) + ~ u ( z ) ) + p(~u ' (z ) + vu(z)) + oz(Ivu(z)l).

Therefore the equations of linearized three-dimensional elasticity, which are obtained from (1.4-1) by replacing T ( z ) by its linear part with respect to Vu(z), are given by

-diva(z) = f(z), z E R, = 0, 5 E ro, (1.4-3)

a(++) = g(z), 2 E rl, where

1 2 ~ ( z ) = X(tre(z))I+ 2pe(z) and e(z) = -(Vu'(z) + Vu(z)) . (1.4-4)

We show in the next section that this linear system has a unique solution in appropriate function spaces.


1.5 A fundamental lemma of J.L. Lions

We first review some essential definitions and notations, together with a fundamental lemma of J.L. Lions (Theorem 1.5-1). This lemma will play a key r61e in the proofs of Korn's inequality in the next Section.

Let R be a domain in IW". We recall that, for each integer m 1, H"(R) and H,"(R) denote the usual Sobolew spaces. In particular,

H1(R) := {w E L2(R); aiv E P p ) , 1 6 i 6 n}, H y n ) := { w E H1(R); &jV E L2(R), 1 < 2, j ,< n},

where aiv and aijv denote partial derivatives in the sense of distributions, and

H,~(R) := { w E ~ ' ( 0 ) ; w = o on r}, where the relation w = 0 on I' is to be understood in the sense of trace. The norm in L2(R) is noted I I . I I L 2 ( n ) and the norm in H"(R), m 2 1, is noted I I . I I H m ( n ) . In particular then,

We also consider the Sobolev space

H- l (R) := dual space of HA(R).

Another possible definition of the space HA (0) being

&(R) = closure of D(R) with respect to I I . I I H l ( n ) , where D(R) denotes the space of infinitely differentiable real-valued functions defined over R whose support is a compact subset of R , it is clear that

w E L2(R) + w E H-l(S2) and &w E H-'(R), 16 i 6 n,

since (the duality between the spaces D(R) and D'(R) is denoted by (., .)):

I(v,cp)I = I J' vcpdzl 6 / /~l lL2(n)l lcpl lH1(R),

I(aiv,cp)I = I - haicp)I = 1 - J' ua,cpdz( 6 ll~IIL~(n)IIcpIIHl(n)

n

R

for all cp E D(R). It is remarkable (but also remarkably difficult to prove!) that the converse implication holds:


Theorem 1.5-1. Let R be a domain in Rn and let u be a distribution o n R. Then

{u E H-l(R) and. 8iv E H-’(R), 1 < i < n} ==+ u E L2(R).

0

This implication was first proved by J.L. Lions, as stated in Magenes & Stampacchia [66, p. 320, Note (27)]; for this reason, it will be henceforth referred to as the lemma of J.L. Lions. Its first published proof for domains with smooth boundaries appeared in Duvaut & Lions [46, p. 1111; another proof was also given by Tartar [84]. Various extensions to “genuine” domains, i.e. , with Lipschitz-continuous boundaries, are given in Bolley & Camus [14], Geymonat & Suquet [52], and Borchers & Sohr [15]; Amrouche & Girault [6, Proposition 2.101 even proved that the more general implication

{w E D’(R) and 8 i w E H”(R), 1 < i 6 n } .j v E Hmfl(R)

holds for arbitrary integers m E Z.

1.6 Existence theory in linearized three-dimensional elasticity

We define a weak solution to the equations of linearized three-dimensional elasticity (Section 1.4) as a solution to the variational equations

(1.6-1)

for all smooth vector fields v : R + R3 that satisfy w = 0 on ro, where

u = X(tr e(u))I + 2pe(u) and e(u) = -(VuT + Vu). 1 2

Note that, because the matrix field u is symmetric, the integrand in the left-hand side can be also written as

u : V v = u : e(v),

1 2

where e(v) := - ( vvT + V W ) .

The existence of a solution to the above variational problem follows from the Lax-Milgram lemma. In order to verify the hypotheses of this lemma, we first need to establish the following classical, and fundamental, inequality:


Theorem 1.6-1 (Korn's inequality). Let R be a domain in R3 and let ro c 8 R be such that arearo 0 . Then there exists a constant C such that

I I 4.) II P(n;$3) 2 CII TJ I I H1 (n;R3)

for di E H & ( R ; R ~ ) := {TJ E H ~ ( R ; R ~ ) ; w = o on ro}. Proof. Several proofs are available in the mathematical literature for this remarkable inequality. We adapt here that given in Duvaut & Lions [46]. We proceed in several steps:

(i) Korn's inequality is a consequence of the identity

d i j w k = &ejk(v) + ajeik(v) - &eij(v)

relating the matrix fields Vv = (ajwi) and e(v) = (eij(v)), where

If v E L2(f12;R3) and e(v) E L2(R2;S3), the above identity shows that &jwk E H-l(R). Since the functions a j v k also belong to the space H-'(R), the lemma of J.L. Lions (Theorem 1.5-1) shows that 8 j V k E L2(R). This implies that the space

E(R; R ~ ) := {v E ~ ~ ( 0 2 ; ~ ~ ) ; e ( w ) E L ~ ( R ; S3)}

coincides with the Sobolev space H'(R; R3). (ii) The space E(R; R3), equipped with the norm

1 1 11 E(n;W3) := ( I v I/ L2(0;R3) + 11 1 1 L2 (n;R3) 7

is clearly a Hilbert space, as is the space H1(R;R3) equipped with the norm

ll4 H1 := 112) I I L z ( n ; ~ 3 ) -t II v v I I L2(n;w3). Since the identity mapping

id : v E H1(R; R3) H v E E ( R ; R3)

is clearly linear, bijective (thanks to the step (i)), and continuous, the open mapping theorem (see, e.g., Yosida [87]) shows that id is also an open mapping. Therefore, there exists a constant C such that

1 1 4 ~ l ( n ; w 3 ) 6 cllvIIE(n;~) for all E E ( R ; R ~ ) ,


ll4l L2(0;Iw3) + 114.) I1 L2(n;83) 2 c-1114H1 (Q;R3)


for all w E H1(R;R3).

w = 0.

any field w E H:n (R; R3) that satisfies e ( w ) = 0 must also satisfy

(iii) We establish that, if w E H&(R;R3) satisfies e(w) = 0, then

This is a consequence of the identity of Part (i), which shows that

&jv, = 0 in R.

Therefore, by a classical result about distributions (see, e.g. Schwartz [80]), the field w must be affine, i.e., of the form w(z) = b + Aa: for all IC E R, where b E R3 and A E M3. Since the symmetric part of the gradient of w , which is precisely e ( w ) , vanishes in 0, the matrix A must be in addition antisymmetric. Since the rank of a nonzero antisymmetric matrix of order three is necessarily two, the locus of all points z satisfying a + Ax = 0 is either a line in R3 or an empty set, depending on whether the linear system a + Aa: = 0 has solutions or not. But a + Ax = 0 on ro and arearo > 0. Hence A = 0 and b = 0, and thus w = 0 in R.

(iv) The Korn inequality of Theorem 1.6-1 then follows by contradiction. If the inequality were false, there would exist a sequence ( w n ) n E ~ in H:n (R; R3) such that

= 1 for all n,

lle(2'n)llL2(n;s13) -+ 0 as n +

Because the set R is a domain, the inclusion H1(R;R3) c L2(O;R3) is compact by the Rellich-Kondrasov theorem. The sequence (on) being bounded in H 1 ( Q R3), it contains a subsequence ( w ~ ( ~ ) ) , where a : N -+

N is an increasing function, that converges in L2(R; R3) as n + m. Since the sequences ( w ~ ( ~ ) ) and (e(w,[,))) converge respectively in

the spaces L2(R;R3) and L2(R;S3), they are Cauchy sequences in the same spaces. Therefore the sequence ( ~ ~ ( ~ 1 ) is a Cauchy sequence with respect to the norm 1 1 . l l ~ ( n ; p ) , hence with respect to the norm 1 1 . IIHl(n;R3) by the inequality established in Part (ii).

The space H;o (0; R3) being complete as a closed subspace of H1(R; R3), there exists w E Hio(R;R3) such that

wu(n) 4 w in H ~ ( R ; R ~ ) .

Since its limit satisfies

e(w) = lim e(vucn)) = 0, n-o3

it follows that w = 0 by Part (iii). But this contradicts the relation 11wIIHl(n;p) = limn+oo = 1, and the proof is complete.

0


The inequality established in Part (ii) of the proof is called Korn's inequality without boundary conditions.

The uniqueness result established in Part (iii) of the proof is called the infinitesimal rigid displacement lemma. It shows that an infinitesimal rigid displacement field, i.e., a vector field w € H1(R; R3) satisfying e(w) = 0, is necessarily of the form

w(z) = a + b A x for all z E 0, where a, b E R3

Remark. In the special case where I'o = dQ, Korn's inequality is a trivial consequence of the identity

Ie(w)12 d z = l 1VvI2 d z for all w E H,1(R;TR3),

itself obtained by applying twice the formula of integration by parts (see Section 1.1). 0

We are now in a position to establish that the equations of linearized three-dimensional elasticity have weak solutions. We distinguish two cases depending on whether arearo > 0 or not.

Theorem 1.6-2. Assume that the Lame' constants satisfy X 3 0 and p > 0 and that the densities of the applied forces satisfy f E L6//5(R;R3) and g E L4/3(l?l;E%3).

If areal70 > 0, the variational problem (1.6-1) has a unique solution in the space

Proof. It suffices to apply the Lax-Milgram lemma to the variational equation (1.6-1), since all its assumptions are clearly satisfied. In particular, the coerciveness of the bilinear form appearing in the left-hand side of the equation (1.6-1) is a consequence of Korn's inequality established in the previous theorem combined with the positiveness of the Lam6 constants, which together imply that, for all w E H;JR; R3),


Theorem 1.6-3. Assume that the Lame' constants satisfy X 3 0 and p > 0 and that the densities of the applied forces satisfy f E L6/5(R;R3) and g E L4/3(dR;R3).

Ifarearo = O andJn f .wdJ:+Jang.wda=O f o r a l l w eH1(R;R3) satisfying e(w) = 0, then the variational problem (1.6-1) has a solution in H1(R; R3), unique up to an infinitesimal rigid displacement field.

Sketch of proof. It is again based on the Lax-Milgram lemma applied to the variational equations (1.6-1), this time defined over the quotient space H1(R;R3)/I&, where I& is the subspace of H1(R;R3) consisting of all the infinitesimal rigid displacements fields. By the infinitesimal rigid displacement lemma (see Part (ii) of the proof of Theorem 1.6-1), Ro is the finite-dimensional space

{ w : R + R3; w(x) = a + b A 2, a, b E R3}.

The compatibility relations satisfied by the applied forces imply that the variational equation (1.6-1) is well defined over the quotient space H1(R;R3)/I&, which is a Hilbert space with respect to the norm

The coerciveness of the bilinear form appearing in the left-hand side of the equation (1.6-1) is then established as a consequence of another Korn's inequality:

lle(qllLyn;s3) 3 Cll4lH1(n2;W3)/Rg for all + E H1(Q R 3 ) / I & .

The proof of this inequality follows that of Theorem 1.6-1, with Part (iii) adapted as follows: The sequence ( w n ) n E ~ is now defined in H1(R; R3)/& and satisfies

/l+nIIH1(R;R3)/Ro = 1 for all 12,

lle(+n)llLyn;s3) + 0 as n + m.

Hence there exists an increasing function a : N 4 N such that the subsequence ( w ~ ( ~ ) ) is a Cauchy sequence in H1(R;R3). This space being complete, there exists w E H1(R; R3) such that

wa(") + w in H1(R;R3),

and its limit satisfies

e ( w ) = lim e(wa(n) ) = 0. 71-00

Therefore w E This implies that

by Part (iii), hence ( w ~ ( ~ ) - w ) -+ 0 in H1(R;R3).

I l ~ a ( n ) I I H ~ ( n ; R 3 ) ~ ~ G I I W ~ ( ~ ) - wIIH1(n2;R3) -+ 0 as n 00,


which contradicts the relation ((i~,(,)llHi(n;p)/~~ = 1 for all n. The variational problem (1.6-1) is called a pure displacement

problem when ro = do, a pure traction problem when rl = dR, and a displacement-traction problem when arearo > 0 and areal71 > 0.

0

Since the system of partial differential equations associated with the linear three-dimensional variational model is elliptic, we expect the solution of the latter to be regular if the data f, g , and dR arc regular and if there is no change of boundary condition along a connected portion of dR. More specifically, the following regularity results hold (indications about the proof arc given in Ciarlet [18, Theorem 6.3-61).

Theorem 1.6-4 (pure displacement problem). Assume that ro = 80. I f f E W"J'(R; R3) and 80 is of class C m f 2 for some integer m 2 0 and real number 1 < p < co satisfying p 3 &, then the solution u to the variational equation (1.6-1) is in the space Wmf2J'(C12;W3) and there exists a constant C such that

l l ~ I I W m + 2 . P ( n ; W 3 ) G C l l f l l W ~ + 2 ~ P ( n ; R 3 ) .

Furthermore, u satisfies the boundary value problem:

-divu(x) = f, x E R, x E 80. U(.) = 0,

Theorem 1.6-5 (pure traction problem). Assume that rl = dR and sn f . w dx + s,, g . w da = 0 for all vector fields v E H1(R; R3) satisfying e(w) = 0. Iff E W">p(R;R3), g E W"+1-1/PJ'(I'~;R3), and dR is of class ern+' for some integer m 2 0 and real number 1 < p < co satisfying p 2 &, then any solution u to the variational equation (1.6-1) is in the space W"f2~p(0;W3) and there exist a constant C such that

Furthermore, u satisfies the boundary value problem:

-divu(z) = f(x), x E 0, x E dR. fT(x)n(x) = g(x),

1.7 Existence theory in nonlinear three-dimensional elasticity by the implicit function theorem

The question of whether the equations of nonlinear three-dimensional elasticity have solutions has been answered in the affirmative when the


data satisfy some specific assumptions, but remains open in the other cases. To this day, there are two theories of existence, one based on the implicit function theorem, and one, due to John Ball, based on the minimization of functionals.

We state here the existence theorems provided by both theories but we will provide the proof only for the existence theorem based on the implicit function theorem. For the existence theorem based on the minimization of functionals we will only sketch of the proof of John Ball (Section 1.8).

The existence theory based on the implicit function theorem asserts that the equations of nonlinear three-dimensional elasticity have solutions if the solutions to the associated equations of linearized three- dimensional elasticity are smooth enough, and the applied forces are small enough. The first requirement essentially means that the bodies are either held fixed along their entire boundary (i.e., = dR), or nowhere along their boundary (i.e., rl = 80).

We restrict our presentation to the case of elastic bodies made of a St Venant-Kirchhoff material. In other words, we assume throughout this section that

X = X(tr E ) I + 2pE and E = - VuT + Vu + VuTVu , (1.7-1)

where X > 0 and p > 0 are the Lam6 constants of the material and IJ : R --f R3 is the unknown displacement field. We assume that ro = dR (the case where rl = dR requires some extra care because the space of infinitesimal rigid displacements fields does not reduce to (0)). Hence the equations of nonlinear three-dimensional elasticity assert that the displacement field u : R 4 R3 inside the body is the solution to the boundary value problem

2 Y )

-div ( ( I + V U ) ~ ) = f in R, u = 0 on dR,

(1.7-2)

where the field X is given in terms of the unknown field u by means of (1.7-1). The existence result is then the following

Theorem 1.7-1. The nonlinear boundary value problem (1.7-1)-(1.7-2) has a solution u E W2>p(R; R3) i f R i s a domain with a boundary dR of class C2, and f o r some p > 3, f E Lp(R;R3) and IlfllLp(n;R3) is small enough.

Proof. Define the spaces

x := (w E w2>p(R; R~); w = o on an}, Y := U ( R ; R3).


Define the nonlinear mapping F : X 4 Y by

F(w) := -div ( ( I + V W ) ~ ) for any w E X ,

where

x = X(tr E ) I + 2 p ~ and E = - v w T + vw + V W ~ V W 2 Y It suffices to prove that the equation

3‘(u) = f

has solutions in X provided that the norm of f in the space Y is small enough.

The idea for solving the above equation is as follows. If the norm of f is small, we expect the norm of u to be small too, so that the above equation can be written as

Since F(0) = 0, we expect the above equation to have solution if the linear equation

F”(0)U = f

has solutions in X . But this equation is exactly the system of equations of linearized three-dimensional elasticity. Hence, as we shall see, this equation has solutions in X thanks to Theorem 1.6-4.

In order to solve the nonlinear equation F(u) = f , it is thus natural to apply the inverse function theorem (see, e.g., Taylor [85]). According to this theorem, if F : X 4 Y is of class C1 and the F’rkchet derivative F’(0) : X 4 Y is an isomorphism (i.e., an operator that is linear, bijective, and continuous with a continuous inverse), then there exist two open sets U c X and V c Y with 0 E U and 0 = F(0) E V such that, for all f E V, there exists a unique element u E U satisfying the equation

F(U) = f .

Furthermore, the mapping

f E V H U E U

is of class C1. It remains to prove that the assumptions of the inverse function the-

orem are indeed satisfied. First, the function F is well defined (i.e., F(u) E Y for all u E X) since the space W’J’(R) is an algebra (thanks to the assumption p > 3). Second, the function .F : X 4 Y is of class


C1 since it is multilinear (in fact, F is even of class P). Third, the Frkchet derivative of .F is given by

F’(0)u = -diva,

where 1 2

c := A(tre) l+ 2pe and e := -(VuT + Vu), from which we infer that the equation F’(0)u = f is exactly the equations of linearized three-dimensional elasticity (see (1.4-3)-(1.4-4) with To = an). Therefore, Theorem 1.6-4 shows that the function F’(0) : X 4 Y is an isomorphism

Since all the hypotheses of the inverse function theorem are satisfied, the equations of nonlinear three-dimensional elasticity (1.7-1)-(1.7-2) have a unique solution in the neighborhood U of the origin in W2iP(Q; R3) if f belongs to the neighborhood V of the origin in L*(Cl;R3). In particular, if S > 0 is the radius of a ball B(0,S) contained in V , then the

0

The unique solution u in the neighborhood U of the origin in W2+’(Cl; R3) of the equations of nonlinear three-dimensional elasticity (1.7-1)- (1.7-2) depends continuously on f, i.e., with self-explanatory notation

f, + f in L*(R;Iw~) j u, --f u in W ~ ~ ~ ( R ; R ’ ) .

problem (1.7-1)-(1.7-2) has solutions for all I l f l l~~cn, < 6.

This shows that, under the assumptions of Theorem 1.7-1, the system of equations of nonlinear three-dimensional elasticity is well-posed.

Existence results such as Theorem 1.7-1 can be found in Valent [86], Marsden & Hughes [SS], Ciarlet & Destuynder [25], who simultaneously and independently established the existence of solutions to the equations of nonlinear three-dimensional elasticity via the implicit function theorem.

1.8 Existence theory in nonlinear three-dimensional elasticity by the minimization of energy (John Ball’s approach)

We begin with the definition of hyperelastic materials. Recall that (see Section 1.3) an elastic material has a constitutive equation of the form

~ ( z ) := ~ f l ( z , v+(z)) for all z E K, where Til : T ( z ) is the first Piola-Kirchhoff stress tensor at 2.

x M$ -+ M3 is the response function of the material and


Then an elastic material is hyperelastic if there exists a function W : s2 x M$ + R, called the stored energy function, such that its response function Tn can be fully reconstructed from W by means of the relation

where denotes the F'rkchet derivative of W with respect to the variable F . In other words, a t each x E D, g ( x , F ) is the unique matrix in M3 that satisfies

for all F E M$ and H E M3 (a detailed study of hyperelastic materials can be found in, e.g., Ciarlet [18, Chap. 41).

John Ball [9] has shown that the minimization problem formally associated with the equations of nonlinear three-dimensional elasticity (see (1.4-1)) when the material constituting the body is hyperelastic has solutions if the function W satisfies certain physically realistic conditions of polyconvexity, coerciveness, and growth. A typical example of such a function W , which is called the stored energy function of the material, is given by

W ( x , F ) = allFllP + bllCofFllq + cI det 3'1' - dlog(det F )

f o r a l l F E M : , w h e r e p > 2 , q > ~ , r > l , a > O , , b > O , c > O , d > O , and 1 1 . 1 1 is the norm defined by IlFll := {tr(FTF)}'12 for all F E M3.

The major interest of hyperelastic materials is that, for such materials, the equations of nonlinear three-dimensional elasticity are, at least formally, the Euler equation associated with a minimization problem (this property only holds formally because, in general, the solution to the minimization problem does not have the regularity needed to properly establish the Euler equation associated with the minimization problem). To see this, consider first the equations of nonlinear three- dimensional elasticity (see Section 1.4):

where, for simplicity, we have assumed that the applied forces do not depend on the unknown deformation a.


A weak solution @ to the boundary value problem (1.8-1) is then the solution to the following variational problem, also known as the principle of virtual works:

for all smooth enough vector fields v : a 4 R3 such that

and the above equation can be written as

= 0 on ro. If the material is hyperelastic, then T#(x, V@(s)) = s ( x , V Q ( x ) ) ,

J’(+)w = 0,

where J’ is the Friichet derivative of the functional J defined by

J ( 9 ) := 1 W ( x , V * ( x ) ) d x - f . 9 d x - 1, g . 9 d u ,

for all smooth enough vector fields 9 : 4 R3 such that 9 = id on ro. The functional J is called the total energy.

Therefore the variational equations associated with the equations of nonlinear three-dimensional elasticity are, at least formally, the Euler equations associated with the minimization problem

J(+) = min J ( 9 ) , *‘EM

where M is an appropriate set of all admissible deformations @ : R + R3 (an example is given in the next theorem).

John Ball’s theory provides an existence theorem for this minimization problem when the function W satisfies the following fundamental definition (see [9]): A stored energy function W : a x Mr”+ 4 R is said to be polyconvex if, for each x f a, there exists a convex function W(x , .) : M3 x M3 x (0, m) 4 R such that

W ( x , F ) = W(x , F , CofF, det F ) for all F E M:.

Theorem 1.8-1 (John Ball). Let R be a domain in R3 and let W be a polyconvex function that satisfies the following properties:

The function W( . , F , H , 6 ) : R 3 R is measurable f o r all ( F , H , 6 ) E

There exist numbers p 3 2, q 3 3, r > 1, a > 0 , and ,f3 E R such M3 x M3 x (0, XI) .

that W ( x , F ) 3 a( llFllP + IICofFIIq + I det FIT) - P

for almost all x E R and for all F E M:.

1.20 Philippe G. Ciarlet, Cristinel Mardare

For almost all x E R, W ( x , F ) + +m i f F E M: i s such that det F + 0.

Let r1 be a relatively open subset of dR, let I'o := dR \ rl, and let there be given fields f E L6l5(R; EX3) and g E L4/3(171; EX3). Define the functional

W ( x , V@(z)) d z - f (z) . @(z) d x -

and the set

M := {@ E W17p(R;R3); Cof(V@) E L4(R;M3), det(V@) E L'((R), d e t ( V 9 ) > 0 a.e. in R, @ = id on ro}.

Finally, assume that arearo > 0 and that infgiiM J ( @ ) < 00.

Then there exists @ E M such that

J ( @ ) = inf J ( 9 ) . * E M

Sketch of proof (see Ball 191 or Ciarlet [18], for a detailed proof). Let @, be a infimizing sequence of the functional J , i.e., a sequence of vector fields Gn E M such that

J(@,) + infgiEM J ( 9 ) < m.

The coerciveness assumption on W implies that the sequences (a,), (Cof(V@,)), and (det(V@,)) are bounded respectively in the spaces W1>p(R; EX3) , L4(R; M3), and L'(R). Since these spaces are reflexive, there exist subsequences (@,(,)), (Cof (V@,(n))), and (det(V@,(,))) such that (3 denotes weak convergence)

+u(n) 3 in w'J'(R; R ~ ) ,

H,(n) := Cof(V9,(,)) 3 H 6m(n) := det(V9,(,)) 2 6

in Lq(R;M3), in L'(R).

For all a E W1J'(R;EX3), H E LQ(R;M3), and 6 E L'(R) with S > 0 almost everywhere in R, define the functional

J(9, H , 6) := W ( Z , V@(x), H ( x ) , S(x)) dx J, where, for each x E R, W ( x , .) : M3 x M3 x (0, m) + IR is the function given by the polyconvexity assumption on W . Since W(x , .) is convex, the above weak convergences imply that


But J ( @ u ( n ) , H u ( n ) , 6 u ( n ) ) = J (@c(n) ) and J (@n) + inf*EM J(*). Therefore J(9, H , 6) = inf*,,M J(*).

vergences above then shows that A compactness by compensation argument applied to the weak con-

H = C o f ( V 9 ) and 6 = de t (V9) .

Hence J (+) = J(9, H , 6). It remains to prove that + E M . The property that W ( F ) 4 +m if

F E M+ is such that det F + 0, then implies that de t (V9) > 0 a.e. in R. Finally, since 9, 2 9 in W1>P(!2;R3) and since the trace operator is linear, it follows that 9 = id on ro. Hence 9 E M .

Since J ( 9 ) = J(9, H,6) = inf,p,,M J(*) , the weak limit 9 of the sequence aU(,) satisfies the conditions of the theorem.

0

A St Venant-Kirchhoff material with Lam6 constants X > 0 and p > 0 is hyperelastic, but not polyconvex. However, Ciarlet & Geymonat [26] have shown that the stored energy function of a S t Venant-Kirchhoff material, which is given by

can be “approximated” with polyconvex stored energy functions in the following sense: There exists polyconvex stored energy functions of the form

Wb(F) = allF112 + bllCofFI12 + cI det FI2 - dlog(det F ) + e

with a > 0, b > 0, c > 0, d > 0, e E R, that satisfy

w ~ ( F ) = W ( F ) + C ? ( I I F ~ F - 1113) .

A stored energy function of this form possesses all the properties required for applying Theorem 1.8-1. In particular, it satisfies the coerciveness inequality:

Wb(F) 3 a(llF112 + /ICofFII’ + (de tF)2) + p, with a > 0 and p E R.

2 Two-dimensional theory

Outline

In the first part of the article, we have seen how an elastic body subjected to applied forces and appropriate boundary conditions can be modeled


by the equations of nonlinear or linearized three-dimensional elasticity. Clearly, these equations can be used in particular to model an elastic shell, which is nothing but an elastic body whose reference configuration has a particular shape.

In the second part of the article, we will show how an elastic shell can be modeled by equations defined on a two-dimensional domain. These new equations may be viewed as a simplification of the equations of three-dimensional elasticity, obtained by eliminating some of the terms of lesser order of magnitude with respect to the thickness of the shell. This simplification i s done by exploiting the special geometry of the reference configuration of the shell, and especially, the assumed “smallness” of the thickness of the shell.

In the next section, we begin our study with a brief review of the geometry of surfaces in R3 defined by curvilinear coordinates. Of special importance are their first and second fundamental forms.

In Section 2.2, we define the reference configuration of a shell as the set in R3 formed by all points within a distance 6 E from a given surface in R3. This surface is the middle surface of the shell and E is its half-thichness. We then define a system of three-dimensional curvilinear coordinates inside the reference configuration of a shell.

In Section 2.3, the equations of nonlinear or linearized three- dimensional elasticity, which were written in Cartesian coordinates in the first part of the article, are recast in terms of these natural curvilinear coordinates, as a preliminary step toward the derivation of two- dimensional shell theories.

In Section 2.5, we give a brief account of the derivation of nonlinear membrane and flexural shell models by letting the thickness E approach zero in the equations of nonlinear three-dimensional elasticity in curvilinear coordinates. The same program is applied in Section 2.6 to the equations of linearized three-dimensional elasticity in curvilinear coordinates to derive the linearized membrane and flexural shell models.

In Sections 2.7-2.10, we study the nonlinear and linear Koiter shell models. The energy of the nonlinear Koiter shell model is defined in terms of the covariant components of the change of metric and change of curvature tensor fields associated with a displacement field of the middle surface of the reference configuration of the shell. The linear Koiter shell model is then defined by linearizing the above tensor fields. Finally, the existence and uniqueness of solutions to the linear Koiter shell equations are established, thanks to a fundamental K o r n inequality o n a surface and to an infinitesimal rigid displacement lemma o n a surface.


2.1 A quick review of the differential geometry of surfaces in R3

To begin with, we briefly recapitulate some important notions of differential geometry of surfaces (for detailed expositions, see, e.g., Ciarlet

Greek indices and exponents (except v in the notation 8,) range in the set {1,2}, Latin indices and exponents range in the set {1 ,2 ,3} (save when they are used for indexing sequences), and the summation convention with respect to repeated indices and exponents is systematically used.

Let w be a domain in EX2. Let y = (y,) denote a generic point in the set 9 and let d, := d/dy,. Let there be given an immersion 0 E C3(9;R3), i.e., a mapping such that the two vectors

~ 2 , 231).

%(Y) := 8 , q Y )

are linearly independent at all points y E G. These two vectors thus span the tangent plane to the surface

s := e(w) a t the point O(y), and the unit vector

is normal to S at the point O(y). The three vectors %(y) constitute the covariant basis at the point 0(y), while the three vectors ai(y) defined by the relations

where dj is the Kronecker symbol, constitute the contravariant basis at the point 0(y) E S. Note that a3(y) = a3(y) and that the vectors aa(y) are also in the tangent plane to S at O(y). As a consequence, any vector field rl : w + R3 can be decomposed over either of these bases as

a2(y) . .j(Y) = q,

r ] = via2 = $ai,

where the coefficients vi and vi are respectively the covariant and the contravariant components of q.

The covariant and contravariant components a,p and affp of the first fundamental f o r m of S , the Christoffel symbols and the covariant and mixed components b,p and bf of the second fundamental f o r m of S are then defined by letting:

a,p := a, . ap, aap := aa ' ap, rzp := a'. dpaa,

b,p := a3 . dpa,, b t := ap'bua.


The area element along S is &dy, where

a := det(a,a).

Note that one also has & = la1 A a2l. The derivatives of the vector fields ai can be expressed in terms of

the Christoffel symbols and of the second fundamental form by means of the equations of Gauss and Weingarten:

a,ap = r ;p , + b,pa3, &a3 = -bLa,.

Likewise, the derivatives of the vector fields aj satisfy

&ar = -r;,,aY + bza3,

&a = -bauau. 3

These equations, combined with the symmetry of the second derivatives of the vector field a, (i.e., a,(auaa) = 6',(&a,)), imply that

These relations are equivalent to the Gauss and Codazzi-Mainardi equations, namely,

where

R:,,, := a,q, - arr;, + rrar;@ - rgarYp are the mixed components of the Riemann curvature tensor associated with the metric (a,@). If R?,, := U"~R?,,~, then one can see that all these functions vanish, save for R12,,. This function is the Gaussian curvature of the surface S , given by

We will see that the sign of the Gaussian curvature plays an important r6le in the two-dimensional theory of shells.


2.2 Geometry of a shell

Let the set w c R2 and the mapping 8 : CJ 4 R3 be as in Section 2.1. In what follows, the surface S = 8(G) will be identified with the middle surface of a shell before deformation occurs, i.e., S is the middle surface of the reference configuration of the shell. The coordinates y1 ,y2 , of the points y E W constitute a system of “two-dimensional” curvilinear coordinates for describing the middle surface of the reference configuration of the shell.

More specifically, consider an elastic shell with middle surface S = O(Z) and (constant) thickness 2~ > 0 , i.e., an elastic body whose reference configuration is the set {fi‘}- := @@‘), where (cf. Figure 2.2-1)

RE := w x ( - E , E ) and 0 ( y , x;) := 8(y) f xga3(y) for all ( y , xj) E a‘. The more general case of shells with variable thickness or with a middle surface described by several charts (such as an ellipsoid or a torus) can also be dealt with; see, e.g., Busse [16] and S. Mardare [67].

Naturally, this definition makes sense physically only if the mapping 0 is globally injective on the set a‘. Fortunately, this is indeed the case if the immersion 8 is itself globally injective on the set Z and E is small enough, according to the following result (due to Ciarlet [20, Theorem 3. I- 11).

Theorem 2.2-1. Let w be a domain in R2, let 8 E C3(G;R3) be a n injective immersion, and let 0 : W x R 4 R3 be defined by

0(y , 23) := B ( y ) + 23a3(y) for all ( y , x3) E w x R.

Then there exists E > 0 such that the mapping 0 i s a C2-diffeomorphism f r o m W x [ - E , E ] onto @(a x [ - E , E ] ) and det(g,,g2,g3) > 0 in w x [-&,&I, where gi := &0.

Proof. The assumed regularity on 8 implies that 0 E C2(W x [ -E , E]; R3) for any E > 0. The relations

-

g, = &0 = a, + xsd,a3 and g3 = a30 = a 3

imply that

det(gl,g2,g3)lZ3=o = det(al ,az,as) > 0 in W.

Hence det(g,, g2, g3) > 0 on W x [ -E, E ] if E > 0 is small enough. Therefore, the implicit function theorem can be applied if E is small

enough: It shows that, locally, the mapping 0 is a C2-diffeomorphism: Given any y E W, there exist a neighborhood U ( y ) of y in G and ~ ( y ) > 0


Figure 2.2-1: The reference Configuration of an elastic shell. Let w be a domain in R2, let RE = w x ( - E , E ) , let 8 E C3((w;R3) be an immersion, and let the mapping 0 : @ + R3 be defined by 0(y, zg) = B(y) +zga3(y) for all (y, zg) E RE. Then the mapping 0 is globally injective on if the immersion 8 is globally injective on W and E > 0 is small enough (Theorem 2.2-1). In this case, the set @(a") may be viewed as the reference configuration of an elastic shell with thickness 2~ and middle surface S = 8 ( W ) . The coordinates (yl,y2,zg) of an arbitrary point zE E are then viewed as curvilinear coordinates of the point 2" = O ( x E ) of the reference configuration of the shell.

-

such that 0 is a C2-diffeomorphism from the set U(y) x [--E(Y),E(Y)] onto O(U(y) x [ - - ~ ( y ) , ~ ( y ) ] ) . See, e.g., Schwartz [81, Chapter 31 (the proof of the implicit function theorem, which is almost invariably given for functions defined over open sets, can be easily extended to functions defined over closures of domains, such as the sets W x [-&,&I; see, e.g., Stein [SZ]).

To establish that the mapping 0 : W x [ - -E ,E] + R3 is injective provided E > 0 is small enough, we proceed by contradiction: If this property is false, there exist E" > 0, (yyn,z;), and (p",;), n 3 0, such that

E, -+ 0 as n -+ 00, y" E W, f7 E W, 6 E,, IZFI < E" ,

(y",z;) # (y", 2;) and O(yn , z;) = 0(p, E?).

Since the set W is compact, there exist y E W and Q E 9, and there


by the continuity of the mapping 0 and thus y = g since the mapping 8 is injective by assumption. But these properties contradict the local injectivity (noted above) of the mapping 0. Hence there exists E > 0

0 such that 0 is injective on the set iZ x [ -E , E ] .

In what follows, we assume that E > 0 is small enough so that the conclusions of Theorem 2.2-1 hold. The reference configuration of the considered shell is then defined by

{fi"}- := 0(2'),

where R' := w x ( -E ,E ) and 6' := @(RE). Let x" = (xg) denote a generic point in the set G" (hence xz = y a ) and let 2" = (2 ; ) denote a generic point in the reference configuration {&}-. The reference configuration of the shell can thus be described either in terms of the "three-dimensional" curvilinear coordinates y1, y2, xj, or in terms of the Cartesian coordinates 2:, 25,2j, of the same point 2' = 0 ( x ' ) E {&}-.

To distinguish functions and vector fields defined in Cartesian coordinates from the corresponding functions and vector fields defined in curvilinear coordinates, we henceforth adopt the following convention of notation: Any function or vector field defined on flE is denoted by letters surmounted by a hat (e.g., ijE is a function defined on &, i" is a vector field defined on !?F, etc.). The corresponding functions and vector fields defined in curvilinear coordinates are then denoted by the same letters, but without the hat (e.g., g" is the function defined on RE by gE(xE) = t j&(P) for all xE E RE, f" is the vector field defined on 0' by f"(zCE) = f (2.) for all xE E RE, etc., the points 2" and zE being related by PE = @(x')).

Let 8; := a/ax: (hence a/axz = d / d y a ) and let 8; := ala2i-f. For each xE E D', the three linearly independent vectors gz(x') := afO(x") constitute the cowarian? basis at the point O ( x E ) , and the three (likewise linearly independent) vectors gj?" (x") defined by the relations gj"(x') . gg(xE) = d i constitute the contravariant basis at the same point. As a consequence, any vector field uE : RE + R3 can be decomposed over either basis as

A &

u" = UZgi?" z Q l E g ; ,

exists an increasing function : N N such that

Hence


where the coefficients uf and uz+ are respectively the covariant and the contravariant components of U' .

The functions gtj(zE) := gf(zE) . g;(z') and gi j>" (zE) := gi>"(x') . gjiE (z') are respectively the covariant and contravariant components of the metric tensor induced by the immersion 0. The volume element in O(nE) is then f l d z ' , where

gE := det(gzj).

For details about these notions of three-dimensional differential geometry, see Ciarlet [23, Sections 1.1-1.31

2.3 The three-dimensional shell equations

In this section, we consider an elastic shell whose reference configuration is (6')- := O(nE) (see Section 2.2), and we make the following assumptions.

The shell is subjected to applied body forces given by their densities 1" : & -+ R3 (this means that 1" d P is the body force applied to the volume d2' at each 2' E 6'). For ease of exposition, we assume that there are no applied surface forces.

The shell is subjected to a homogeneous boundary condition of place along the portion @(yo x [ -E, €1) of its lateral face 0 ( d w x [ -E , E]), where yo is a measurable subset of the boundary aw that satisfies lengthy0 > 0. This means that the displacement field of the shell vanishes on the set

The shell is made of a homogeneous hyperelastic material, thus char- @(yo x [-&,El).

acterized by a stored energy function (see Section 1.8)

: M3 -+ R.

Such a shell problem can thus be modeled by means of a minimization problem (Section 1.8), which is expressed in Cartesian coordinates, in the sense that all functions appearing in the integrands depend on three variables, the Cartesian coordinates 2' = (2; ) of a point in the reference configuration {&}- of the shell. We now recast this problem in terms of the curvilinear coordinates zE = (zz) describing the reference configuration {&}- = @(a') of the same shell. This will be the natural point of departure for the two-dimensional approch to shell theory described in the next sections.

More specifically, the minimization problem consists in finding a minimizer 8' : {h'}- -+ R3 of the functional s' (see Section 1.8) defined bv

129 An Introduction to Shell Theory

over a set of smooth enough vector fields 9" = {A"}- R3 satisfying &"(2') = 2" for all 2" E @(yo x [-&,&I). Recall that the functional k is the total energy of the shell.

This minimization problem can be transformed into a minimization problem posed over the set a", i.e., expressed in terms of the "natural" curvilinear coordinates of the shell, the unknown +' : a" --f R3 of this new problem being defined by

+&(xE) = 2&(iE) for all 2" = xE E a'. If E > 0 is small enough, the mapping 0 is a C'-diffeomorphism of a" onto its image {AE} - = @(a") and det(VEO) > 0 in a' (Theorem 2.2- 1). The formula for changing variables in multiple integrals then shows that (P" is a minimizer of the functional J" defined by

~"(9") := 1,. %(V9"(x")(VEO(xE))-') det V'O dz"

f"(xE) . 9 " ( x E ) det V"O dx", - In.

where the matrix field V"9" : a' + M3 is defined by V"W = (aj$;) (cf. Section 1.1) and the vector field f" : n" + R3 is defined by

f E ( x E ) := ]"(P) for all 2' = ~ ( x ' ) , x~ E RE.

Note that the function det V"O is equal to the function fl, where gE = det(gtj); cf. Section 2.2.

Consider next a linearly elastic shell with Lam6 constants X > 0 and p > 0. In this case, the minimization problem associated with the equations of linearized three-dimensional elasticity (Section 1.4) consists in finding a minimizer & : {A"}- + R3 over a set of smooth enough vector fields 9' = {&}- -+ R3 satisfying G"(2") = 2" for all 2" E @(yo x [ -&,&I ) of the functional k defined by

where

X 8 4 i @ ( ~ ) = -(tr(FT + F - 21))' + ~ I I F ~ + F - 21112 for all F E ~3

(this stored energy function for a linearly elsatic material easily follows from the equations of linearized three-dimensional elasticity given in Sec- tion 1.4). Its expression shows that the functional J' is well defined if 9' E Hl(A"; R3).


As in the nonlinear case, this minimization problem can be recast in curvilinear coordinates. As such, it consists in finding a minimizer W : D" + R3 over the set of all vector fields * I E E H1( f iE; R3) satisfying W = 0 on yo x [ - E , E ] of the functional J" defined by

As usual in linearized elasticity, it is more convenient to express this energy in terms of the displacement field U" : 2" ---f R3, defined by

w(x" ) = ~ ( x " ) + u E ( x E ) for all xE E 52".

Likewise, let v E : nE 4 R3 be such that W = 0 + v'. Then a straightforward calculation shows that

r/ir( VEQE ( VEO)- ' ) = AijkelE efj (v ' )e ie (v" ),

1 2

e:j(vE) := - (dfvE . g ; + d;v' .g : ) .

The functions AijkeiE and e:j (u") denote respectively the contravariant components of the three-dimensional elasticity tensor in curvilinear coordinates, and the covariant components of the linearized strain tensor associated with the displacement field v". It is then easy to see that U"

is a minimizer over the vector space

v(V) := { u " = ufgi+; uf E H'(R"), u: = o on 7 0 x ( - E , E ) } ,

of the functional J" defined by

This minimization problem will be used in Section 2.6 as a point of departure for deriving two-dimensional linear shell models.

2.4

In a two-dimensional approach, the above minimization problems of Sec- tion 2.3 are "replaced" by a, presumably much simpler, two-dimensional problem, this time "posed over the middle surface S of the shell". This means that the new unknown should be now the deformation cp : F -+ R3 of the points of the middle surface S = t9(U), or, equivalently, the displacement field c : U + R3 of the points of the same surface S (the

The two-dimensional approach to shell theory


deformation and the displacement fields are related by the equation ‘p = 8 + 6); cf. Figure 2.4-1.

The two-dimensional approach to shell theory yield a variety of two- dimensional shell models, which can be classified in two categories (the same classification applies for both nonlinear and linearized shell models) :

A first category of two-dimensional models are those that are obtained from the three-dimesional equations of shells “by letting E go to zero”. Depending on the data (geometry of the middle surface of the shell, boundary conditions imposed on the displacement fields, applied forces) one obtains either a membrane shell model, or a flexural shell model, also called a bending shell model. A brief description of these models and of their derivation is given in Sections 2.5 and 2.6.

h Y

Figure 2.4-1: An elastic shell modeled as a two-dimensional problem. For E > 0 “small enough” and data of ad hoc orders of magnitude, the three-dimensional shell problem is “replaced” by a “two-dimensional shell problem”. This means that the new unknowns are the three covariant components Ci : GJ + E% of the displacement field Ciai : GJ + R3 of the points of the middle surface S = O@). In this process, the “three-dimensional” boundary conditions on !?o need to be replaced by ad hoc L‘twO-dimensional’’ boundary conditions on yo. For instance, the “boundary conditions of clamping” Ci = = 0 on 70 (used in Koiter’s linear equations; cf. Section 2.8) mean that the points of, and the tangent spaces to, the deformed and undeformed middle surfaces coincide along the set O(y0).


A second category of two-dimensional models are those that are obtained from the three-dimensional model by restricting the range of admissible deformations and stresses by means of specific a priori assumptions that are supposed to take into account the “smallness” of the thickness (e.g., the Cosserat assumptions, the Kirchhoff-Love assumptions, etc.). A variety of two-dimensional models of shells are obtained in this fashion, as, e.g., those of Koiter, Naghdi, etc. A detailed description of Koiter’s model is given in Sections 2.7 and 2.8.

2.5 Nonlinear shell models obtained by r-convergence

Remarkable achievements in the asymptotic analysis of nonlinearly elastic shells are due to Le Dret & Raoult [64] and to F’riesecke, James, Mora & Miiller [49], who gave the first (and only ones to this date) figorous proofs of convergence as the thickness approaches zero. In so doing, they extended to shells the analysis that they had successfully applied to nonlinearly elastic plates in Le Dret & Raoult [62] and F’riesecke, James & Miiller [48].

We begin with the asymptotic analysis of nonlinearly elastic membrane shells. H. Le Dret and A. Raoult showed that a subsequence of deformations that minimize (or rather “almost minimize” in a sense explained below) the scaled three-dimensional energies weakly converges in W1>p(R;R3) as E 4 0 (the number p > 1 is governed by the growth properties of the stored energy function). They showed in addition that the weak limit minimizes a “membrane” energy that is the r-l imit of the (appropriately scaled) energies. We now give an abridged account of their analysis.

Let w be a domain in R2 with boundary y and let 8 E C2(G; R3) be an injective mapping such that the two vectors a,(y) = d,8(y) are linearly independent at all points y = (ye) E SLi.

Consider a family of elastic shells with the same middle surface S = 8(G) and whose thickness 2~ > 0 approaches zero. The reference configuration of each shell is thus the image @@&) c R3 of the set s1 c R3 through a mapping 0 :

By Theorem 2.2-1, if the injective mapping 8 : W 4 R3 is smooth enough, the mapping 0 : 4 R3 is also injective for E > 0 small enough and y1, y2, x$ then constitute the “natural” curvilinear coordinates for describing each reference configuration O(@) .

Assume that all the shells in the family are made,of the same hyperelastic homogeneous material (see Section 1.8), satisfying the following properties :

The stored energy function I%’ : M3 4 R of the hyperelastic material

-&

+ R3 defined in Section 2.2.


satisfies the following assumptions: There exist constants C > 0, a > 0, p E EX, and 1 < p < ca such that

It can be verified that the stored energy function of a St Venant- Kirchhoff material, which is given by

satisfies such inequalities with p = 4.

Remark. By contrast, the stored energy function of a linearly elastic material, which is given by

P 4

@ ( F ) = -1IF + FT -

where JJFJJ := (tr FTF}'12, satisfies the first inequality with p = 2, but not the second one. 0

It is further assumed that, for each E > 0, the shells are subjected in their interior to applied body forces of density f" = ffgZ+ : fi" 4 R3 per unit volume, where ff E Lq(fi") and $ + i = 1, and that these densities do not depend on the unknown deformation. Applied surface forces on the "upper" and "lower" faces of the shells could be likewise considered, but are omitted for simplicity; see in this respect Le Dret & Raoult [64] who consider a pressure load, an example of applied surface force that depends on the unknown deformation.

Finally, it is assumed that each shell is subjected to a boundary condition of place along its entire lateral face O(y x [-&, &]), where y := dw, i.e., that the displacement vanishes there.

The three-dimensional problem is then posed as a minimizat ion prob- l em in terms of the unknown deformation field

a'(.") := O(x") + U 8 ( Z E ) , ZE E a", of the reference configuration, where U" : fiE 4 R3 is its displacement


field (Section 2.3): It consists in finding a" such that

a" E M(R") and J"(a") = inf J E ( S E ) , where

M(R') := {a" E W11p(R";IW3); 9" = 0 on y x [ -E, E ] } ,

W € M ( W )

- In. f" . 9" det V"0 dx".

This minimization problem may have no solution; however, this is not a shortcoming as only the existence of a "diagonal infimizing family", whose existence is always guaranteed, is required in the ensuing analysis (Theorem 2.5-1).

The above minimization problem is then transformed into an analogous one, but now posed over the fixed domain R := W X ] - 1, 1[. Let x = ( 2 1 , 2 2 , 2 3 ) denote a generic point in a and let & := 8/8xi. With each point x E a, we associate the point xE E a" through the bijection

- 7r" : 2 = ( 2 1 , 2 2 , 2 3 ) E R -+ 2" = ( 2" i ) = ( 2 1 , 2 2 , E Z g ) E n".

We then define the unknown scaled deformation a(&) : a + R3 by letting

@ ( E ) ( Z ) := @ " ( x E ) for all 2" = T'Z, z E a. Finally, we assume that the applied body forces are of order O(1)

with respect to E , in the sense that there exists a vector field f E L2(R; R3) independent of E such that

fE(x") = f ( 2 ) for all zE = T"Z, IL: E a.

Remark. Should applied surface forces act on the upper and lower faces of the shells, we would then assume that they are of order O(E) with respect to E . 0

In what follows, the notation (bl; b2; b3) stands for the matrix in M3

These scalings and assumptions then imply that the scaled deforma- whose three column vectors are bl, b2, b3 (in this order).

tion % ( E ) satisfies the following minimization problem:

a(&) E M(&; 0) and J ( E ) ( @ ( E ) ) = inf J ( E ) ( ~ ) , where

M ( E ; R) := {9 E W1>p(R;W3); 9 = + O ( E ) on y x [-I, I]}, *'EM("; n)

1 J(E)(*) := 1 I@( (819; 8 2 s ; -&*)(G(E))-') & det G(E) dz

R r

- /n f . +det G(E) dz,


where the vector field + o ( E ) : fi + R3 is defined for each E > 0 by

+o(~)(z) := @(xE) for all xE = T'Z, x E '32,

and the matrix field G(E) : --+ M3 is defined by

G(E)(z) := VeO(xE) for all xE = T ' X , x E n. The scaled displacement

U ( E ) := +(&) - a+)(&)

therefore satisfies the following minimization problem:

U ( E ) E W(R;IR3) and J(E)(u(E)) = inf J(E)(w), where

w ( R ; R ~ ) := {w E w'>P(s~;IR~); w = o on y x [-I, 11) v€w(n;R3)

1 J(E)(w) := h % ( I + (&w; d2w; -&w)(G(&))-') det G(E) dx

5 f . ('Po(&) + w) det G(E) dx.

Central to the ensuing result of convergence is the notion of quasi- convexity, due to Morrey [71, 721 (an account of its importance in the calculus of variations is provided in Dacorogna [38, Chap 51): Let M m x n denote the space of all real matrices with m rows and n columns; a function I@ : M m x n 4 R is quasiconvex if, for all bounded open subsets U c R", all F E M m x n , and all 6 = (&)gl E W,'l"(U;R"),

where V6 denotes the matrix (aj<i) E M"'". Given any function W : MImXn.+ R, its quasiconvex envelope QW : MImxn + R is the function defined by

QW := sup{$ w m x n 4 R; X is quasi-convex and X < W ) .

Remark. An illuminating instance of actual computation of a quasiconvex envelope is found in Le Dret & Raoult [63], who explicitly determine the quasiconvex envelope of the stored energy function of a St Venant- Karchhoff material.

Also central to the ensuing analysis is the notion of r-convergence, a powerful theory initiated by De Giorgi [40, 411 (see also De Giorgi &


F’ranzoni [43]); an illuminating introduction is found in De Giorgi & Dal Maso [42] and thorough treatments are given in the books of Attouch [8] and Dal Maso [39]. As shown by Acerbi, Buttazzo & Percivale [l] for nonlinearly elastic strings, by Le Dret & Raoult [62, 641 for nonlinearly elastic planar membranes and membrane shells, by F’riesecke, James & Muller [48] for nonlinearly elastic flexural plates, and by F’riesecke, James, Mora & Muller [49] for nonlinearly elastic flexural shells, this approach has thus far provided the only known convergence theorems for justifying lower-dimensional nonlinear theories of elastic bodies. See also Ciarlet [19, Section 1.111 for an application to linearly elastic plates.

We then recall the fundamental definition underlying this theory: Let V be a metric space and let J ( E ) : V -+ R be functionals defined for all E > 0. The family ( J ( E ) ) ~ > o is said to r-converge as E -+ 0 if there exists a functional J : V -+ R U {+co}, called the F-limit of the functionals J(E), such that

V ( E ) -+ v as E -+ 0 + J ( v ) < liminf J(E)(v(E)) ,

on the one hand and, given any v E V, there exist V ( E ) 6 V, E > 0, such that

V ( E ) -+ v as E -+ 0 and J ( v ) = lim J(E)(v(E)) ,

E+O

E+O

on the other. As a preparation to the application of I?-convergence theory, the

scaled energies J(E) : M(R) -+ R found above are first extended to the larger space LP(R; EX3) by letting

J ( E ) ( w ) if w E M ( Q ) , f c o if w E LP(R;R3) but w $ M(R). i J ( E ) ( W ) =

Such an extension, customary in F-convergence theory, has inter alia the advantage of “incorporating” the boundary condition into the extended functional.

Le Dret & Raoult [64] then establish that the fami ly ( J ( E ) ) ~ > ~ of extended energies F-converges as E -+ 0 in LP(R; EX3) and that i ts r - l imi t can be computed by means of quasiconvex envelopes.

More precisely, their analysis leads to the following remarkable convergence theorem, where the limit minimization problems are directly posed as two-dimensional problems (part c)); this is licit since the solutions of these limit problems do not depend on the transverse variable (part (b)). Note thaq, while minimizers of J(E) over M ( Q ) need not exist, the existence of a “diagonal infimizing family” in the sense understood below is always guaranteed because inf,,M(n) J ( E ) ( w ) > -co.

In what follows, the notation (bl; b2) stands for the matrix in M3x2 with bl, b2 (in this order) as its column vectors and &dy denotes as usual the area element along the surface S.


Theorem 2.5-1. Assume that the applied body forces are of order O(1) with respect to E , and that there exist C > 0 , a > 0, p E R, and 1 < p < 00 such that the stored energy function W : M3 3 R satis$es the following growth conditions:

IW(F)I < C(I+ IF\”) W(F) b a\FI” + p

for all F E M3,

for all F E M ~ ,

(W(F) - W/(G) I 6 C(I + l ~ l p - l + I G \ ~ - ~ ) I F - GI for all F , G E ~ 3 .

Let the space M(R) be defined by

M(R) := {v E W11p(R;R3); v = 0 on y x [-I, I]},

and let (U(E)),>O be a “diagonal infimizing family” of the scaled energies, i.e., a family that satisfies

U ( E ) E M(C2) and J(E)(u(E)) 6 inf J ( E ) ( v ) + h(&) for all E > 0,

where h is any positive function that satisfies h(&) + 0 as E -+ 0 . Then: (a) The family ( U ( E ) ) , > ~ lies in a weakly compact subset of the space

(b) The limit u E M(R) as E -+ 0 of any weakly convergent subsequence of ( U ( E ) ) ~ > O satisfies 8321 = 0 in R and is thus independent of the transverse variable.

S-, u dx3 satisfies the following minimiza- t ion problem:

VEM(n)

Wl~”(R;R3).

1 (c) The vector field 6 :=

C E W ~ 1 p ( w ; R 3 ) and j ~ ( 6 ) = inf j ~ ( r l ) , 17EW;’P(W;P3)

where

G(Y) := (ai(~); a 2 ( ~ ) ; as(^)),

the vectors a i ( y ) forming for each y E 2 the covariant basis at the point O(y) E S , and QWo(y , .) denotes for each y E G the quasiconvex envelope of W o ( Y , .). 0


It remains to de-scale the vector field C. In view of the scalings performed on the deformations, we are naturally led to defidng for each E > 0 the limit displacement field C" : Z -+ EX3 of the middle surface s by

C" := 6. It is then immediately verified that C" satisfies the following minimiza- t ion problem (the notations are those of Theorem 2.5-1):

C" E W~'*(w;EX3) and j$(C") = inf jG(r]), where 1)EW,1'P(W;R3)

The unknown r] in the above minimization problem appears only by means of its first-order partial derivatives a,r] in the stored energy function

r ] E W1Jyw;R3) --f €&*& (a1 + 8177; a2 + a2r]>>

found in the integrand of the energy jh.

in the sense that Assume that the original stored energy function is frame-indifferent,

I@(RF) = I@(F) for all F E M3 and R 6 0;.

This relation is stronger than the usual one, which holds only for F E M3 with det F > 0 (see Ciarlet [18, Theorem 4.2-11); it is, however, verified by the kinds of stored energy functions to which the present analysis applies, e.g., that of a St Venant-Kirchhoff material. Under this stronger assumption, Le Dret & Raoult [64, Theorem 101 establish the crucial properties that the stored energy function found in jh, once expressed as a function of the points of S, is frame-indifferent and that it depends only on the metric of the deformed middle surface. For this reason, this theory is a frame-indifferent, nonlinear "membrane" shell theory .

It is remarkable that the stored energy function found in jh can be explicitly computed when the original three-dimensional stored energy function is that of a St Venant-Kirchhoff material; see Le Dret & Raoult [64, Section 61.

Again for a St Venant-Kirchhoff material, Genevey [50] has furthermore shown that, when the singular values of the 3 x 2 matrix fields (8,vi) associated with a field r] = viai belong to an appropriate compact subset of R2 (which can be explicitely identified), the expression


j & ( q ) takes the simpler form

where aapuT .- .- %a@aUT + 2p(aa‘-7aP7 + aOrTaPU),

x + 2p a,O(q) := a,(e + viai) . a,(e + qjaj).

This is precisely the expression of j & ( q ) that was found by Miara [69] to hold “for all fields q” (i.e., without any restriction on the fields q such as that found by Genevey [50]), by means of a formal asymptotic analysis. This observation thus provides a striking example where the limit equations found by a formal asymptotic analysis “do not always coincide” with those found by means of a rigorous convergence theorem.

Le Dret & Raoult [64, Section 61 have further shown that, if the stored energy function is frame-indifferent and satisfies l@(F) 3 $ ( I ) for all F E M3 (as does the stored energy function of a St Venant-Kirchhoff material), then the corresponding shell energy i s constant under compres- sion. This result has the striking consequence that “nonlinear membrane shells o#er no resistance to crumpling. This is an empirical fact7 wit- nessed by anyone who ever played with a deflated balloon” (to quote H. Le Dret and A. Raoult).

We now turn our attention to the asymptotic analysis, by means of I?- convergence theory, of nonlinearly elastic flexural shells. In its principle, the approach is essentially the same (although more delicate) as ‘that used for deriving the nonlinear membrane shell equations. There are, however, two major differences regarding the assumptions that are made at the onset of the asymptotic analysis.

A first difference is that the applied body forces are now assumed to be of order C ~ ( E ~ ) with respect to E (instead of 0(1)), in the sense that there exists a vector field f E L2 (a; Pi3) independent of E such that

fg(xE) = E ~ ~ ( x ) for all x E = T‘X E s2.

A second difference (without any counterpart for membrane shells) is that the set denoted MF(w) in the next theorem contains other fields than 8 (the interpretation of this key assumption is briefly commented upon after the theorem).

Under these assumptions, F’riesecke, James, Mora & Muller [49] have proved the following result. The notations as($), asp($), and hap($)


used in the next statement are self-explanatory: Given an arbitrary VFC- tor field + E H2(u;JR3),

Theorem 2.5-2. Assume that the applied body forces are of order C ~ ( E ~ ) with respect t o E . Assume in addition that the stored energy function W : M3 4 R satisfies the following properties: It is measurable and of class C2 in a neighborhood of Ot, it satisfies

* ( I ) = 0 and i@(RF) = $ ( F ) for all F E M3 and R E O:, and, finally, it satisfies the following growth condition: There exists a constant C > 0 such that

IL+(F)I c inf I F - ~1~ for all F E M ~ . REO?

Finally, assume that the set

M F ( w ) := {+ E H2(w;JR3); a,p(+) = aap in w ;

+ = 8 and as(+) = a3 on YO},

contains other vector fields than 8 . Then the scaled energies J ( E ) , E > 0 , constitute a family that I?-

converges as E + 0 in the following sense: A n y "diagonal infimizing family" (defined as in Theorem 2.5-1) contains a subsequence that strongly converges in H1(R; R3).

Besides, the limit of any such subsequence is independent of the

transverse variable, and the vector field cp :=

where

An introduction to Shell Theory 141

The assumption that the set M F ( u ) contains other vector fields than 8 means that there exist nonzero displacement fields viai of the middle surface 6(9) that are both inextensionaZ, in the sense that the surfaces 8(Z) and +(G), where + := 8 + viai, have the same metric (as reflected by the assumption a,p(+) = a,p in w ) , and admissible, in the sense that the points of, and the tangent spaces to, the surfaces 8(Z) and +(Z) coincide along the set 6(yo) (as reflected by the boundary conditions + = 8 and a3(+) = a3 on 70).

i t follows that the “de-scaled” unknown deformation pE : Z + R3 of the middle surface of the shell is a minimizer over the set M F ( w ) of the functional j g defined by

When the original three-dimensional stored energy function is that of a St Venant Kirchhof material, the expression j$(+) takes the simpler form

where

interestingly, exactly the same expression jg(+) was found for all + E M F ( w ) by means of a formal asymptotic analysis by Lods & Miara [65], as the outcome of sometimes exceedingly delicate computations. This observation is thus in sharp contrast with that made for a membrane shell, whose limit equations cannot always be recovered by a formal approach, as noted earlier.

Remark. Although r-convergence automatically provides the existence of a minimizer of the r-limit functional, the existence of a minimizer of the functional j~ over the set M F ( u ) can be also established by means of a direct method of calculus of variations; cf. Ciarlet & Coutand [24]. c3


2.6 Linear shell models obtained by asymptotic analysis

In this section, we briefly review the genesis of those two-dimensional linear shell theories that can be found, and rigorously justified, as the outcome of an asymptotic analysis of the equations of three-dimensional linearized elasticity as E -+ 0 .

The asymptotic analysis of elastic shells has been a subject of considerable attention during the past decades. After the landmark attempt of Goldenveizer [53], a major step for linearly elastic shells was achieved by Destuynder [44] in his Doctoral Dissertation, where a convergence theorem for “membrane shells” was “almost proved”. Another major step was achieved by Sanchez-Palencia [77], who clearly delineated the kinds of geometries of the middle surface and boundary conditions that yield either two-dimensional membrane, or two-dimensional flexural, equations when the method of formal asymptotic expansions is applied to the variational equations of three-dimensional linearized elasticity (see also Cail- lerie & Sanchez-Palencia [17] and Miara & Sanchez-Palencia [70]).

Then Ciarlet & Lods [27, 281 and Ciarlet, Lods & Miara [31] carried out an asymptotic analysis of linearly elastic shells that covers all possible cases: Under three distinct sets of assumptions on the geometry of the middle surface, on the boundary conditions, and on the order of magnitude of the applied forces, they established convergence theorems in H I , in L2, or in ad hoc completion spaces, that justify either the linear two-dimensional equations of a “membrane shell”, or those of a “generalized membrane shell”, or those of a ‘‘j?exural shell”.

More specifically, consider a family of linearly elastic shells of thiclc- ness 2~ that satisfy the following assumptions: All the shells have the same middle surface S = 8(C) c R3, where w is a domain in R2 with boundary y, and 8 E C 3 ( Z ; R 3 ) . Their reference configurations are thus of the form O(fi“), E > 0, where

RE := w x (-&,&)

and the mapping 0 is defined by

@ ( y , x ~ ) := 6 ( y ) + x;a3(y) for all ( y , ~ : ) .

All the shells in the family are made with the same homogeneous isotropic elastic material and that their reference Configurations are natural states. Their elastic material is thus characterized by two Lam6 constants X > 0 and p > 0.

The shells are subjected to body forces and that the corresponding applied body force density is O(EP) with respect t o E , for some ad hoc power p (which will be specified later). This means that, for each E > 0,


the contravariant components f z ? " E L2(RE) of the body force density f' = fi)'gS are of the form

f i l E ( y , E Z ~ ) = & p f i ( y , z3) for all (y, z3) E R := w x 1-1, I[,

and the functions f i E L2(R) are independent of E (surface forces acting on the "upper" and "lower" faces of the shell could be as well taken into account but will not be considered here, for simplicity of exposition). Let then the functions pi>' E L2(w) be defined for each E > 0 by

J -€

Finally, each shell is subjected to a boundary condition of place on the portion @(yo x [ -E, €1) of its lateral face, where yo is a fixed portion of y, with lengthy0 > 0.

Then the displacement field of the shell satisfies the following minimization problem associated with the equations of linearized three- dimensional elasticity in curvilinear coordinates (see Section 2.3):

uE E V ( V ) and JE(uE) = min JE(v'), where V " E V ( W )

V ( R E ) := {vE = v;gi+; w; E H ~ ( R ) , u: = o on TO x ( - E , E ) } .

For each E > 0, this problem has one and only one solution u' E V ( R ) . For any displacement field q = V i a i : w ---f R3, let

1 y d v ) = ~ ( d p v . acu + aav. ap) and pOlg(r l ) = ( % g q - r&a,q) . a3

denote as usual the covariant components of the linearized change of metric, and linearized change of curvature, tensors.

In Ciarlet, Lods & Miara [31] it is first assumed that the space of linearized inextensional displacements (introduced by Sanchez-Palencia PI)

v,(w) := { q = viai; qa E ~ ' ( w ) , 773 E H ~ ( w ) ; vi = dyv3 = 0 on yo, ycyp(q) = 0 in w}

contains non-zero functions. This assumption is in fact one in disguise about the geometry of the surface S and on the set 70 . For instance, it is satisfied if S is a portion of a cylinder and O(70) is contained in one or two generatrices of S , or if S is contained in a plane, in which case the shells are plates.


Under this assumption Ciarlet, Lods & Miara [31] showed that, i f the applied body force density is O(e2) with respect to E , then

-!- 1' uE dz: -+ C in H 1 (w; R3) as E + 0, 2 E -&

where the limit vector field satisfies the equations of a linearly elastic "flexural shell" , viz.,

:= &ai belongs to the space V F ( W ) and

E" 3 S, a a p c T p u T ( < ) p f f p ( v ) h d y = b p i l ~ v i h d y

for all 77 = viai E VF(W). Observe in passing that the limit 6 is indeed independent of E , since both sides of these variational equations are of the same order (viz., E ~ ) , because of the assumptions made on the applied forces.

Equivalently, the vector field 6 satisfies the following constrained minimization problem:

5 E V F ( W ) and G(5) = inf&(v),

where

for all v = viai E VF(W) , where the functions

are precisely the familiar contravariant components of the shell elasticity tensor.

If V F ( W ) # { 0 } , the two-dimensional equations of a linearly elastic 'Vexural shell" are therefore justified.

If V F ( W ) = ( 0 } , the above convergence result still applies. However,

the only information it provides is that - u' d&J -+ 0 in H1(w; R3) as E -+ 0. Hence a more refined asymptotic analysis is needed in this case.

A first instance of such a refinement was given by Ciarlet & Lods [27], where it was assumed that yo = y and that the surface S is elliptic, in the sense that its Gaussian curvature is > 0 everywhere. As shown in Ciarlet & Lods [27] and Ciarlet & Sanchez-Palencia [35], these two conditions, together with ad hoc regularity assumptions, indeed imply that V p ( w ) = (0).

2E SE -&


In this case, Ciarlet & Lods [28] showed that, i f the applied body force density is 0(1) with respect to E , then

/' u: dxg + <a in H1(w) and - ug dx; --f c3 in L 2 ( w ) as E --f 0, 2 E p E 2 E J' - E

where the limit vector field C := &ai belongs t o the space

V M ( W ) := (77 = viai; va E H,l(w), ~3 E ~ ~ ( w ) ) ,

and solves the equations of a linearly elastic "membrane shell", viz . ,

&aapUTyur ( C ) ~ a p (q)& d~ = pi"vi& d y

for all q = viai E V M ( W ) , where the functions a"Pu7,yap(q) ,a , and pi)€ have the same meanings as above. If yo = y and S i s elliptic, the two-dimensional equations of a linearly elastic "membrane shell" are therefore justified. Observe that the limit C is again independent of E ,

since both sides of these variational equations are of the same order (viz., E ) , because of the assumptions made on the applied forces.

Equivalently, the field C satisfies the following unconstrained minimization problem:

where &(v) := 2 1 J I _ E U U " " ' m T ( q ) y a p ( l ) ~ d Y - l p z 3 E v i & d y .

Finally, Ciarlet & Lods [30] studied all the "remaining" cases where VF(w) = { 0 } , e.g., when S is elliptic but lengthyo < lengthy, or when S is for instance a portion of a hyperboloid of revolution, etc. To give a flavor of their results, consider the important special case where the semi-norm

becomes a n o r m over the space

W ( W ) := { q E H ' ( ~ ; I w ~ ) ; q = o on yo).

In this case, Ciarlet & Lods [30] showed that, i f the applied body forces are "admissible" in a specific sense (but a bit too technical to be described here), and i f their density is again 0(1) with respect t o E , then


where M V L ( w ) := completion of W ( w ) with respect to ] . I w .

Furthermore, the limit field C E V # , ( w ) solves "limit" variational equations of the form

E B ~ ( c ~ , q) = ~ g ( q ) for all q E v ~ ( w ) ,

where B L is the unique extension to V # , ( w ) of the bilinear form BM defined by

i.e., EBM is the bilinear form found above for a linearly elastic "membrane shell", and L g : V& (w) -+ R is an ad hoc linear form, determined by the behavior as E -+ 0 of the admissible body forces.

In the "last" remaining case, where V F ( W ) = ( 0 ) but 1.1: is not a norm over the space W ( w ) , a similar convergence result can be established, but only in the completion V,(W) with respect of ( . I w of the quotient space W ( w ) / W , ( w ) , where W , ( w ) = {q E W ( w ) ; 'yap(q) = O in w } .

Either one of the above variational problems corresponding to the %e- maining" cases where V F = {0} constitute the equations of a linearly elastic "generalized" membrane shell, whose two-dimensional equations are therefore justified.

The proofs of the above convergence results are long and technically difficult. Suffice it to say here that they crucially hinge on the Korn inequality "with boundary conditions" (Theorem 2.9-3) and on the Korn inequality ((on an elliptic surface" (end of Section 2.9).

Combining these convergences with earlier results of Destuynder [45] and Sanchez-Palencia [75, 76, 781 (see also Sanchez-Hubert & Sanchez- Palencia [79]), Ciarlet & Lo& [28, 291 have also justified as follows the linear Koiter shell equations studied in Sections 2.8 to 2.10, again in all possible cases.

Let C' denote for each E > 0 the unique solution (Theorem 2.10-2) to the linear Koiter shell equations, viz., the vector field that satisfies

C" E V ( w ) = { q = viai; va E ~ ' ( w ) , q 3 E H'(w); vi = ~ ~ 7 3 = o on 'yo},

. i l M

1 W { & ~ " " " ' ~ P T ( C ' ) ~ ~ ~ ( ~ ) + ~ U " " " ' p u " ' ( C " ) p , B ( q ) } ~ d ~


or equivalently, the unique solution to the minimization problem

C' E V ( w ) andj(6') = inf j (7) v E V ( w )

where

Observe in passing that, for a linearly elastic shell, the stored energy function found in Koiter's energy, viz.,

E3 77 - { iaapcT rv.r(7))raa(rl) -I- ~aa@'PuT(17)Pa/3(7)}

i s thus exactly the s u m of the stored energy func t ion of a linearly elastic "membrane shell" and of that of a linearly elastic 'yexural shell".

Then, f o r each category of linearly elastic shells (membrane, gener- 1 PE 1 alized membrane, or flexural), the vector fields C' and -

where u' denotes the solution of the three-dimensional problem, have exactly the same asymptotic behavior as E + 0 , in precisely the same func t ion spaces that were found in the asymptotic analysis of the three- dimensional solution.

It is all the more remarkable that Koiter's equations can be fully justified f o r all types of shells, since it is clear that Koiter's equations cannot be recovered as the outcome of an asymptotic analysis of the three-dimensional equations, the two-dimensional equations of linearly elastic, membrane, generalized membrane, or flexural, shells exhausting all such possible outcomes!

So, even though Koiter's linear model i s not a limit model, it i s in a sense the "best" two-dimensional one for linearly elastic shells!

One can thus only marvel at the insight that led W.T. Koiter to conceive the "right" equations, whose versatility is indeed remarkable, out of purely mechanical and geometrical intuitions!

We refer to Ciarlet [20] for a detailed analysis of the asymptotic analysis of linearly elastic shells, for a detailed description and analysis of other linear shell models, such as those of Naghdi, Budiansky and Sanders, Novozilov, etc., and for an extensive list of references.

2.7 The nonlinear Koiter shell model

In this section, we begin our study of the equations proposed by W.T. Koiter for modeling thin elastic shells. These equations are "two-dimen-


sional”, in the sense that they are expressed in terms of two curvilinear coordinates used for defining the middle surface of the shell.

To begin with, we describe the nonlinear Koiter shell equations, so named after Koiter [59] , and since then a two-dimensional nonlinear model of choice in computational mechanics.

Given an arbitrary displacement field 77 := viai : 3 4 R3 of the surface S with smooth enough components vi : W + R, let

aap(v) := aa(v) . ap(v), where aa(v) := aa(6 + v), denote the covariant components of the first fundamental form of the deformed surface (6 + ~)(a). Then the functions

denote the covariant components of the change of metric tensor associated with the displacement field 77 = vzaz of S.

If the two vectors aa(v) are linearly independent at all points of w, let

1 bap(v) := ~ aa,(@ + v). {a1(v) A a2(v)l,

477) := det(aap(v)),

m where

denote the covariant components of the second fundamental form of the deformed surface (6 + v)(W). Then the functions

Rap(v) := bap(v) - bap

denote the covariant components of the change of curvature tensor field associated with the displacement field v = vzaz of S. Note

Note that both surfaces 6(W) and (6 + q)(W) are equipped with the same curvilinear coordinates y1, y2.

As a point of departure, consider an elastic shell made of a St Venant- Kichhoff material modeled as a three-dimensional problem (Section 2.3). The nonlinear two-dimensional equations proposed by Koiter [59] for modeling such an elastic shell are then derived from those of nonlinear three-dimensional elasticity on the basis of two a priori assumptions: One assumption, of a geometrical nature, is the Kirchhoff-Love assumption. It asserts that any point situated on a normal to the middle surface remains on the normal to the deformed middle surface after the deformation has taken place and that, in addition, the distance between such a point and the middle surface remains constant. The other assumption, of a mechanical nature, asserts that the state of stress inside the shell

that m = la1 (v) A a2 (v) I .

An Introduction to Shell Thcory 149

is planar and parallel to the middle surface (this second assumption is itself based on delicate a priori estimates due to John [56, 571).

Taking these a priori assumptions into account, W.T. Koiter then reached the conclusion that the displacement field 6" = <fai of the middle surface S := O(Z) of the shell, where the functions <: are unknowns, should be a stationary point, in particular a minimizer, over a set of smooth enough vector fields q = viai : 55 --j R3 satisfying ad hoc boundary conditions on 7 0 , of the functional j defined by (cf. Koiter [59, eqs. (4.2), (8.1), and (8.3)]):

- ~ P z ~ " v i & d y ,

where the functions a'lpur and pi>" E L 2 ( w ) are the same as in Section 2.6, i.e., they are defined by

The above functional j is called Koiter's energy for a nonlinear

The stored energy function W K found in Koiter's energy j is thus elastic shell.

defined by

&3 apuTRu, (q)Rap(77)

& w K ( q ) = - ~ ~ f l ~ ~ ~ ~ ~ ( r ~ ) ~ ~ ~ ( q ) + 2

for ad hoc vector fields q = viai. This expression is the sum of the "membrane" part

& W M ( V ) = 5a '""" 'Gu7(~)Gap(~)

and of the 'yexural" part

E3 W F ( r l ) = --aa~u"'Ru'(rl)R,p(rl). 6

Another closely related set of nonlinear shell equations "of Koiter's type" has been proposed by Ciarlet [all. In these equations, the denominator Ja(rl) that appears in the functions R,p(q) = bCYp(q) - b,p is simply replaced by 6, thereby avoiding the possibility of a vanishing denominator in the expression WK(T,I). Then Ciarlet & Roquefort


[34] have shown that the leading term of a formal asymptotic expansion of a solution to this two-dimensional model, with the thickness 2s as the "small" parameter, coincides with that found by a formal asymptotic analysis of the three-dimensional equations. This result thus raises hopes that a rigorous justification, by means of r-convergence theory, of a nonlinear shell model of Koiter's type might be possible.

2.8 The linear Koiter shell model

Consider the Koiter energy j for a nonlinearly elastic shell, defined by (cf. Section 2.7)

- pi'Erli& d y ,

for smooth enough vector fields r] = viai : W --+ R3. One of its virtues is that the integrands of the first two integrals are quadratic expressions in terms of the covariant components Gap(r]) and R,p(r]) of the change of metric, and change of curvature, tensors associated with a displacement field r] = viai of the middle surface S = O(W) of the shell. In order to obtain the energy corresponding to the linear equations of Koiter [60], which we are about to describe, it suffices, " b y definition", to replace the covariant components

1 G,p(r]) = 2(aap(r ] ) - asp) and Radr]) = b a d r ] ) - bap,

of these tensors by their linear parts with respect t o r] , respectively denoted r a p ( r ] ) and pap(r]) below. Accordingly, our first task consists in finding explicit expressions of such linearized tensors. To begin with, we compute the components rap ( r ] ) .

Theorem 2.8-1. Let w be a domain in R2 and let 8 E C2(W;R3) be an immersion. Given a displacement field r] := viai of the surface S = O(w) with smooth enough covariant components v i : W -+ R, let the function -yap(r]) : W 4 R be defined by

where a,p and aap(r ] ) are the covariant components of the first fundamental f o rm of the surfaces O(D) and (@+r])(g), and [. . denotes the


linear part with respect t o 11 in the expression [. . . ] . Then

1

1

1

~ ( v ) = 2 (apv * a, + a,v. ap) = ypa(v)

= #l'"lp + vpla) - bapv3

- - - ( ~ o T , + a a v ~ ) - r&vu - bap773,

where the covariant derivatives v,lp are defined by valp = dpv, - r",vu. In particular then,

v, E H1(w) and v 3 E L 2 ( w ) * ~ ~ ~ ( 7 7 ) E L2(u) .

Proof. The covariant components a,p(q) of the metric tensor of the surface (0 + v)(W) are by definition given by

then show that

hence that

The other expressions of 'yap(v) immediately follow from the relation

aav = &(viai) = (&vu - - b,,v3)a' + (&v3 + bzv7)a3,

itself a consequence of the Gauss and Weingarten equations (see Section

&ar = -r;uau + bLa , &a = -b,,a".

2.1) 3

3

17

The functions yap(q) are called the covariant components of the linearized change of metric tensor associated with a displacement 77 = viai of the surface S.

We next compute the components pap(v).

The relations


Theorem 2.8-2. Let w be a domain in R2 and let 8 E C3(W;R3) be an immersion. Given a displacement field r,~ := Viai of the surface S = 8(W) with smooth enough and "small enough" covariant components r/i : w + R, let the functions pap(q) : 2 + R be defined by

P a d r l ) := [bap(r l ) - bapl?

where b,p and b,p(v) are the covariant components of the second fundamental form of the surfaces 8(w) and (8 + v)(W), and [. . denotes the linear part with respect to rl in the expression [. . . I . Then

Pap( r l ) = (aaprl - c p a u r l ) . a3 = ppa(r l ) - - r/3l0$ - b;b~Or/3 + b;r/glp + b z r / ~ j a f b;l,r/~

- aapr/3 - rgpaur /3 - b",cpr/3 -

+bg(dpr /U - r;Ur/7) + b;(a,r/T - r z ~ r / U )

+(aab$ f rLUb; - r;pbz)r/T,

where the covariant derivatives qalp, r/31,0, and bzla are defined b y

I n particular then,

7, E and 73 E H 2 ( 4 =+ Pap(77) E L 2 ( 4 .

The functions b;l, satisfy the symmetry relations

b;la = b:/p.

Proof. For convenience, the proof is divided into five parts. In parts (i) and (ii), we establish elementary relations satisfied by the vectors ai and a2 of the covariant and contravariant bases along S.

(i) The two vectors a, = da8 satisfy (al A a2( = &, where a =

Let A denote the matrix of order three with al, a2, a 3 as its column det (asp ) .

vectors. Consequently,

= /a1 A a21. ai A a 2

la1 A a21 det A = (al A az) . a3 = (a1 A a2) .

Besides, (det A)2 = det(ATA) = det(aap) = a ,


since a, . ap = a,p and a, . a3 = Sa3. Hence lal A a21 = &. (ii) The vector fields ai and aa are related by a1 A a3 = -*a2 and

To prove that two vector fields c and d coincide, it suffices to prove a3 A a 2 = -&a’.

that c . ai = d . ai for i E {1,2,3}. In the present case,

(a1 A as) . al = 0 and (a1 A a3) . a3 = 0, (a1 A as) . a 2 = -(a1 A a 2 ) . a3 = -fi,

since &a3 = al A a2 by (i), on the one hand; on the other hand,

-&a2 . a1 = -&a2. a3 = 0 and - &a2 . a 2 = -&, since ai .aj = 6;. Hence a1 A a 3 = -&a2. The other relation is similarly established.

(iii) The covariant components b,p(q) satisfy

b,p(rl) = b,p + (daprl - rZPa,rl) a3 + h.o.t.,

where ((h.0.t.” stands for “higher-order terms”, i.e., terms of order higher than linear with respect to 7. Consequently,

~ , p ( r l ) := [bap(r l ) - bap]lin = (daprl - r ~ ~ & r l ) . a3 = pp,(rl).

Since the vectors a, = a,9 arc linearly independent in W and the fields rl = via, are smooth enough by assumption, the vectors &(9 + q) are also linearly independent in w provided the fields r,~ are “small enough”, e.g., with respect to the norm of the space C1(i;S;R3). The following computations arc therefore licit as they apply to a linearization around rl = 0.

Let

1 ,-= - (&ap + aaprl) . (a1 A a 2 + a1 A 8217 + dlrl A a 2 + h.0.t.) m

where

Then


since baj3 = daa0 • a3 and daa0 = r^aC T + bap&3 by the formula ofGauss. Next,

(r^aCT + ba0as) • (ai A d2v)

= r ^ a 2 • (ai A d2ri) - bapd2r] • (ai A a3)

• a 2 ) ,

since, by (ii), a2 • (ax A d2r)) = -d2ij • (ai A a2) = -y/ad2r\ • a3 andai A a3 = —-y/aa2; likewise,

^ A a2) = Va(-r^g#i77 • a3 + b^dxT} • a1).

Consequently,

i ^ 1 + a-^ • aCT) + ( ^ ^ - r£/A»7) • a3 + h.o.t.} .

There remains to find the linear term with respect to q in the expan-

sion —1 - —-(11 + . . . ) . To this end, we note that

det(A + H) = (detA

with A := (asp) and A + H := (a,p(q)). Hence

H = aprl . a, + a,~ . ap + h.o.t.,

since [a,p(V) — aap]lin = aprl . a, + 8,~. • a0 (Theorem 2.8-1). Therefore,

a(v) = det(a,p(V)) = det(a,p)(l + 28,~. • a" + h.o.t.),

since A-' = (amp); consequently,

- - 1 --(11 - a,~ . a a + h.0.t.).

Noting that there are no linear terms with respect to q in the product(1 - a,V . a a)( l + 8,~ . a ' ) , we find the announced expansion, viz.,

b,p + {dapv ~ r&a,rl) . a3 + h.0.t.

(iv) The components pap(ij) can be also written as

where the functions %|a/3 and b^\a are defined as in the statement of thetheorem.


The Gauss and Weingarten equations, viz.,

da*T = -T^a" + Vaa3,

daa3 = -batTaT,

imply that

then that

da0V • a3 = da

~\~{pa0ri3 + [dQ

= b^(d0r)a — I

since

We thus obtain

{(<H,-r^-Wfc)«

:^K+^ Q ? ? T )a 3 .a 3 + (

a17 • a3 = ( - r ^ T a T + b^a3)

da&3 • a3 = ~batTaa • a3

i" +

a3

= 0.

'3 + &^

(5/3% -f

; + V07]r

,bT0)Vr i

= b^,

\a3rja ,

~b0Vr

)daa3

-b0da

)a3}-a3

•a3

VT,

• a3

While this relation seemingly involves only the covariant derivativesrj3\af3 and rj^p, it may be easily rewritten so as to involve in additionthe functions r\r\a and bp~,. The stratagem simply consists in using therelation TT

acrbaprjr - T

aaTbT

0r]a = 0! This gives

(v) The functions bila are symmetric with respect to the indices Qand /3.

Again, because of the formulas of Gauss and Weingarten, we canwrite

0 - da0aT - d0aa

T = da ( - I ^ X + b}e?) - d0 {-TTaaar + br

aa3)

= -(9arT

0a)^ + T ^ r ^ a " - r ^ 6 > 3 + (dabT

0)a3 ~ b0baaa°


Consequently,

0 = (&paT - 8paaT) . a3 = dab; - dpb7, + r' a" b" P - I7iUb;, on the one hand. On the other hand, we immediately infer from the definition of the functions b i l , that we also have

and thus the proof is complete.

The functions p a ~ ( q ) are called the covariant components of the linearized change of curvature tensor associated with a displacement q = viai of the surface S. The functions

r/31ap = 8,0773 - r;p8,773 and b;la = dab; + rLUbz - respectively represent a second-order covariant derivative of the vector field viai and a first-order covariant derivative of the second fundamental form of S , defined here by means of its mixed components b;.

Remarks. (1) The functions bap(v) are not always well defined (in order that they be, the vectors a,(q) must be linearly independent in a), but the functions pap(q) are always well defined.

(2) The symmetry pap(q) = pp,(q) follows immediately by inspection of the expression pap(q) = (d,pq-r'",8,q).a3 found there. By contrast, deriving the same symmetry from the other expression of p,p(q) requires proving first that the covariant derivatives bzla are themselves symmetric with respect to the indices a and ,D (cf. part (v) of the proof of Theorem 2.8-1). 0

While the expression of the components pap(q) in terms of the covariant components qi of the displacement field is fairly complicated but well known (see, e.g., Koiter [60]), that in terms of q = viai is remarkably simple but seems to have been mostly ignored, although it already appeared in Bamberger [lo]. Together with the expression of the components ~ ~ p ( q ) in terms of q (Theorem 2.8-l), this simpler expression was efficiently put to use by Blouza & Le Dret [13], who showed that their principal merit is to afford the definition of the components ~ ~ p ( q ) and pa0 (q) under substantially weaker regularity assumptions on the mapping 8.

More specifically, we were led to assume that 8 E C 3 ( W ; R3) in The- orem 2.8-2 in order to insure that pap(q) E L 2 ( w ) if q = viai with 7, E H1(w) and 773 E H2(w) . The culprits responsible for this regularity are the functions b';la appearing in the functions pap(q). Otherwise


Blouza & Le Dret [13] have shown how this regularity assumption on O can be weakened if only the expressions of yap(q) and pap(q) in terms of the field q are considered.

We arc now in a position to describe the linear Koiter shell equations. Let yo be a measurable subset of y = dw that satisfies lengthyo > 0, let 8, denote the outer normal derivative operator along dw, and let the space V(w) be defined by

~ ( w ) := { q = viai; va E ~ ' ( w ) , q 3 E ~ ~ ( w ) , q i = dvr/3 = o on yo}.

Then the displacement field C" = <Fai of the middle surface S = O(is) of the shell (the covariant components <; are unknown) should be a stationary point over the space V ( w ) of the functional j defined by

for all q = viai E V(w). This functional j is called Koiter's energy for a linearly elastic shell.

Equivalently, the vector field <" = <fai E V(w) should satisfy the variational equations

We recall that the functions

denote the contravariant components of the shell elasticity tensor (A and p are the Lam6 constants of the elastic material constituting the shell), yap(q) and pap(q) denote the covariant components of the linearized change of metric, and change of curvature, tensors associated with a displacement field q = viai of S , and the given functions pi,& E L 2 ( w ) account for the applied forces. Finally, the boundary conditions qi = 8,773 = 0 on yo express that the shell is clamped along the portion O(y0)

of its middle surface (see Figure 2.4-1). The choice of the function spaces H1 (w) and H 2 (w ) for the tangential

components qa and normal components 73 of the displacement fields q = viai is guided by the natural requirement that the functions yap(q)


and pag(q) be both in L 2 ( w ) , so that the energy is in turn well defined for 71 E V ( w ) . Otherwise these choices can be weakened to accommodate shells whose middle surfaces have little regularity (see Blouza & Le Dret [131).

Remark. Koiter’s linear equations can be fully justified by means of an asymptotic analysis of the “three-dimensional” equations of linearized elasticity as E + 0; see Section 2.6. For more details, see Ciarlet [20,

0 Chapter 71 and the references therein.

2.9

Our objective in the next sections is to study the existence and uniqueness of the solution to the variational equations associated with the linear Koiter model. To this end, we shall see (Theorem 2.10-1) that, under the assumptions 3X + 2p > 0 and p > 0, there exists a constant ce > 0 such that

Korn’s inequalities on a surface

c [ t a p ( 2 < ceaa~uT(y) tuTtap a,P

for all y E and all symmetric matrices ( t a p ) . When lengthyo > 0, the existence and uniqueness of a solution to this variational problem by means of the Lax-Milgram lemma will then be a consequence of the existence of a constant c such that

2 1 /2 { c Ilvall%l(w) + llv3IlLyd)} < c{ c IlryaP(rl)llLz(w)

+ c I I~~~(~])JI ;+,)

a %P

for all rl = qiai E ~ ( w ) . %P

Such a key inequality is an instance of a Korn inequality on a surface. The objective of this section is to establish such an inequality.

To begin with, we establish a Korn’s inequality on a surface, “without boundary conditions”, as a consequence of the lemma of J.L. Lions (cf. Theorem 1.5-1). We follow here Ciarlet & Miara [33] (see also Bernadou, Ciarlet & Miara [la]). Theorem 2.9-1. Let w be a domain in R2 and let 8 E C3(W;R3) be an injective immersion. Given 77 = viai with qa E H 1 ( w ) and 73 E H2(w) , let

1 ya)’ap(rl) := { z ( a p r l . am + aarl. a@)} E ~2(w),

pao(rl) := { (aaprl - rgpaDr l ) . a3) E ~’(w)


denote the covariant components of the linearized change of metric, and linearized change of curvature, tensors associated with the displacement field r ] = viai of the surface S = @(a). Then there exists a constant c0 = co(w,@) such that

f o r all q = viai with qa E H1(w) and 7 3 E H 2 ( w ) .

Proof. The “fully explicit” expressions of the functions rap(q) and p a p ( r ] ) , as found in Theorems 2.8-1 and 2.8-2, are used in this proof, simply because they are more convenient for its purposes.

(i) Define the space

W ( w ) := { q = viai; va E L ~ ( w ) , ~ ~ E ~ l ( w ) ,

raa(r]) E L2(w)1 P a d r ] ) E L 2 ( 4 } .

Then, equipped with the norm ~ ~ ~ ~ ~ w ( w ) defined by

11r]11W(w) := { c Ilva11~22(w) + llv311:l(w) + c llraP(r]>112L2(w)

+ c llPaP(r])l12L2(w)}1’21

a aY,P

%P

the space W ( w ) is a Hilbert space. The relations “yap(r ] ) E L2(w)” and “ p a p ( r ] ) E L2(w)” appearing in

the definition of the space W ( w ) are to be understood in the sense of distributions. They mean that a vector field r ] = viai, with qa E L2(w) and 7 3 E H1(w), belongs to W ( w ) if there exist functions in L 2 ( w ) , denoted r a p ( r ] ) and pap(?), such that for all cp E D ( w ) ,

1 / YCYP(r])(PdY = - s, { Z(VPd*cp + va%(P) + r : p w + bapV3cp} dY, W

P ~ ~ ( v ) ( P ~ Y = - / {8av3%3cp + JZp%v3cp + b3,p773cp W

+ vu-a,(b:cp) + b:r;,vTcp

+ vTaa(b;cp) + b;r:,vocp

- (8, b i + r:, bz - r’zp b:) qT p} dy .


Let there be given a Cauchy sequence (qk)& with elements qk = vtai E W ( w ) . The definition of the norm (( . ( (w(w) shows that there exist va E L 2 ( w ) , 173 E H1(w) , yap E L 2 ( w ) , and pap E L 2 ( w ) such that

7; 4 va in ~ ' ( w ) , T$ -+ 73 in ~ ' ( w ) , 7ap(vk) -+ -Yap in L2(4, pap(vk) -+ Pap in L 2 ( 4

as k -+ 00. Given a function cp E D(w) , letting lc -+ 00 in the relations

L p ( k yap(q )cpdw = . . . and ap qk)cpdw = . . .

then shows that yap = yap(q) and pap = pap(q).

(ii) The spaces W(w) and {q = viai; va E H1(w),r/3 E H2(w)}

Clearly, {q = %ai; va E H 1 ( w ) , 773 E H2(w)} c W ( w ) . To prove the coincide.

other inclusion, let q = via2 E W ( w ) . The relations

1 sap(rl) := pave + 43va> = yap(r1) + r",v, + bapv3

then imply that sap(?) E L 2 ( w ) since the functions continuous on w. Therefore,

and b,p are

aova E H-l (w) , ap(aova) = {aps,o(rl) + @7sap(v) - aasp&d) E f . - l (w),

since x E L 2 ( w ) implies a,x E H-'(w). Hence aova E L2(w) by the lemma of J.L. Lions (Theorem 1.5-1) and thus qa E H1(w) .

The definition of the functions pap(q), the continuity over W of the functions bop, bg, and ad;;, and the relations pap(q) E L2(w) then imply that dap773 E L2(w), hence that q 3 E H2(w) .

(iii) Korn's inequality without boundary conditions. The identity mapping L from the space { r ] = viai; 77, E H 1 ( w ) , r/3 E

E l 2 ( # ) } equipped with the norm

rl = viai cc IlvallLl(w) + 11v311L2(w)}1'2 a

into the space W ( w ) equipped with ( 1 . ( ( w ( ~ ) is injective, continuous, and surjective by (ii). Since both spaces are complete (cf. (i)), the open mapping theorem then shows that the inverse mapping L - ~ is also continuous or equivalently, that the inequality of Korn's type without boundary conditions holds. 0


In order to establish a Korn’s inequality “with boundary conditions” , we have to identify classes of boundary conditions to be imposed on the fields 71 = viai, with r ] , E H 1 ( w ) and 773 E H 2 ( w ) , in order that we can “get rid” of the norms ) ) ~ ~ , I I L Z ( ~ ) and in the right-hand side of the above inequality, i.e., situations where the semi-norm

becomes a norm, which should be in addition equivalent to the norm

rl = via2 cc llvall;l(w) + l177311;2(w)}1’2. Q

To this end, we need an infinitesimal rigid displacement lemma “on a surface” (the adjective “infinitesimal” reminds that only the linearized parts r,p(r]) and p a p ( q ) of the “full” change of metric and curvature tensors i ( aap ( r l ) - a,p) and (b,p(q) - b,p) are required to vanish in w ) , which is due to Bernadou & Ciarlet [ll, Theorems 5.1-1 and 5.2-11; see also Bernadou, Ciarlet & Miara [12, Lemmas 2.5 and 2.61 and Blouza & Le Dret [13, Theorem 61.

Part (a) in the next theorem is an infinitesimal rigid displacement lemma on a surface, “without boundary conditions”, while part (b) is an infinitesimal rigid displacement lemma on a surface, “with boundary conditions”.

Any proof of this theorem is, at least to some extent, delicate. The one given here is not the shortest, but it is a natural one: It relies on the classical, and much easier to prove, “three-dimensional infinitesimal rigid displacement lemma in Cartesian coordinates” used in part (iii) of the next proof (a “direct” proof, such as the one originally found by Bernadou & Ciarlet [ll] , is surprisingly “technical”).

In the next proof, the functions Eij(G) and eij(v) are the Cartesian and covariant components of the “three-dimensional” linearizecstrain tensor respectively associated with a displacement field G i 8 : (0)- 4 EX3 and a displacement field vigi : EX3; the functions are the Christoffel symbols (of the second kind) associated with the mapping 0 : 2 ---f {0}-; finally, the functions vuiilj are the covariant derivatives

0 of the vector field vig’ : 0 + EX3.

Theorem 2.9-2. Let there be given a domain w in EX2 and an injective immersion 8 E C3 (w; EX3).

h

. -

(a) Let q = viai with 7, E H 1 ( w ) and 773 E H 2 ( w ) be such that

rap(r l ) = pap(77) = 0 in w .


Then there exist two vectors a, b E R3 such that

q ( y ) = a + b A O(y) for all y E a. (b) Let yo be a dy-measurable subset of y = dw that satisfies length

yo > 0 and let a vector field q = viai with va E H1(w) and 773 E H2(w) be such that

yap(q) = ,oaa(q) = 0 in w and qi = dVv3 = 0 on yo.

Then q = 0 an w .

Proof. The proof is divided into five parts, numbered (i) to (v).

(i) I n parts (i) to (iii), R denotes a domain. in R3 and 0 : a + R3 denotes a C2-diffeomorphism from fi onto its image @(a). Consequently, the three vectors gi(x) := &0(3:), where di = d/axi, are linearly independent at all points 3: = (xi) E 3 and the three vectors gi(x) defined by gi(z)gj(z) = 6; are likewise linearly independent at all points 3: E a. Note also that gi E C1(a;R3) and gz E C1(a;Iw3).

Let Gz denote the b@s of R3, let 2i deno_te the Cartesian coordinates of a point 2 E EX3, let di := a/d2i, and let R := @(a). With any vector field v = uig’ : R -+ R3 with ui E H’(R), we then associate a vector field 3 = Gig : {6}- + R3 by letting

. -

Gi ($ )2 = vi(x)gi(x) for all 2 = ~ ( x ) , x E G.

It is then clear that Gi E H 1 ( 6 ) . We now show that, for all x E 0,

gjGi(2) = (uUk,~e[~”li[~elj)(”), 2 = q x ) , where

and

denotes the i - th component of gk(z) over the basis {Gl,G2,G3}. In what follows, the simultaneous appearance of 2 and x in an equal-

ity means that they are related by 2 = O(x) and that the equality in question holds for all 3: E R.

Let O(3:) = O‘(cc)Gk and 6(2) = @(2)ei, where 6 : 6 -+ R3 denotes the inverse mapping of 0 : R -+ R3. Since 6(0(3:)) = cc for all cc E R, the chain rule shows that the matrices VO(x) := (a j@(z)) (the row index is k) and V6@) := (dkO’(2)) (the row index is i ) satisfy

u . . .- 8.v. - r ? . v and r?. := gp . &gj, 2113 .- 3 2 ‘3 P ‘3

[gk(x)]z := g“3:) . Gi

h h .

V6(2)V@(x) = I,


or equivalently,

The components of the above column vector being precisely those of the vector gj(x), the components of the above row vector must be those of the vector gi(x) since gi(x) is uniquely defined for each exponent i by the three relations gi(x) . gj(x) = d;, j = 1 , 2 , 3 . Hence the k-th component of gZ(x) over the basis { & , G 2 , & } can be also expressed in terms of the inverse mapping 0, as:

h

h A .

[gi(x)]k = dkO"2).

We next compute the derivatives &gq(x) (it is easily seen that the fields gq = gqTgT are of class C1 on R since 0 is assumed to be of class C2) . These derivatives will be needed below for expressing the derivatives GGi(2) as functions of x (recall that Gi(2) = uk(z)[gk(x)]i). Recalling that the vectors gk(x) form a basis, we may write a priori

amq (.) = -r&(x)gk (x) 1

thereby unambiguously defining functions expressions in terms of the mappings 0 and 0, we observe that

: R + R. To find their A

r;k(x) = r ; m ( x ) q = r;m(x)gm(x) .gk(x) = -aegq(x) .gk(x). A h

Hence, noting that &(gq(x) . gk(x)) = 0 and [gq(x)lP = aPOq(2), we obtain

h h

r&(x) = gq(x) . aegk(x) = ap@q(2)dlk@p(2) = r&(x) .

Since 0 E C2(R;R3) and 6 E C'(6;R3) by assumption, the last

We are now in a position to compute the partial derivatives @$i(?)

as functions of x, by means of the relation Gi((a) = vk(z)[gk(~)]i. TO this end, we first note that a differentiable function w : R --t IFS satisfies

relations show that E C0(R). h

al,w(6(2)) = aew(X)5j@((a) = aew(x)[ge(41j,

by the chain rule. In particular then,

5&2) = 5,vk(6(2))[g~(z)]i + vq(z)8j[gq(6(2))li = aevk(~)[ge(~)lj[s"~)li + 4 4 (~e[S"4li) [ge(41j = (aevk(4 - r:k(+%(x)) [Sk(41i[ge(x)lj,


since &gg(x) = - r&(x )gk (x ) . We have therefore shown that

@i(Z) = v k l , e ( 4 [gk(41i[ge(x) l j ,

2 . ' k l l e ( 4 := aevk(x) - q k ( x ) V q ( Z ) ,

where

and [gk(x)] i and r&(z) are defined as above.

(ii) With any vector field w = vigz : R + R3 with vi E H1(R), we next associate the functions eij(w) E L2(R) defined by

'

1 1 2

e i j (v ) := -(villj + v . 311 i = ,(ajvi + aivj) - r:jvup,

and, with any vector field G = Gigi : 6 + EX3 with Gi E H 1 ( 6 ) we associate the functions Z i j ( G ) E L 2 ( 6 ) defined by

1 - h

2 h e i j ( 3 ) := -(ajGi + aiGj).

If the Jields w and 3 are related as in (i), it then immediately follows f rom (i) that

Z i j (G)(Z) = (eke(w)[gk]i[ge]j) (x) for all 2 = ~ ( x ) , x E R.

(iii) Let a vector field w = vigi with vi E H1(R) be such that

eij(w) = 0 in R.

Then there exist two vectors a, b E R3 such that the associated vector field viga is of the f o r m

w(x) = a + b A O(z) for all x E a. Next, let l?o be a dr-measurable subset of the boundary 6'0 that satis-

f ies arearo > 0, and let a vector field w = vigi with vi E H 1 ( 0 ) be such that

eij(w) = 0 in R and w = 0 on Fo.

Then w = 0 in R. It follows from part (ii) that

Then the identity

An Introduction to Shell Theory

further shows that

165

By a classical result from distribution theory (Schwartz [80, p. SO]), each function Gi istherefore a polynomial of degree < 1 in the variables zj, since the set R is connected. There thus exist constants ai and bij

such that q ( z ) = ai + b i j 2 j for all 2 = (2i) E R.

But Eij ( G ) = 0 also implies that bij = -bji. Hence there exist two vectors a, b E R3 such that (2 denotes the column vector with components 2i)

G~(z)c? = a + b A 2 for all 5 E 6,

h h h


vi(z)gi(z) = a + b A O ( x ) for all x E R.

Since the set where such a vector field G i 2 vanishes is always of zero area unless a = b = 0 (as is easily proved; see, e.g., Ciarlet [18, Theorem 6.3-4]), the assumption arearo > 0 implies that G = 0.

(iv) W e now let R := w x ( - E i 3 , & g ) ,

O ( y , xg) := qY) + zgag(y) for all x = (xi) := ( y , zg) E 2,

where EO > 0 has been chosen in such a way that the mapping 0 is a C2-diffeomorphism from into its image @(a) (Theorem 2.2-1). With any vector field v = Viai with covariant components r ] , in H1(w) and r]3

in H2(w) , let there be associated the vector field v = vigi defined on 2

and we let the mapping 0 : 2 4 R3 be defined by

by v z (Y , z3 )g i (Y , z3 ) = r ] i (Y )az (Y ) - 53(&xr]3 + b:r]u)(Y)aa(Y)

fo r all ( y , xg) E R, where the vectors gi are defined by gz . gj = 6;. Then the covariant components I J ~ of the vector field vigz are in

H 1 ( R ) and the corresponding functions e i j ( v ) E L2(R) defined as in part (ii) are given by

eayP(v) = ,Yaap(77) - z3PaL3(77)


Note that, as in the above expressions of the functions e,yp(v), the dependence on x3 is explicit, but the dependence with respect to y E W is omitted, throughout the proof. The explicit expressions of the functions yap(q) and p,y~(q) in terms of the functions qi (Theorems 2.8-1 and 2.8-2) are used in this part of the proof.

To prove the above assertion, we proceed in two stages. First, given functions v,, X , E H1(w) and 773 E H2(w) , let the vector field v = vigi be defined on a by

vigi = viai + x3 x,aa. Then the functions vi are in H1(R). Besides, the functions eiilj(v) defined as in part (ii) are given by

covariant derivatives of the fields qiai and Xiai with X3 = 0. To see this, we note that

a,a3 = -bza ,

by the formula of Weingarten (see Section 2.1). Hence, the vectors of the covariant basis associated with the mapping 0 = 8 + x3a3 are given

g , = a, - x3b:au and g 3 = a s .

The assumed regularities of the functions qi and X , imply that

ui = ( v j g j ) . g i = ( V j a j + x3Xaa") . g i E H 1 ( R )

since g i E C'(a). The announced expressions for the functions eij(v) are then obtained by simple computations, based on the relations qj =

{ a j ( v d ) ) ' g i (part (i)) and e i j ( v ) = i(uillj + vjlli).

by

Second, we show that, when

xa = -(&PI3 + bEVu) ,

the functions e i j ( v ) above take the expressions announced in the statement of part (iv).


We first note that X, E H1(w) (since bg E C1(a)) and that ea3(w) =

0 when X, = -(&vs + b g q g ) . It thus remains to find the explicit forms of the functions e,p(w> in this case. Replacing the functions X, by their expressions and using the symmetry relations bglp = b;Ja (Theorem 2.8-2), we find that

1 S{XaI,O + XOIa - b:(77rlp - bpu773) - b ; b T / , - bryT773)}

- - -'%lap - bz'%lP - b;'%Ia - bil,77~ f bgbop773,

i.e., the factor of z3 in e,p(w) is equal to -pap(q) . Finally,

- b F u l p - b;+, - - bE (r]31pg + b&3r/~ + h z v ~ l p ) + bg (1; /31,~ + b,"j,% + b:qula)

= bE (PP&?) - b&U + bpTLT773) + bg ( P A ? ) - b,"77+ + b,"bC7?73)

= b ; P p d r l ) + b;P,T(r]) - 2b:b;yUT(rl),

i.e., the factor of $ in e,p(w) is indeed as announced.

(v) Let the set fi = w x ( - - E o , E ~ ) and let the vector field w = vigi with wi E N1(R) be defined as in part (iv). By part (iv), the assumption that yap(r]) = pap(r]) = 0 in w implies that

e i j ( v ) = 0 in R.

Therefore, by part (iii), there exist two vectors a, b E R3 such that

vi(y, x3)gi(y7 z3) = a + b A {B(y) + z3ady)) for all ( ~ , 2 3 ) E TI.

Hence

and part (a) of the theorem is established. Let next yo c y be such that lengthy0 > 0. If in addition qi =

&773 = 0 on yo, the functions xa = -(8,773 + bgq,,) vanish on yo, since 773 = dv773 = 0 on yo implies 8,773 = 0 on yo. Part (iv) then shows that

v; = (ujgJ) . gi = (7j.J + z3Xaaa) . gi = 0 on := 70 x [--ED, E ~ ] .

Since arearo > 0, part (iii) implies that w = 0 in G, hence that q = 0 0

We are now in a position to prove the announced Korn's inequality

on J and part (b) of the theorem is established.

on a surface, "with boundary conditions".


This inequality was first proved by Bernadou & Ciarlet [Ill. It was later given other proofs by Ciarlet & Miara [33] and Bernadou, Ciarlet & Miara [12]; then by Akian [4] and Ciarlet & S. Mardare [32], who showed that it can be directly derived from the three-dimensional Korn inequality in curvilinear coordinates (this idea goes back to Destuynder [45]); then by Blouza & Le Dret 1131, who showed that it still holds under a less stringent smoothness assumption on the mapping 8. We follow here the proof of Bernadou, Ciarlet & Miara [12].

Theorem 2.9-3. Let w be a domain in R', let 8 E C3(L7;R3) be an injective immersion, let yo be a dy-measurable subset of y = dw that satisfies lengthyo > 0, and let the space V ( w ) be defined as:

V ( W ) := { q = viai; qa E ~ ' ( w ) , r ] 3 E H ' ( w ) , v ~ = dVr/3 = 0 on yoyo).

Given q = viai with qa E H 1 ( w ) and q 3 E H'(w), let

1 ~ ~ p ( r l ) := {,(am. a, + 8 , ~ . a p } E ~ ~ ( w ) ,

~ a D ( r l ) := { (aapv - ~ 0 8 ~ ~ ) . as} E ~ ~ ( w )

denote the covariant components of the linearized change of metric and linearized change of curvature tensors associated with the displacement field q = viai of the surface S = 8(W). Then there exists a constant c = c(w, yo, 0) such that

for all q = viai E ~ ( w ) .

proof. Let the space v ( w ) := { q = qiai; qa E H 1 ( w ) , v 3 E f f 2 ( ~ ) } be equipped with the norm

-

If the announced inequality is false, there exists a sequence ($)El of vector fields q'" E V ( w ) such that


Since the sequence is bounded in P ( w ) , a subsequence (qg(k))r=l (u : N 4 N is an increasing function) converges in e ( w ) by the Rellich-KondraSov theorem. Furthermore, each sequence ( ~ ~ p ( r f ( ~ ) ) ) F ~ and ( p a p ( v u ( k ) ) ) & also converges in L2(w) (to 0, but this information is not used at this stage) since

The subsequence (v"('))g1 is thus a Cauchy sequence with respect to the norm

hence with respect to the norm I I . I I H l ( w ) x H l ( w ) x H 2 ( W ) by Korn's inequa2- ity without boundary conditions (Theorem 2.9-1).

The space V ( w ) being complete as a closed subspace of the space V ( w ) , there exists v E V ( w ) such that

vU(lc) + v in P ( w ) and the limit v satisfies

I l" laP(rl) l lL2(w) = 2:: IITap(v'(k))IIL~(w) = 0,

IIPaP(v)IIL2(W) = k-+w lim I IPap(vu(k) ) l lL2(W) = 0.

Hence 7 = 0 by Theorem 2.9-2. But this contradicts the relations

4 k ) Ilv I I H ~ ( ~ ) ~ H ~ ( ~ ) ~ H ~ ( ~ ) = 1 for all /C 2 1,

and the proof is complete.

If the mapping 0 is of the form O(yl, y y ~ ) = (yl, y2,O) for all ( y l , y2) E i3, the inequality of Theorem 2.9-3 reduces to two distinct inequalities (obtained by letting first qa = 0, then 773 = 0):

for all q3 E H2(w) satisfying 773 = dvv3 = 0 on yo, and


for all 17, E H 1 ( w ) satisfying q, = 0 on yo. The first inequality is a well-known property of Sobolev spaces. The second inequality is the two-dimensional K o r n inequality in Cartesian coordinates. Both play a central rijle in the existence theory for linear two-dimensional plate equations (see, e.g., Ciarlet [19, Theorems 1.5-1 and 1.5-21).

As shown by Blouza & Le Dret [13], Le Dret [61], and Anicic, Le Dret & Raoult [7], the regularity assumptions made on the mapping 8 and on the field 11 in both the infinitesimal rigid displacement lemma and the Korn inequality on a surface of Theorems 2.9-2 and 2.9-3 can be substantially weakened.

It is remarkable that, for specific geometries and boundary conditions, a Korn inequality can be established that only involves the linearized change of metric tensors. More specifically, Ciarlet & Lods [27] and Ciarlet & Sanchez-Palencia [35] have established the following Korn inequality “on an elliptic surface”:

Let w be a domain in R2 and let 8 E C2’’(W;IW3) be an injective immersion with the property that the surface S = 8(a) is elliptic, in the sense that all its points are elliptic (this means that the Gaussian curvature is > 0 everywhere on S) . Then there exists a constant CM = C M ( W , 8) > 0 such that

for all 11 = qiai with q, E H:(w) and q3 E L2(w).

Remarks. (1) The norm )lr/311H2(W) appearing in the left-hand side of the Korn inequality on a “general” surface (Theorem 2.9-3) is now replaced by the norm 11173IIL~(~). This replacement reflects that it is enough that qa E H 1 ( w ) and 173 E L2(w) in order that yap(^) E L2(w) , where Q = Viaz. As a result, no boundary condition can be imposed on 7l3.

(2) The Korn inequality on an elliptic surface was first established by Destuynder [45, Theorems 6.1 and 6.51, under the additional assumption that the Co(5)-norms of the Christoffel symbols are small enough. 0

Only compact surfaces defined by a single injective immersion 8 E C3((w) have been considered so far. By contrast, a compact surface S “without boundary’’ (such as an ellipsoid or a torus) is defined by means of a finite number I 2 2 of injective immersions 8i E C3(Wi), 1 6 i 6 I , where the sets wi are domains in EX2, in such a way that S = UiE1 Bi(wi). As shown by S. Mardare [67], the Korn inequality “without boundary conditions” (Theorem 2.9-1) and the infinitesimal rigid displacement


lenima on a surface "without boundary conditions" (Theorem 2.9-2) can be both extended to such surfaces without boundary.

2.10 Existence, uniqueness, and regularity of the solution to the linear Koiter shell model

Let w be a domain in R2, let yo be a measurable subset of y = d w that satisfies lengthyo > 0, let d, denote the outer normal derivative operator along dw, let 8 E C3(W; R3) be an immersion] and let the space V ( w ) be defined by

V(w) := { q = viai; qa E ~ l ( w ) , q ~ E H ~ ( w ) ; qi = d,q3 = o on yo),

where yo is a dy-measurable subset of y := dw that satisfies lengthyo > 0. Our primary objective consists in showing that the bilinear form B : V(w) x V(w) 4 IR defined by

for all ( c , ~ ) E V(w) x V(w) is V(w)-elliptic. As a preliminary, we establish the uniform positive-definiteness of

the elasticity tensor of the shell, given here by means of its contravariant components aapuT (note that the assumptions on the Lam6 constants, viz., 3 X + 2p > 0 and p > 0, are weaker than those usually made for elastic materials).

Theorem 2.10-1. Let w be a domain in EX2, let 6 E C3(W;IR3) be an injective immersion, let amp denote the contravariant components of the metric tensor of the surface 8(Z), let the contravariant components of the two-dimensional elasticity tensor of the shell be given by

and assume that 3 X + 2 p > 0 and 1-1 > 0 . Then there exists a constant c, = c,(w, 8, A, p ) > 0 such that

for all y E W and all symmetric matrices ( t a p ) .

Proof. In what follows, M2 and S2 respectively designate the set of all real matrices of order two and the set of all real symmetric matrices of order two.


(i) To begin with, we establish a crucial inequality. Let x and p be two constants satisfying x + p > 0 and p > 0. Then there exists a constant y = y(x,p) > 0 such that

y t r (BTB) < X(trB)2 + 2ptr(BTB) for all B E M2.

If x 2 0 and p > 0, this inequality holds with y = 21. It thus remains to consider the case where - p < x < 0 and p > 0. Given any matrix I3 E M2, define the matrix C E M2 by

C = A B := X(tr B)I + 2pB.

The linear mapping A : M2 ---f M2 defined in this fashion can be easily inverted if x -t- p # 0 and p # 0, as

Noting that the bilinear mapping

(BIG') E M2 x & -+ B : C := t r (BTC)

defines an inner product over the space M2, we thus obtain

X(tr B)2 + 2p tr(BTB) = (AB) : B = C : A-'C

for any B = A-lC E M2 if - p < x < 0 and p > 0. Since there clearly exists a constant ,b' = p(x,p) > 0 such that

B : B < ,BC : C for all B = A - l C E MI2,

the announced inequality also holds if -p < x < 0 and p > 0, with y = (2pp)-l in this case.

(ii) We next show that, for any y E W and any nonzero symmetric matrix ( t a p ) ,

a"fluT(Y)tuTtap 3 yaau(Y)aP7(y)to,t,p > 0,

where y = y(x, p ) > 0 is the constant found in (i). Given any y E W and any symmetric matrix ( t a p ) , let

A(!/) = (a%)) and T = ( t a p ) ,

let K ( y ) E S2 be the unique square root of A ( y ) (i.e., the unique positive- definite symmetric matrix that satisfies ( K ( Y ) ) ~ = A ( y ) ) , and let

B ( y ) := K(y)TK(y) E S2.


Then

By the inequality established in part '(i), there thus exists a constant a ( X , p ) > 0 such that

1 - a a O n ~ 2 (y)tU,tap 2 a t r (B(Y)TB(Y))

if x + p > 0 and p > 0, or equivalently, if 3X + 2p > 0 and p > 0

(iii) Conclusion: Since the mapping

is continuous and its domain of definition is compact, we infer that

Hence ltapI2 6 aau(Y)aflT(Y)t,Ttap

a,P and thus c ItapI2 6 ceaapUT(y)t,,tap

%B

for all y E iJ and all symmetric matrices ( t a p ) , with ce := (yb)-l. 0

Combined with Korn's inequality "with boundary conditions" (The- orem 2.9-3), the positive definiteness of the elasticity tensor leads to the existence of a weak solution, i.e., a solution to the variational equations of the linear Koiter shell model.

Theorem 2.10-2. Let w be a domain in R2, let yo be a subset of y = dw with length yo > 0, and let 8 E C3(w; R3) be an injective immersion. Finally, let there be given constants X and p that satisfy 3X + 2p > 0 and p > 0, and functions pa>€ E L T ( w ) f o r some r > 1 and p3+ E L1(w) .

Then there is one and only one solution <" = <fai to the variational problem:

<' E V ( W ) = {Q = Viai; qa E H1(w),q3 E H2(w) , qi = dyq3 = o on yo},


where

aap*T = J±t_aa0a°r + 2ll(aa°a^ + aaTa^),

1

= -(dpr] • aa + dar) • a/3) and pap{r}) = (da0Ti - T^daW • a3-

The field C e V(u) is also the unique solution to the minimizationproblem:

where

- /J UJ

Proof. As a closed subspace of the space V(UJ) := {n = ^a l ; r]a GH1(w),773 e H2(LJ)} equipped with the hilbertian norm

:= { Z^ ll7?"!1 /9

the space V(W) is a Hilbert space. The assumptions made on the map-ping 8 ensure in particular that the vector fields ai and ai belong toC2(w;R3) and that the functions aapuT, Fgp, and a arc continuous onthe compact set w. Hence the bilinear form defined by the left-hand sideof the variational equations is continuous over the space V(W).

The continuous embeddings of the space H1(W) into the space Ls(w)for any s 3 1 and of the space H2(w) into the space C°(W) show that thelinear form defined by the right-hand side is continuous over the samespace.

Since the symmetric matrix (u,p(y)) is positive-definite for all y E a,there exists a0 such that a(y) 3 a0 > 0 for all y E i;s.

Finally, the Korn inequality "with boundary conditions" (Theorem2.9-3) and the uniform positive definiteness of the elasticity tensor of theshell (Theorem 2.10-1) together imply that

2

min {e, y j c j V 2 ^ ^ \\va\\m {uj)

for all 17 = viai E V(w). Hence the bilinear form B is V(w)-elliptic.


The Lax-Milgram lemma then shows that the variational equations have one and only one solution. Since the bilinear form is symmetric, this solution is also the unique solution of the minimization problem stated in the theorem. 0

The above existence and uniqueness result applies to linearized pure displacement and displacement-traction problems, i.e., those that correspond to lengthyo > 0.

We next derive the boundary value problem that is, at least formally, equivalent to the variational equations of Theorem 2.10-2. In what follows, 71 := 7\70, (ua) is the unit outer normal vector along 7, 71 := - 2 4 , 7-2 := u1, and arx := rJax denotes the tangential derivative of x in the direction of the vector (ra).

Theorem 2.10-3. Let w be a domain in R2 and let 0 E C3(W;R3) be a n injective immersion. Assume that the boundary y of w and the functions pi>€ are smooth enough. If the solution C" = <:ai t o the variational equations found in Theorem 2.10-2 i s smooth enough, then C" i s also a solution to the following boundary value problem:

where

E3 nap := EaapqrYu7(<") and mag := -aapqr 3 Pur (C") ,

and, for a n arbitrary tensor field with smooth enough covariant components t a p : w 4 R,

t a q p := apt ap + rpaqtPq + r$qtaq, t a p l a p := a a ( t a p l p ) + r;q(taPlp).

Proof. For simplicity, we give the proof only in the case where 70 = y, i.e., when the space V ( w ) of Theorem 2.10-2 reduces to

V ( W ) = (7 = %ai; V a E H,l(w),r/3 E H,2(w)}.

The extension to the case where lengthy1 > 0 is straightforward.


In what follows, we assume that the solution C' is "smooth enough" in the sense that n a p E H1(w) and map E H2(w).

(i) We first establish the relations

Let A denote the matrix of order three with a1,a2,a3 as its column vectors, so that f i = det A (see part (i) of the proof of Theorem 2.10- 2). Consequently,

since 8,ap = r&aU + bapa3 (see Section 2.1).

(ii) Using the Green formula in Sobolev spaces (see, e.g., NeEas [73]) and assuming that the functions nap = npa are in H 1 ( w ) , we first transform the first integral appearing in the left-hand side of the variational equations. This gives, for all q = viai with 71, E H;(w) and q 3 E L 2 ( w ) , hence a fortiori for all q = viai with va E H t ( w ) and r/3 E H i ( w ) ,

(iii) We then likewise transform the second integral appearing in the


left-hand side of the variational equations, viz.,

forhall q = qzat with qa E H i ( w ) and q 3 E H i ( w ) . Using the symmetry map = m p a , the relation ap ,h = f i r & (cf. part (i)), and the same Green formula as in part (ii), we obtain

&(apmaP + r&,map + l?:pmuP)daq3 dy

Consequently,

+ S, &{maplap - b ~ b , p m a p } m d y .

Using in this relation the easily verified formula

(b,"mUP)lp = (bFI,)mup f b,Q(mUplp)


and the symmetry relations bFlu = bglp (Theorem 2.8-2), we finally obtain

(iv) By parts (ii) and (iii), the variational equations

1 1 w { aaPuTru~(<">rap(~) + p a p u T p u T ( S E ) p a p ( ~ ) - piiEqi}&dy = 0

imply that

+l &{bapnap + bgbupmaP - m a p l a p + p 3 ' € } q 3 d y = 0

for all 7 = viai with va E H;(w) and 7 3 E Hi(w) . The announced 0 partial differential equations are thus satisfied in w.

The functions nap = &aapuTynT(SE)

are the contravariant components of the linearized stress resul- tant tensor field inside the shell, and the functions

&3 POT (C" ) ,aP = -affPu.r

3 are the contravariant components of the linearized stress couple, or linearized bending moment, tensor field inside the shell.

The functions

t a p l a = a@ + rpantPu + r P P u tau,

t ap lap = aa(t"PIp) + r:,(taPlp),

which have naturally appeared in the course of the proof of Theorem 2.10-3, constitute examples of first-order, and second order, covariant derivatives of a tensor field defined o n a surface, here by means of its contravariant components t a p : i;s + R.

Finally, we state a regularity result that provides an instance where the weak solution, viz., the solution of the variational equations, is also a classical solution, viz., a solution of the associated boundary value problem. The proof of this result, which is due to Alexandrescu [5], is long and delicate and for this reason is only briefly sketched here.


Theorem 2.10-4. Let w be a domain in R2 with boundary y and let 9 : w 4 R3 be an injective immersion. Assume that, for some integer m 3 0 and some real number q > 1, y is of class 9 E Cmf4(W; EX3), palE E Wm+liq(~) , and p3>€ E W",q(w) . Finally, assume that yo = y. Then the weak solution C' = <fai found in Theorem 2.10-2 satisfies

-

<: E W"+3,q(w) and <: E W"+4+7(w)

Sketch of proof. To begin with, assume that the boundary y is of class C4 and the mapping 9 belongs to the space C4(sS; R3).

One first verifies that the linear system of partial differential equations found in Theorem 2.10-3 (which is of the third order with respect to the unknowns <: and of the fourth order with respect to the unknown <:) is uniformly elliptic and satisfies the supplementing condition on L and the complementing boundary conditions, in the sense of Agmon, Douglis & Nirenberg [3].

One then verifies that the same system is also strongly elliptic in the sense of NeEas [73, p. 1851. A regularity result of NeEas [73, Lemma 3.2, p. 2601 then shows that the weak solution C" = <fa2 found in Theorem 2.10-2, with components cz E Hi(w) and 53" E Hi(w) since yo = y by assumption, satisfies

<: E H 3 ( w ) and <; E H 4 ( w )

if pa)" E H 1 ( w ) and p3>" E L2(w).

sociated linear operator then implies that A result of Geymonat [51, Theorem 3.51 about the index of the as-

<: E W 3 > 4 ( w ) and c: E W414(w)

if pa+ E W1i'J(w) and p3lE E LQ(w) for some q > 1. Assume finally that, for some integer m 3 1 and some real number

q > 1, y is of class Cm+4 and 9 E C"+4(sj;R3). Then a regularity result of Agmon, Douglis & Nirenberg [3] implies that

C" a E W"+3i4(w) and <: E W"f4)q(~) .

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185

Some New Results and Current Challenges in the Finite Element Analysis of Shells

Dominique Chapelle INRIA-Rocquencourt

B. P. 105, 78153 Le Chesnay Cedex, France E-mail: Dominique. Chapelle@inria. fr

Abstract

Finite element procedures are widely used in engineering practice to analyse the behaviour of shell structures. However, the concept of “shell finite element” is still somewhat fuzzy, as it may correspond to very different ideas and techniques in various actual implementations. In particular, a significant distinction can be made between shell elements that are obtained via the discretization of shell models, and shell elements-such as the general shell elementsderived from 3D formulations using some kinematical assumptions, without the use of any shell theory. Our first objective in this paper is to give a unified perspective of these two families of shell elements. This is expected to be very useful as it paves the way for further thorough mathematical analyses of shell elements. A particularly important motivation for this is the understanding and the treatment of the deficiencies associated with the analysis of thin shells (among which the locking phenomenon). We then survey these deficiencies, in the framework of the asymptotic behawiours of shell models. We conclude the article by giving some detailed guidelines to numerically assess the performance of shell finite elements as regards these pathological phenomena, which is essential to design improved procedures.

1 Introduction

Finite element procedures are widely used to analyse the behaviour of shell structures in various areas of engineering (e.g. in the automotive, aerospace and civil engineering industries), and such analyses have been conducted ever since the early development of finite element methods. Over the time, however, and across the various disciplinary cultures involved, the concept of “shell finite element” has referred to very different ideas and techniques, and this situation has resulted into an incredibly abundant literature and also to some significant degree of confusion. In

186 Dominique Chapelle

this respect, we can identify in the “zoology” of shell finite elements a particularly distinct “line of division” between finite element methods that result from the discretization of shell mathematical models (namely essentially two-dimensional models with unknowns given on the midsurface of the shell, see e.g. (Ciarlet 2000)) and finite element techniques obtained from various other-usually engineering-rooted-considerations. Roughly speaking, the finite element methods derived from a shell model draw their justification from the assumed relevance of such models, the solution of which they can be shown to approximate within a certain accuracy using numerical analysis techniques, see (Bernadou 1996). By contrast, other shell finite element methods are grounded on mechanical considerations used “at the element level”, see for example (Bathe 1996) and the references therein. Although they are usually numerically tested using various benchmark problems, what is typically lacking for such finite element procedures is the proof that the finite element solutions converge to some known limit solution when the mesh is being refined. Indeed, the characterization of such a limit solution would be extremely valuable to evaluate the whole approach.

One of the earliest techniques used in shell analysis consists in simply combining plate bending and membrane behaviours in flat elements, the collection of which approximates the geometry of the actual shell structure. This is known as the facet-shell element approach. Examples of this category of elements can be found in (Batoz and Dhatt 1992). In (Bernadou, Ducatel and Trouv6 1988), a basic facet-shell procedure is analysed and proved to be a non-conforming discretization scheme of a shell model. However, it is also shown that the corresponding consistency error is non-converging, and this result is consistent with previous observations of the lack of convergence exhibited by these methods, see e.g. (Irons and Ahmad 1980). Additional deficiencies associated with the facet type approximation of the geometry are identified in (Akian and Sanchez-Palencia 1992). Various further refinements of the facet-shell techniques have been introduced, but whether or not these developments actually counteract the above-mentioned deficiencies is not clear (and indeed not proven). Moreover, by construction the coarse approximation of the geometry involved in these procedures implies that, even if a successfully converging scheme were to be found, the accuracy of the approximation would be rather poor.

On the other hand, shell finite element procedures obtained by directly discretizing a shell model can be thoroughly analysed and explicit error estimates are then obtained under some well-identified conditions (pertaining to the regularity of the exact solutions, the type of numerical integration used, etc.), see (Bernadou 1996) and the references therein. The major drawback of this approach is, however, that building a shell finite element procedure on a set shell model is clearly restrictive as to

Some New Results and Current Challenges ... 187

the variety of situations that can be analysed. In particular, most shell theories are based on a linear elastic behaviour, and adapting the models to account for an arbitrary (3D) constitutive equation-r for a general non-linear behaviour-is not straightforward. Note that this drawback in fact also applies to facet shell elements.

By contrast, a third methodology has been developed to specifically allow for unrestricted versatility as regards shell modeling capabilities. This methodology is based on the idea of “degenerating” a 3D solid finite element into a shell element by using some kinematical a s sumpt ions for describing the variation of displacements across the thickness of the shell structure. Since the only model used is the 3D model in consideration, general 3D constitutive laws can be employed. This is indeed why these finite elements are called general shell e l emen t s (and sometimes also “degenerated solid elements”), see the seminal work of (Ahmad, Irons and Zienkiewicz 1970), and also (Bathe 1996) and (Bischoff and Ramm 1997), and the references therein, for later generalizations of these concepts. This approach is widely used in practice and inumerable evi- dences of its effectiveness can be found in the literature, see for example (Bathe 1996). However, due to its specific construction that does not rely on any shell theory, a mathematical analysis of such a procedure is very difficult to achieve.

The first objective of this article is to present some recent results that give a unified perspective of general shell elements in the framework of shell models and their discretizations. More specifically, we show that we can identify a well-defined shell model that “underlies” general shell elements, i.e. such that the finite element solutions converge to the exact solution of this model when the mesh is refined. In addition, we compare this model to other classical shell models by means of an asymptotic analysis. This unification of concepts paves the way for further mathematical analyses of these highly attractive numerical procedures.

A particularly important motivation for a thorough mathematical analysis of shell finite elements is the understanding and the treatment of some serious numerical pathologies that are known to occur when the thickness of the shell structure is “small” (to fix the ideas, these phenomena typically become very significant as soon as the ratio of the thickness over the other characteristic dimensions of the structure is of the order of 1%, which is a situation commonly encountered in practice). The sources of these difficulties, among which the by-now classical numerical locking, are now well-identified, and they are closely related to the complexity and diversity of the asymptotic behaviours exhibited by shells, see (Chapelle and Bathe 1998). From a numerical analysis point of view, however, a finite element procedure with proven reliability with respect to the variation of the thickness parameter is still to be found. On the other hand, the literature abounds with allegedly reliable or locking-free


shell finite elements, but the assessment performed to reach this conclusion is often inadequate or incomplete, when not altogether irrelevant. The second main objective of this article is thus to survey the issue of the reliability of shell finite elements with respect to variations of the thickness parameter. Of course, this issue must be put into perspective with the variations of behaviours induced in the exact solutions by variations of the thickness, namely with the analysis of the asymptotic behaviours of shell models.

This article is organized in the following manner. In Section 2 we present the two families of shell finite elements that we want to analyse and compare, namely the finite elements obtained by discretizing shell models (Section 2.1), and the general shell elements (Section 2.2). Section 3 is then dedicated to the issue of the reliability of shell finite elements. We start by surveying the asymptotic behaviours of shell models (Section 3.1) before focusing on computational issues (Section 3.2), and we conclude in Section 3.3 by presenting some guidelines for assessing and improving the reliability of shell finite elements.

In all our discussions we focus on linear formulations. Indeed, the difficulties that we want to analyse are already present in linear problems, hence it is natural to treat them in this framework first. Another simplification that we use is to assume that the thickness is constant (denoted by E ) over the whole shell structure. This assumption is made only in order to simplify the formulas and the discussions, and does not represent a restriction in our analysis, see (Chapelle and Bathe 2003) for a general analysis with arbitrary thickness.

Our notation is based on the notation used in (Ciarlet 2001), see also (Ciarlet 2000), hence we do not repeat the same definitions here and we only define the new notation (mainly related to finite element concepts). In addition, we use the symbol C to denote a generic positive constant that, unless otherwise stated, is independent of all the parameters appearing in the same equation (and in particular of the mesh parameter h and of the thickness parameter E ) , with the classical convention that C may take different values at different occurences (including when appearing several times in the same equation).

2 Two families of shell finite elements

2.1 Discretizations of classical shell models As discussed in (Ciarlet 2000), classical shell models are mathematical models in which the unknowns are defined on the midsurface of the shell body or, equivalently, on the 2D reference domain w from which the midsurface is obtained via the mapping 8 (that we call the chart). In general, these models are based on kinematical assumptions that are used


to describe the displacements of points located on material fibers that are orthogonal to the midsurface in the original configuration. Shell models can then be (roughly speaking) divided into two categories according to the type of kinematical assumption made.

1. When the assumption is that any normal fiber remains straight and unstretched during the deformation, the displacements in the whole shell body are completely described by the datum-n w-of a displacement field and a rotation field. Namely, a displacement at any point in RE is given by

where s, as the vector that represents the effect of the rotation of the normal fiber, is a vector tangent to the midsurface at the point of coordinates ( ~ 1 ~ ~ 2 ) ~ i.e. s = saaa. We call this kinematical assumption the Reissner-Mindlin kinematical assumption, and the shell models that are based on this assumption the Reissner- Mindlin type models, see (Reissner 1945) and (Mindlin 1951). As an example of this category of shell models, we will consider the model summarized in (Naghdi 1963), that we call the Naghdi model.

2. When the assumption is that any normal fiber remains straight, unstretched and normal t o the deformed midsurface (the so-called Kirchhoff-Love kinematical assumption) , the displacements can be described by a displacement field only, i.e.

where r](zl ,z~) is defined on w. Note that, comparing (2.1) and (2.2), the additional assumption on the preserved orthogonality of fibers is equivalent to the rotation vector being given by

sa = - (773p + b 3 U ) . (2.3)

We will call these shell models the Kirchhoff-Love type models. The shell model proposed in (Koiter 1965) belongs to this category and will be henceforth used as an example (we refer to this model as the Koiter model).

An important difference between these two categories of models is the respective regularities of their solution spaces (that we also call the displacement spaces). Namely, for the Naghdi model the natural diplace- ment space is

VN = {(v, s) E [H1(u)15} n BC, (2.4)


and VK = {q E [~'(w)]~ x H'(w)} n BC (2.5)

for the Koiter model, where BC symbolically denotes the essential boundary conditions prescribed. Of course these boundary conditions must, in each case, be compatible with the nature of the functional space. Note that the discrepancy in the regularity of the transverse component of the displacement field ( u s ) can be interpreted as the consequence of the additional constraint (2.3) acting on the solution space VN where

Clearly, in order to obtain well-posed variational formulations, some appropriate boundary conditions need be prescribed in the displacement space. For the Naghdi and Koiter models (and indeed also for similar models from the two above categories), it can be shown that the corresponding bilinear forms are coercive and bounded on their respective displacement spaces provided that the boundary conditions are sufficient to prevent any rigid body motion of the shell body, see (Bernadou 1996) and the references therein. Note that this holds in particular when the displacement field is set to zero on some part of the boundary dw that does not correspond to a straight segment in the physical space. When the boundary conditions are compatible with the functional space considered and such that rigid body motions are prevented, we will say that we have admissible boundary conditions.

As a consequence, the discretization of classical shell models by conforming finite element methods is rather straightforward, at least in principle. The main difficulty lies in the required use of C1-conforming finite elements for the transverse displacements in models of Kirchhoff-Love type. These issues are thoroughly addressed in (Bernadou 1996). Let us emphasize that these finite element procedures are based on meshes that are constructed in the reference domain w, and on computations that require an extensive use of the chart 6. Since conforming methods are used, a priori error estimates follow from interpolation bounds and are of the type

s € [Hl(w)]2.

for the Naghdi model, and

for the Koiter model, where the constants C and the orders of convergence k depend on the regularity of the exact solution and on the finite element shape functions considered. We emphasize that these estimates are not independent of the thickness parameter E , as the constants C in (2.6)-(2.7) above depend on E for two reasons:

Some New Results and Current Challenges . . . 191

the coercivity and continuity constants of the corresponding bilin-

0 the regularity bounds are not independent of E in general.

This issue will be further addressed in Section 3.2. In practice, in spite of the extensive use of shell finite elements in

engineering, the type of finite element procedure that we have been describing in this section is not very often encountered.

ear forms depend on E ;

2.2 General shell elements Most shell finite element procedures used in engineering practice fall into this category of general shell elements. Unlike the previously discussed procedures, general shell elements are not derived from a shell model, but instead from a 3D variational formulation. A typical general shell element procedure is constructed as follows.

1. Consider a general 3D variational formulation posed on the 3D geometrical domain defined by the mapping

@ ( 2 1 , 2 2 , 2 3 ) = 0 ( 5 l , Z 2 ) +23(113(21 ,52) , (2.8) ( 2 1 , 2 2 , 2 3 ) E W x ] -&,&[,

and infer a modified variational problem from the stress assumption

(2.9)

B 3 D ( ~ , ~ ) = L 3 D ( ~ ) , VV E V 3 D , (2.10)

D33 - = 0.

We symbolically denote this modified problem by

where B30 and L30 respectively represent the external and internal virtual works, and V3D denotes the appropriate functional space, taking into account the essential boundary conditions.

2. Consider a 2D mesh of w given in the form of a set of points (the nodes) in w and of a connectivity that defines the elements. Isoparametric elements are used, hence this defines a one-to-one mapping inside each element between the (51, 5 2 ) coordinates and the local coordinates ((1, &), in the form

(2.11)

where the functions X i are the shape functions associated to the k nodes of the element considered, and the quantities (zf), 5;)) denote the nodal coordinates. Of course, the shape functions depend

192

3.

4.

Dominique Chapelle

on the polynomial order chosen and on the geometric type of the element (namely quadrilateral or triangular). Using this mapping, we define the interpolation operator Z such that

i = l

in every element of the mesh, for any continuous scalar or vector function 4. Consider displacement functions defined over RE that satisfy Re- issner-Mindlin kinematical assumptions at all the nodes of the mesh, and are interpolated inside the elements using the above shape functions, namely

k

~ ( 2 1 , 2 2 , 2 3 ) = C Xi(E1, E 2 ) ( ~ ( 2 ) + Q S ( ~ ) ) , (2.13) i=l

where s(2) corresponds to a rotation at node i , hence it satisfies

,(i) . at) = 0. (2.14)

We call the space of functions of this type that satisfy the boundary conditions prescribed on the structure (note that boundary conditions can be prescribed here as they would be for a Reissner-Mindlin type model).

Consider the problem Find u E V2D such that

BiD(u, w) = L ” h ( W ) , v w E v y , (2.15)

where BiD and LiD are obtained from B30 and L30 by using, instead of the exact chart 0, the following approximation

@ h ( 2 i 7 ~ z , 2 3 ) =Z(~)(~i,~z) + 23Z(a3)(2i72z), (2.16)

to be compared with (2.8).

The general shell element procedure consists in solving Problem (2.15), hence the unknowns (degrees of freedom) are the 3 displacement components and the 2 rotation components at the nodes. Note that the degrees of freedom are similar to those of a classical shell finite element procedure obtained by discretizing a Reissner-Mindlin type model. However, it appears from the above description that general shell elements have two key advantages:


0 They are not obtained from a specific shell model, but from a general 3D formulation, hence they can be easily used for arbitrary 3D material laws and also for non-linear formulations. This is indeed why these procedures are called general shell elements.

0 As implied by the use of the approximate chart Oh, the derivation of the associated matrix problem does not require the datum of the exact midsurface chart 8 , but only the position of the points and the value of the normal vectors at the nodes. This is extremely convenient in practice as most structures analysed are the result of design procedures (CAD systems) that do not provide these charts.

The connections between general shell elements and classical shell elements (i.e. those obtained by discretizing classical shell models) have remained unclear for many years. Moreover, this issue has probably been a source of major misunderstanding between the engineering community (primarily using and developing general shell elements) and the numerical analysis community (considering only classical shell elements). These connections are now much better identified, see in particular (Chapelle and Bathe 2000). Indeed, for a given 3D material law we can consider the problem Find (C, r ) in VN such that

Clearly, this defines a shell model of Reissner-Mindlin type, since the unknowns are a displacement field and a rotation field, both being given on the midsurface (or in the reference domain). This shell model is the natural candidate for being the model that underlies general shell elements, i.e. the model that provides the solution to which finite element solutions converge when the mesh parameter h (namely the diameter of the largest element) tends to zero. With a view to analysing Prob- lem (2.15) as an internal approximation of Problem (2.17), we define Vj as the scalar finite element space associated with the isoparametric discretization above, i.e. a function 4 is in vj if and only if 4 = z(4). Note that Vj c H1(w). Then E V2D is equivalent to

with the three conditions

C1. q . Bi E Vj and i . Bi E Vj for i = 1,2,3;

C2. 5 . a 3 = 0 at all the nodes of the mesh;

C3. 7 and S satisfy the proper boundary conditions.


Denoting by T the operator that projects a vector field defined over w onto the tangential plane of the midsurface at each point, i.e.

T ( 5 ) = 5 - (5 ' u3)u3, (2.19)

we define the finite element space VNh as the set of functions (q,r(B)) such that (11,s) satisfies the three conditions (Cl,C2,C3) above. Clearly VNh is a (finite-dimensional) subspace of VN. Note that the use of the projection operator is required to that purpose because we do not directly have 5 in the tangential plane at every point of w . Using the equivalence

T ( 5 ) = s * 5 = Z(s) (2.20)

valid for any (q, ~ ( 5 ) ) in VNh, we can then reformulate the general shell element procedure as follows. Find (<,r) in VNh such that

B ~ D ( ~ + 5 3 Z ( r ) , q f Z 3 Z ( 3 ) ) = L3hD(q+Z3Z(s)), v(q, s ) E VNh. (2.21)

Comparing now this equation with the continuous problem (2.17), we can say that the general shell element procedure corresponds to a discretization scheme of the continuous problem with approximate bilinear and linear forms, see e.g. (Ciarlet 1978), and we observe that consistency errors come from two sources: the approximation of the geometry corresponding to Eq. (2.16) and the presence of the interpolation operator in (2.21).

When choosing specific 3D formulations, the connection between the shell model represented by (2.17) and the corresponding general shell element procedure can be further analyzed. In particular, when considering Hooke's constitutive law we obtain, as shown in (Chapelle and Bathe 2000),

Note that we assume that the Lam6 constants do not'depend on E .

Taking into account the Reissner-Mindlin kinematical assumption we have


where we recall that yuT, pFT and 703 respectively represent the components of the tensors of membrane strains, bending (or flexural) strains and shear strains in the Naghdi model, and tcuT corresponds to a new tensor defined by

1 2 t c U T ( s ) = -(b:s,lT f b:s,lU)' (2.25)

In addition, we consider the external virtual work given by

(2.26)

i.e. f represents the applied distributed force, and we assume f E [L2( f lE) ]3 . We then have the following result, under standard assumptions regarding the mesh and the finite element shape functions used, see (Chapelle and Bathe 2000).

Proposition 1. Provided that the boundary conditions are admissible, Problem (2.17) has a unique solution. Moreover, this solution is the limit of the solutions of Problem (2.15) when the mesh parameter h tends to zero.

This result shows that (at least in the case of Hooke's law) the shell model given by (2.17) really is the mathematical model that underlies general shell element procedures. It is then important to investigate the relations between this shell model (that we call the basic shell model, see (Chapelle and Bathe 2003)) and classical shell models. If we formally transform the expression (2.22) by truncating the expansions in powers of x 3 of the expressions appearing under the integral sign, taking

0 the first-order approximation of the strains, namely

eUl/T(rl f x 3 s ) yUT(rl) - x3 p ~ ( ~ I s ) (2.27) { e p , , d r l + 23s) = TP3(rlI S > I

0 the zero-order approximation of all geometric coefficients, namely

9 affflu.T, gap M asp, & = & 7 (2.28)

we obtain exactly the bilinear form of the Naghdi model (with c = 8 in the equations recalled in(Ciar1et 2000)), see (Chapelle and Bathe 2000), and also (Delfour 1998) where various combinations of truncations are considered and analysed. This suggests that some close connections exist between the basic model and the Naghdi model. We will further substantiate these connections in Section 3.1 by showing that the two models are asymptotically equivalent as regards their asymptotic solutions when the thickness parameter tends to zero.


As a conclusion, we can say that, although they do not seem (at first sight) to bear much resemblance to classical shell elements, general shell elements are-in fact-non-conforming discretization schemes of a shell model of Reissner-Mindlin type which is itself very closely connected to the Naghdi model. This is extremely important, as it allows to analyse finite element procedures of both “families” using similar tools. As a matter of fact, further advances in the analysis of some specific general shell procedures have already been obtained with this approach (cf. e.g. (Malinen 2000)), in particular as regards the reliability of shell finite elements which is the subject of our next section.

3 Computational reliability issues for thin shells

The essential motivation underlying the formulation of shell models and shell finite elements is the reduction of the cost of the analysis (in particular the computational cost), compared to the cost of a full 3D analysis, by instead considering a 2D problem. This motivation is based on the assumption (or the “hope”) that the accuracy of the solution obtained only depends on criteria that prevail in 2 0 analysis, i.e. typically on the fineness of the 2D mesh regardless of the third dimension, namely the thickness of the structure. This means that we implicitly expect uniform convergence of the finite element solution with respect to the thickness parameter, i.e. error bounds of the type

where we now use E as a superscript to indicate that the solutions depend on this parameter, and where C (and also k) should not depend on E . We denote the norm symbol with a V’ to signify that we do not necessarily require that uniform convergence hold in the Sobolev space in which the problem is originally set (for example [H1(w)]’ x H’(w) for Kirchhoff- Love type models), but that we may tolerate uniform convergence in weaker norms to be defined.

Unfortunately, it was soon recognized that standard finite element methods (such as the ones that we described in the previous section) do not provide such uniform bounds in general. The reason for this will be analysed in Section 3.2. To that purpose it is necessary to start by surveying the behaviour of the exact solution of the shell model when the thickness parameter tends to zero, namely the asymptotic behaviours of shell models, which is the objective of the forthcoming section. This analysis will, in particular, allow us to specify the norms in which we seek uniform estimates for approximate solutions.

Some New Results and Current Challenges ...

Asymptotic behaviours of shell models

197

3.1 Most classical shell models can be written in the generic following form: Find ZE E V such that

E 3 B F ( Z E , E ) + EBM(Z&, E ) = L E ( E ) , YE E V. (3.2)

The meaning of the symbols appearing in this formulation is as follows.

2": the unknown solution, namely the displacement of the midsurface for a Kirchhoff-Love type model, or this displacement and the rotation of the normal fiber for a Reissner-Mindlin type model;

E : a corresponding test function that we call a "displacement" (even though it also contains a rotation for a Reissner-Mindlin type model);

V: the solution space (we recall that the definition of this space takes into account the essential boundary conditions) ;

BF: a scaled representation of the bending (flexural) energy;

B M : a scaled representation of the membrane energy for a Kirchhoff-Love type model, or of the membrane energy and the shear energy for a Reissner-Mindlin type model;

L E ( E ) : the external virtual work associated with E .

For example, for the Koiter model we have, as seen in (Koiter 1965),

B i m , T ) , (77, s ) ) = (3.6)

Sw[""P~77aP(6)7u7(.I) + CPaaP7a3(C, T h P d r l , 4lJ;EdY.

We further emphasize that, in (3.2), the expressions of the bilinear forms BF and BM do not depend on the thickness parameter E . We also introduced E as a superscript in the right-hand side of the formulation because it is unlikely %o obtain a weU-posed asymptotic behaviour while keeping the loading constant over the whole sequence of problems. More


specifically, what we will be looking for in the asymptotic analysis is a scaling of the right-hand side in the form

L'(E) = E ~ G ( E ) , (3.7)

where G is a function of V' independent of E and p is a real number, for which the scaled external work G(ZE) converges to a finite and non- zero limit when E tends to zero. In this case, we will say that the given scaling provides an admissible asymptotic behaviour. Of course, this is equivalent to requiring that the scaled internal work E~--PBF(Z", 2") + E'--PBM(Z", 2") have a finite non-zero limit, since

E 3 - ~ ~ F ( 2 " , Z E ) + E 1 - ~ ~ M ( ~ ~ , 2") = ~ ( 2 " ) . (3.8)

We can then show the following result, see (Blouza, Brezzi and Lovadina 1999), (Baiocchi and Lovadina 2002) and (Chapelle and Bathe 2003).

Proposition 2. There is at most one exponent p that provides an admissible asymptotic behaviour. I n addition, i f such a number exists we have

1 G p G 3 . (3.9)

Remark 1. I n some cases the convergence may also be such that the sequence (2") tends to some limit in V , but we want to allow for more general situations in which a weaker convergence property may hold.

As discussed in (Sanchez-Hubert and Sanchez-Palencia 1997) (see also (Ciarlet 2000) for the Koiter model), classical shell models feature asymptotic behaviours that dramatically differ according to the contents of the subspace:

V F = { E E v I BM(E,E) = 0). (3.10)

This subspace contains the displacements that have zero membrane energy (and also zero shear energy for Reissner-Mindlin type models). For this reason, these displacements are called pure- bending displacements (since only the bending energy is non-zero). As a matter of fact, due to the severe kinematical constraints corresponding to zero membrane strains (3 equations versus 3 components of displacements), it may very well happen that

VF = ( 0 ) . (3.11) In this case we say that pure bending is inhibited. This situation induces an asymptotic behaviour very different from when pure bending is not inhibited. The existence of non-zero pure bending displacements is governed by the geometry of the midsurface and the boundary conditions prescribed on the structure (we further analyse this issue below).


We start by considering the case of non-inhibited pure bending. It can then be shown that a well-defined asymptotic behaviour exists for p = 3. In this case, the variational formulation can indeed be re-written as

(3.12)

Since E is a small parameter, this equation can be interpreted as the penalized f o r m of a problem in which the solution Zo is constrained to lie in the subspace VF. Namely, the candidate limit problem for (3.12) is Find Zo in V, such that

1 &2

BF(Z&, E ) -t --BM(Z“, E ) = G(E) , VE E V.

BF(ZO, E ) = G(E) , V E E VF. (3.13)

Note that this problem is well-posed, since it corresponds to the restriction to VF, a closed subspace of v, of the minimization problem equivalent to the original variational formulation. We then have the following result, see (Chenais and Paumier 1994), and also (Pitkaranta 1992) and (Sanchez-Palencia 1992).

Proposition 3. Assume that pure bending i s no t inhibited. Then , setting p = 3, ZE converges strongly in V to Zo, the solution of (3.13). I n addition we have

(3.14) 1

lim --Bn,r(ZE, ZE) = 0. E-7-0 &2

Equation (3.14) shows that, in the scaled deformation energy, the part corresponding to the membrane energy (and also the part corresponding to the shear energy when applicable) tends to zero, while the part that corresponds to bending deformations tends to a finite value. This is why this asymptotic behaviour is also called a bending-dominated behaviour.

Note that, when pure bending is not inhibited, Proposition 3 shows that we have an admissible asymptotic behaviour for the scaling p = 3 provided that Zo i s non-zero. If Zo = 0, i.e. if and only if

G ( E ) = 0, VE E VF, (3.15)

an other admissible scaling (for 1 < p < 3) may exist. This, however, does not correspond to a “physical situation” as small perturbations in the geometry (producing variations in VF) or in the loading lead to a violation of (3.15), hence to effects of smaller asymptotic order, namely of “larger magnitude” (corresponding to a bending-dominated behaviour, i.e. p = 3). Therefore we do not further consider this situation.

By contrast, when pure bending is inhibited (which is a situation more common than the previous one, although both may be encountered in practice, see (Ciarlet 2000) and (Chapelle and Bathe 2003)),


the asymptotic behaviour is very different and more complex. We consider the tentative scaling p = 1. The variational formulation then gives

(3.16) B M ( Z € , E ) + E 2 B ~ ( Z E , E ) = G ( E ) , b'E E v,

(3.17)

defines a norm over V which, of course, satisfies

l l E l l M 6 cllEllV, vE E vi (3.18)

but is not equivalent to the original norm (cf. examples of the Koiter and Naghdi models above). We call this new norm the membrane energy norm. We then recognize in (3.16) a singular perturbation problem. Defining VM as the space obtained by completion of V for the membrane energy norm, we introduce the variational problem Find Z M in VM such that

Bll,r(Z', E ) = G ( E ) , E VM. (3.19)

Clearly, this problem is well-posed provided that G E (VM)'. Note that this corresponds to a restriction on the loading since (VM)' C V'. An equivalent manner to write this condition is

(G(E)I < C[Bn/r(E, E)] ' , YE E V (3.20)

since, by density considerations, this is equivalent to the same condition holding for any E E VM. Under this assumption we have an admissible asymptotic behaviour as shown in the following proposition, see (Lions 1973).

Proposition 4. Assume that pure bending is inhibited and that G E (VM)'. Then, setting p = 1, 2' converges strongly in VM to Z M , the solution of (3.19). In addition we have

lim E ' B F ( z ~ , zE) = 0. (3.21)

Since, due to (3.21), the bending part of the scaled energy tends to zero while the remaining part (the membrane part, and also the shear part for Reissner-Mindlin type models) tends to a finite value, this asymptotic behaviour is called a membrane-dominated behaviour. Note that (under the assumptions of the proposition) we always have an admissible asymptotic behaviour since Z M cannot be zero (unless, of course, G is zero).

We have therefore identified two very distinct categories of admissible asymptotic behaviours for classical shell models, namely the bending- dominated behaviour and the membrane-dominated behaviour. It is

E'O

and we note that


the contents of the subspace VF that determine into which category a given problem falls. For each of these asymptotic behaviours, convergence of the solutions is obtained in a given norm, hence it is natural to seek uniform convergence of finite element solutions in the same norm (cf. next section). Other asymptotic behaviours may exist and the above discussion shows that they can arise only when pure bending is inhibited and G $! (VM)’) (disregarding cases when (3.15) holds, for the above-mentioned reason). These situations are more complex than the two “clear-cut” asymptotic behaviours that we have discussed, and they have not been completely elucidated yet, although significant advances have been made in their analysis, see in particular (Pitkaranta and Sanchez-Palencia 1997) (cf. also Section 3.3).

We now consider the case of the basic shell model, namely the shell model that underlies general shell element. We recall that the general asymptotic analysis performed above is not directly applicable to this model since it cannot be written in the form (3.2). In particular the membrane and bending deformation energies are coupled in the term

(3.22)

as shown in (2.24). Another significant difference is that the loading acting on the structure is three-dimensional, in the form

(3.23)

In order to perform an asymptotic analysis consistent with the general one above, we consider the asymptotic assumption

f‘ = &P-lg, (3.24)

recalling that the external work involves an integration over the thickness. Furthermore, we assume that g is regular enough to provide the following expansion

g ( x i , Z 2 , x 3 ) =gO(x1,x2) + x 3 9 1 ( 5 i , x 2 ) -k ( x 3 ) 2 g 2 ( x ~ , 5 z , 5 3 ) , (3.25)

where go and g1 are in L2(w), while 9 2 is a bounded function. We denote by (C“, T “ ) the solution of the basic shell model, i.e. Find (C“,rE) E V l such that

(3.26) 3 0 E B (6 + Z 3 T E , r ] + 2 3 S ) =


with (3.28)

and the parameter c set to 8 in Bg, see (Ciarlet 2000). Calling VF the pure-bending displacement subspace relative to this Naghdi problem (3.27), and V$ the completed space by the membrane energy norm when pure bending is inhibited, we then have the following asymptotic properties (assuming that the chart is smooth), see (Chapelle and Bathe 2000).

Proposition 5 . .If pure bending is no t inhibited in the Naghdi problem (3.27), then for the scaling p = 3 both sequences (CE,rE) and ( g " , T E ) converge in VN t o the same limit (Co,ro) which is the element of VF that satisfies

@%C0,r0), (11, s ) ) = G(rl), q r l , .) E VF". (3.29)

Alternatively, if pure bending is inhibited and if G E (V,")', then for the scaling p = 1 both sequences (C" , r E ) and (e, T E ) converge in V{ to the same limit (C',rM) which is the element of VG that satisfies

Therefore, the solution of the basic shell model converges to the same limit solutions, and under the exact same assumptions, as the solution of the Naghdi problem (3.27). To summarize these properties, we will say that the basic shell model and the Naghdi model are "asymptotically equivalent".

We close this section by providing some guidelines for the analysis of the subspace of pure bending displacements, since the contents of this subspace have been shown to crucially determine the asymptotic behaviour of the shell problems. First of all we note that, for both Kirchhoff-Love type and Reissner-Mindlin type models, pure bending displacements have identically zero membrane strains. In addition, for a Reissner-Mindlin type model pure bending displacements also have zero shear strains. However, the condition of zero shear strains can be written in the form of an explicit expression giving the rotation fhield q in terms of the displacement field s , namely

= - ( ~ 3 , a + b&). (3.31)

We also introduce the following Naghdi problemFind such that


By contrast, the condition of vanishing membrane strains

YaL?(rl) = 0, va, P = 1 , 2 , (3.32)

make up an exactly determined differential system (three equations versus three unknowns) that must be satisfied by the displacement field only. Hence this is clearly the crucial condition that determines the contents of the pure bending displacement subspace, which is why elements of this subspace are also called inextensional displacements (we recall that the membrane strain tensor is the tensor of linearized change of metric of the midsurface, see (Koiter 1965)), even in the case of Reissner-Mindlin type models. In addition, System (3.32) enjoys the following remarkable property (cf. (Sanchez-Palencia 1989a), (Sanchez-Palencia 1989b) and the references therein).

Proposition 6. The differential nature (elliptic, parabolic or hyperbolic) of System (3.32) is the same as the geometric nature of the midsurface at the point in consideration. Furthermore, when such a concept is relevant (namely in the hyperbolic and-by extension-parabolic case) the characteristics of the system are also the asymptotic lines of the surface.

This property is extremely valuable to analyse the contents of the subspace of pure bending displacements. In particular it shows how the geometry (i.e. the type of the midsurface) and the boundary conditions can affect the contents of the subspace (cf. e.g. (Chapelle and Bathe 1998) for examples). We refer to (Sanchez-Hubert and Sanchez-Palencia 1997) fo a more detailed analysis of this issue.

3.2

We recall that our objective is to obtain finite element procedures that behave uniformly well with respect to the thickness parameter, i.e. for which the finite clement solution Z;l satisfies an estimate of the type

Asymptotic reliability of shell finite elements

(3.33)

where the norm used remains to be specified. In the light of the above discussion regarding the asymptotic behaviours of shell models, it is natural to require that

1. In a bending-dominated case such a uniform estimate hold in the norm of the displacement space V ;

2. In a membrane-dominated case such a uniform estimate hold in the membrane energy norm.


We consider a standard finite element procedure obtained by discretizing the variational formulation (3.2), namely the problem Find Z;I E v h such that

E3~F(z:, E ) + & B ~ ( z ; I , E ) = L E ( E ) , VE E v h , (3.34)

where Vj denotes the finite element displacement space used. For a membrane-dominated problem we-in fact-need to consider the scaled problem

BM(Z;I, E ) + E'BF(Z;I, E ) = G ( E ) , VE E Vj. (3.35)

We can then show that we have a uniform convergence property, as proved in (Chapelle and Bathe 1998).

Proposition 7. Assume that there exist two interpolation operators Zh and Jh , defined respectively an VM and V , and both with values an vh,

VE E VM, VE E V , (3.36) V E E VM,

VE E V , lirn IIE - &(E)IlV = 0 , VE E V . (3.37) h-0

Then for any fixed E~~~ > 0 we have

lirn sup {llZE - Z;I\\M + &IJZE - Z;Illv} = 0. (3.38)

If we consider only the membrane energy norm in this uniform convergence result, this bound is weaker than an estimate of the type (3.33) in the membrane energy norm. However, this is clearly the best estimate that we can obtain without uniform regularity results on the exact solutions, and such regularity results are not available. We further point out that the interpolation assumptions (3.36) and (3.37) are rather un- restrictive. For example the Clement operator satisfies these two sets of assumptions when VM is a Sobolev space on which this operator is properly defined.

By contrast, in a bending-dominated situation, the problem to be considered is instead

h-o EE10,Emasl

1 BF(Z;I, E ) + ~ B M ( Z ; I , E ) = G ( E ) , VE E vh. (3.39)

The major difficulty associated with this type of finite element formulation is the classical numerical locking phenomenon. As is well-known


from other penalized formulations (such as nearly incompressible elasticity, beams, plates. . . ), locking occurs at its worst in situations where

v h n VF = (0). (3.40)

Indeed, Proposition 3 can now be applied with v h instead of V. We thus have that, keeping the finite element subspace (in particular the mesh) fixed, when E tends to zero the finite element solution 2; converges to the solution of the limit problem Find 2: in vh n VF such that

BF(Z;I, E ) = G ( E ) , YE E Vh n VF, (3.41)

and of course the solution of this problem is the zero displacement if (3.40) holds. Even though the thickness is always finite in practice, this implies that no uniform error bound of the type (3.33) holds in this case, and more specifically that, for a given mesh, the finite element solution gets “smaller and smaller” when the thickness decreases (whereas the exact solution does not vanish). Hence, this is a purely numerical artefact that makes the structure appear stiffer as it gets thinner, which explains the terminology “numerical locking”. The specific difficulty with shells is that, unlike for beams or plates, the pathological situation expressed by (3.40) is the common rule, as illustrated in the following result, for which we also give the proof because it is simple and illuminating as to how the geometry influences the asymptotic properties (cf. also (Choi’, Palma, Sanchez-Palencia and Vilarino 1998) and (Sanchez-Hubert and Sanchez-Palencia 1997) for other results concerning (3.40)).

Proposition 8. Consider a regular hyperbolic shell fixed on some part of its boundary, and a finite element scheme, in the framework of (3.39), in which the displacement components ~ 1 , 172 and 173 are approximated using continuous piecewise-polynomial functions. Assume that n o element edge in the mesh is part of an asymptotic line. Then (3.40) holds.

Proof. We will show that, for any inextensional displacement 7, if 7 is zero on some edge of any element of the mesh, then 7 is identically zero over the whole element. This being granted, the result is immediately obtained by “propagating” the zero displacements from the boundary conditions to the whole domain.

Consider an arbitrary element of the mesh, with 7 set to zero on any of its edges, then we have two possibilities:

a) Either the lattice of asymptotic lines originating from this edge covers the element completely (like in Figure 1-a) and the conclusion directly follows from Proposition 6.

b) Or the element is only partially covered (Figure 1-b), but in this case we still have ~1 = ~2 = 173 = 0 on a part of non-zero area of


Figure 1 Asymptotic lines and inhibited region for an element

the element. Recalling that the displacement components are given by polynomial functions, this implies that they are zero over the whole element. 0

In order to make the statement of this property simpler, we assumed that no single element edge corresponds to an asymptotic curve. It is obvious, however, that this result carries over to any case in which the same propagation technique can be applied, which is much more general. A practical consequence of this discussion is that, when using benchmark problems to numerically detect locking phenomena in shells, we should not align the mesh with the asymptotic lines of the midsurface when such lines exist, in order to avoid inhibiting the locking mechanism identified in the above proof. This holds in particular for cylindrical test problems for which the (very strong) temptation to align the mesh along the axis should definitely be resisted if one wants to draw conclusions of somewhat general significance (note that, in industrial computations, aligning meshes along lines of specific geometric properties is impossible in general).

We further emphasize that Proposition 8 reveals a significant difference between finite element methods for shells and for other structures. For beams and plates, it is indeed always possible to avoid (3.40) by raising the polynomial degrees of the discretization spaces, see in particular (Arnold 1981). Here our argument is independent of the polynomial degree, hence locking always occurs using displacement- based finite element formulations. But of course, for a given thickness, the errors in


the solution may be acceptably small if the order of the polynomials is sufficiently high and the mesh is sufficiently fine.

In order to circumvent the locking phenomenon, it is by now classical to resort to mixed formulations, in which an additional unknown- corresponding to the “delinquent” term of the energy-is introduced, see (Brezzi and Fortin 1991, Bathe 1996). For example, for Reissner- Mindlin plates this additional unknown corresponds to the shear stress. For shell formulations, it is thus natural to introduce the membrane stresses (and the shear stresses for Reissner-Mindlin type models) as auxiliary unknowns. For instance, for the Koiter shell model the additional unknown is the membrane stress tensor, namely the tensor with contravariant components

(3.42)

If we symbolically denote by ECE the group of additional stress unknowns, and by E the corresponding test function, the mixed formulation equivalent to (3.12) then reads

(3.43) B F ( Z € , E ) + M ( E , F) = G(E), VE E V , { M(Z‘, Z) - &2D(EE, S) = 0, V E E L2,

where the second equation expresses the stress-strain relationship for the additional unknowns, and L2 denotes the product space obtained by taking L 2 ( w ) for each component of the stress term. By construction, we have that D(E,E)+ defines a norm which is equivalent to the L2- norm, and that the mixed bilinear form M is continuous over V x L2, see e.g. (Arnold and Brezzi 1997). Then, both fields (displacements and stresses) are discretized in the finite element procedure. In general, discontinuous shape functions are used for the stresses, in order to allow the elimination of the stress degrees of freedom at the element level. The resulting finite-element formulation can then be written as

1 B F ( Z ~ , E ) + y B h ( Z g , E ) = G(E), V E E V h ,

& (3.44)

where Bh is a bilinear form that “resembles BM”, but for which the constraint B&(E, E ) = 0 is “more easily satisfied” for finite element displacements than Bn/r(E, E ) = 0. Note that, from this point of view, this strategy is fairly close to the engineering-based idea of reduced integration. In fact, in some cases it has been possible to substantiate some earlier-proposed reduced-integration schemes using the theory of mixed methods, cf. e.g. (Lyly 2000). When the analysis of the mixed formulation can be achieved, it indeed provides a proof that uniform convergence with respect to E holds (in the norm of V). This analysis,


however, relies on a crucial inf-sup condition which can be written in the general form

where s h denotes the discrete stress space and constant independent of h and E. Defining the semi-norm

is a strictly positive

(3.46)

we have the equivalent-more classical-xpression of the inf-sup condition

(3.47)

This inf-sup condition requires a deep adequation between the displacement and stress finite element spaces in order to be fulfilled. For shell formulations, the problem of finding a couple of displacement/stress finite element spaces in order to satisfy the corresponding inf-sup condition is still open (cf. (Arnold and Brezzi 1997) for an attempt in this direction).

In the case of nearly-incompressible elasticity where locking also arises, due to the kinematical constraint introduced by the incompress- ibility condition, a numerical procedure-called the inf-sup test-has been introduced in (Chapelle and Bathe 1993) t o test whether the inf- sup condition is satisfied without resorting to an actual mathematical proof. This procedure has been shown to give very reliable results (i.e. by clearly indicating whether the condition is satisfied or not) on numerous finite element procedures for which theoretical results where available, see also (Iosilevich, Bathe and Brezzi 1997) for plate formulations. A similar test has been devised and used in (Bathe, Iosilevich and Chapelle 2000) for shells. The MITC shell elements (MITC4, MITCS and MITC16 elements, cf. (Bathe and Dvorkin 1986) and (Bucalem and Bathe 1993)), which are based on a mixed formulation, have passed this numerical testing procedure, whereas the corresponding standard finite elements (that are also called displacement-based methods to distinguish them from mixed methods) have all failed as expected. However, the absence of theoretical results pertaining to shell inf-sup conditions does not allow a thorough validation of this test. Moreover, two other reasons make it difficult to advocate the shell inf-sup test as strongly as for nearly-incompressible elasticity:

1. the dramatic dependance of the shell behaviour on the geometry (and the boundary conditions), which implies that many confignra- tions


should probably be considered in order to reach an acceptable level of confidence (only one configuration was used for nearly- incompressible elasticity);

2. the inf-sup condition that the test is based upon is in fact stronger than the inf-sup condition corresponding to the shell mixed formulation, which means that a given finite element procedure could satisfy the actual inf-sup condition and yet fail in the numerical test.

Given the considerable difficulty to prove an inf-sup condition for shell finite elements, another interesting idea, proposed and analysed in (Bramble and Sun 1997), is to relax the inf-sup condition while tolerating some degree of non-uniformity in the convergence of the approximate solutions. Namely a mixed method is used for which a stability condition weaker than the inf-sup condition (3.45) is shown to hold, and a uniform error estimate is then obtained under the condition

h 6 C&. (3.48)

This means that locking should not appear unless a very small thickness (or a coarse mesh) is used. However, a thorough numerical testing of this finite element procedure is lacking, as regards locking as well as other difficulties featured by mixed finite elements for shells (see below).

Finally, as regards the mathematical analysis of the locking of shell problems, to our best knowledge the only shell finite element method for which an analysis valid for general geometries has provided a uniform error estimate is the method given in (Chapelle and Stenberg 1998). This method corresponds to a mixed-stabilized formulation in which additional stabilizing terms are added to the mixed formulation in order to by-pass the inf-sup condition.

In the framework of shell models, mixed formulations feature another - still more preoccupying-difficulty, which is that they are clearly tailored to the approximation of a problem in which a constraint is enforced by penalization. But for shell problems to fall into that category, according to our above discussion the asymptotic behaviour must be of the bending-dominated type. On the other hand, if the problem solved using a mixed method happens to be of the membrane-dominated type, instead of (3.44) we in fact need to consider

B&(Zi, E ) + c 2 B ~ ( Z ; I , E ) = G ( E ) , VE E Vh. (3.49)

Hence, compared to Equation (3.35) satisfied by the exact solution, the mixed formulation leads to a perturbation of the leading t e rm of the formulation. Therefore, convergence (and in particular uniform convergence) cannot be expected in this case unless the consistency of B&


with respect to BM is strictly controlled. This condition is very difficult to enforce, in particular because mixed methods are not designed to that purpose. As a matter of fact, for the above-mentioned mixed- stabilized scheme that works well in bending-dominated cases, uniform convergence is not obtained in membrane-dominated situations.

Therefore, it appears that we are “trapped” in a dilemma caused by the dramatic discrepancy between the two major asymptotic states of shell problems. Displacement-based finite elements work well in membrane-dominated situations but are subject to numerical locking in bending-dominated situations, in general. On the other hand, if we try to “fix” the-bending-dominated behaviour by using mixed methods (or some kind of reduced integration strategy), there is a major risk of seriously deteriorating the numerical solution in membrane-dominated cases. As a matter of fact, we do not know of any shell finite element procedure that would have been proven to work well in both regimes.

3.3 Guidelines for assessing and improving the reliability of shell finite elements

Whereas the complete mathematical analysis required to design and justify a general uniformly-convergent finite element method for shell problems still seems out of reach, we believe that it is crucial, in this reliability quest, to combine the insight provided by the analysis of the mathematical models and the elements of analysis available on their discretizations on the one hand, with extensive numerical testing on the other hand. In this interaction, benchmarks are both an essential interface and an invaluable means to assess the performance and the reliability of shell finite element methods, hence also to improve these methods.

Of course, in order to provide a rigorous and meaningful assessment tool, benchmarks must be very carefully chosen, and properly used. To that purpose, some detailed guidelines are given in (Chapelle and Bathe 1998), which rely on the following principles:

A basic set of benchmarks must contain instances of the two fundamental asymptotic behaviours of shell structures; Several geometries must be considered, since the asymptotic behaviour is highly sensitive on the geometry of the midsurface (for example a set of benchmarks only based on cylinders is not satis- factory) ;

0 Numerical computations must be performed and reported upon for several values of h (since we are concerned with the whole convergence behaviour and not only with the accuracy in one specific instance) and for several values of E (in order to assess the robust- ness of the convergence with respect to this parameter).


P = PO cos(2sz)

computational domain

Figure 2 Cylinder loaded by periodic pressure

In (Chapelle and Bathe 1998), a basic set of benchmarks is also proposed based on these principles. It consists of the following test problems:

0 Circular cylinder loaded by periodic pressure, cf. Figure 2. De- pending on the boundary conditions we can obtain either one of the two main asymptotic behaviours (bending-dominated behaviour if both ends are left free, well-posed membrane-dominated behaviour if both ends are clamped). This problem was originally proposed in (Pitkaranta, Leino, Ovaskainen and Piila 1995) which also presented a procedure to obtain numerical solutions of arbitrary accuracy'.

0 Partly clamped hyperbolic paraboloid. The shell shown in Figure 3 is clamped on one side and loaded, e.g., by self weight. It can be shown that pure bending is not inhibited, cf. (Chapelle and Bathe 1998). We do not have an analytical solution for this problem (see (Choi' 1999) where the exact solution of the limit problem is analysed), but we can use reference solutions obtained with a very fine mesh to compute error measures.

0 Clamped hemispherical cap. It can be shown that pure bending is inhibited for this problem (as a matter of fact, non-inhibited

.

'The values of displacements and deformation energies given in (Pitkaranta et al. 1995) should be multiplied by (1 - v2) and (1 - v2)', respectively.


Figure 3 Hyperbolic paraboloid

pure bending is obtained for a hemispherical cap only when the boundary is completely free, see (Ciarlet 2000), and also (Chapelle and Bathe 1998)). If we choose an axisymmetric loading such as the one shown in Figure 4 we can compare the results of the shell analysis with the results of some (reliable) axisymmetric analysis scheme.

We also emphasize that a deep insight on the behaviour of the exact solutions is an essential prerequisite in the assessment procedure, i.e. in the proper use of benchmarks. A particularly important feature of these solutions is the development of boundary layers which vary in amplitude and width when the thickness of the shell varies. If these boundary layers are not properly taken into account in the choice of the discretization scheme (in particular by refining the mesh when and where appropriate), they can strongly affect the convergence behaviour and induce wrong conclusions in the reliability assessment. This insight, available for plates (see e.g. (Haggblad and Bathe 1990, Arnold and Falk 1996, Dauge and Yosibash 2000)), is now becoming available for shells also through some more recent works (Pitkaranta, Matache and Schwab 1999, Karamian, Sanchez-Hubert and Sanchez-Palencia 2000).

Once a good set of benchmarks has been selected, it is crucial to measure the error made in the numerical computations by using proper error measures. We now review and evaluate some possible choices, including the currently most widely used.

1. Pointwise displacements. This error measure is probably the most It is, extensively used, especially in the engineering literature.

Some New Results and Current Challenges . . .

t 2 p = po cos(2a)

213

Figure 4 Hemispherical shell under axisymmetric loading

however, likely to be misleading, since the pointwise convergence of finite element methods is not guaranteed, in general;

2. Energy norm, namely [E'BF(E, E ) + BM(E, E ) ] 3 (note that we can scale the energy by any given power of E since we are concerned with relative errors). This cannot be a good choice, since it is the norm in which the displacement-based finite element solution is the best approximation, whereas this solution locks in bending- dominated situations;

3. Modified energy norm, namely [E'BF(E, E ) + B!&,(E, E ) ] $ . This choice is advocated in (Malinen and Pitkaranta n.d.), in particular. Its major drawback, beside the fact that its use with the exact solution (or any reference solution) is delicate (since the BL-form is usually tailored to one particular discretization), is that the use of a non-physical modification of the energy makes it difficult to interpret convergence in this norm;

4. Energy variation, namely I& - Ehl where & is the energy that corresponds to the exact (or reference) solution, namely

E = & 2 B F ( Z E , 2') + B M ( Z ' , Z'), (3.50)

and &h the energy of the finite element solution, namely

Eh = E 2 B F ( 2 i , 2;) + BM(Zi, 2;) (3.51)


for displacement-based methods and

for mixed methods. For displacement-based methods, it can be shown that this measure equals the square of the error in the energy norm, but this docs not hold for mixed methods, which makes the interpretation of this indicator very difficult (see also (Malinen and Pitkaranta n.d.));

5, Asymptotic convergence norms, namely-as discussed in the introduction of this section-the norms in which exact solutions arc shown to converge to limit solutions, i.e. the norm of the displacement space for bending-dominated problems and the membrane energy norm for membrane-dominated problems. For practical purposes the membrane energy norm can be directly computed using its definition (and the corresponding stiffness matrices). For the norm of the displacement space (typically a combination of Sobolev norms) it is more convenient to use an equivalent norm constructed with the deformation energy, for example [L2BF(E, E ) + B M ( E , E ) ] 4 where L is a characteristic dimension of the structure (the diameter, for instance). Note that, when using general shell elements, these computations are not straightforward since the membrane and bending energies are combined in the basic shell model. However, as a consequence of our above discussion (cf. Section 3.1), since the Naghdi model is asymptotically equivalent we can use the corresponding energy expressions (and stiffness matrices) to compute adequate norms.

Using the norms in which asymptotic convergence is obtained for the exact solutions, it is then natural to seek finite element procedures for which uniformly-decreasing errors can be measured on benchmarks such as those proposed above, when boundary layers (namely, the “nonsmooth part” of the exact solutions) are properly resolved. Indeed, for a displacement-based finite element procedure used in a membrane- dominated case it can be shown that, see (Chapelle and Bathe 1998),

I/ZE - z;llM + 41ZE - ZFlllV (3.53) < C inf (112‘ - ell^ +&lIZE - Ellv},

E E v h

which shows that a displacement-based finite element procedure is uni- f o r m l y optimal with respect to E for the norm 1 1 . / 1 ~ f e l l . IIv. Therefore, if the mesh is refined so that the boundary layers of a given test problem arc properly resolved for this norm, uniform optimal convergence is obtained. This is illustrated in Figures 5 to 7, where the case of


l L

0 1

E B : 001 e h

0001

0 0001 1

t /R=l/ lO + ! t/R=lll00 -- x -

t/R=111000 x Quadratic rate

7

7

7

I

10 100

a clamped circular cylinder is considered (as described in Fig. a ) , see (Chapelle, Oliveira and Bucalem 2003). The Naghdi model is used, and Q2 displacement-based elements are employed to discretize the problem. Figure 5 shows the convergence curves obtained for the membrane norm and for decreasing values of E (t denotes the actual thickness of the shell, namely 2e) when as many layers of elements are used within the layer of width along the boundaries (i.e. the boundary layers) as in the rest of the domain. This can be shown to be the correct way to resolve the boundary layers, see (Pitkaranta et al. 1995). We can see that the convergence rate is optimal for all values of the thickness, and that there is very little sensitivity of the convergence behaviour on the value of E .

By contrast, if no particular treatment of the boundary layers is applied, we obtain the convergence curves displayed in Figure 6 where some significant sensitivity with respect to E is observed. Also, if the original norm of V is used instead of the membrane norm (Figure 7) we again detect some marked sensitivity. All these numerical results are clearly consistent with Equation (3.53).

Likewise, for bending-dominated problems classical results for mixed formulations show that, when the inf-sup condition is satisfied, we have the following optimal error bound (see (Brezzi and Fortin 1991)):

Figure 5 Clamped cylinder: adapted mesh, membrane norm

216

1

0.1

E b : 0.01 e UI

0.001

0.0001

E b L

!? W

Dominique Chapelle

Figure 6 Clamped cylinder: non-adapted mesh, membrane norm

1

0.1

0.01

0.001

0.0001 1 10

N=llh 100

Figure 7 Clamped cylinder: adapted mesh, original norm

Some New Results and Current Challenges . . , 217

Of course, as previously mentioned we do not know of any shell finite element procedure that is proven to satisfy the inf-sup condition. Note however that a procedure which passes the inf-sup test on a given test problem automatically satisfies the estimate (3.54) for this same test problem and o n the sequence of meshes used for the inf-sup test (as it is the inf-sup values that enter in the bounding constants). Moreover, this estimate is very useful to determine how the meshes should be refined in order to ascertain uniformly converging errors in the numerical results obtained with finite element schemes that “would satisfy” the inf-sup condition (hence locking-free finite elements) , even though this inf-sup condition cannot be proven. Namely, the regularity of the stresses also needs to be considered to control the error in the semi-norm 1 . I S (note that this semi-norm is always bounded by the L2-norm, cf. (Chapelle and Stenberg 1998)).

Finally, we emphasize that all the above-proposed benchmarks are based on problems for which a well-defined asymptotic limit exists, and more specifically for which the asymptotic behaviour is either membrane- dominated or bending-dominated. As previously discussed, other types of asymptotic behaviours (as defined in Section 3.1) do exist, and in particular in cases when pure bending is inhibited and “G $ (VM)’”, see in particular (Pitkaranta and Sanchez-Palencia 1997) and (Pitkaranta et al. 1995) for examples. In these cases, however, it does not seem to be possible to identify a norm for which an asymptotic limit exists, hence it is difficult to see what kind of uniform convergence could be sought in finite element solutions. This is why the use of such problems cannot be recommended to assess the reliability of finite element procedures. We point out that the “Scordelis-Lo roof” (for which the geometry and the boundary conditions are defined in Fig. 8, and the loading corresponds to self-weight), which appears to be one of the most widely used test problems in the engineering literature, belongs to this category, as it can be shown that it corresponds to a situation of inhibited pure bending with “G $ (VM)”’, see (Chapelle and Bathe 1998) (in fact, it is the loading in the vicinity of the free boundaries which causes this difficulty). Nevertheless, some recent works have shown that in some cases (more specifically when the shell midsurface is parabolic or hyperbolic) it is possible to split the solution into several components, namely a smooth part (in which the membrane energy is asymptotically dominant like in membrane-dominated situations) and a singular part, where the singular part consists of strong boundary layers or of “generalized layers” that lie along asymptotic lines of the surface, see (Pitkaranta et al. 1999) and also (Karamian et al. 2000). Moreover the displacements that develop in these layers appear to be very similar to pure bending displacements. Even though more analysis is still needed, these recent results give hope that finite element procedures which would perform well (i.e. converge


clamped boundaries

\ I \ I \ I

Figure 8 Scordelis-Lo Roof

uniformly) in bending-dominated and membrane-dominated situations would also perform well in these more complex situations, i.e. converge uniformly for each of the above-mentioned components separately.

Acknowledgement

The author is particularly indebted to Professor Klaus-Jiirgen Bathe as many ideas and results presented in this article have originated from a fruitful collaboration with him, which is attested by the joint publications listed in the references.

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222

A Differential Geometry Approach to Mesh Generation

Pascal Frey Laboratoire J.L. Lions, Universite‘ Pierre et Marie Curie

175, rue du Chevaleret, 7501 3 Paris, France

Abstract

In this chapter, we present a mesh adaptation scheme based on a geometric error estimate. The term geometric refers here to the manner we are considering the numerical solution of a partial derivative equation. Indeed, the solution is envisaged as a manifold embedded in a higher dimensional space, for instance a two-dimensional Cartesian surface embedded in EX3. This allows to study the variations of the solution in terms of the calculus of variations and more precisely to take advantage of the powerful tool of differential geometry to properly characterize the surface.

Theoretical issues about error estimate in H 1 seminorm or in Lp norms are discussed, the main concepts of mesh generation and adaptation are briefly presented and practical issues related to the implementation of the numerical algorithms are discussed. Numerical examples from various fields of applications are presented to emphasize the potential of this approach.

Introduction

“In a f ew decades, the numerical simulations appeared a privileged tool of investigation in science and technology. Their purposes is t o reproduce by calculation the behavior of a system usually described by a model based o n partial derivative equations. The rise of the numerical simulations thus reinforce the need for the mathematical study (analysis) of these equations and for their numerical resolution.”

Excerpt from the inaugural lesson of Professor Pierre Louis Lions at the Collkge de France, Paris, june 2003.

This short excerpt of the inaugural lesson of Pr. P. L. Lions shows clearly that many, if not any, problems resulting from physics, chem- istry, mechanics, meteorology, engineering sciences, finance and, even more recently, from biological sciences are based on partial differential

A Differential Geometry Approach to Mesh Generation 223

equations (PDEs in short). The same applies for industrial fields (aero- nautics, space agencies, car manufacturers, etc.). Actually, all these phenomena involve values that are solutions of PDE systems.

Let us recall that a differential equation is an equation containing in addition to the independent variables (and functions of these variables), derivatives or differentials of these functions. If the functions that app- pear in the differential equation depend on an independent variable, the equation is then a general ordinary differential equation (ODE), On the other hand, if the equation contains partial derivatives of functions of independent variables with respect to some of these variables, then the equation is a partial differential equation (PDE) or, more precisely, a differential equation with partial derivatives.

In general, additional information on the boundary of the domain of interest (in space and time) are associated with the PDE. This information is usually called an initial or final condition (thus referring to time) or a boundary condition (referring to space).

Notice here the fundamental difference between ODE’S and PDE’s. The general solution of an ODE involve constants whereas the general solution of a PDE contains arbitrary functions.

Here, are written some traditional equations of the physics:

ut + cu, = 0 , ut + uu, = 0 ,

ut + UU,.&Uzz = 0 , utt - c2u,, = 0,

-nu = f,

linear (1st order) transport equation, Burgers equation (non linear, 1st order), Burgers with dissipation term, wave propagation (linear, 2nd order), Poisson equation (equilibrium problems),

-A% = f, biharmonic equation, u,,uYY - uZY = f(z, y), Monge-Amphre equation.

The goal of numerical simulations. The classical objective of a numerical simulation is to predict by calculation the behavior of the system under investigation that is described by a mathematical model. Usually, the solution u of such problem is not known analytically. Hence, for each PDE or each model, the aim is to compute a numerical solution function u h as accurately as possible (ie., close to the analytical solution or, at least, such that the difference J J u - u h J J , in a given norm, can be bounded). It remains to be defined the way the computer will carry out the computation, what is usually called an algorithm or a numerical method.

In this context, a fundamental notion emerges: the notion of a well- posed problem in the sense of Hadamard. Such a problem is characterized by the existence of a unique solution and it enjoys a property of stability that allows to control the solution by the data of the problem in certain norms. There is a continuous dependence of the solution with re-

224 Pascal Frey

spect to the initial data. Consequently, it is of the utmost importance to raise the question of knowing whether the problem is indeed well-posed before even considering computing an approximation of its solution.

Formally speaking, - - the resolution of a general PDE problem can be sketched as Figure 1:

Mathematical (maths, physics..) Analysis Analysis

(1) (2) (3)

Algorithm

Figure 1 Flowchart of the numerical solving of PDE problems.

Stage (1) establishes the mathematical translation of a given problem. The so-called model is a simplified description of the phenomenon expressed as a set of differential equations supplied with boundary conditions. Stage (2) has as an ambition to study the validity of the models by exhibiting properties satisfied by the solution of the equations. This is indeed a tedious task, notably because of the non-linear models arising in most of the physical problems. Stage (3) is made necessary because we can only consider a finite (limited) number of numerical values. It is thus a matter of studying the behavior of the approximations when the parameter of the discretization (h or t ) tends toward 0 and of comparing these approximate solutions to the exact solution. Hence, it is important to prove mathematically that the numerical results are indeed accurate approximations of the solution. Stage (4) involves the notions of algorithm, complexity as well as programming and is usually referred to as numerical simulation.

In this paper, we will mainly set the emphasis on the steps (3) and (4).

Problem statement. Let us consider a PDE problem on a closed bounded domain R c Rd and let us denote by u the exact solution of this problem. The finite element method used to solve such a problem relies on the approximation of the solution using polynomial functions on each element K of a triangulation Th of R. For linear elliptic problems, it is easy to prove the convergence of this method, when the maximal diameter h of the elements tends towards zero. Similarly, it is possible to compute an a priori error estimation of the difference between the exact solution u and the numerical (discrete) solution uh, that involves h as well as the global regularity of u.

On the other hand, a new tool has rapidly proved essential in the context of numerical simulations: the adaptation of the domain discretization (the triangulation or the mesh Th) to the solution. Why this


method is it become popular ? Well, for the rather simple following reasons. Given a fixed number of degrees of freedom, mesh adaptation allows to improve the accuracy of the numerical solution u h by adjusting the mesh density locally. Or conversely, given a desired accuracy value, it allows to dramatically reduce the number of degrees of freedom of the discrete problem as compared to a constant size mesh.

Indeed, in numerous test cases and realistic problems, the desired solution u presents some singularities in various regions of the domain and numerical experiments tend to show that the approximation error decreases as the triangulation is refined (elements are subdivided into smaller ones) at these places. However, these regions are usually not known a priori. Therefore, one needs a tool to measure the discrepancy between the exact and the numerical solution. The aim of an error estimqte or an error indicator is to accurately identify the regions where the solution is discontinuous and to provide useful information about the local mesh density to reduce the error. This information will be then used in combination with a mesh refinement technique to construct, in a few iterations only, a mesh adapted to the solution of the problem at hand, on which the approximation error is now below the target value.

Approximation of the solution. Let us go back for a short while to the concept of finite element approximation solutions for elliptic problems. More precisely, the variational method places the search for the solutions of elliptic equations within the general framework of Hilbert spaces. Hence, such solutions appear as minimizing functions, on a subspace or a given convex domain, with respect to a quadratic form representing the energy. The variational formulation has numerous advantages among which we rank the following:

i) from the theoretical point of view, it allows for the existence of a weak solution of the problem (this solution satisfies the equation and the boundary conditions only in a weak sense, i.e., almost everywhere),

ii) it places the problem of searching solutions within the very convenient framework of the best approximation in the sense of the norm associated with a scalar product defined in Hilbert spaces.

For instance, let us consider the variational formulation of an elliptic problem:

‘3nd u E Banach space X such that, Vu E X , u(u, u) = (f, u) ”, where u(. , .) is a continuous elliptic bilinear form on X x X and (., .) denotes the duality product between X f and X (f is supposed to be in X f ) . And let us consider the finite element approximation of the problem as well:

226 Pascal Frey

‘snd u h E X h c X such that , Vuh E X h , a(uh,uh) = ( f , u h ) ” . We introduce a regular set (Th)r, of triangulations of 0, the functions of X h are polynomials on each element K E Th.

An a priori error estimation can be written as follows:

IIu - Wl lX 6 F ( h , u, f )

its existence ensures the convergence of the methods when the solution u is sufficiently regular. Such an estimation corresponds for instance to the estimation of the energy: it ensures that the energy of the solution is controlled by the energy of the right-hand side term f. On the other hand, an a posteriori error estimation is of the following form:

Ilu - U h ll < G(h, U h , f )

in which the right-hand side term can be explicitly computed this time as it involves the numerical solution obtained on the given triangulation of the domain.

Let q be the second term of the a posteriori estimation, the so-called error estimate. It allows to control as well as to ensure the overall stability of the method. By assumption, it can be written as:

the quantities q(K) , K E Th allow to adaptively refine the mesh. Ideally, one would like to have inequalities like:

crl(K) 6 IIU-uhllX(K) 6 rl(K)>

where X ( K ) denotes the space of the restrictions to K of the functions of space X .

An interpolation error estimate. Several error estimates have been proposed in the context of the finite element methods, essentially based on the calculus of residuals and the resolution of a global or local problem. They can be considered as efficient, reliable and robust [18].

However, the solution u h obtained using finite element methods is usually not interpolant: the numerical solution Uh at the nodes 2 h of Th does not coincide with the exact value u(zh) at these nodes. Moreover, it is not even possible to guarantee that the numerical solution uh coincide with the exact solution u in at least one point in each element K E Th. All that contributes to the difficulty of explicitly computing the error llu - uh((. Another concern is that this computation is specific t o each


type of equation and must be remade with each change of equation. Hence, to achieve a higher level of abstraction, and to deviate from the physical problem (related to the mathematical operators), we advocate here the use of an indirect approach to compute the error.

Let V be a real Hilbert space and let vh c V be a subspace of V of finite dimension. Here, we consider that the weak formulation on V :

“find u E V such that for all w E V , a ( u , v ) = L(v)”

holds, here u(u, v ) is a bilinear continuous and coercitive (a-elliptic) form on V and L(v ) is a linear continuous form on V . And we would like to solve the projected problem:

“find U h E vh such that f o r all V h E vh, U ( U h , V h ) = L(vh).”

We have the following result regarding the error between the analytical and the numerical solution (known as the CBa’s lemma):

where M > 0 (resp. a) is the constant related to the continuity (resp. the ellipticity) of the bilinear form u(., .).

Now, let PhU denotes the linear interpolate of the solution u on each element K E Th and let us consider the interpolation error IIu - Phull, given Th a triangulation of the domain R E EXd. The numerical solution uh is known to converge toward the exact solution u as the diameter h tends toward zero. hence, one can deduce that the approximation error tends toward the interpolation error. According to Cea’s lemma for elliptic problems [8] , the following inequality holds:

11u - Uhl lX < c llu - PhUllX

Thus, controlling the interpolation error leads to controlling the approximation error as well. Now, the former adaptation problem can be rewritten as follows:

“Given a linear interpolation Phu of u with respect to a mesh Th, construct a mesh TL # Th on which the interpolation error Ilu-Phull is bounded in all directions by a given tolerance value. ”

From the practical point of view, this problem involves:

- the construction of an error estimate for the interpolation error, - the definition of an anisotropic metric tensor field,

- the adaptation of mesh Th to the metric specifications.

228 Pascal Frey

Although it does not appears clearly in the previous statement, this problem is closely related to another approximation problem: the generation of a piecewise linear approximation for a surface manifold embedded in EX3. This paper will attempt to reveal how these two problems are intimately related.

Paper outline. All the items listed in the previous paragraph will be reviewed in the following sections. In Section 1, we will provide some definitiogs about basic notions of triangulations and metric tensors. Sec- tion 2, we will introduce a geometric error estimate for the interpolation error and suggest various manners of computing approximations of second order derivatives of a discrete function uh. Section 3, we will focuss on mesh adaptation issues and on controlling the geometric accuracy in surface mesh generation. Finally, Section 4, we will give and comment numerical examples to emphasize the potential of differential geometry applied to mesh generation.

1 Preliminary definitions

In this section, we will provide the reader with some basic notions and definitions about mesh generation, metric tensors and differential geometry.

1.1 Triangulations and meshes Definition 1.1 (simplex). Let us consider d + 1 points aj = ( ~ i j ) $ ~ E Rd, 1 < j 6 d + 1, not all in the same hyperplane, i.e., such that the matrix A of order d + 1

is invertible. A simplex K whose vertices are the aj is the convex hull of the a j . Every point x E EXd, x = (xi) is fully specified b y the data of d f 1 scalar values X j = Xj(x) that are solutions of the following linear system, its associated matrax being A:

j d j=1

The A,(.) are called the barycentric coordinates of point x with respect to a j .


Let us consider a bounded domain R c Rd and denote 69 its boundary. We like to introduce Oh = U K as a simplicial set: the union of simplices of dimension d. Then, we define the notion of covering up.

Definition 1.2. a h is a simplicia1 covering up of provided that:

i) ah is homotopic to R, ii) VP vertex of dRh, P E Oh, iii) 8, n KJ = 0, vKi, K~ E R h , i # j ,

iv) K # 0, VK E Rh.

Condition i) indicates that R and Rh must have the same topology. Conditions iii) and iv) have been introduced to proscribe overlapping elements as well as degenerated (null volume) elements. Given these two definitions, it is now possible to define a triangulation of R.

Definition 1.3. A simplicia1 set T is a triangulation of the domain R if:

i) T is a simplicia1 covering up of R, ii) the intersection of two simplices is either empty or reduced to a

d - 1-simplex.

We like to call k-face a simplex of dimension Ic < d - 1, a 0-face is a single point. A triangulation is also called a conformal covering up of R, the condition ii) providing the conformity property (Figure 2).

Figure 2 Conforming (left) and non-conforming triangulations (right).

Remark 1.1. The previous definition does not give any hint about the existence nor about the uniqueness of the triangulation of the domain R.

The Euler formula and the Dehn-Sommerville relations relate the number of k-faces in a triangulation of R.

230 Pascal Frey

Definition 1.4. The Euler characteristic x of a triangulation T is given by the alternate sum:

d

k=O

where n k , k < d denotes the number of its k-faces.

Remark 1.2. When the triangulation is homeomorphic to the topological sphere, its Euler characteristic is equal to 1 + (-l)d. I n two dimensions, the following relation holds:

n , - n e + n f = 2 - c

where c corresponds to the number of connected components of 8R. More precisely, if the domain R includes no hole, then n, - n, + nf = 1. I n three dimensions, the above formule becomes:

n, - n, + nf - n, = 2 - 29

where g stands for the genus of the surface (Le., the number of holes). Hence, the triangulation of a close surface is such that n, - na + nf = 2.

Among the various possible types of triangulations, one triangulation is of particular interest in computational geometry: the Delaunay triangulation.

Definition 1.5. Let T be a triangulation of R; T is the Delaunay triangulation of R if the open ball circumscribed to any of its element contains no vertex of T .

This criterion, the so-called Delaunay criterion or the empty sphere criterion, means that the open ball associated with each element K does not contain any vertex of T but the vertices of K . A fundamental theoretical issue follows then:

Theorem 1.1. The Delaunay triangulation of a given set of points in general positions' exists and is unique.

Proof: this result is obtained trivially by involving the duality with the Voronoi' diagram associated with the set of points. The existence is given by construction and the uniqueness property is achieved since the points are in general position by assumption, thus considering a simplicia1 triangulation.

Remark 1.3. I f the points are not all in general positions, the Delaunay triangulation can still be defined using the previous definition, although it will contain non-simplicia1 polytopes.

'Points in general positions: no more than d + 1 cocyclic or cospherical points.


In the context of finite element methods, it is required to generate a discretization of the domain R. The latter is usually a conforming covering-up or a conforming triangulation of R. We will call such a triangulation a mesh. We denote by Th a mesh of a domain R, thus relating the letter h to the local diameter of the elements, the inradius or the length of the largest edge, for instance. It is also possible to consider non-simplicia1 meshes or mixed meshes, containing polyt.opes of different dimensions.

Defining the connectivity of a mesh as the type of connection between its vertices, allow us to introduce the notion of structured meshes, where the connectivity is like the Cartesian grid using in finite differences schemes, and unstructured meshes, exhibiting any non Cartesian type of connectivity. It is easy to see that structured meshes are usually convenient for very simple geometries while unstructured meshes are more flexible to handle complex geometries. By extension, one can also define the connectivity of a mesh element in terms of the connections between its vertices.

1.1.1 Numerical simulations

In numerical simulations based on the finite element method, computations usually involve the following steps:

i) mesh generation: managing the geometric approximation (spatial discretization) of the domain boundaries,

ii) interpolation: for constructing finite elements,

iii) matrix assembly: discretization of the equations,

iv) numerical resolution of the resulting (linear) system.

In particular, it is necessary to define all nodes, and not only the vertices, that will be associated with the degrees of freedom. For Lagrange PI elements, nodes and vertices are exactly coincident. Then, boundary and/or initial conditions must be assigned to mesh elements, providing physical information and attributes to the solver.

To summarize, let us recall that creating a mesh of a computational domain requires at first discretizing the domain boundaries as accurately as possible and then creating vertices within the covering-up of the domain. These two stages can be carried out separately or simultaneously. Obviously, additional concerns must be stressed, regarding in particular the existence of the triangulation, the accuracy and the quality of the discretization.

232 Pascal Frey

1.1.2

Despite many conceptual differences, as mesh generation methods have been developed in different contexts and are usually targeted at different fields of applications, we suggest the following classification [20]:

MM manual or semi-automatic methods: usually applicable to geometrically simple domains. Enumeration methods (mesh entities are user-specified) and explicit methods (taking advantage of the geometry of the domain) are representative of this class;

PM parameterization (mapping) methods: the mesh is the result of the inverse transformation mapping of a regular lattice of points in a parametric space to the physical (real) space. Two types of such methods can be identified: algebraic interpolation methods (transfinite interpolation from boundary curves) and solution- based methods (numerical resolution of a PDE on the computational domain);

DD domain decomposition methods: the mesh is the result of a top- down analysis consisting in splitting the domain into smaller geometrically simpler domains, close to a domain of reference. Both block decomposition methods and spatial decomposition methods (e.g. octree) belong to this class;

IM point insertion incremental methods: consist in creating and inserting vertices in the domain. Advancing-front and Delaunay-based methods are representatives of this class. Both methods rely on a given discretization of the domain boundaries;

CM constructive methods: the final mesh results from the merging of several submeshes using topological or geometrical transforma- tions, each submesh may have been created by any of the previous methods.

Remark 1.4. The identification of the method suitable for solving the mesh generation problem at hand is a dificult and tedious task. Usually the domain geometry as well as the field of application are the key points t o take into account for chosing a method.

A brief survey of mesh generation methods

1.1.3 Mesh adaptation

Solution-adaptive mesh generation is a very promising and challenging technique that is targeted at improving the accuracy of the numerical solution at a lower computational cost. It relies on a more efficient (optimal) point distribution, regarding the number as well as the locations of mesh vertices ( i e . , the density), and also on the shape of the mesh elements (isotropic vs. anisotropic).


After a numerical computation has been performed on a given initial mesh Ti, the numerical solution is analyzed using an a posteriori error estimate and the result of this analysis is converted into a metric field providing shape and size specifications. A new mesh Ti+l is then created using an adaptive technique from the mesh Ti, in order to equidistribute the approximation/interpolation error over the mesh elements. The whole procedure is repeated until convergence of this scheme, the local error being then bounded by above on each element.

The overall mesh adaptation scheme can be summarized as follows:

1. construction of an initial mesh To (i = 0 ) ,

2. computation of the numerical solution ui on Ti, 3. estimation of the local error lleill and definition of a metric field

4. reconstruction of a mesh Ti+l according to M i , 5. interpolation (projection) of the solution ui on Ti+l, 6. resume the adaption loop: i = i + 1, go to 2.

Mi,

where IleilI represents either the approximation error IIu - uhll or the interpolation error 11u - IIhuII. Mesh adaptation can be carried out by using either a global approach (advancing-front or Delaunay-based method, for instance) or a local reconstruction (optimization) method or a combination of global-local techniques.

In addition to adapting the mesh element size, the so-called h-method, an alternative consists in adapting the order of the interpolation for the finite elements (the so-called pmethod), using Lagrange Pk elements, lc > 1, in some regions.

Remark 1.5. The mesh adaptation scheme described here is well suited for steady-state problems, but requires consequent modifications for un- steady (time dependent) problems [l].

1.2 Notion of metric tensor Lengths and distances play important roles in a mesh generation process. From the mathematical point of view, the length of a vector or the distance between two points in space are based on the definition of the inner product2. Algebraic results indicate that the inner product is related to a positive-definite nondegenerate symmetric bilinear form. More precisely,

2The scalar (or dot) product of two vectors u = [ui,. . . , u,], v = [vi , . . . , v,] E n

i=l EXn, (u,v) = C uivi, is the canonical inner product of Euclidean space.

234 Pascal F’rey

Definition 1.6. Considering a vector space V over the scalar field F , the inner product can be defined as a map: (., .) : V x V ---f F, satisfying the axioms:

i) conjugate symmetry:

V x , y , E V , ( x , y) = ( y , x ) , -

hence, V x E V, ( x , x ) E F ,

ii) linearity (in the first variable):

iii) non negativity (positive definiteness):

vx E v , ( x , x ) 2 0

iv) non degeneracy:

( x , y ) = O , V y E V i i f x = O .

Obviously, if F = R then the first property is the symmetry of the inner product: ( x , y ) = ( y , x ) . Every inner product space has a natural norm defined as: )1x11 = m; the non negativity axiom of the definition making this definition always possible. The norm of a vector x is simply the expression of the length of vector x.

In this section, we will review the definition of a metric tensor and of the associated scalar product. Then, we will present metric-related operations.

1.2.1

In mathematics, a metric or a distance function is a function which defines a distance between the elements of a set; such a set is then called a metric space. This notion has been first introduced by M. Frkhet in his work in 1906 [19] . As a consequence, a metric induces a topology on the set. Here, we will adopt the differential geometry notion of metric tensor which refers to a structure defined only on a vector space, also known as a Riemannian metric.

Definition 1.7. A metric tensor i s a symmetric tensor field g of rank 2 that is used to measure distance in a space. So, given a smooth manifold, it is a (covariant) tensor field of type (0,2) on the manifold’s tangent spaces.

Metric, scalar product and distance

At any point in the manifold, this tensor takes a pair of vectors in the tangent space to that point and associates a real number; at p E M , the


bilinear form gp : TpM x T,M + R is symmetric and positive definite. It can be seen as an inner product. If a system of local coordinate xi is fixed, the metric tensor appears as a matrix, denoted g = ( g i j ) and has a unique expression:

n

g = gi jdx i 8 d x j , where gij = g (d L, . i+l dXi ’ dXj

While the distinction between covariant and contravariant indices is of importance for general tensors, these two notions are strictly equivalent for tensors in three-dimensional Euclidean space, known as Cartesian tensors.

A Riemannian metric tensor on a manifold M is not a distance metric on M . However, on a connected manifold every Riemannian metric tensor becomes a metric space and induces a distance metric on M , given by:

where the infimum is taken over all the rectifiable curves y : [0, 11 + M with y(0) = z and $1) = y . The curves which locally join their points along shortest paths are then called geodesics. In other words, in a Riemannian manifold, it becomes easy to define the length of a segment [a , b] of a curve parameterized by t as:

1 = ] d-dt. a

Remark 1.6. If we consider the two-dimensional Euclidean metric tensor, we can write:

9 = [;el ,

and the length of a curve simply becomes the well-known formula:

At this point, we refer the reader to Ciarlet [ll] and to the chapter devoted to differential geometry in this book for more details about this notion of metric tensor and, in particular, the covariant components of a metric tensor.

236 Pascal Frey

As an example, in the vector space R", the general form (., . )M defined by:

(X,Y)M = x T M y , k Y E R",

where M represents any positive definite matrix, is an inner product. For a Hermitian matrix (symmetric in the real case), the condition

x T M x > 0 , v x # 0 E Rn

is only satisfied if the matrix M is positive definite, i.e., if its n eigenvalues are positive. This can be interpreted as an orthogonal transformation, or as the dot product of the results of directionally differential scaling of the two vectors x and y, with positive scale factors and orthogonal directions of scaling. With these two definitions, it is easy to introduce:

Definition 1.8. Given the inner product (., . ) M in the vector space Rn, we define the Euclidean n o r m of any vector in E%" as:

1121111.1 = &xGj = d z i z ;

it actually measures the length of vector x according t o metric M .

From the axioms of Definition 1.6, it can be proven the following inequalities, known as the Cauchy-Schwarz inequality and the triangle inequality:

Definition 1.9. Let (., .) be a n inner product defined on vector space V , then we have:

l(X,Y)l G ll~ll.11Yll , vx7Y E v , with the equality only if x and y are linearly dependent (colinear), and we have:

112 + YII 6 0 ~ 1 1 + IIYII , yx, Y E v . Then, according to the previous inequalities, the norm defined by the

inner product is indeed a norm on V .

1.2.2 Metric decomposition

We know that a positive definite (non degenerated) matrix M is the matrix of a positive quadratic form f on R. Here, we use the characterization of a diagonalizable matrix.

Definition 1.10, If V is afinite-dimensional vector space, then a linear map T : V 4 V is called diagonalizable, if there exists a basis of V with respect t o which T is represented by a diagonal matrix.


The following sufficient condition is often used: the map T is diagonalizable if it has exactly n = d i m ( V ) distinct eigenvalues, i.e., its characteristic polynomial has n distinct roots. Thus, we can write that

Definition 1.11. A square matrix A is diagonalizable if it is similar to a diagonal matrix or i f it exists an invertible matrix P such that P-IAP is a diagonal matrix.

It follows that a symmetric positive definite matrix3 M can be decomposed in the special form:

M = R A R - ~ ,

where R is a matrix composed of the eigenvectors and A is the matrix constructed from the corresponding eigenvalues. Notice that this decomposition is not unique, as any eigenvector of the eigen subspace is associated to an eigenvalue, the columns of R and A can be permuted. However, it is possible to make this decomposition to be unique. For instance in R2, if we have a double eigenvalue A, then M = XI2. It is sufficient to take R = Id and A = M . In the case of two distinct eigenvalues, R can be seen as a rotation (orthogonal) matrix of angle 0, 0 E [O, -1. In R3, R can be considered as a combination of rotation matrices.

IT

2

1.2.3 Geometric representation

Definition 1.12. The closed unit ball in a normed vector space V , supplied with the norm 1 1 . 1 1 , is:

1. E v 7 11xl1 < 1) '

B y extension, i f V is a linear space with a real quadratic f o rm f : V --$ R, then { x E V , f ( x ) = 1) is called the unit sphere of V .

Remark 1.7. The geometric shape of the unit ball is totally dependent on the choice of the norm.

In R3, a metric M will be represented by its unit ball, an ellipsoid defined as: E = p E R 3 / l l O p l l ~ = d m = 1}, where 0 denotes the center of the ellipsoid. The main axes of the ellipsoid are given by the eigenvectors of the matrix M and the corresponding radii are given by the square root of the inverse of the eigenvalues of M .

{

3For the sake of simplicity, we will associate a metric tensor M with a symmetric positive definite matrix also denoted M . This indicates that M is the matrix representation of the linear map T with respect to some basis.

238 Pascal Frey

Remark 1.8. Given a n ellipsoid E of center 0, it exists a unique metric tensor M such that the geometric locus of the points satisfying the relation ll0pll = 1 describes the ellipsoid. Similarly, ma in axes are given by the eigenvectors and radii are associated with the eigenvalues of M .

This will enable us to consider the notions of metric and its geometric representation as equivalent. Hence, to analyze a metric tensor (or a matrix), it is possible to analyze its unit ball with respect to the metric.

1.2.4 Metric intersection

Let suppose that two metric tensors are specified at a point p E Rd. For various reasons, including mesh generation purposes, we would like to deal with a single metric at the vertex. To this end, we will define an intersection procedure. Considering the problem from the geometric point of view, we would like to define the largest ellipsoid E included in the intersection of the two ellipsoids El and E2 associated with the two metric tensors:

We consider two quadratic forms q and q' on a real vector space. We have seen that a matrix M representing a metric can be associated with a quadratic form. If q is positive definite, it is possible to find a basis of the vector space in which the matrices associated with q and q' are represented respectively by I, and D, a diagonal matrix of M,(R). This is the aim of the simultaneous reduction of the quadratic forms. In other words, the idea is to find a basis (el,e2,e3) in which M = M(q) and N = M(q' ) are congruent to a diagonal matrix and then to deduce the metric tensor of the intersection. To achieve this result, the matrix N' = M-' N is introduced. The matrix N' is diagonalizable in R and admits a basis of eigenvectors (orthogonal) with respect to q. In this basis, the terms of the diagonal matrices associated to the metric tensors M and N are obtained using the Rayleigh formula:

T X i = eTMei and pi = ei N e i , i = 1 , 3 .

Denoting by P = (e l e2 e3) the invertible matrix of GL3(R) formed by the eigenvectors e i , i = 1 , 3 of the matrix N'; hence, we have:


Finally, the intersection matrix Mn defined as:

rnax(X1,Pl) 0 0 Mn = (PT)-' ( 0 m=(Xz,CLz) 0 ) p - l .

0 0 max(X3,/13)

is a symmetric positive definite matrix as:

det Mn = (P-')z det A where A = (max(Xi, pi ) ) .

Hence Mn is a metric tensor according to Definition 1.7.

Remark 1.9. i) this definition does not necessarily imply that the eigenvectors of any of the two metrics M or N are preserved.

ii) in the case of isotropic metric tensors, M = diag(X1) and N = diag(Xz) , then Mn = diag(X) where X = max(X1,Xz).

iii) i f N' admits a triple eigenvalue a, then i f a 6 1, we have N' = M and i f a > 1, then N' = N .

Figure 3 using the simultaneous reduction of two quadratic forms.

Example of metric intersection in two and three dimensions

1.2.5 Metric interpolation

Let us consider a parametrization of the segment pq as c : [0,1] + R2, c( t ) = (1 - t ) p + t q (a mesh edge, for instance) and suppose that Mp and M, are two symmetric positive definite matrices representing two metric tensors associated with the endpoints. We are looking for the metric tensor at t , hence for a matrix Mt defined along the segment c( t ) for any value of the parameter t E [0,1]. The definition of this matrice Mt involves the interpolation of the eigenvalues of the matrices M p and

240 Pascal F’rey

M,. This procedure allows to define a continuous metric field along the segment. To this end, we suggest a linear interpolation scheme, for which the metric at any point c( t ) is given by:

Mt = ( ( 1 - O < t < l .

Finding the interpolated metric tensor Mt requires to express the two matrices in a basis {ei} in which both are congruent to a diagonal matrix and then to deduce the metric tensor at point t. In other words, this scheme is similar to reducing simultaneously the two quadratic forms associated with the metrics. Denoting by P = (el e2 e3) the invertible matrix of GL3(R) formed by the eigenvectors ei of the matrix N = Mp-lMql we compute (Xi)i=1,3 and (pi)i=1,3 the eigenvalues of Mp and M, in the basis {ei}i=1,3. Then, we define hi = l / f i and hi = 1 1 6 , for i = 1 , 3 . Finally, the metric tensor Mt is given by:

7 0 0 HI ( t ) 1

M t = ( P T ) - l ( ; $ ; )F1 O < t < l ,

m where Hi(t)i=1,3 are affine functions such that: Hi( t ) = (1 - t)hb + th$. Figure 4 shows the result of linear interpolation of metric tensors in two dimensions.

Figure 4 Example of metric interpolation in two dimensions.

1.3 A differential geometry primer In this short section, we will introduce the main concepts about the differential geometry of surfaces required to understand the point of view adopted in the following section. The reader interested by differential geometry is invited to read the chapter of this book devoted to this topic or to consult reference books like [ l l , 16,291. In particular, detailled proofs of the main results given here can be found in these textbooks.

Differential geometry is often seen as the calculus on manifolds or as the study of the different aspects of curvature. This topic emerged with


a paper of K.F. Gauss entitled Disquisition generales circa superficies curvas (General investigations of curved surfaces) published in 1827, in which this great mathematician explains that the whole geometry of a curved surface is fully determined from the formula for the line element ds , this in turn is determined by metric. Intuitively, the notion of curvature is related to the deviation from flatness at any point in a differential manifold. More generally, the differential study of surfaces aims at finding invariants that do not depend on the choice of the parameterization. One of the most important concept in differential geometry is that of tangent space.

Definition 1.13. Let U denotes a n open set in R2 and p = (u, v ) E R2 be a point of the set U .

i) A differentiable mapping f : U -+ R3 such that dfu : TpR2 + Tf(p)R3 is injective f o r all p E U is a parameterized surface. A mapping f satisfying this condition is called regular.

ii) The two-dimensional linear subspace d fp (Rg) c Tf(p)R3 is called the tangent space of f at p and is denoted as Tp f . Elements of Tp f are called tangent vectors off at p .

Tangent vectors are the partial derivatives of the position function. The tangent space at p is a vector space of dimension two, isomorphic to R2. Let f : U -+ R3 be a surface. A vector field along f = (f : U -+ R3} is a differentiable mapping X : U -+ R3. A vector field X along f takes values in the tangent space of R3 restricted to the surface f;- X ( p ) E TpR3.

Definition 1.14. The vector field:

f u x f u

I fu x f u l n :=

i s called the Gauss unit normal field along f. The mapping n : U -+ S2 c R3 is also referred to as the Gauss map. The moving frame ( f u , f u , n) is called the Gauss frame of the surface f : U -+ R3.

The surface f is regular at p = (u, v ) E U if and only if fu x f u # 0, and p is then a regular point. Note that ( f u , f u ) is a basis of the vector space TpR3.

1.3.1

One of the most important structures carried by a surface is the first fundamental form, which we will now introduce. The natural inner product of U c R3 induces on each tangent plane TpS of a regular surface U an inner product (., . ) p . In other words, the tangent plan inherit from the Euclidean structure of the ambient space.

The first fundamental form; area

242 Pascal Frey

Definition 1.15. The inner product of U c R3 which is a symmetric bilinear form induces a quadratic form on Tpf c Tf(p$R3 M R3 by restriction. This f o r m is called the first fundamental form and is denoted by g p or Ip:

The inner product o n Tf(p!IK3 composed with the linear map df, : R2 -+

Tf(p )R3 induces a quadratzc form o n TpR2 which is also called the first fundamental f o r m and also denoted by g or I .

These two definitions can be identified when identifying TpR2 with Tp f by means of the isometry df, : TpR2 M R2 + R2 M Tp f . Geomet- rically, this form allows to make measurements on the surface such as lengths of curves and areas of regions, for instance.

Definition 1.16. The matrix representation of the first fundamental form, with respect t o the basis f u l , fu2 is denoted by:

I,(W) = ( w , w ) p = 1w12 2 0 . (2)

(9 id = ( d f u i 7 fU.1) '

Commonly, we use Gauss' tensor notation: E = g(fu, fu ) , F = dfU,fw) , G = s(fv,fw), where arc@) = f,i(p) . f ,k(p) . According to the previous definition, (g ik (p) ) is also equivalent to the matrix notation I (e i , e k ) of I with respect to the canonical basis (e l , e2) of TpR2, and we have:

Proposition 1.1. The first fundamental f o r m I of a surface f : U --+ R3 is positive definite.

Let f : U --+ R3 be a surface. A curve on the surface f is a curve c : I -+ R3 which can be written as f o u, where u : I + U c R2 is a curve in U . Then, we define a tangent vector C ( t ) to f at u(t):

C ( t ) = c tii 0 u(t) , i

and the length of c(t) is given by:

I C( t ) 1 = c gij (u( t ) ) 2 (t ) t i j (t ) i>j

Definition 1.17. For a curwe c ( t ) = f o u(t) on a surface f , the arc- length parameter s ( t ) is uniquely defined by:


From the definition, we can see that the length 1 of an arc of curve on the surface joining two points p = (u(to), v(t0)) and q = (u( t l ) , v ( t1 ) ) is given by the formulas:

tl

= Lo dEU2 + 2FUV + GV2 dt .

We can also refer to ds2 = Edu2 + 2Fdudv + Gdv2 as a Riemannian metric. As mentioned, the first fundamental form allows areas to be computed on a surface.

Definition 1.18. The element of area d A of a regular surface contained in the coordinate neighborhood of the parametrization X : U c R2 -+ S is given by the positive number:

d A = IXu A X , ( d u dv = d E G - F 2 d u dv .

The area does not depend on the choice of the parametrization. How- ever, on a curved surface, the inner product induced by the Riemannian metric on the tangent space at every point changes as the point moves on the surface. This is the main concept behind the definition of a Rie- mannian manifold. Hence, the first fundamental form is not sufEcient to characterize the surface.

The vector field of fu x f, is a normal vector field along f as (f (u), X ( u ) ) E Tf(u)R3 is orthogonal to T, f for all u E U . Hence, it determines a normal orientation of the tangent plane. Considering a differentiable field of unit normal vectors on an open set U c S , i.e., a differentiable mapping n : U -+ R3 associating to each p E U a unit normal vector n ( p ) E R3 to S at q, we come to the following definition:

Definition 1.19. A regular surface S c R3 is orientable i f and only if there exists a differentiable field of unit normal vectors n : S -+ R3 on S .

Proposition 1.2. If a regular surface is given by S = { p E R3; f ( p ) = a} , where f : U c R3 -+ R is diflerentiable and a is a regular value of f , the surface S is orientable.

1.3.2

Let f : U + R3 be a surface and let consider the Gauss map n : U -+ S2 c R3, p H n ( p ) , for which n maps p into the unit normal vector n ( p ) E S2 to f at p . We introduce the following definitions:

The second fundamental form; curvatures

244 Pascal F’rey

Definition 1.20. The quadratic form:

-dn, ‘ df, : TpR2 x TpR2 -+ R

is called the second fundamental form off at p and is denoted by 11 or II,. The linear mapping L, := -drip o df;’ : T,f -+ T,f is called the Weingarten map. The matrix representation of IIp with respect to the canonical basis { e i } of TpR2 and the associated basis { f..} of T, f is:

( h i k ) = (-n.i . f u k ) = (n . f u i u k ) ,

or, i f using Gauss’ notation:

(b 3 = G:: 2) ‘ Now, we will show the relation between the second fundamental form

and the local curvatures of a surface. More precisely,

Definition 1.21. Let X E T,f, 1x1 = 1 be a unit tangent vector on a surface f. The normal curvature in the direction f X is the number:

.(X) = .(-X) = I I ( X , X ) .

Definition 1.22. Let f : U + R3 be a surface and let Sif = { X E T, f lI,(X, X ) = 1) denotes the unit circle in T, f . A vector X O E 5’; f is said to be a principal direction i f X O is a critical point of the function:

X E S j f H . ( X ) = r I ( X , X ) E R .

By extension, if X O is a principal direction, then the associated value .(Xo) is called a principal curvature of f at p . Principal curvatures are characterized by the result:

Proposition 1.3 (Rodriguez). Let X E S i f . Then X is a principal direction i f and only if X is an eigenvector of the Weingarten map:

L, = -dn, o df-’ : Tp f + T, f ,

and the associated eigenvalues are the principal curvatures.

We will denote ~1 and 6 2 the largest and smallest eigenvalues, respectively.

Corollary 1.1. If 11 is proportional to I (11 = .I), every direction is a principal direction; else there exists exactly two (up to sign) principal directions orthogonal to each other.

A Differential Geometry Approach t o Mesh Generation 245

We will now use the principal curvatures to introduce two fundamental notions.

Definition 1.23. Let f : U + EX3 be a surface. The Gauss curvature and the mean curvature off are the following functions o n U :

Proposition 1.4. The curvature functions are determined by the equation:

det(K id + d n o df -I) = K’ - 2Htc + K ,

where the left-hand side is the characteristic polynomial of the Wein- garten map L, in K . Consequently, 2 H ( p ) = TrL, and K ( p ) = detL, and we have:

We will use these definitions to characterize points on the surface. More precisely,

Definition 1.24. Let f : U + R3 be a surface. A point po is called an umbilic if ~ ~ ( p o ) = ~ ~ ( p o ) . I n addition, if K I C ~ O ) = ~ ~ ( p o ) = 0 , PO is called a planar point. A surface is said to be planar i f n ( p ) is constant. I t i s spherical i f there exists xo E R3 such that J f ( p ) - xo) = p, a positive constant.

Proposition 1.5. A surface is composed of only umbilic points i f and only i f it is planar or spherical.

Definition 1.25. A surface f : U + R3 is:

elliptic > O

hyperbolic < 0. parabolic at po E U i f detII,, is = 0

Regarding the intrinsic properties of a surface, we can state:

Definition 1.26. Let c ( t ) = f ou(t) be a curve o n a surface f : U --+ R3; c ( t ) is a geodesic if V C ( t ) / d t = 0 .

Theorem 1.2. Let X E Tpo f be a tangent vector to a surface f . Then for suficiently small E > 0 there exists a unique geodesic c ( t ) = f o u(t) , It1 < E satisfying the initial conditions u(0) = po and C(0) = X .

246 Pascal Frey

As a consequence, all the nonconstant geodesics on a sphere are great circles.

To give compatibility equations for the theory of surfaces, we introduce the six Christoffel symbols:

where gij,k = dg i j /duk . This will allow us to write the following result:

Proposition 1.6 (Integrability conditions). The Gauss formula and Codazzi-Mainardi equations provide relations between g i k , h i k , g i k , l , hik, l

and l?Fj , z :

r F , k - r2,j + C(rfjr2 - rfkr;) = C(hijhk1 - h i k h j z ) g z , (3) I I

Finally, we can give the fundamental result, known as Bonnet’s theorem:

Theorem 1.3 (Fundamental theorem of surface theory). Let U be an open, simply connected subset of W2. Suppose I, and II, are quadratic forms on TpW2, p E U , whose coeficients (g ik (p) ) and (h i k (p ) ) are differentiable functions of p . If I, is positive definite and the Gauss and Codazzi- Mainardi equations are satisfied, then:

i) there exists a surface f : U -+ W3 whose first and second fundamental forms are I, and IIp.

ii) any two surfaces f and f ” defined on U which have the same first and second fundamental forms differ by an isometry:

f = B o f , B an isometry of W3.

The theorem asserts that the knowledge of the first and second fundamental forms determines a surface locally. This result concludes our short introduction to the differential geometry of curves and surfaces. We will use these concepts in Section 3, when dealing with the tedious problem of meshing surfaces.

2 A geometric error estimate

Anisotropic error estimation has been studied extensively in the last two decades. In this section, we briefly review the concept of error estimation


and we propose an approach to obtain an anisotropic bound for the interpolation error suitable for linear simplicia1 elements in any dimension. In addition] we will explain how this bound will provide a characterization of the optimal mesh elements] both in terms of shape and of size. We assume the reader to be familiar with the concept of Hilbert spaces and with variational (weak) formulation for elliptic problems.

2.1 A priori error analysis We will primarily focuss on error estimation in the context of finite element methods (or FEM in short). The concepts presented in this section can however be transposed to other classical discretization methods such as finite differences or finite volumes, for instance.

2.1.1 Interpolation error in L2 norm and H 1 seminorm

Let a be a bounded domain of Rd and let u be the solution of an elliptic differential problem on under Dirichlet boundary conditions on dol the domain boundary. For an elliptic problem of order 2, for which we can write a variational formulation:

where V is a Hilbert space, a is a bilinear continuous elliptic form, 1 is a linear continuous form on V , we will recall the classical error bound given by Cea’s lemma [9] and already mentioned in the introduction of this chapter.

‘fJind u E v such that a(u,v) = l(v), vw E V ”

A bilinear form a(., .) is continous means that:

3A4 > 0, such that (u(w,w)( < A4 ( (w( ( ( ( ~ 1 1 , vw,w E V .

and a( . , ,) is a-elliptic or coercive if

3a > 0, such that la(wl w)I < Q 11w((2 l v ~ E V

Let Vh be the internal approximation space constructed with a given finite element method (Vh c V ) , the following inequality holds:

where uh denotes the discrete solution associated with Vh and I ] . ( ( represents the norm defined in the space V . Moreover] if the form a is symmetric, a(u - uh,wh) = 0, Vwh E Vh indicates that uh is the projection of u on Vh according to the scalar product a4 and represents the

4Recall that for real vector spaces, the positive-definite nondegenerate sesquilinear form called an inner product is actually a positive-definite nondegenerate symmetric bilinear form.

248 Pascal F'rey

best approximation in a subspace of finite dimension. In such a case, a better upper bound can be provided:

Remark 2.1. The previous inequalities are only valid i f the same forms a and 1 are used in the exact and approached problems, respectively. In particular, this requires to compute the integrals exactly.

The main idea of a priori error estimates consists in bounding the approximation error of the exact solution by the interpolation error by chosing the PhU interpolate of u as specific 2/h in the finite element space

Let us denote by lI.l1~,2,n the H 1 norm on the domain 0. We will provide an upper bound on the approximation error 1Iu - Uhl11,2$ involving the interpolation error IIu - Phu111,2,$. At this stage, it remains to evaluate the interpolation error in the L2 and H1 norms depending on the choice of the finite elements used for the approximation [lo].

Let assume the domain 0 is covered by a mesh Th. The first point is to notice that the norm on 0 can be obtained from the norms on each element of the mesh Th; in other words, the interpolation error can be written as:

vh.

Now, let us consider the affine transformation from a reference triangle to an arbitrary simplicia1 element K . Let B denotes the matrix of this transformation, the following relations hold:

where h (resp i) and p (resp. 6) represent the longest edge and the diameter of the inscribed circle of the element (resp. reference element) (Figure 5).

If the mesh Th is uniform ( i e . , of constant size), then it exists a constant a! such that:

,,We use the following inequalities, for all w E C1:

IlVfjll < IlBll l l V 4 and IlVvll 6 IIB-lll IlVfjll ,


Figure 5 Reference element and arbitray triangle.

to deduce the inequalities] for all w E Cm:

and thus, given the integral semi-norms l w l m , 2 , ~ defined as:

we obtain, for all w E H"(K)

We use then the inequality:

and we consider the interpolation error in the reference element for polynomials elements of degree k :

as well as the Relations (6) and the hypothesis (7) to deduce the following fundamental result.

Theorem 2.1 (General). For finite elements of degree k (Pk or &k) and for an exact solution u, suficiently smooth (u E H'++'(R)), the following bounds of the norms L2 and H 1 hold:

where the constant C is independent of h.

250 Pascal Frey

2.1.2

Let consider a given bounded interval [a, b] subdivided into N sub- intervals Ti = [xi-l,xi]. Here, Vh is the space of the piecewise affine continous functions on each segment Ti. We will use the upper bound on the approximation error introduced previously (Relation (5)):

PI elements in one dimension

M IIU - Uh111,2 < - llu - %1)1 ,2 , VVh E vh a

Chosing for vh the interpolate Phu of u in the finite element space Vh, allows us to bound the approximation error Jlu - uhJJ by a constant multiplied by the interpolation error IIu - PhuII. This leads to the following results:

Theorem 2.2. Denoting h =

tablish, Vx E [a, b] , the following inequalities:

max it- { 1,. . , N }

(xi - xi-11, it i s possible to es-

h2

h 2 xt-[a,b]

I.(.) - Phu(x)I 6 - max (u”(x)[,

Iu’(x) -PAu(x)I < - rnax Iu”(x)I,

8 xE[a,b]

hence, IIu - Phu111,2 6 C h max Iu”(x)I,

and finally 11’11 - uh111,2 < C h max Iu”(z)I.

(8)

(9)

E [a, bl

xE[a,bl

Proof: the sketch of the proof is classically based on Taylor expansions: .

(Xi-1 - U(5i-1) = U(X) + (Xi-1 - X ) U ’ ( X ) + 2 u“ (Ei ) i

U(Zi) = U(X) + (Xi - X ) U ’ ( X ) + (Xi - X ) z U ” ( V i ) .

In each interval [zi--1,zi], we classically denote wi-1 and wi the two basis functions of v h associated with the points xi-1 and xi, respectively. Their restriction in the interval [ x ~ - I , x ~ ] can be written as follows:

Then, we have wi-i(z) + wi(z) = 1 and x ~ - ~ w ~ - I ( z ) + ziwZ(z) = X, for all z in [zi--l,zi], as well as the relation:

PhU(z) = U(Xi-l)Wi-l(X) + U(Zi)Wi(X), thus, it is easy to deduce that:

( P h U Y ( Z ) = U(Xi-l)WL(X) + U(zi)w:(X).


Using the previous Taylor expansions yields to:

PhU(Z) = u(z) + #Xi-’ 1 - z) 2 wz-l(z)u”(&) + (Xi - z)2wi(z)u”(?7i)]

and (Phu)‘(z) = u’(z) + -[(zi-~ 1 - z) 2 1 ~ ~ - ~ ( z ) u ” ( E i ) + (xi - ~ ) ~ w : ( z ) u ” ( 7 7 i ) ] . 2

On the one hand, we write the inequality:

and we deduce the upper bound:

1 2 h2 2 8 -[(Zi-- l - z) wi-l(z) + (Xi - z)”Wi(z)] 6 -,

and, on the other hand, we write:

1 - z)2w:-1(z)u”(Ei) + (3% - z)2w:(z)21”(77i)] 6

Remark 2.2. I n each element Ki of length hi local error bounds are obtained according to:

IIu - Phu111,2,K, < Chi max Iu”(z)I xEKi

Hence, it is possible to specify the size hi for each element Ki to globally equidistribute the interpolation error as follows:

0 in L2 norm: hi c( ( rnaxIu”(z) I ) - i

0 in H 1 norm: hi 0: (max Iu”(z)I)-’.

It can be shown that each of these choices minimizes the L2 and H 1 global errors, respectively. For instance for the L2 error, using the relation:

xEKi

xEKi

1 2

PhU(2) = u(z) + -[(5i-1 - .)2Wi-l(.).”(&) + (Xi - 2 ) 2 w i ( 2 ) u ~ ~ ( r ] i ) ] ,

252 Pascal F’rey

the interpolation error IIu - PhUllo,a,n is bounded by the term:

The explicit computation of [(~i-~-x)~wi_~(x)+(xi-x)’w~(x)] leads to the simple expression: (xi - x)(x - xi-1). By introducing hi = xi - xi-1, we obtain finally the following bound:

The minimum is obtained by computing the gradient of this error with respect to the points xi. This leads to the following relation:

hi /g = hz+i /-, vi = 1,. . . , N - 1

Remark 2.3. Assuming u is a polynomial of degree 2, then u’ is an a f ine function and u” becomes a constant. I n such a case, the maximum value of the interpolation error of the solution u is obtained for the midpoint of the interval. Indeed, the derivative of the interpolate of u, denoted PAu, is a constant function which is equal to the value of u’ at the midpoint of the interval. I n other words, at the midpoint of [xi-l, xi], we have a maximum value of the interpolation error on u and, simultaneously, an interpolation error equal to 0 on u‘. This explains why it is interesting to estimate the derivatives of the solution at the midpoints of the intervals.

2.1.3

Now, let us consider a polygonal domain R c R2 covered by a simplicia1 mesh Th. w e take for the approximation space vh, the space of piecewise affine continuous functions on each simplex Ki. Again, we will make use of the fundamental result given by Relation (5):

Lagrange PI elements in two dimensions

Here, in the same way as we did in dimension one, we chose for vh the interpolate PhU of u in the finite element space vh. Then, the approximation error IIu - Uhll can be bounded by a constant multiplied by the interpolation error IIu - PhuII.


Let us denote by h the longest edge of a triangle, by 00 the smallest internal angle among all mesh simplices, and by D2u the Hessian matrix of a C2 function u defined on R2:

and by llD2u(x, y) 1 1 its spectral following result, echoing Theorem 2.2:

Theorem 2.3. Vx, y, E R, the following inequalities hold:

It is now possible to enounce the

hence, IIu - P h ~ 1 1 1 , 2 6 C h SUP IID2u(x,~)II

and finally, llu - ~ h l l 1 , 2 6 C h SUP l l ~ 2 ~ ( x , ~ ) l l

(11)

(12)

X,YER

X,YEn

Proof. (see [24] for more details) in each triangle Ki, we denote classically by X i the barycentric coordinates, that are the restrictions to the triangle Ki of the basis functions of v h associated with the three vertices. Then, for all ~ , y E Ki,

3 3 3

cxi(Z,Y)=l, cZiX2(Z,Y)=Z, CYzxi(Z,Y)=Y, i=l a= 1 i=l

and 3

P h U ( Z , Y) = c U(Zi, Yi)Xi(Z,Y),

V ( P h U ) ( Z , Y) = C U ( Z i , Yi)VXi(Z, Y) .

i=l

and, consequently we deduce easily that:

3

i=l

Using the following Taylor expansions:

_____

5We recall that the spectral norm of a matrix A is the largest singular value of A or the square root of the largest eigenvalue of the positive-semidefinite matrix AA".

254 Pascal Rey

we are able to obtain:

d __f

&(x, y)MAi.D2u(& q)MAa (14) 1 i=3

Ph'LL(X, y ) = u(x, y ) + - i=l

and, using Relation (13), we have finally the equation:

Obtaining the error bounds is now a simple exercise. Indeed, it can easily be shown that:

hence,

On the other hand, the error on the gradient value can be bounded by above:

Noticing that A j A k

I V X i ( X , Y ) I = - 2lTI

where h~ denotes the length of the longest edge in the mesh, where IT1 denotes the area of T and OT represents the smallest interior angle in the mesh, we use the following inequality:

4 IT1 2 h$ sin(&)

to obtain: n

and the desired result:

Remark 2.4 (Optimality). I n the upper bound on the error, the dominant term is related to the gradients. I t will be minimal for a minimal

value of -. Among all triangles having a longest edge of length

h, the triangle for which this term is minimal is that f o r which sin80 is maximal. Hence, we will consider the equilateral triangle as the optimal triangle. I t is also the triangle of maximal surface inscribed within a given circle.

h szn(80)


Let us suppose now that u is a polynomial function of degree 2. Then, the gradient of u is an affine function and D2u becomes a constant matrix. Moreover, considering a homogeneous and isotropic problem, the Hessian matrix D2u is a scalar matrix. In each simplex, the maximum value of the interpolation error of the solution u is achieved at the center of the circumscribed circle (the circumcenter). The gradient VPhU of PhU is a constant function equal to the value of V u at the center of gravity of the triangle. Hence, when considering an equilateral triangle, at the circumcenter (which coincide with the center of gravity), we have simultaneously a maximum value of the interpolation error of u and a minimal (null) value of the interpolation error on Vu. This analysis explains why it is desirable to estimate the derivatives of the solution u at the center of gravity of the elements.

On the other hand, for non isotropic problems in which the solution changes rapidly in one privileged direction, the previous analysis proves to be at least insufficient. A more clever analysis consists in looking for the eigenvalues and the eigenvectors of the Hessian matrix D2u in each triangle and to consider as optimal triangle the triangle of maximal surface inscribed within an ellipse. To this end, we rely on Relations (14) and (15). In anisotropic problems, the Hessian matrices diagonalized in their eigenbasis are not scalar anymore, but only diagonal. Thus, to ensure positive values, we are taking the absolute values of the eigenvalues. Hence, this defines a new scalar product and consequently a new metric for which the concentric circles of the error isovalues become concentric ellipses of fixed excentricity. If we denote by:

the new symmetric positive-definite matrix associated with the scalar product, the following expressions still hold:

as well as:

Again, we observe that the error on u is null on the ellipse of fixed excentricity circumscribed to the triangle and is maximal at its center. It is exactly the opposite for the error on Vu, which is null at the center and maximal at the triangle vertices. Hence, it is the error on the gradients that will dictate the optimal shape of the triangles in the mesh. The

256 Pascal Frey

interpolation error on u depends only on the size of the elements. Re-garding the approximation error, as it is known by the H1 norm, it willthen nds on both the size and the shape of the triangles (Figurd 6).

Figure 6 lues of the error and optimal triangle in the isotropic caseand i n in the non isotropic case (right-hand side).

2.2 Mesh adaptation using the error estimateWe will now approach the numerical and algorithmic aspects related tothe implementation of the error bounds previously established, in thecontext of mesh adaptation for numerical simulations.

2.2.1 Anisotropic formulation of the interpolation error

In the previous section, we have established a bound of the interpolationerror of the solution u at each mesh point. More generally, let K denotea mesh element in any dimension, the global bound on the interpolationerror can be expressed as:

\\u-Phu\\oo,K max max{v,]D2u(x,y)\v),y£K v<ZK

(16)

where cd is a constant only related to the dimension of the domain(specifically, c2 = 219 and c3 = 9/32). However, from a pra pointof view, this bound is not fully usable as it involves the estim of twomaxima that cannot be computed numerically. To overcome this prob-lem, we will first replace the second maximum (on the vector v c K)by introducing esh edges. Indeed, any vector v c K can be expreas a linear combination of the edges of the simplex K. Hence, it followsthe result:

VvC-K, maxe€EK

257

where EK denotes the set of edges in the element K . Then, the previous bound can be rewritten as follows:

A Differential Geometry Approach to Mesh Generation

But again, the right-hand side term is numerically tedious to compute. The maximum over the metric field 1D2ul is not known. Overcoming this problem requires to be able to find a metric tensor FK, hence a symmetric positive definite matrix GG on the element K such that:

and such that the region defined by: { ( w , FK w ) / 'v'w c K } has a minimal surface. We will show hereafter how to compute the metric tensor & when approximating the metric ID2ul. Then, we obtain the explicit bound:

IIu -

Hence, the interpolation error EK on the element K is evaluated as follows:

N

EK = 1121 - PhulI,,~ = cd e E E K m a x ( e , M K e ) . (17)

Remark 2.5. i) This expression indicates that the interpolation error o n K is proportional t o the s q u a r e d t h e length of the longest edge of K measured according to the matrix M K .

ii) W e could also notice that the interpolation error is related to the lengths of the edges of a mesh. Hence, controlling the lengths of the mesh edges will enable us to control the interpolation error o n each element. This will reveal extremely useful for mesh generation purposes.

If we consider a linear solution in one dimension (a line), only two points are sufficient to describe exactly the solution. On the other hand, if the solution is represented by a curve (a polynomial function), the number of points and edges required to exactly describe it is an infinite set. To represent the curve with a given accuracy would require the knowledge of the intrinsic properties (the local curvature) of this curve. In other words, the node distribution must be proportional to the local curvature, the coefficient of proportionality being related to the desired accuracy (a tolerance value). The answer to this problem requires the computation or the knowledge of the first-order derivatives of the underlying parametrized arc to identify the extrema, as well as the variation of the derivatives, namely the second-order derivatives.

In two dimensions, the solution to this problem can be seen as a parametric Cartesian surface embedded in IR3 : C = {(z,y,a(z,y)}.

258 Pascal Frey

Similarly, the control of the piecewise linear approximation of the surface C requires, at least, to introduce second-order derivatives (the Hessian) of u in the evaluation. In such a case, the second-order derivatives allow us to introduce anisotropy.

In three dimensions, the solution can be seen as a hypersurface embedded in El4. And we can use the same reasoning to introduce second- order derivatives in the expression of the interpolation error. This can explain why the error estimate for the interpolation error is naturally called a geometric error estimate.

2.2.2 Problem statement

In most meshing problems, the aim is to equidistribute the interpolation error in each direction over the mesh elements in order to control the approximation error. Let E denote the maximal error value on each mesh element K . According to Relation (17), the interpolation error on the edges of an element K is such that:

- EK = llu - PhuII,,K = cd max (e , MK e ) , b'e E EK.

eEEK

cd - &

If we introduce the matrix MK = -MK, where E = Ilu-Pphu((,,~, then the edges must be such that:

(e,MKe) = 1 ( ( e ( ( M K = 1, b'e E E K .

In other words, the interpolation error on an element K is of order E

if the length of all its edges is equal to 1 when computed according to the matrix M K . Obtaining a mesh on which the interpolation error is equidistributed is a problem equivalent of constructing a mesh in which all edges have a length according to MK equal to 1, or at least numerically very close to 1. Such a mesh will be then called a unit mesh.

However, as the matrix M K indicated in the above relation is not known, we have yet to explain how to define it analytically and eventually how to compute it numerically.

2.2.3 Metric construction

A metric tensor, as a symmetric tensor field, over a domain R c Rd is a data of infinite dimension that cannot be used as such for numerical purposes. Hence, we will use a discrete approximation of the metric tensor, considering the mesh as the underlying support. More precisely, we will consider a metric defined at each mesh vertex and represented by a symmetric positive definite matrix, because it is usually more convenient


for finite element simulations where the unknowns and the vertices coincide. This metric field is made continuous using a classical interpolation scheme.

Let E denote the maximal interpolation error accepted over the mesh and let hmin (resp. h,,,) represent the minimal (resp. maximal) edge size. Then, according to Relation (17), we define at each mesh vertex the anisotropic mesh metric as:

and R is the matrix composed of the eigenvector, the di represent the eigenvalues of the Hessian matrix D 2 u ( x , y ) ; cd is a constant value related only to the dimension. Notice that this metric tensor is naturally anisotropic, the main directions are given by the eigenvectors and the sizes by the positive eigenvalues di. Remark 2.6. i) introducing a minimal and a maximal sizes is a practical way of avoiding unrealistic metn'cs. It is also a way of controlling the time-stepping in a flow solver.

ii) the new scalar product provided b y the metric tensor is related to the Hessian of the variables of the problem at hand. Hence, it is highly desirable to have a good estimation of the Hessian in order to define an accurate metric tensor.

iii) to define an isotropic metric, one can simply consider:

d = max(di) and M = c:) , a=l,d

in this case, the interpolation error is still bounded by E , although it will not necessarily be equidistributed.

2.2.4

The expression of the bound on the interpolation error, Relation (16), involves the Hessian of the exact solution of the problem and this matrix appears also in the metric construction, Relation (17). However, as the exact solution is not known, only a discrete solution uh, piecewise linear in this case, can be obtained using for instance a finite element method. Thus, the Hessian of u h is null in the interior of an element and is a Dirac mass on its boundary. We have then to reconstruct an approximation of the Hessian of the solution u based on the discrete solution uh.

Evaluation of the Hessian matrix

260 Pascal F’rey

In order to evaluate the right-hand side of Relation (16), the idea is to reconstruct a C2 continuous solution u*, based on the discrete solution uh, and to evaluate the approximation error as the norm of the difference between the reconstructed solution u* and the discrete solution uh:

((u - uh(( - ((u* - uh(( .

Moreover, as the solution u* coincide with uh at the mesh vertices, we can write IIu* - uhll = IIu* - Phu*II. By a,nalogy with Relation (16), the interpolation error can be bounded by above in the following manner:

Ib* - where, in this case, all the terms present in the right-hand side of the inequality are well known or can be computed.

Remark 2.7. In order to compute the interpolation error, at i s not necessary to explicitly know u* o n fl, it is suficient here to reconstruct its Hessian matrix.

We will now describe two different approaches to numerically com-

0 on the first hand, we uses a double L2 projection and a Clkment interpolation operator to reconstruct the Hessian matrix. This approach usually provides a good accuracy for the Hessian, although it does not guarantee that the reconstructed Hessian is associated with u*, as u* may not be defined;

0 on the other hand, we use a Taylor expansion and a resolution of the linear system obtained by a least-square approximation. As such, it allows to reconstruct a Hessian matrix associated with a solution u*.

pute the Hessian value of the solution:

a) A dual L2 projection scheme. Let consider a mesh Th of a domain fl C @. We denote by vh = {w € Hi(fl2)such thatwllc E p k }

the approximation space associated with a Lagrange Pk finite element, where Pk is the space of polynomials with scalar coefficients of degree less or equal to k . At each vertex p i , 1 < i < n, we associate a basis function (pi E v h and we denote as si the support of (pi: si = supppi .

At first, we recall the Clkment interpolate. Let consider the local L2 projection operator. The idea consists in projecting w E L2(fl) on Pk independantly on each Si c Th. More precisely, for w E L2(fl), we define now E L2(R) as follows:

A Differential Geometry Approach to Mesh Generation 26 1

The Clkment operator lIc : L2 - Vh is defined as:

Now, it becomes possible to define an operator to reconstruct the gradient as follows:

We still have to explain how to reconstruct a PI gradient from uh. As u h is the PI discrete solution on each element, its gradient is a constant value on each element and it is not necessarily continuous. We consider the case where k = 0 and we apply the L2 projection operator to vuh E L'(S2). For each Si, we have by definition, b'p E Po:

where /KI (resp. ISil) is the measure of the surface of K (resp. Si). Notice that this scheme is simply a reconstruction using a weighted average, the weight being related to the surface of the elements. Hence, for each vertex p i , the operator to reconstruct the gradient leads to:

or similarly:

c IK IWhIK)

lSil (19)

KES, V R U h ( P i ) =

At this stage, having reconstructed the gradient of the solution, we can apply the same procedure to each component of this gradient, in order to reconstruct the Hessian matrix. The operator to reconstruct the Hessian matrix is given by the following relation:

H * ( U h ) + H*(WJT HR(Uh) = 2

262 Pascal Frey

where the matrix H * ( u ~ ) corresponds to:

with for each component xj, 1 < j 6 d:

Remark 2.8. This approach is proved to be equivalent to a method in which we apply a dual L2 projection globally on the space of Lagrange PI finite element functions with a treatment of the mass matrix by mass lumping.

b) A least-square approximation of the Hessian matrix. Let p be a vertex of the mesh Th and let B(p) be the ball of p , the set of mesh vertices connected to p . Without loss of generality, we suppose that Card(B(p ) ) = n and n 2 3; this is usually true in two dimensions for an arbitrary unstructured mesh. Considering a Taylor expansion of u* at a vertex pi E B ( p ) and truncating it a t the second order leads to the following relation:

This relation can be also developed as follows:

PPi = (;;) , V U * ( P ) = (;) , O”*(p) = ( y ) .

This leads to a usually over-determined system of the form:

A X = B , with X T = ( a b c )

where A is a (n x 3) matrix function of (xi, yi) and B is a vector of dimension n given by the right-hand side of the Relation (ZO), and function


of (a , /3, x i , yi, u*(p), u*(p i ) ) . This system can be solved using a least- square approximation. The latter consists in minimizing the distance between the vectors AX and B of W” by minimizing the square of the Euclidean norm of their difference, as follows:

”Find X E EX3, such that llAX - B1I2 = inf llAY - B)I2.”

It can be shown that the solution of this problem is the solution of the linear 3 x 3 system of normal equations [9]:

Y EP3

A ~ A X = A ~ B .

This final system is solved using a classical method, for instance a Gauss pivoting algorithm.

Remark 2.9. i) I n some peculiar cases, the system can be under- determined (if C a r d ( B ( p ) ) < 3). Additional vertices connected to the vertices of B(p) can be taken into account to increase the number of equations.

ii) this computation requires the knowledge of u* and of Vu* at the mesh vertices. To solve this problem, we consider that u* and U h coincide at the mesh vertices. To evaluate Vu*, several methods can be envisaged, for instance:

- a local L2 projection as described before, - the same approach but with truncating the Taylor expansion at the

first order and solving a over-determined linear system of two unknowns using a least-square approximation.

2.2.5

We now turn to a practical application of this concept of error estimate based on the interpolation error for numerical simulations in computational fluid mechanics (CFD).

Physical phenomena can involve large scale variations (e.g. multi- scale phenomena, recirculation, weak as well as strong shocks). It is thus difficult to capture the weakest phenomena, even using mesh adaptation. It is even harder to capture such phenomena when strong shocks are located in the flow, because they often ”hide” weak phenomena. However, capturing weak phenomena is crucial for abtaining an accurate solution and a good understanding of the behavior of the physical phenomenon. Such accuracy requires taking into account all phenomena interactions in the main flow area.

A local error estimate can overcome this problem. Following the ideas suggested by [6,26], Relation (16) is normalized using the local absolute value of the variable u:

An error estimate for CFD problems

264 Pascal Frey

where IuI, = max(lu/,cIIullm,n), with 6 < '1 a constant. The term ~llull,,n introduces a cutoff to avoid numerical problems.

On the other hand, in the context of anisotropic mesh adaptation for compressible flows, capturing weak phenomena by means of Relation (21) leads to mesh the strong shock regions with isotropic elements. This is mainly due to the discretization of the solution that introduces virtual oscillations in the direction parallel to the shock. These oscillations have a magnitude of the same order as weak phenomena. Various techniques have been proposed to overcome this problem (see [l, 271 for instance). Here, we advocate to filter the oscillations with the local gradient of the solution in order to preserve the anisotropy. To this end, we propose the following error estimate:

(22) where h is the diameter ( i e . , the length of the longest edge) of element K and 0 < y < 1 is a parameter that is considered close to or equal to zero if strong shocks are involved in the flow, 7 = 1 - y. To achieve isotropic adaptation, y is set to 1.

Remark 2.10. To obtain a metric that takes into account all variables, we use the intersection procedure described in the previous Section, in order to define a single metric that accounts for all. This is made possible because the previous error estimation provides a relative error measure, thus dimensionless variables.

3 Mesh adaptation using a geometric error estimate

Over the last few years, a large number of papers have been published presenting mesh adaptation techniques for numerical simulations based on finite element or finite volume methods. As mentioned previously, the error estimates involve the parameter h, the element size, in convergence and accuracy results. This advocates adapting the size of the mesh elements as suggested in the previous section in order to equidistribute the interpolation error. This approach is usually known as h-adaptation. However, other approaches suggest to adapt the approximation space and, more precisely, to increase the degree of the polynomials in the space 9. This approach is in turn known as padaptation.

In this section, we will attempt to explain how the theoretical concepts of metric tensor and geometric error estimate introduced in the previous sections can be used to create meshes adapted to the simulation at hand. Indeed, mesh adaptation allows to improve the numerical accuracy of the solution as well as the convergence rate.


3.1 Classical vs. adaptive Delaunay mesh generation

Since the pioneering work of Delaunay [14], Delaunay triangulations have been extensively studied, especially in computational geometry [4]. In- deed Delaunay triangulations are of great interest as they can serve to support efficient and versatile mesh generation methods [22].

3.1.1 The Delaunay triangulation

In Section 1, we have defined the notion of Delaunay triangulation of the convex hull of a set of points S c Rd and Theorem 1.1 gives a fundamental result about the existence and uniqueness of such a triangulation. The following lemma introduces a local property of Delaunay triangulations.

Lemma 3.1 (Delaunay lemma). Let T be an arbitrary triangulation of the convex hull of a set of points S c Rd in general position. If f o r each and every pair of adjacent simplices in T , the Delaunay criterion (the empty sphere criterion) holds, then this criterion holds globally and T is a Delaunay triangulation.

Proof. (following Lawson 1311) At first, we can notice that the criterion is symmetric. Let us consider a pair of adjacent simplices in EXd sharing a k-face of dimension d - 1 and let us denote p (resp. q ) the vertex opposite to this face in each simplex, and let us consider B ( p ) as the open ball circumscribed to the simplex containing p . Then, according to the assumption: q $l! B(p) H

p $! B ( q ) . Now, it is easy to note that if q $! B ( p ) then B(p) n Hq C B(q) n Hq and B(q) n H p c B(p) n H p , where H , represents the half-space containing

limited by the hyperplane support the common k-face. This allows us to conclude. Let suppose that the Delaunay criterion is true for each and any configuration of two adjacent simplices in the triangulation T and that a point x E T exists that is enclosed in the circumsphere of a simplex KO not adjacent to any of the simplices having z as vertex. We consider the point g barycenter of KO and the line gz that intersects an ordered set of simplices Ki, i = 1, n, according to the intersection point from g to z. We know that z $! B(K,-i) as K,-1 is adjacent to K,. Denoting by zi the vertex of Ki not common with Ki-1, we define Hi the half-space containing Ki and not containing Ki-1. By definition, x E H,. As z E Hi+1 and as B(zi) n Hi+1 c B(zi+l) n Hi+i then x E B(xi+l). The result follows: as z E B(zo), then z E B(z,-i) which is in contradiction with the assumption that z $l! B(zn-1). 0

What must be emphasized here is that a local property for any pair of two adjacent simplices leads to a global property for the whole triangulation. This gives a practical way of checking whether the triangulation T is a Delaunay triangulation or not.

266 Pascal Frey

Now, we provide an incremental yet practical method for constructing a Delaunay triangulation.

Theorem 3.1 (Delaunay kernel). Let Ti be the Delaunay triangulation of the convex hull of a set of point S c Rd and let p $ S be a vertex enclosed in Ti. The Delaunay kernel procedure can be written as:

Ti+l = Ti \ C ( P ) + B(P) 9 (23)

and provides Ti+l, a Delaunay triangulation of the convex hull of S U { p } .

Proof: (can be found in [20]) based on the duality with the VoronoY diagram. In this fundamental result, C(p) stands for the cavity of point p: the set of simplices in Ti such that their circumspheres contain point p and B ( p ) denotes the ball of point p: the set of simplices formed by joining p to the external faces of C(p) (Figure 7). Practically, this result ensures that the cavity C(p) is a star-shaped polytope with respect to point p .

Figure 7 Construction of the Delaunay triangulation using the Delau- nay kernel (bold lines indicate the external edges of C(P) in two dimensions).

3.1.2 Constrained triangulation

As mentioned previously, the Delaunay triangulation of a set of points S E Rd is purely a triangulation of the convex hull of the set S. However,


in numerical simulations, at least two specific problems arise, related to the necessity of

i) inserting specific entities in the triangulations (a given set of edges defining the domain boundary, for instance) and

ii) creating additional vertices in the triangulation that are not part of the initial set S. This is the case when the initial points all belong to the domain boundary, internal points must be inserted within the domain to enable numerical computations and provide information about the evolution of the solution within the domain.

Regarding the last requirement, the following result will provide the existence of a triangulation:

Lemma 3.2. Let Ti be an arbitrary triangulation and let p E E X d be a point enclosed in Ti, p not being a vertex of Ti. Then, a valid conforming triangulation Ti+1 having p as vertex can be created using the Delaunay kernel, Theorem 3.1.

Proof: this is simply the extension of Theorem 3.1 when the initial triangulation is not a Delaunay triangulation. Notice that Ti+l is not a Delaunay triangulation either. 0

Let consider a pair of adjacent triangles K1 = ( a , b, c) and K2 = ( a , d , b ) sharing a common edge ab. If the quadrilateral formed by the four vertices is convex, an alternate configuration of adjacent triangles K; = (a ,d , c ) and Ki = ( c , d , b) exists and is a valid conforming triangulation of the set of points. Going from (K1,Kz) to (K; ,K i ) is the purpose of the edge flip operator (Figure 8).

d d b

Figure 8 Edge flip operation in two dimensions.

Proposition 3.1. Given a set of non-intersecting edge segments in EX', it is always possible to generate a triangulation containing these segments as element edges. This triangulation can be obtained from any arbitrary triangulation by using a edge-flip operator in a iterative manner.

Proof: see [22] for a formal proof. 0

268 Pascal F'rey

3.1.3 Delaunay-based meshing algorithm

The previous results can be combined and used to develop a Delaunay- based mesh generation method. Let consider a closed bounded domain R c Rd and its boundary dR. We suppose here that there exists a piecewise afine discretization & of dR composed of non-intersecting k- faces of dimension d- 1. Here, the set S corresponds to the set of vertices of 1. The overall scheme of this method can be sketched as follows, it involves the successive steps:

1. construction of a triangulation of the bounding box of & (for instance composed of only 2 triangles in R2);

2. generation of a Delaunay triangulation of the points of S using the Delaunay kernel and enforcement of the possibly missing entities among the required entities of the set &, using Lemma 3.2;

3. creation and insertion of additional vertices pi in the triangulation using Theorem 3.1, resulting in the triangulation Th of the domain.

Beware that the resulting triangulation Th may not be a Delaunay triangulation as entites have been enforced using the edge flip operation. Although many numerical problem arise when implementing of the algorithm (especially in three dimensions), we will attempt to focuss here on a few issues that are of interest in the context of mesh adaptation for numerical simulations and especially on the creation of internal points. The next section will present a method for dealing with the discretization of dR for R c R3, a two-dimensional manifold embedded in R3.

3.1.4

Defining the cavity C ( p ) of a point p requires identifying all simplices having a circumsphere that contains point p . The radius of the sphere circumscribed to a simplex K corresponds to the distance between any vertex pi of K and the circumcenter 0 of K : r K = d(p i ,O) = IIOpiII. The circumcenter 0 can be computed as the solution of a linear system bf the form:

Creation and insertion of internal points

Hence, given a point p , an element K belongs to the cavity C ( p ) iif the following relation holds:


a ( p , K ) being called the Delaunay measure of point p with respect to the simplex K .

Inserting a point p in a given (Delaunay) triangulation Th requires at first to identify all simplices that belong to the cavity C ( p ) . Th' is can be achieved by finding the simplex or, in some peculiar cases, the set { K E Th , p E K } of all simplices containing p . Then, using adjacency relationships, it is possible to enumerate the set C(p) (see [20] for more details on this procedure). Although Theorem 3.1 ensures that the cavity C ( p ) is star-shaped polytope with respect to point p , this must be check and eventually enforced numerically.

The extension to the anisotropic case consists in introducing a metric tensor, hence a symmetric positive definite matrix Mp at each point p E R". This will allow us to consider the Euclidean norm of any vector in R" given the inner product (., . ) M . Hence, all distance checks involved in the Delaunay measure will be replaced by length checks according to the given matrix Mp, namely:

Obviously, taking into account only the metric tensor at the given point p is not sufficient and accurate in many applications. It is therefore highly recommended to consider all matrices related to the vertices of element K [23]. By doing so however, leads to solving a non-linear system of equations; overcoming this problem is tedious and beyond the scope of this paper. Let us simply mention the following result in two dimensions:

Proposition 3.2. The Delaunay measure, a ~ ~ ( p , K ) < 1, provides a valid anisotropic Delaunay kernel.

Proof: it consists in checking that the so-defined cavity is a star-shaped poly-

In mesh adaptation, the metric or the matrix M is usually defined at the vertices of the current mesh Th, on which the approximate solution uh as been computed. In this sense, a discrete metric field can be defined on Th using an a priori geometric error estimate as discussed in Section 2. One question left is related to the decision of creating and eventually inserting a new point p or removing an existing vertex q in the triangulation Th, in order to equidistribute the interpolation error. To answer this question in a practical manner, we advocate a naive algorithm based on the edge length analysis or, as these two notions are strictly equivalent, the computation of distances between vertices.

According to Section 2, the geometric error estimate provides a symmetric positive definite matrix M at each and any mesh vertex so as to equidistribute the interpolation error over the mesh elements, given a

gon with respect to point P (see [ 2 2 ] ) . 0

270 Pascal Frey

user-specified tolerance value. We have already stated that this problem is equivalent to constructing a unit mesh.

To this end, all mesh edges are analyzed and their lengths with respect to the metric tensor M are compared to one. If for a given edge, the length is much larger than one, then the edge is subdivided into sub- segments of unit length. On the other hand, short edges, having a length much smaller than one, are collapsed by merging the two endpoints into a single vertex. The result of this operation is to remove the edge and the set of its connected simplices, provided the resulting triangulation remains valid. From a practical point of view, point insertion and point deletion operations cannot be based on the sole check I labl l~ = 1 on the length of any edge ab, as it is too conservative, numerically speaking. Hence, we introduce the notion of a metric conforming edge:

Definition 3.1. Given a matrix M , a mesh edge ab is said to be matrix conforming iif:

Once a new point p E K has been created and inserted in a triangulation Th, the associated metric tensor Mp is obtained using an interpolation scheme based on the metric tensors of point p i E K (Section 1).

3.2 Surface mesh generation

This is by far the most tedious problem among all mesh generation problems. Remember that the classical mesh adaptation scheme described previously assumed given a triangulation & of the boundary do of a domain R c W d . The aim of this section is precisely to explain how to generate the triangulation & for a closed bounded domain R in R3. The boundary dR is then a two-dimensional manifold embedded in R3.

Among the many algorithms designed for surface mesh generation, we will focuss here on two issues: (i) the control of the geometric approximation for Cartesian surfaces and (ii) the adaptation of discrete piecewise linear surfaces.

3.2.1 Problem statement: the concept of geometric mesh

Given a differentiable function f : R -+ R, a linear approximation of f can be obtained by using Taylor formula for a point x close to a point a and by dropping second-order terms:


It represents a tangent line approximation as the right-hand side term matches the equation of the tangent line to the graph of f at point (a , f ( a ) ) . This result can be extended to Banach spaces ( i e . , complete normed vector spaces), for which we can write:

f ( x ) = f ( a ) + of (a)(. - a ) , where of(.) represents the Frkchet derivative o f f at point a.

In the context of surface meshing, it is usually sufficient to consider the notion of geometric continuity at the orders 0 and 1, regarding local Taylor expansions. Given a surface C, a surface triangulation Th is said to be U-order geometric if each element is close to the surface. This corresponds indeed to an approximation property. It will ensure that the gap between the piecewise affine approximation and the underlying surface is bounded by above. On the other hand, the 1-order geometric property is satisfied when each linear element is close to the tangent plane at each point of the surface. This smoothness property is related to the regularity of the surface approximation. Formally speaking, this property is locally related to the measure of the gap between the normal to the element and the normal at any point of the surface discretized by the element.

For the sake of clarity, let us simply remark that these axioms allow to answer the following questions: (i) how close to the surface C is the triangulation Th? and (ii) how regular (smooth) is the approximation Th? Additional concerns may arise when dealing with affine elements, mainly related to creating a minimal number of elements as well as creating well-shaped elements, in view of numerical simulations for instance.

Let R be a closed bounded domain of R2 and let consider the map- ping6 a : R + R, x = (u, v) H a(u, v). We have the following result:

Proposition 3.3 (O-order axiom). The interpolation error lla - Phall can be bounded by above as follows:

(24)

(25)

2 9 x E K 2

IIa - PhC(lco,K 6 - 1 (aa', D2fl(x)aa')I 7

IILT - Pha(Ico,K 6 - max max (% IP"(4Iv) , 9 x E K uCK

where Pha denotes the linear interpolate of LT on an element K E Th, a represents any vertex of element K and a' is the intersection point between the line a x and the edge of K opposite to vertex a.

6Actually, we shall consider here an immersion 0 : R -+ R3 and perform the relevant analysis using the covariant basis. We refer the reader to the chapter of this book dealing with differential geometry by P.G. Ciarlet and to his book [Ill for more details on this approach. We adopt here a less general point of view.

272 Pascal Frey

Proof: the sketch of the proof is based on classical Taylor expansions. We define the site a of point x E K as the element vertex closer to x than any other vertex of K . We develop the expression of u - Phu at site a of K , with respect to any point x = (u, v) E K , then (u - Phu)(a) is equal to:

(0 - PhO)(Z) + (xu, V(0 - PhU)(Z)) + (1 - t ) ( a x , D2+ + t x a ) a+. I’ We are looking for an upper bound on u - Phu, hence for an extremum of this function or a point x where the extremum is attained, ie., such that: V(u - Phu)(z) = 0. This leads to solving the following equation:

(v, V(O - P h u ) ( ~ ) ) = 0, VV C K

As the surface triangulation T h is supposed interpolant, (u - Phu)(a) = 0 and at point x, we have:

(u-Pha)(x) = - ( l - t ) ( a x , D 2 a ( z + t s a ) a x ) d t , (26) I’ I(. - PhfT)(Z)l = I /‘(I - t ) (ax ,D2u(x + t z a ) a x ) d t l . (27)

0

Let assume that x E K , x is not located along the element edges. We denote by a’ the intersection point between the line ax and the edge opposite to a in K . Then, it exists X E R such that ax = Xaa‘, thus:

I(u-Phu)(x)) = IS’(l - t )X2(aa’ ,D2u(z+txa )aa‘ )d t~ 0

2 3

As a is the site of x, X < - and we find then the following bounds:

and finally: 2 9 yEaa’

l(u - P~(T)(Z)~ < - max I(aa’, ~ ‘ u ( y ) aa’)~

or similarly:

2 9 y E K

I(. - ~hu)(x)I 6 - max I ( a a ’ , ~ ~ a ( y ) a a ’ ) ~ .

This result provides a global upper bound on the interpolation error on u at point z E K . Now, let us consider the L, norm of the interpolation error llu - Phull,,~, we have then:

A Differential Geometry Approach t o Mesh Generation 273

We are willing to introduce a metric tensor in the previous inequality, thus meaning that the Hessian matrix must be converted into a symmetric positive definite matrix. Being already symmetric, the Hessian matrix can be decomposed as follows:

D2a = R (2 :2) R-l ,

where R corresponds to the matrix of the eigenvectors and the di are the eigenvalue. Let us denote 1D2cr1 the symmetric positive definite matrix:

this allows to obtain the following inequality:

Indeed, looking at the eigenbasis of the matrix ID2al and denoting by (211, wz) the components of any two-dimensional vector v, the following relation holds:

l d l d + d2w221 6 Id1v11 + ldzv221,

I(w,D2a(w1,wz) w)l 6 (w, JD2a(w1,v2)l w).

from which we can conclude that:

Notice that in Relation (as), ID2a(y)I is a symmetric positive definite matrix, hence the right-hand side term can be written as follows:

Relation (29) involves an extremum value that is obviously not known a p r i - ori, thus making the evaluation of the interpolation error rather tedious. To overcome this problem, we introduce the following bound on the interpolation error:

IIa - P ~ ~ ( I ~ , K 6 - max max(w, I D ' ~ ( Y ) I w) . 2 9 Y E K v C K

Indeed, this last inequality is more coarse than Relation (29) as it does not involve points a and a' and hence is now independent of any point x E K . Obviously, the same conclusion holds when considering vertex b or vertex c of K in the analysis.

To be complete, it remains to consider the case where the extremum x is not attained within the element K but on its boundary, along an edge of K .

Considering Taylor expansions with an integral rest of a - PhU at both vertices a and b with respect to any arbitrary point x along the edge ab, it comes by summation:

2 74 Pascal Frey

where x1 = z + t xa and 2 2 = x + t xb. As z E [a, b] then 3X E [0,1] such that ax = Xab and bx = (1 - X)ba. Hence, following the same analysis, it is easy to show that lo - Pho)(x)I is bounded by above as:

1 4 y E A B

= - (A’ + (1 - A ) ~ ) max ~ ( a b , ~ ~ o ~ ( y ) a b ) l .

1 2 As the polynomial P(X) = X2 + (1 -

1 8 Y E A B

where ID20(y)I is defined as previously, Relation (28). Finally, this analysis leads to work out the following bound for the interpolation error:

has a maximum at A = -, we have:

I(U - PN)(~)I < - max(ab, I~’o(y)IabI),

1 8 Y E A B

l(o - ~ h o ) ( x ) ( ~ , K < - max(ab, l ~ ~ o ( y ) ( a b ) .

As f < $, we will then keep and retain Relation (29) as the upper bound of the interpolation error. 0

As stated before, the problem of generating a geometric mesh Th of a domain R consists in defining an optimal mesh which would be as smooth as possible. To this end, we suggest here two approaches to create an optimal mesh. The first, based on the local deformation, allows us to account for isotropic adaptation. The second, based on the intrinsic properties of the surface (i. e., the principal curvatures and directions) allows us to deal with anisotropic mesh generation.

3.2.2

The main idea consists in characterizing the deviation of the surface mesh Th in the vicinity of a vertex p with respect to the tangent plane Tp to the surface at this vertex. This deviation can be measured by considering the Hessian along the normal to the surface at this vertex, hence the second fundamental form of the surface.

Let us consider a vertex p as the origin of the local frame p = (O,O, a(0,O)). A Taylor expansion at the second order at a point (u, v) in the vicinity of vertex p can be written as follows:

Local deformation of a surface

1 2 a(u, v) = a(0, 0) + uuu + a v v + - (auuu2 + 2uuvuw + uvvw2) + o(u2 + 2) ,

where u, and a,, represent the first and second partial derivatives of u with respect to the variable u. Given n o the unit normal vector to the


surface at point p , then the distance E ( P ) between the point u, w, o(u, w) and Tp the tangent plane at point p can be expressed as follows:

&(PI = (720, (u,u, 4 % - (0, O , a ( O , O ) ) ) = (no1 (u, w I o ( % w) - 0(0 ,0) ) ) .

As no is orthogonal to both the vectors (1,0, a,) and (0,1, a,) and if we introduce the vector eg = (0, 0, l), we come easily to the relation:

which is proportional to the second fundamental form of the surface for u2 + w2 sufficiently small. Hence, we suggest the following definition.

Definition 3.2. The discrete local deformation of the surface at a vertex p an a mesh Th is defined as:

E ( P ) = max(np, P P ~ ) VPi E B(P) . (30)

This deformation measures the maximal distance between the vertices pi adjacent to p and the tangent plane Tp at vertex p .

Erom the given upper bound Relation (29), we can assume that the local deformation at each vertex p is proportional to the square of the element size in the vicinity of p . Hence, the optimal element in the mesh Th of dR must have a local size proportional to the inverse of the square root of ~ ( p ) :

where E represents the maximal desired tolerance value and k is the average size of the elements in the vicinity of p .

As pointed out, this measure is only suitable for isotropic mesh adaptation as Relation (30) corresponds to a scalar value. In order to have an anisotropic control of the mesh elements, we will now introduce another approach based on the intrinsic properties of the surface.

3.2.3

Let consider a vertex p = (u, w, a(u, w)) on an arbitrary surface triangulation T h and let p 1 , p2 denote the minimal and maximal radii of curvature at p and el , e2 the corresponding unit vectors along the principal directions of curvature.

As mentioned, the element size h ( p ) , defined in the tangent plane Tp, must be locally proportional to the minimal radius of curvature to guarantee that the local deviation remains bounded by above: h ( p ) = ap1, a E R. Let consider the local frame F = ( p , e l , e2) in Tpr then h ( p )

Local curvature of a surface

276 Pascal Frey

can be written as a linear combination of el and e2 in F hlel + h2e2.

Then, the size specification can be formulated as follows:

to prescribe a unit mesh. This equation corresponds to a circle of unit radius centered at point p in Tp. Projecting this circle onto the plane of 0, allow us to obtain the size specification at an arbitrary point x E Q. If 211 and 212 stand for the two orthogonal projections of el and e2, then the size specification in the usual frame (z, i, j ) is given by the equation:

where P = (w1 w2) and hi denotes the coordinates in (z ,z , j ) of the projection of h in the plane of 0. This equation defines an anisotropic metric tensor at point x E R.

One could also consider to define the metric tensor so as to obtain anisotropic triangles on the surface. In such a case, Relation (31) must be rewritten as:

with a and r ] two coefficients related to the minimal and maximal radii of curvature. This equation corresponds to an ellipse in the tangent plane Tp that fully encloses the circle defined by Equation (31).

3.2.4 Evaluation of the intrinsic properties of a discrete surface

The expressions obtained so far involve computing the principal curvatures and the principal directions of curvature. Relation (29) contains the Hessian of the function IJ that is supposed to be known analytically. However, when the surface is not analytically defined but is only known given a triangulation Th, the problem of estimating the second derivatives becomes accute. We are then looking for a numerical scheme to evaluate the intrinsic properties of this discrete surface. To this end, we will now briefly review some of the methods that have been proposed to deal with this numerical evaluation problem.


a) A local circle fitting. This approach is based on the Euler and Meusnier theorems for finding an estimation of the principal curvatures [7]. Indeed, let us first notice that we have the following equation for the normal curvature:

1 1 2 2 ~c., = -(KI + 1c.2) - - ( K I - 1c.~)(c0s28cos2a + sin28sin2a),

where 8 represent the angle between any arbitrary direction u in the tangent plane Tp at vertex p and the principal direction that corresponds to 1c.1 and a is the angle between the tangent direction corresponding to the normal curvature r;, and the direction u.

This equation can be reformulated by taking into account the equations of three circles passing through vertex p and two of its adjacent neighbors as:

r ; ,=a -bcos2a+cs in2a , where coefficients a , b and c can be determined by solving the linear system corresponding to the circle equations using a least square fit. Finally, the principal curvatures and directions are given by the following set of equations:

IQ = a f Jb2fc2 and 8 = 2

b) The Gauss-Bonnet relation. Let consider a vertex p and its ball B ( p ) . The Gauss-Bonnet theorem leads to the following equation for the gaussian curvature K, given a surface triangulation Th:

n-1

/Js, K d S = 27r - ai , i=O

where IS1 represents the sum of the areas of triangles Ki E B ( p ) , ai is the angle between two successive edges ppi, Vpi E B ( p ) and n = Card(B(p)). Assuming K is constant in a small neighborhood leads to write:

n-1 27r - c ai

K = 3 i=o s . c) A local quadric surface fitting. Similar to the evaluation of the Hessian matrix (Section 2.2.4), this approach is based on a least square approximation of a quadric surface. More precisely, a t each vertex p E Th, a unit normal vector np is computed as the average value of the adjacent face normals nKi, Ki E B(p):

, with n = Card(B(p))

2 78 Pascal Frey

following an idea suggested by Hamann 1251, the numerical scheme approximates a small neighborhood of the mesh around vertex p by an osculating paraboloid (the quadric fitting). Locally, the principal curvatures of the surface are supposed to coincide with the principal curvatures of the paraboloid. This C2 continuous quadric surface centered at vertex p is represented in the explicit form z = z(u, v). Vertex p is transformed along with its adjacent vertices to the origin p = (O,O, 0) of the local frame 3 = ( p , r 1 , r 2 , n p ) such that np coincide with the z axes, r1 and 7-2 being two orthogonal vectors of the tangent plane Tp. At a regular surface point ( x , y , z ) we can write the canonical equation of the surface:

f(z, y , Z) = ax2 + 2bzy + cy2 - z = 0 ,

where a, b, c are coefficients to be numerically computed as the solution of a linear system of equations.

Considering that all pi E B ( p ) belong to the quadric surface leads to writing a usually over-determined system of the form A X = B. Solving this system is equivalent to solving the minimization problem:

n

i= 1

which, in turn, leads to solving the linear system:

2x3y .zY2

2 x 3 y 4 x 2 y 2 2xy3 b = 2 x y z . (xr i2 2xy3 y4 ) (I) (::I)

The coefficients of the first and second fundamental form of the quadric surface are such that:

I ( v ) = E d u 2 + 2Fdudv + Gdv2 and II(w) = L d u 2 + 2 M d u d v + N d v 2 ,

where v = A71 +pr2 represents any vector of the tangent plane Tp. Given the parameterization of the paraboloid, is then easy to compute these coefficients and we found at p = (0,O):

E = 1 + ( 2 ( a u + bv))2 = 1, L = 2a, F = ~ ( U U + bv)(bu + CW) = 0 , G = 1 + (2(bu + CV))’ = 1,

M = 2b, N = 2c.

Finally, the Gaussian and the mean curvatures are computed as:

K = 4 a c - b 2 and H = a + c .

Computing ti1 < t i 2 the main curvatures requires finding the roots of the polynomials:

t i 2 - K,% + H = 0 .


Once the two principal curvatures and radii of curvature have been computed, Euler's theorem will give access to the principal directions of curvature [all.

Several heuristic methods have been proposed to provide information about the local neighborhood of a surface point p and, in particular, to help decide whether the point p is elliptic, parabolic or hyperbolic. We consider the previous approaches sufficiently accurate in the context of numerical simulations.

4 Applications

Finally, we would like to illustrate the potential of a mesh generation approach based on the differential geometry concepts we have introduced in the previous sections on various examples covering analytic (academic) as well as realistic applications.

4.1 An academic example As a first example, we consider a two-dimensional analytical example. We assume that at any point p of the two-dimensional parametric space Q = [-4,4] x [-4,4] is given a symmetric positive definite quadratic form represented by a 2 x 2 matrix Mp. This form allows us to prescribe the size and the stretching of the mesh elements tangentially to the algebraic curves of equations:

r : x3 - y2 + 2 - 3xZ = 0,

X being considered here as the parameter. At any point p E R, the anisotropic metric is represented by the matrix Mp defined as follows:

thus prescribing the sizes hl and h2 given by the set of equations:

min(0.2(X - 1)3 + 0.005, l), if X 2 1, + 0.05, l), if X < 1, h l ( X , Y ) = { min(0.2(1 -

min(O.z(X - q3 + 0.2, I), if X 2 1, h 2 ( X , Y ) = { min(0.2(1 - x ) ~ + 0.2, I), if x < 1,

and R is the rotation matrix of angle 6 given by the formula:

-2Y 3(x2 - A) ' 6 = arctan

280 Pascal Frey

Figures 9 and 10 present the initial and adapted meshes generated using an adaptation scheme based on the given metric. Although the quadratic form is defined analytically at each and any point of the domain R, we deliberately used it in a discrete way: the only data supplied to the mesh generation algorithm was the matrix M p defined at the mesh vertices. In other words, we have used an interpolation scheme (cf. Section 1.2.5) to compute the matrix at any vertex location when inserting a new vertex in the existing mesh. In this sense, the metric field can be considered as a discrete field. The only difference with a numerical simulation is that, in this case, the exact and numerical solutions u and U h coincide exactly at all mesh vertices.

Figure 9 Anisotropic mesh adaptation for an analytic metric specified on a two-dimensional domain R = [-4,4] x [-4,4]. From top to bottom and from left to right: adapted meshes at iterations 0 (initial), 1 and 6 and corresponding isovalue curves of the maximal eigenvalue of the matrix Mp.


Figure 10 Anisotropic mesh adaptation for an analytic metric specified on a two-dimensional domain R = [-4,4] x [-4,4]. Enlargements of the adapted mesh at the final iteration.

4.2 Curvature-driven evolution flows Now we will focuss on a physical dynamic problem in two dimensions. To this end, let us briefly introduce the basic concepts of level sets functions. To fix the ideas, let consider for instance a multi physics problem involving two fluids in parallel motion with two different velocities and densities, yielding an unstable interface for all speeds.

More precisely, we can consider the general motion of an interface between two media in the normal direction under an external velocity field F ( K ) related to the curvature of the interface. Following Sethian's work 1361, let suppose 70 = $3'0) is a starting curve in R2 and let C(t) ,O < t < T denotes a family of closed C2 curves embedded in W2 parameterized by y(z,t). Let assume that this family of curves is generated by moving 70 along its normal vector field with a speed F related to the local curvature: y(x, t ) = F(lc).n, n being the unit normal vector. We can embed the interface curve as the zero level set of a higher dimensional function u, thus let u(z, t = O ) , z E R" be defined by:

u(z,t = 0 ) = k d

where d represents the signed distance to the interface, the curve 70. Derivating this equation permits to find a partial differential equation from the evolution of u:

u~(z, t ) + Vu * y(z, t ) = 0 , u(z, 0) = yo given,

known as the level set equation. Introducing the Hamiltonian

H ( x , t , VU) = F ( K ) / V U /

282 Pascal Frey

leads to consider the so-called parabolic Hamilton-Jacobi equation:

U t ( Z , t ) + H(x1 t , V U ) = 0 (2 , t ) E RxIO, T[ . (33)

A correct definition of the solution of such equation for which we have existence] unicity and stability results, is given by the notion of viscosity solution. The standard theory of viscosity solution has been introducing by Crandall and Lions [12,32] for first order Hamilton-Jacobi equations. Let us briefly introduce this powerful concept.

Definition 4.1. A function u : Rx]O,T[-+ R is a viscosity solution of Equation (33) i f u is a sub- (resp. super-) solution: for all test function v E C1, at any point ( 2 0 , t o ) E Rx]O, T[ where u--2, has a local maximum (resp. min imum) then:

The following lemma shows that the definition of viscosity solution is consistent with that of classical solutions.

Lemma 4.1. A classical solution of Equation (33) is a viscosity solution, and a continuously diflerentiable viscosity solution is a classical solution.

For such equation, the weak solutions are in W1@ and the viscosity solution represents the physical (good) solution among the weak solutions.

In order to deal with topological changes to the dynamically evolving interface, the problem is to embed the interface of an open region R as the level set of a smooth (at least Lipschitz continuous) higher dimensional function u(xl t ) . The level set function u is closely related to the definition of an implicit function and enjoys the following properties:

u(z1 t ) > 0, vx $! n1 u(x, t ) < 0 , Qz E R, { u(2, t ) = 0, vx E 8 0 .

Describing the interface in this way makes possible the natural change of topology of the interface. The interface y(xl t ) is evolved in time by an external velocity field V(xl t ) according to the simple advection equation:

U t ( Z , t ) + v ' Vu(z, t ) = 0 .

Let consider here the case where the normal velocity is the mean curvature -6. Therefore] the level set equation for motion by mean curvature


is given by the following equation?:

when noticing that the normal and the mean curvature can be written in terms of u as follows:

n = - v u and K = div (5) .

lVul

However, this equation is stiff and thus tedious to solve numerically. The conventional manner of solving this type of equation would be using an implicit scheme, a t least for linear equations. Unfortunately as Equation (34) is non-linear it would not be straightforward to implement an implicit method. On the other hand, with explicit methods, small timesteps are required to maintain stability. To overcome these problems, a semi-implicit method for computing mean curvature flows which is stable for large time steps has been suggested by Smereka [37], among others.

In principle, the level function can be chosen as a Lipschitz continuous function. However, the so-called signed distance function is known to produce more robust numerical results, to improve mass conservation and to reduce errors in the computations of geometrical quantities such as the interface curvatures. But, partly because of numerical diffusion, the level set function u must be reinitialized to be the signed distance to the interface. This operation can be achieved by solving to steady state (for infinite fictious time) the equation [38]:

ut + sgn(uO)(lVul - 1) = 0 ,

where t represents the fictitious time. Depending on the numerical scheme used to solve Equation (33), this reinitialisation process must be carried out as often as each time step.

As the interface may change or deform in time, mesh adaptation seems a natural way of dealing with the accuracy problem as well as with capturing the interface shape. In other words, the mesh must be refined near the interface while coarsening in regions farther away. Moreover,

'In the classical setup, mean curvature flow is a geometric initial value problem. Starting from a smooth initial surface 70 E Rd, the solution y( t ) evolves in time so that at each point its normal velocity vector is equal to its mean curvature vector. For surfaces of co-dimension one, a completely different approach, represents the evolving surfaces as the level set of an auxiliary function solving an appropriate nonlinear differential equation. This level-set approach has been extensively analytically and numerically developed since the seminal works of Dervieux [15], Sethian [36] and Osher [33] .

284 Pascal Frey

the local mesh size is directly related to the local curvature of the zero levelset curve 7 0 . Thus, we impose a local size to be proportional to the radius of curvature at the interface and a smooth gradation while moving away from the interface. In order to achieve anisotropy, the size in the tangent direction is set as a coefficient times the minimal size in the normal direction. This allows to define a metric tensor at each point of the domain R, the eigenvectors being colinear to the normal and tangent vectors to the curve, respectively.

To illustrate the level set method, we consider a domain R = [0,1] x [0,1] and an analytical curve 7 0 as represented Figure 11, rotating under the divergence free velocity field: V ( x , y) = (-y, x) E W2. We solve the pseudo-advection problem using PI Lagrange finite elements and the method of characteristics that reduces the partial differential equation to an ordinary differential equation. On this Figure, it can be seen that two regions are refined corresponding to the initial tn and final position tnfl of the curve 70, at the beginning and at the end of the time stepping. This will ensure that the numerical solution is not lost or diffused when solving the advection problem; the solution at tn+' is computed on a refined grid with a maximal accuracy. Mesh refinement is achieved using the metric intersection procedure described Section 2, that takes both metrics into account for all points in the domain.

We provide another example of an evolution flow, on a domain R = [0,1] x [0,1] and a curve yo evolving under the divergence free velocity field: V ( x , y) = (y3, -x3) E R2 (Figure 12).

4.3 A CFD example in two dimensions

The next example is related to a Euler computation at Mach 3 in a scramjet configuration and can be considered as typical of numerical simulations in compressible fluids, involving highly anisotropic phenomena (shocks). Here, the aim is to capture the behavior of the physical phenomenon and to emphasize the reduction of the number of degrees of freedom obtained thanks to the anisotropy. The geometry of the computational domain is shown in Figure 13. Numerically, 9 adaptations have been performed, each 400 time steps of the Euler solver. The density variable has been chosen to adapt the meshes with the following parameters: E = 0.02, hmin = 0.01 m and h,,, = 2 m. The error estimate described Section 2 has been used to prescribe the anisotropic metric tensor at each mesh vertex.


I .

Figure 11 Curvature-based mesh generation for a levelset approach. Top: anisotropic mesh adapted to the local curvature for two consecutive time steps(1eft); levelset curves of the distance function at time t (right). Middle: zoom on the curvature adaptated meshes. Bottom: evolution of the levelset curve u = 0 for two consecutive time steps.

286 Pascal Prey

Figure 12 Curvature-driven mesh generation for a levelset approach.Left: anisotropic meshes adapted to the local curvature for two consec-utive time steps tn and tn+1; right: corresponding levelset curves of thedistance function at times tn and tn+1.


Figure 13 Computational Fluid Dynamics: Euler computation at Mach 3 in a scramjet configuration. Initial and adapted meshes at the iterations 2 and 9 and corresponding isodensity distributions.

4.4 Curvature-based surface mesh adaptation Let us consider an analytical solution corresponding to a Cartesian hypersurface in JR4 defined on the domain CJ = [-1, lI3 as:

f~(x, y, z ) = tanh ((x + 1.3)20 (y - 0.3)’~) .

The first case consists in adapting the mesh to the metric field prescribed by the geometric error estimate. Here again, the exact solution and the numerical solution coincide at the mesh vertices, allowing us to define a discrete metric field over the mesh. However, when introducing new mesh vertices, the related metric is obtained using the interpolation scheme described Section 2. The results, Figure 14, allow us to check the efficiency of Hessian based mesh adaptation.

In addition, a second test has been performed with the same analytically defined solution on a supertoroid geometry, Figure 15. The aim was to combine the curvature-based metric defined using a least-squares approximation of a quadric surface (Section 3.2.4) with the metric provided by the error estimate. The expected result is to obtain mesh refinement in highly curved regions as well as in regions where the gradient of the SO-

lution is important. Figure 15 clearly shows that the objective has been reached using the metric intersection procedure. More details as well as numerical evaluation of this approach can be found in Reference [2].

4.5

Finally, we would like to show an example of anisotropic mesh adaptation in three dimensions. This example concerns a classic numerical simulation of transonic air flow around the ONERA M6 wing. A Euler

Mesh adaptation in three dimensions

288 Pascal Frey

Figure 14 Example of anisotropic surface mesh adaptation. Top: initial isotropic mesh and final anisotropic mesh; bottom: corresponding isovalues of the analytical solution.

Figure 15 Another example of anisotropic surface mesh adaptation: initial geometry-based mesh and final anisotropic mesh.


solution is computed for Mach number equal to 0.8395 with an angle of attack of 3.06 degrees. This transonic simulation case gives raise to a well-known lambda-shock. The initial mesh is a relatively coarse mesh containing 7,815 vertices, 5,848 boundary triangles and 37,922 tetrahedra. The variable used to adapt the mesh is the Mach number. The mesh has been adapted 9 times, every 250 time steps. Figure 16 shows the anisotropic adaptation. The final mesh contains 23,516 and 132,676 tetrahedra. In this example, the maximal aspect ratio achieved for the anisotropic elements is about 10, thus revealing the difficulty of achieving strong anisotropy numerically. Reasons explaining this loss of stretching can be found in [27].

Figure 16 Onera M6 Wing test case for CFD computations. Initial and final anisotropic meshes (iterations 0 and 9) and cut through the volume mesh.

290 Pascal Frey

5 Conclusions

In this chapter, we have presented a differential geometry approach to mesh generation and mesh adaptation. This approach is based on the definition of a tensor metric at each vertex of a given triangulation Th in Rd. The symmetric positive definite matrix associated with a quadratic form is used to prescribe the element size as well as the anisotropic stretching of the mesh elements. In the context of numerical simulations, this matrix is defined based on a geometric error estimate related to the Hessian of the numerical solution. Several application examples have been described to emphasize the efficiency of this approach with regard to the numerical solution as well as to the computational cost. More details about mesh generation and mesh adaptation can be found in reference [20].

Acknowledgments

I am greatly indebted to Professors P.G. Ciarlet (dept. of Mathematics, City University of Hong Kong) and M. Jambu for their invitation in participating in the CIMPA school-ISFMA symposium on differential geometry: theory and applications, as well as to Professor Li Ta-Tsien (dept. of Mathematics, Fudan University, Shanghai) for providing a more than friendly and convenient environment to the lecturers. Over 50 participants of more than ten different countries have attended the summer school; they must all be thanked for their kindness and assiduity.

Other people shall also be mentioned here to which i am indebted, in particular J.H. Saiac (CNAM, Paris) and F. Alauzet (INRIA, Rocquen- court) for their valuable discussions and suggestions about the topics covered in this chapter, as well as V. Ducrot (LJLL, Paris) for providing me with the example of curvature-based meshes for levelsets methods.

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[34] 0. Pironneau, The finite element method for fluids, Wiley (1989). [35] P.A. Raviart and J.M. Thomas, Introduction h l'analyse nume'rique des

e'quations a m ddriue'es partielles, Masson, Paris (1983). [36] J.A. Sethian, Curvature and the Evolution of Fronts, Communications of

Mathematical Physics, 101:4 (1985). [37] P. Smereka, Semi-implicit level set method for curvature and surface dif-

fusion motion, Journal of Scientific Computing, 19: 1-3, 439-456 (2003). [38] M. Sussman, P. Smereka, and S. Osher, A level set approach for comput-

ing solutions to incompressible two-phase How, J. Comput. Phys., 114: 146159 (1994).

0 i f f e r e n t i a I G e 0 1 e t ry : Theory a n d A p p l i c a t i o n s

Series in Contemporary Applied Mathematics CAM 9

This book gives the basic notions of differential geometry, such as the metric tensor, the Riemann curvature tensor, the fundamental forms of a surface, covariant derivatives, and the fundamental theorem of surface theory in a self- contained and accessible manner. Although the field is often considered a “classical” one, it has recently been rejuvenated, thanks to the manifold applications where it plays an essential role.

The book presents some important applications to shells, such as the theory of linearly and nonlinearly elastic shells, the implementation of numerical methods for shells, and mesh generation in finite element methods.

This volume will be very useful to graduate stydents and researchers in pure and applied mathematics.

Higher Education Press www.hep.corn.cn

World Scientific www.worldscientific.com

6598 hc ISBN-I3 978-981-277-146-9

differential geometry, theory and applications (contemporary applied mathematics) (philippe g...

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