A new rotation-free isogeometric thin shell formulation and acorresponding continuity constraint for patch boundaries

Thang X. Duong, Farshad Roohbakhshan and Roger A. Sauer 1

Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH AachenUniversity, Templergraben 55, 52056 Aachen, Germany

Published2 in Computer Methods in Applied Mechanics and Engineering,DOI: 10.1016/j.cma.2016.04.008

Submitted on 29. December 2015, Revised on 7. April 2016, Accepted on 8. April 2016

Abstract: This paper presents a general non-linear computational formulation for rotation-freethin shells based on isogeometric finite elements. It is a displacement-based formulation thatadmits general material models. The formulation allows for a wide range of constitutive laws,including both shell models that are extracted from existing 3D continua using numerical inte-gration and those that are directly formulated in 2D manifold form, like the Koiter, Canham andHelfrich models. Further, a unified approach to enforce the G1-continuity between patches, fixthe angle between surface folds, enforce symmetry conditions and prescribe rotational Dirichletboundary conditions, is presented using penalty and Lagrange multiplier methods. The for-mulation is fully described in the natural curvilinear coordinate system of the finite elementdescription, which facilitates an efficient computational implementation. It contains existingisogeometric thin shell formulations as special cases. Several classical numerical benchmark ex-amples are considered to demonstrate the robustness and accuracy of the proposed formulation.The presented constitutive models, in particular the simple mixed Koiter model that does notrequire any thickness integration, show excellent performance, even for large deformations.

Keywords: Nonlinear shell theory, Kirchhoff–Love shells, rotation–free shells, Isogeometricanalysis, C1-continuity, nonlinear finite element methods

List of important symbols

1 identity tensor in R3

a determinant of matrix [aαβ]A determinant of matrix [Aαβ]aα co-variant tangent vectors of surface S at point x; α = 1, 2Aα co-variant tangent vectors of surface S0 at point X; α = 1, 2aα,β parametric derivative of aα w.r.t. ξβ

aα;β co-variant derivative of aα w.r.t. ξβ

aαβ co-variant metric tensor components of surface S at point xAαβ co-variant metric tensor components of surface S0 at point Xaαβγδ contra-variant components of the derivative of aαβ w.r.t. aγδb determinant of matrix [bαβ]B determinant of matrix [Bαβ]b curvature tensor of surface S at point x

1corresponding author, email: sauer@aices.rwth-aachen.de2This pdf is the personal version of an article whose final publication is available at www.sciencedirect.com

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b0 curvature tensor of surface S0 at point XB left Cauchy-Green tensor of the shell mid-surfacebαβ co-variant curvature tensor components of surface S at point xBαβ co-variant curvature tensor components of surface S0 at point X

bαβ contra-variant components of the adjugate tensor of bαβB matrix of the coefficients of the Bernstein polynomials for element Ωe

c bending stiffnesscαβγδ contra-variant components of the derivative of ταβ w.r.t. aγδC right Cauchy-Green tensor of the shell mid-surface

C right Cauchy-Green tensor of a 3D continuumC∗ right Cauchy-Green tensor of a shell layerCe Bezier extraction operator for finite element Ωe

d shell director vectordαβγδ contra-variant components of the derivative of ταβ w.r.t. bγδδ... variation of ...ε penalty parameterE Green-Lagrange strain tensor of the shell mid-surface

eαβγδ contra-variant components of the derivative of Mαβ0 w.r.t. aγδ

f prescribed surface loadsfα in-plane components of f

fαβγδ contra-variant components of the derivative of Mαβ0 w.r.t. bγδ

f e• discretized finite element force vectorF deformation gradient of the shell mid-surface

F deformation gradient of a 3D continuumg determinant of matrix [gαβ]G determinant of matrix [Gαβ]gα, g3 current tangent and normal vectors of a shell layer; α = 1, 2Gα, G3 reference tangent vectors and normal of a shell layer; α = 1, 2gc G1-continuity and symmetry constraintsgαβ co-variant components of the metric tensor of S∗Gαβ co-variant components of the metric tensor of S∗0Gext external virtual workGint internal virtual workΓγαβ Christoffel symbols of the second kind

H mean curvature of surface S at xH0 mean curvature of surface S0 at Xi identity tensor in SI identity tensor in S0

I1 first invariant of C

I1 first invariant of CJ surface area change

J volume change of a 3D continuumJa Jacobian of the mapping P → SJA Jacobian of the mapping P → S0

ke• finite element tangent matricesκ Gaussian curvature of surface S at xκ pull-back of the curvature tensor bK relative curvature tensorK surface bulk modulus

K bulk modulus of 3D continua

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L patch boundary on which edge rotation conditions are prescribedλi principal stretches of surface S at xµ surface shear modulusµ shear modulus of 3D continuaµ0 moment tensor in the reference configurationµ moment tensor in the current configuration

Mαβ, Mαβ0 contra-variant bending moment components

n, n surface normals of S at x

N , N surface normals of S0 at XN array of NURBS-based shape functions

Ne array of B-spline basis functions in terms of the Bernstein polynomials

N eA B-spline basis function of the Ath control point; A = 1, ..., n

Nαβ total, contra-variant components of σν unit normal on ∂Sξ out-of-plane coordinateξα in-plane coordinates; α = 1, 2p external pressureP parametric domain spanned by ξ1 and ξ2

P shell material pointΠL potential of the constraint used to enforce edge rotation conditionsϕ deformation map of surface Sq Lagrange multiplier for the continuity constraintρ areal density of surface SS current configuration of the shell surfaceS0 reference configuration of the shell surfaceS∗ current configuration of a shell layerS∗0 reference configuration of a shell layer∂S boundary of SS second Piola-Kirchhoff stress tensor of the shell

S second Piola-Kirchhoff stress tensor of a 3D continuumSα contra-variant, out-of-plane shear stress componentsσ Cauchy stress tensor of the shellσ Cauchy stress tensor of a 3D continuumσαβ stretch related, contra-variant components of σt current shell thicknessT reference shell thicknessT traction acting on a cut ∂S normal to νT α traction acting on a cut ∂S normal to aα

τ unit direction along a surface boundaryτ Kirchhoff stress tensor of a 3D continuumταβ contra-variant components of the Kirchhoff stress tensor of the shellταβ in-plane components of τv velocity, i.e. the material time derivative of xV space for admissible variations δxwA NURBS weight of the Ath control point (= FE node); A = 1, ..., nW strain energy density function per reference area

W strain energy density function per reference volumex current position of a surface point on SX initial position of x on the reference surface S0

x current position of a material point of a 3D continuum

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X reference position of a material point of a 3D continuumxe array of all nodal positions for finite element Ωe

Xe array of all nodal positions for finite element Ωe0

Ωe current configuration of finite element eΩe

0 reference configuration of finite element e

1 Introduction

This work presents a new rotation-free isogeometric finite element formulation for general shellstructures. The focus here is on solids, even though the formulation generally also applies toliquid shells. The formulation is based on Sauer and Duong (2015), who provide a theoreticalframework for Kirchhoff–Love shells under large deformations and nonlinear material behaviorsuitable for both solid and liquid shells.

From the computational point of view, among the existing shell theories, the rotation-freeKirchhoff–Love shell theory is attractive since it only requires displacement degrees of freedomin order to describe the shell behavior. The necessity of at least C1-continuity across shellelements and their boundaries is the main reason why this formulation is not widely usedin practical finite element analysis. Although various efforts have been made for imposingC1-continuity on Lagrange elements (see e.g. Onate and Zarate (2000); Brunet and Sabourin(2006); Stolarski et al. (2013); Munglani et al. (2015) and references therein), the proposedcomputational formulations are usually either expensive or difficult to implement. Due to thiscost and complexity, finite shell elements derived from Reissner–Mindlin theory, which requireonly C0-continuity but need additional rotational degrees of freedom, are more widely used(Simo and Fox, 1989; Simo et al., 1990; Bischoff and Ramm, 1997; Yang et al., 2000; Bischoffet al., 2004; Wriggers, 2008). It is worth noting that there are some other formulations thatare different from the above prevailing approaches, like extended rotation-free shells includingtransverse shear effects (Zarate and Onate, 2012), rotation-free thin shells with subdivisionfinite elements (Cirak et al., 2000; Cirak and Ortiz, 2001; Green and Turkiyyah, 2005; Cirakand Long, 2010), meshfree Kirchhoff–Love shells (Ivannikov et al., 2014) and discontinuousGalerkin method for Kirchhoff–Love shells (Noels and Radovitzky, 2008; Becker et al., 2011).

Isogeometric analysis (IGA), initially introduced by Hughes et al. (2005), has become a promis-ing tool for the computational modeling of shells. For instance, Benson et al. (2010); Thaiet al. (2012); Dornisch et al. (2013); Dornisch and Klinkel (2014); Kang and Youn (2015); Leiet al. (2015a) study various Reissner–Mindlin shells with isogeometric analysis. Uhm and Youn(2009) introduce a Reissner–Mindlin shell described by T-splines. The hierarchic family of isoge-ometric shell elements presented by Echter et al. (2013) includes 3-parameter (Kirchhoff–Love),5-parameter (Reissner–Mindlin) and 7-parameter (three-dimensional shell) models. Solid-shellelements based on isogeometric NURBS are investigated by Bouclier et al. (2013a,b); Hosseiniet al. (2013, 2014); Bouclier et al. (2015); Du et al. (2015). The shell formulation of Bensonet al. (2013) blends both Kirchhoff–Love and Reissner–Mindlin theories.

Particularly for rotation-free thin shells, which are the focus of this research, isogeometric anal-ysis is a great help. This is due to the fact that the IGA discretization can provide smoothnessof any order across elements, allowing an efficient yet accurate surface description, which issuitable for thin shell structures. The first work on combining IGA with Kirchhoff–Love the-ory is presented by Kiendl et al. (2009). Later, Kiendl et al. (2010) use the bending stripmethod to impose the C1-continuity of Kirchhoff–Love shell structures comprised of multiplepatches. Nguyen-Thanh et al. (2011) then propose PHT-splines for rotation-free shells. Theapproach is tested for a linear shell formulation and it demonstrates the advantage of providing

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C1-continuity for thin shells. In the rotation-free shell formulation suggested by Benson et al.(2011), the Kirchhoff–Love assumptions are satisfied only at discrete points, so that the requiredcontinuity can be lower. Nagy et al. (2013) propose an isogeometric design framework for com-posite Kirchhoff–Love shells with anisotropic material behavior. Goyal et al. (2013) investigatethe dynamics of Kirchhoff–Love shells discretized by NURBS. Nguyen-Thanh et al. (2015) pro-pose an extended isogeometric element formulation for the analysis of through-the-thicknesscracks in thin shell structures based on Kirchhoff–Love theory. Deng et al. (2015) suggest arotation-free shell formulation equipped with a damage model. For thin biological membranes,a thin shell formulation is developed by Tepole et al. (2015). Riffnaller-Schiefer et al. (2016)present a discretization of Kirchhoff–Love thin shells based on a subdivision algorithm. WeakDirichlet boundary conditions of isogeometric rotation-free thin shells are considered by Guoand Ruess (2015b). Lei et al. (2015b) introduce a penalty and a static condensation method toenforce the C0/G1-continuity for NURBS-based meshes with multiple patches. Recently, isogeo-metric collocation methods have been also introduced for Kirchhoff–Love and Reissner–Mindlinplates as an alternative for isogeometric Galerkin approaches (Kiendl et al., 2015a; Reali andGomez, 2015).

Recently, Kiendl et al. (2015b) extended the proposed formulation of Kiendl et al. (2009) to non-linear material models. By using numerical integration over the thickness of 3D continua, theextended formulation admits arbitrary nonlinear material models. However, the formulation ofKiendl et al. (2015b) leads to special relations for the membrane stresses and bending moments.In general, such a (fixed) relation is rather restricted. An example is cell membranes composedof lipid bilayers (see e.g. Sauer and Duong (2015)). In this case, the material behaves like afluid in the in-plane direction, i.e. without elastic resistance, while in the out-of-plane directionthe material behaves like a solid with elastic resistance. Hence, the membrane and bendingresponse may range from fully decoupled to very complicated relations. Additionally, for fluidmaterials, it is desired to include other conditions such as area incompressibility and stabilizationtechniques. Therefore, a further extension of the formulation of Kiendl et al. (2015b) is needed.

Besides, in computation of thin shells, it is beneficial to accommodate material models thatare directly constructed in surface strain energy form, like the Koiter model (Ciarlet, 2005),Canham model (Canham, 1970) or Helfrich model (Helfrich, 1973). In these models, contraryto some of the approaches mentioned above, no numerical integration is required such that thecomputational time reduces drastically.

In this paper, we develop a general nonlinear IGA thin shell formulation. The surface formula-tion, presented in Sec. 2.6, admits any (non)linear material law with arbitrary relation betweenbending and membrane behavior, while the models of Kiendl et al. (2009, 2010); Nguyen-Thanhet al. (2011); Echter et al. (2013); Guo and Ruess (2015a); Lei et al. (2015a) are based on lin-ear 2D stress-strain relationships. The 3D formulation, presented in Sec. 3, can be reducedto Benson et al. (2011); Kiendl et al. (2015b) as special cases. However, the model of Bensonet al. (2011) is restricted to single patch shells and the continuity constraint of Kiendl et al.(2015b) is different than our constraint. Further, our approach to model symmetry and clampedboundaries is distinct from all the existing rotation-free IGA shell formulations. In summary,our formulation contains the following new items:

• The bending and membrane response can be flexibly defined at the constitutive level.• It admits both constitutive laws obtained by numerical integration over the thickness of

3D material models and those constructed directly in surface energy form.• It can thus be used for both solid and liquid shells.• It is fully described in curvilinear coordinates, which includes the constitutive laws, FE

weak form and corresponding FE matrices.

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• It includes a consistent treatment for the application of boundary moments.• It includes an efficient finite element implementation of the formulation.• It includes a unified treatment of edge rotation conditions such as the G1-continuity be-

tween patches, symmetry conditions and rotational boundary conditions.

The remaining part of this paper is organized as follows: Sec. 2 summarizes the theory ofrotation-free thin shells, including the kinematics, balance laws, strong and weak forms ofthe governing equations, as well as remarks on constitutive laws. In Sec. 3, we present aconcise and systematic procedure to extract shell constitutive relations from existing 3D materialmodels. Sec. 4 discusses the finite element discretization as well as the treatment of symmetry,surface folds and G1-continuity constraints for multi-patch NURBS. Several linear and nonlinearbenchmark tests are presented in Sec. 5 to illustrate the capabilities of the new model. Theseexamples consider solid shells. Liquid shells will be presented in future work (Sauer et al., 2016).Sec. 6 concludes the paper.

2 Summary of rotation-free thin shell theory

This section summarizes the theory of nonlinear shells in the unified framework presented inSauer and Duong (2015) and references therein. Here, the shells are treated mathematically as2D manifolds. Later, in Sec. 3, the kinematics and constitutive formulations are defined in a 3Dsetting considering different shell layers through the thickness. These two different approachesare schematically illustrated in Fig. 1.

2.1 Thin shell kinematics and deformation

Figure 1: Mapping between parameter domain P, reference surface S0 and current surface Sof a Kirchhoff–Love shell. The boundaries of the physical shell are shown by solid red lines. Ashell layer is denoted by S∗ and S∗0 in the current and reference configuration, respectively, andis shown by dashed blue lines.

As shown in Fig. 1, let the mid-surface S of a thin shell be described by a general mapping

x = x(ξα) , α = 1, 2, (1)

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where x denotes the surface position in 3D space and ξα are curvilinear coordinates embeddedin the surface. They can be associated with a parameter domain P. Differential geometry allowsus to determine the co-variant tangent vectors aα = ∂x/∂ξα on S, their dual vectors aα definedby aα · aβ = δαβ , and the surface normal n = (a1 × a2)/||a1 × a2||, so that the metric tensor

with co-variant components aαβ = aα ·aβ and contra-variant components [aαβ] = [aαβ]−1 of thefirst fundamental form are defined. The relation between area elements on S and the parameterdomain is da = Ja dξ1 dξ2 with Ja =

√det aαβ.

The bases a1, a2, n and a1, a2, n define the usual identity tensor 1 in R3 as 1 = i+n⊗nwith i = aα ⊗ aα = aα ⊗ aα. The components of the curvature tensor b = bαβ a

α ⊗ aβ aregiven by the Gauss–Weingarten equation

bαβ = n · aα,β = −n,β · aα , (2)

where a comma denotes the parametric derivative aα,β = ∂aα/∂ξβ. With the definition of the

Christoffel symbol Γγαβ = aγ · aα,β, the so-called co-variant derivative is defined as aα;β :=

aα,β − Γγαβ aγ = (n ⊗ n)aα,β. The mean and Gaussian curvature of S can be computed fromthe first and second invariants of the curvature tensor b, respectively, as

H :=1

2tr b =

1

2bαα =

1

2aαβ bαβ , (3)

and

κ := det b =b

a, (4)

wherea = det[aαβ] , b = det[bαβ] . (5)

The variations of the above quantities such as δaαβ, δaαβ, δbαβ, δbαβ, δH and δκ can be founde.g. in Sauer and Duong (2015).

To characterize the deformation of S due to loading, the initially undeformed configuration ischosen as a reference configuration and is denoted by S0. It is described by the mapping X =X(ξα), from which we have Aα = ∂X/∂ξα, Aαβ = Aα ·Aβ, [Aαβ] = [Aαβ]−1, Aα = AαβAβ,N = (A1 × A2)/||A1 × A2||, the initial identity tensor I := Aα ⊗ Aα = Aα ⊗ Aα, where1 = I +N ⊗N , and the initial curvature tensor b0 := BαβA

α ⊗Aβ.

Having the definition of aαβ and bαβ, the deformation map x = ϕ(X) is fully characterized bythe two following quantities:1. The surface deformation gradient, which is defined as

F := aα ⊗Aα . (6)

Accordingly, the right Cauchy-Green surface tensor, C = F TF = aαβAα ⊗ Aβ, its inverse

C−1 = aαβAα ⊗Aβ, and the left Cauchy-Green surface tensor, B = FF T = Aαβ aα ⊗ aβ, aredefined. They have the two invariants I1 = C : I = B : i = Aαβ aαβ and J2 = detC = detB.Therefore, similar to the volumetric case, the surface Green-Lagrange strain tensor can bedefined as

E = EαβAα ⊗Aβ :=

1

2(C − I) =

1

2(aαβ −Aαβ)Aα ⊗Aβ , (7)

which represents the change of the metric tensor due to the surface deformation.2. The symmetric relative curvature tensor, which is defined as

K = KαβAα ⊗Aβ := κ− b0 = (bαβ −Bαβ)Aα ⊗Aβ, (8)

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with

κ := F T bF = bαβAα ⊗Aβ , (9)

which furnishes the change of the curvature tensor field, following the terminology of Steigmann(1999b).

2.2 Stress and moment tensors

In order to define stresses at a material point x of the shell surface S, the shell is cut intotwo at x by the parametrized curve C(s).3 Further, let τ := ∂x/∂s be the unit tangent andν := τ × n be the unit normal of C at x. Then, one can define the traction vector T = T α ναand the moment vector M = Mα να on each side of C(s) at x, where να = ν · aα. Accordingto Cauchy’s theorem, these vectors can be linearly mapped to the normal ν by second ordertensors as

T := σTν ,

M := µT ν ,(10)

whereσ := Nαβ aα ⊗ aβ + Sα aα ⊗ n ,

µ := −Mαβ aα ⊗ aβ(11)

are the Cauchy stress and moment tensors, respectively.

It should be noted that the Cauchy stress σ in Eq. (11.1) is generally not symmetric. Fur-thermore, as shown e.g. in Sauer and Duong (2015), a suitable work conjugation (per referencearea) arising in the presented theory is the quartet

Wint := ταβ aαβ/2 +Mαβ0 bαβ , (12)

where Wint is the local power density and we have defined

ταβ := Jσαβ ,

Mαβ0 := JMαβ .

(13)

Further,

σαβ := Nαβ − bβγMγα (14)

are the components of the membrane stress responding to the change of the metric tensor.4

We also note that the mathematical quantities introduced in Eqs. (10) and (11) stem from theCosserat theory for thin shells (Steigmann, 1999b). As shown e.g. by Steigmann (1999a) andSauer and Duong (2015), these quantities can be related to the effective traction t and physicalmoment m transmitted across C(s). Namely for m, the relation is given by

m := n×M = mν ν +mτ τ , (15)

wheremν := Mαβ να τβ ,

mτ := −Mαβ να νβ ,(16)

are the normal and tangential bending moment components along C(s). For the effective tractiont, one can show that (Steigmann, 1999b; Sauer and Duong, 2015)

t := T − (mν n)′ , (17)

where (•)′ denotes the derivative with respect to s.

3s denotes the arc length such that ds = ‖dx‖.4Note that σαβ 6= aα · σ aβ , instead aα · σ aβ = Nαβ .

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2.3 Balance laws

To derive the governing equations, the surface S is subjected to the prescribed body forcef = fα aα + pn on S and the boundary conditions

u = u on ∂uS ,n = n on ∂nS ,t = t on ∂tS ,

mτ = mτ on ∂mS ,

(18)

where u is a prescribed displacement; n is a prescribed rotation, t := tα aα is a prescribedboundary traction and mτ is a prescribed bending moment. The equilibrium of the shell is thengoverned by the balance of linear momentum together with mass conservation, which gives

T α;α + f = ρ v ∀x ∈ S , (19)

while the balance of angular momentum leads to the condition that σαβ, defined in Eq. (14), issymmetric and the shear stress is related to the bending moment via

Sα = −Mβα;β . (20)

We note that Mαβ is also symmetric but Nαβ is generally not symmetric.

2.4 Weak form

The weak form of Eq. (19) can be obtained by contracting Eq. (19) with the admissible variationδx ∈ V and integrating over S, (see Sauer and Duong (2015)). This results in

Gin +Gint −Gext = 0 ∀ δx ∈ V , (21)

where

Gin =

∫S0δx · ρ0 v dA ,

Gint =

∫S0

1

2δaαβ τ

αβ dA+

∫S0δbαβM

αβ0 dA =

∫S0δEαβ τ

αβ dA+

∫S0δKαβM

αβ0 dA ,

Gext =

∫Sδx · f da+

∫∂tS

δx · t ds+

∫∂mS

δn ·mτ ν ds+ [δx ·mν n].

(22)

Remark 1: Here, the last term in Gext is the virtual work of the point load mν n at cornersof ∂mS (in case mν 6= 0), and the second last term in Gext denotes the virtual work of themoment mτ ν. It is worth noting that this is the consistent way to apply bending moments onthe boundary of Kirchhoff–Love shells.

2.5 Linearization of the weak form

For solving the nonlinear equation (21) by the Newton–Raphson method, its linearization isneeded. For the kinetic virtual work, one gets

∆Gin =

∫S0δx · ρ0 ∆v dA , (23)

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in which ∆v depends on the time integration scheme used. For dead loading of f , t and mτ ν,one finds5

∆Gext =

∫∂Smτ δaα ·

(νβ n⊗ aα + να aβ ⊗ n

)∆aβ ds (24)

and for the internal virtual work term

∆Gint =

∫S0

(cαβγδ 1

2δaαβ12∆aγδ + dαβγδ 1

2δaαβ ∆bγδ + ταβ 12∆δaαβ +

eαβγδ δbαβ12∆aγδ + fαβγδ δbαβ ∆bγδ + Mαβ

0 ∆δbαβ

)dA ,

(25)

where the material tangent matrices are defined as

cαβγδ := 2∂ταβ

∂aγδ, dαβγδ :=

∂ταβ

∂bγδ,

eαβγδ := 2∂Mαβ

0

∂aγδ, fαβγδ :=

∂Mαβ0

∂bγδ.

(26)

They are given in Sauer and Duong (2015) for various material models. Here it is noted thatdαβγδ = eγδαβ posses minor symmetries, while cαβγδ and fαβγδ posses both minor and majorsymmetries.

2.6 Constitution

In this framework, σαβ and Mαβ are determined from constitutive relations for stretching andbending. For hyperelastic shells, we assume there exists a stored energy function in the form

W = W (E,K), (27)

where E and K are defined in Eqs. (7) and (8). The Koiter model (see Eq. (35)) is a simpleexample of this form. It is easily seen that W can be equivalently expressed as a function of Cand κ, or as a function of aαβ and bαβ, since Aα is constant. If the material has symmetries,e.g. isotropy, the stored energy function can also be expressed as a function of the invariantsI1, J , H and κ, defined in Sec. 2.1 (see e.g. Steigmann (1999a)). Thus, the following functionalforms are equivalent to Eq. (27).

W = W (C,κ) = W (aαβ, bαβ) = W (I1, J,H, κ) . (28)

Given the stored energy function W , the common argument by Coleman and Noll (1964) leadsto the constitutive equations (see Sauer and Duong (2015))

ταβ = 2∂W

∂aαβ,

Mαβ0 =

∂W

∂bαβ.

(29)

Remark 2: The constitutive equations (29) are defined in curvilinear coordinates. Therefore,they either give the components of the classical second Piola-Kirchhoff stress and moment

S :=∂W (E,K)

∂E= 2

∂W

∂aαβAα ⊗Aβ =: ταβAα ⊗Aβ ,

µ0 := −∂W (E,K)

∂K= − ∂W

∂bαβAα ⊗Aβ =: −Mαβ

0 Aα ⊗Aβ,

(30)

5For live mτ ν, see Eqs. (127) and (128).

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of the surface manifold. Alternatively, they give the components of the Kirchhoff stress andmoment defined by pushing forward S and µ0 as

F S F T := ταβ aα ⊗ aβ ,

F µ0 FT := −Mαβ

0 aα ⊗ aβ = J µ.(31)

Here, we note that ∂(•)∂X in Eq. (30) denotes the derivative of a scalar-valued function by an

arbitrary second order tensor X (see e.g. Itskov (2009)).

Remark 3: In this framework, the stress σαβ and moment Mαβ are both constitutively de-termined from a surface stored energy function W . The advantage of this setup is that it canaccept general constitutive relations, which implies that the relation between the membrane andbending response can be flexibly realized at the constitutive level and it is not restricted withinthe formulation. The stretching and bending response may range from fully decoupled, like inthe Koiter material model (Eq. (35)), to a coupled relation, such as in the Helfrich materialmodel (see e.g. Sauer and Duong (2015)). Due to this flexibility, the presented formulation issuitable for both solid and liquid shells and is also convenient for adding kinematic constraints(e.g. area constraint) or stabilization potentials (Sauer et al., 2016).

Remark 4: In the presented formulation, we note that the definition of the shell thickness isnot needed for solving Eq. (21), which is often referred to as a “zero-thickness” formulation.Consequently, the stress and moment in Eq. (29) can be computed without defining a thick-ness. However, this does not imply that thickness effects have been neglected or approximated.Instead, they are in some sense hidden in the constitutive law of Eq. (27). If desired, they canbe determined as noted in Remark 5. But this connection is not a requirement as it is in thecase of constitutive laws derived from 3D material models (see Sec. 3.4).

In the following, we consider some example constitutive models suitable for solid shells todemonstrate the flexibility of the formulation. The application of the formulation to liquidshells is considered in future work (Sauer et al., 2016).

2.6.1 Initially planar shells

For initially planar shells, one can consider the model of Canham (1970) for the bending con-tribution, while for the membrane contribution, the stretching response can be modeled with anonlinear Neo-Hookean law. This gives

W =Λ

4(J2 − 1− 2 lnJ) +

µ

2(I1 − 2− 2 lnJ) + c J

(2H2 − κ

), (32)

where c, Λ and µ are 2D material constants. From Eqs. (29) and (32), we find the stress

ταβ =Λ

2

(J2 − 1

)aαβ + µ

(Aαβ − aαβ

)+ c J (2H2 + κ) aαβ − 4c J H bαβ , (33)

and the moment

Mαβ0 = c J bαβ , (34)

which is linear w.r.t. the curvature. The tangents for this are given in Sauer and Duong (2015).We note that since Eq. (32) is given in surface strain energy form, the stress and moment canbe computed directly from Eqs. (33) and (34) without needing any thickness integration.

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2.6.2 Initially curved solid shell

For initially curved shells, the surface strain energy model proposed by Koiter (Ciarlet, 2005;Steigmann, 2013) can be considered. It is defined in tensor notation as

W (E,K) =1

2E : C : E +

1

2K : F : K , (35)

with the constant fourth order tensors

C = Λ I I + 2µ (I ⊗ I)s ,

F =T 2

12C .

(36)

Since I I : X = (trX) I and (I ⊗ I)s : X = sym(X), with X being an arbitrary secondorder tensor, it follows from Eqs. (35) and (36) that

ταβ = Λ trEAαβ + 2µEαβ ,

Mαβ0 =

T 2

12(Λ trK Aαβ + 2µKαβ),

(37)

where Kαβ := Aαγ Kγδ Aδβ, Eαβ := Aαγ Eγδ A

δβ, and

trK = Kαβ Aαβ , trE = Eαβ Aαβ . (38)

From Eq. (37), we further find dαβγδ = eαβγδ = 0 and

cαβγδ = ΛAαβ Aγδ + µ(Aαγ Aβδ +Aαδ Aβγ

),

fαβγδ =T 2

12cαβγδ .

(39)

Remark 5: The set of parameters Λ and µ can be determined in different ways. Firstly, theycan be determined directly from experiments, i.e. without explicitly considering the thickness.Secondly, they may be obtained by analytical integration over the thickness of the simple 3DSaint Venant–Kirchhoff model (see e.g. Ciarlet (2005) and Sec. 3.4). In this case, µ and Λ aregiven by

Λ := T2 Λ µ

Λ + 2 µ, µ := T µ, (40)

where Λ and µ are the classical 3D Lame constants in linear elasticity. Thirdly, they can bedetermined by numerical integration over the thickness of a general 3D material model.

Remark 6: Note that, in Eq. (35), the stretching and the bending behavior can be specifiedseparately. For instance, the first term can be replaced by a nonlinear Neo-Hooke model (seeEq. (32)), i.e.

W =Λ

4(J2 − 1− 2 lnJ) +

µ

2(I1 − 2− 2 lnJ) +

1

2K : F : K , (41)

in order to capture large membrane strains. In this case ταβ follows from the front part ofEq. (33), while Mαβ

0 is given by Eq. (37.2).

12

3 Shell constitution derived from 3D constitutive laws

Provided the displacement across the shell thickness complies with the Kirchhoff–Love assump-tion, a constitutive relation W = W (E,K) for shells can also be extracted, without approxi-mation, from classical three-dimensional constitutive models of the form W = W (C), where Cis the right Cauchy-Green tensor for 3D continua. This procedure corresponds to a projectionof 3D models onto surface S, or to an extraction of 2D surface models out of 3D ones. This ap-proach goes back to Hughes and Carnoy (1983); De Borst (1991); Dvorkin et al. (1995); Klinkeland Govindjee (2002); Kiendl et al. (2015b), and it is presented here to show how it relates toour formulation.

3.1 Extraction procedure

As shown in Fig. 1, a shell material point P can be described w.r.t. the mid-surface in thereference configuration as (Wriggers, 2008)

X(ξα, ξ0) = X(ξα) + ξ0N(ξα) , (42)

and in the current configuration as

x(ξα, ξ) = x(ξα) + ξ d(ξα) , (43)

where ξ0 ∈ [−T/2, T/2] is the thickness coordinate of the shell and d is the director vector whichhas three unknown components in general. For Kirchhoff–Love theory, which is considered here,d := n and ξ = λ3 ξ0, where λ3 denotes the stretch in the normal direction. Thus, the tangentvectors at P are expressed w.r.t. the basis formed by the tangent vectors on the mid-surface as

gα := x,α = aα − ξ bγα aγ ,

g3 := x,ξ = d ,

Gα := X ,α = Aα − ξ0BγαAγ ,

G3 := X ,ξ0 = N = G3 ,

(44)

and the metric tensors at P can also be expressed in terms of the metric tensors on the mid-surface as

gαβ := gα · gβ = ga aαβ + gb bαβ ,

gαβ := gα · gβ = ga aαβ + gb bαβ ,

Gαβ := Gα ·Gβ = GAAαβ +GB Bαβ ,

Gαβ := Gα ·Gβ = GAAαβ +GB Bαβ ,

(45)

where gα = gαβ gβ, [gαβ] = [gαβ]−1 and likewiseGα. Due to Eq. (44), the coefficients in Eq. (45)are found to be

ga := 1− ξ2 κ , gb := −2 ξ + 2H ξ2 ,

GA := 1− ξ20 κ0 , GB := −2 ξ0 + 2H0 ξ

20 ,

(46)

andga := s−2 (ga + 2H gb) , gb := −s−2 gb ,

GA := s−20 (GA + 2H0GB) , GB := −s−2

0 GB .(47)

13

Here,

s :=√g/a = 1− 2H ξ + κ ξ2 , s0 :=

√G/A = 1− 2H0 ξ0 + κ0 ξ

20 , (48)

denote the so-called shifters, which are the determinants of the shifting tensors i − ξ b andI − ξ0B, respectively, and we have defined

G := det[Gαβ], A := det[Aαβ], B := det[Bαβ], g := det[gαβ] . (49)

From Eqs. (46), (47), and (48), we thus find the variations (considering ξ fixed)

δga = ξ2 κ aγδ δaγδ − ξ2 bγδ δbγδ ,

δgb = −ξ2 bγδ δaγδ + ξ2 aγδ δbγδ ,(50)

where bαβ := 2H aαβ − bαβ. With these we get

δgαβ = ga δaαβ + gb δbαβ + aαβ δga + bαβ δgb . (51)

Further, we can represent the right 3D Cauchy-Green tensor w.r.t. the basis G1, G2, N as(e.g. see Wriggers (2008))

C = CαβGα ⊗Gβ + Cα3

(Gα ⊗N +N ⊗Gα

)+ C33N ⊗N . (52)

We note here that Gα accounts for the surface stretch due to the initial shell curvature (seeEq. (44.3)) and the basis G1, G2, N on a shell layer at ξ defines the usual identity tensorin R3 as

1 = I +N ⊗N , I := Gα ⊗Gα = Aα ⊗Aα. (53)

Further, the 3D Kirchhoff stress tensor can be written as

τ = ταβ gα ⊗ gβ + τα3(gα ⊗ g3 + g3 ⊗ gα

)+ τ33 g3 ⊗ g3 . (54)

For the Kirchhoff–Love shell we have Cαβ = gαβ, while Cα3 = gα3 = gα · g3 = 0 and C33 = λ23.

Thus the variation of C is

δC = δC∗ + 2λ3 δλ3N ⊗N , C∗ := gαβGα ⊗Gβ . (55)

With this, the total strain energy of the shell can be expressed w.r.t. the mid-surface as

W =

∫Ω0

WdV =

∫S0

∫ T2

−T2

W dξ0 dG =

∫S0

∫ T2

−T2

s0 W dξ0 dA , (56)

where dG is the infinitesimal area element at layer ξ along the thickness, and dA is the in-finitesimal area element on the mid-surface. Thus, the surface energy of the shell (per unitarea) follows from the thickness integration

W =

∫ T2

−T2

s0 Wdξ0 = W (aαβ, bαβ) , (57)

and its variation reads

δW =

∫ T2

−T2

s0 δWdξ0 =

∫ T2

−T2

s0

(1

2ταβ δgαβ + τ33 λ3 δλ3

)dξ0 , (58)

14

where

ταβ := 2∂W

∂gαβ; τ33 :=

1

λ3

∂W

∂λ3. (59)

Here, similar to Remark 2, ταβ are the components of either the second Piola-Kirchhoff stresstensor S or the Kirchhoff stress τ , as

ταβ = gα · τ gβ = Gα · S Gβ . (60)

For shells, since it is common to assume a plane stress state, we have the condition

τ33 =1

λ3

∂W

∂λ3= 0 . (61)

Substituting Eqs. (51) and (61) into Eq. (58) we get

δW =1

2

∫ T2

−T2

s0

[ga τ

αβ + εa ξ2 κ aαβ − εb ξ2 bαβ

]dξ0 δaαβ ,

+1

2

∫ T2

−T2

s0

[gb τ

αβ − εa ξ2 bαβ + εb ξ2 aαβ

]dξ0 δbαβ ,

(62)

with εa := ταβ aαβ and εb := ταβ bαβ. Since

δW =∂W

∂aαβδaαβ +

∂W

∂bαβδbαβ , (63)

we find

ταβ = 2∂W

∂aαβ=

∫ T2

−T2

s0

[ga τ

αβ + εa ξ2 κ aαβ − εb ξ2 bαβ

]dξ0 ,

Mαβ0 =

∂W

∂bαβ=

1

2

∫ T2

−T2

s0

[gb τ

αβ − εa ξ2 bαβ + εb ξ2 aαβ

]dξ0.

(64)

It should be noted here that s0, ga, gb, εa and εb are all functions of ξ0.

Remark 7: So far, Eq. (64) is exact, since we have not made any approximations apart fromthe Kirchhoff–Love hypothesis, i.e. d = n, and the plane stress assumption. For a curved thinshell (T R, where T is the thickness and R is the radius of curvature) one may approximatethe variations δga ≈ 0 and δgb ≈ 0 in Eq. (51). In this case the two rear terms in Eq. (64)vanish so that

ταβ =

∫ T2

−T2

s0 (1− ξ2 κ) ταβdξ0 ,

Mαβ0 =

∫ T2

−T2

s0 (−ξ +H ξ2) ταβdξ0 .

(65)

This case will be examined in the examples in Sec. 5. If only the leading terms in front of ταβ

in Eq. (65) are considered, it is further simplified into

ταβ =

∫ T2

−T2

ταβdξ0 ,

Mαβ0 = −

∫ T2

−T2

ξ ταβdξ0 .

(66)

15

These expressions are commonly used in shell and plate formulations, see e.g. Kiendl et al.(2015b); Nguyen-Thanh et al. (2011). Here the sign convention for Mαβ

0 follows Sauer andDuong (2015); Steigmann (1999b).

Remark 8: The stretch through the thickness, λ3, can be determined in various ways. Itcan be included in the formulation as a separate degree of freedom. Another common andstraightforward approach is to derive it from the plane stress condition (61). The resultingequation is usually nonlinear and can be solved numerically (see e.g. Hughes and Carnoy (1983);De Borst (1991); Dvorkin et al. (1995); Kiendl et al. (2015b)), or analytically for some specialcases (e.g. see Sec. 3.2 for a Neo-Hookean model). In particular, if incompressible models areused, i.e. J = 1, then λ3 can be determined analytically (e.g. see Sec. 3.3). Sometimes alsoλ3 ≈ 1 is assumed. In this case, condition (61) is generally not satisfied anymore. Instead τ33

should be treated as the Lagrange multiplier to the constraint λ3 = 1.

Remark 9: For an efficient implementation, we consider that ταβ is expressible in the form

ταβ = τg gαβ + τGG

αβ . (67)

Substituting this into Eq. (64), and taking into account Eq. (45), we get the surface stress andmoment written in the form

ταβ = τa aαβ + τb b

αβ + τAAαβ + τB B

αβ ,

Mαβ0 = M0

a aαβ +M0

b bαβ +M0

AAαβ +M0

B Bαβ ,

(68)

so that only the (scalar) coefficients need to be computed by thickness integration as

τa =

∫ T2

−T2

s0

(ga τg g

a + εa ξ2 κ)

dξ0 , τA =

∫ T2

−T2

s0 ga τGGA dξ0 ,

τb =

∫ T2

−T2

s0

(ga τg g

b − εb ξ2)

dξ0 , τB =

∫ T2

−T2

s0 ga τGGB dξ0 ,

(69)

and

M0a =

1

2

∫ T2

−T2

s0

[gb τg g

a + (εb − 2H εa) ξ2]

dξ0 , M0A =

1

2

∫ T2

−T2

s0 gb τGGA dξ0 ,

M0b =

1

2

∫ T2

−T2

s0

(gb τg g

b + εa ξ2)

dξ0 , M0B =

1

2

∫ T2

−T2

s0 gb τGGB dξ0 .

(70)

The linearization then follows the representation of Eq. (141) with the scalar coefficients com-puted by integration over the thickness.

Remark 10: We note that since the integrands in Eq. (64) and even in its reduced forms (65),(66), are rather complex w.r.t. ξ, we need numerical integration in general. Analytical inte-gration may only be possible for special cases, such as the Saint Venant–Kirchhoff constitutivemodel discussed in Sec. 3.4. Thus, it is obvious that the models constructed in 2D manifoldform, such as the Koiter (Sec. 2.6.2), Canham (Sec. 2.6.1) and Helfrich constitutive models (seee.g. Sauer et al. (2016)), are much more efficient than models requiring numerical thicknessintegration.

In the following, we will present the extraction example for Neo-Hookean materials to demon-strate the procedure. We will also show that by analytical integration through the thickness ofthe 3D Saint Venant–Kirchhoff strain energy, the Koiter model is recovered.

16

3.2 Extraction example: compressible Neo-Hookean materials

The classical 3D Neo-Hookean strain-energy per reference volume is considered in the form

W (I1, J) =Λ

4(J2 − 1− 2 ln J) +

µ

2(I1 − 3− 2 ln J), (71)

where I1 and J are the invariants of the 3D Cauchy-Green tensor C. Due to Eq. (52), they canalso be expressed in terms of the kinematic quantities on the mid-surface, i.e.

I1 := C : 1 = I∗1 + λ23, J :=

√det C = J∗ λ3 , (72)

where

I∗1 := C∗ : I = gαβ Gαβ , J∗ :=

√detC∗ =

√g

G(73)

denote the invariants of C∗ defined in Eq. (55). Therefore, their variations are

δI1 = δI∗1 + 2λ3 δλ3 , δI∗1 = Gαβ δgαβ , (74)

and

δJ = λ3 δJ∗ + J∗ δλ3 , δJ∗ =

J∗

2gαβ δgαβ. (75)

λ3 can be determined using the plane stress condition Eq. (61),

∂W

∂λ3=

Λ

2

[J∗2 λ3 −

1

λ3

]+ µ

(λ3 −

1

λ3

)= 0, (76)

which is solvable analytically w.r.t. λ3 as

λ23 =

Λ + 2 µ

Λ J∗2 + 2 µ. (77)

From Eq. (59.1) we then find

ταβ = µ Gαβ − µ Λ + 2 µ

Λ J∗2 + 2 µgαβ. (78)

The stress in Eq (78) is then substituted into Eq. (64), (65) or (66). Numerical integration isstill required to evaluate those. The corresponding linearized quantities and tangent matricesare provided in Appendix C.

3.3 Extraction example: incompressible Neo-Hookean materials

The Neo-Hookean strain-energy per reference volume in this case is defined as

W (I1, J , p) =µ

2(I1 − 3) + p (1− J), (79)

where p is the Lagrange multiplier associated with the volume constraint. From Eq. (79), wethus find

ταβ = 2∂W (gαβ, λ3, p)

∂gαβ= µGαβ − p Jgαβ. (80)

17

Further, the plane stress condition implies ∂W/∂λ3 = 0 and the incompressibility constraintrequires λ3 := 1/J∗, which together allow us to determine the Lagrange multiplier analyticallyas

p =µ

J∗2. (81)

Substituting this into Eq. (80) we obtain

ταβ = µ

(Gαβ − 1

J∗2gαβ), (82)

for Eq. (64) or (65), (66). Numerical integration is also required here.

3.4 Extraction example: Saint Venant-Kirchhoff model

We now consider the Saint Venant–Kirchhoff model,

W (C) :=1

2E : C : E , C := Λ 1 1 + 2 µ (1⊗ 1)s . (83)

In this case, the Koiter model of Eq. (35) can be recovered by using Eq. (57) and Eq. (61)together with the following ingredients:

• s0 ≈ 1, ξ ≈ ξ0, and only the leading terms w.r.t. ξ in Eq. (46) and Eq. (47) are taken intoaccount, so that Eq. (83) reduces to

W (C∗) :=1

2E∗ : C∗ : E∗ , C∗ := Λ I I + 2 µ (I ⊗ I)s , (84)

whereE∗ := E − ξK. (85)

• The bending and stretching response is considered fully decoupled, so that all the mixedproducts, such as trE trK and tr (EK), are disregarded in the strain energy function.Eq. (84) then further reduces to

W (E,K, ξ) =Λ

2(trE)2 + µ tr (E2) + ξ2 Λ

2(trK)2 + ξ2 µ tr (K2) . (86)

Integrating the above energy over the thickness [−T/2, T/2] gives

W (E,K, T ) =T

2Λ (trE)2 + T µ tr (E2) +

T 3

24Λ (trK)2 +

T 3

12µ tr (K2) . (87)

• Finally, the plane stress assumption is used throughout the thickness. Accordingly, the3D Lame constant Λ in Eq. (87) is replaced by its plane-stress counterpart (2 Λ µ)/(Λ+2 µ)(see e.g. Hackl and Goodarzi (2010)), which results in the Koiter model of Eqs. (35) and (40).

Alternatively (Steigmann, 2013), the Koiter model can also be derived systematically as theleading-order model from the Taylor expansion of W with the aid of Leibniz’ rule for smallthickness.

4 FE discretization

In this section, we first discuss the discretization of the weak form (21) and its linearization onthe basis of IGA to obtain FE forces and tangent matrices. We will then discuss the continuityconstraint between patches, patch folds, symmetry and rotational Dirichlet boundary conditions.

18

4.1 C1-continuous discretization

In order to solve Eq. (21) by the finite element method, the surface S is discretized usingthe isogeometric analysis technique proposed by Hughes et al. (2005). Thanks to the Bezierextraction operator Ce introduced by Borden et al. (2011), the usual finite element structure isrecovered for NURBS basis function NAnA=1, where n is the number of control points definingan element e. Namely,

NA(ξ, η) =wA N

eA(ξ, η)∑n

A=1wA NeA(ξ, η)

, (88)

where N eAnA=1 is the B-spline basis function expressed in terms of Bernstein polynomials as

Ne(ξ, η) = Ceξ B(ξ) ⊗ Ce

η B(η), (89)

with N eA to be the corresponding entries of matrix Ne. For T-splines, the construction of the

isoparametric element based on the Bezier extraction operator can be found in Scott et al.(2011).

4.2 FE approximation

The geometry within an undeformed element Ωe0 and deformed element Ωe is interpolated from

the positions of control points Xe and xe, respectively, as

X = N Xe , x = N xe , (90)

where N(ξ) := [N11, N21, ..., Nn1] is defined based on the NURBS shape functions of Eq. (88).It follows that

δx = N δxe ,

aα = N,α xe ,

δaα = N,α δxe ,

aα,β = N,αβ xe ,

aα;β = N;αβ xe ,

(91)

where N,α(ξ) := [N1,α1, N2,α1, ..., Nn,α1], N,αβ(ξ) := [N1,αβ1, N2,αβ1, ..., Nn,αβ1], NA,α =∂NA/∂ξ

α, NA,αβ = ∂2NA/(∂ξα∂ξβ) (A = 1, ..., n) and6

N;αβ := N,αβ − Γγαβ N,γ . (92)

Inserting those expressions into the formulas for δaαβ and δbαβ (Sauer and Duong, 2015) gives

δaαβ = δxTe

(NT,α N,β + NT

,β N,α

)xe , (93)

andδbαβ = δxT

e NT;αβ n . (94)

4.3 FE force vectors

Substituting Eqs. (93) and (94) into Eq. (21) gives

nel∑e=1

(Gein +Geint −Geext) = 0 ∀ δxe ∈ V , (95)

6A tilde is included in N;αβ to distinguish it from N,αβ .

19

where nel is the number of elements and

Gein = δxTe f ein , with f ein :=

∫Ωe0

NT ρ0 v dA . (96)

Similarly,Geint = δxT

e

(f eintτ + f eintM

), (97)

with the internal FE force vectors due to the membrane stress ταβ and the bending momentMαβ

0

f eintτ :=

∫Ωe0

ταβ NT,α aβ dA (98)

and

f eintM :=

∫Ωe0

Mαβ0 NT

;αβ ndA . (99)

Here the symmetry of ταβ has been exploited. The external virtual work follows as (Sauer et al.,2014; Sauer and Duong, 2015)

Geext = δxTe

(f eext0 + f eextp + f eextt + f eextm

), (100)

where the external FE force vectors are

f eext0 :=

∫Ωe0

NT f0 dA ,

f eextp :=

∫Ωe

NT pnda ,

f eextt :=

∫∂tΩe

NT tds ,

f eextm := −∫∂mΩe

NT,α ν

αmτ nds .

(101)

Here f0 is a constant surface force and p is an external pressure acting always normal to S(Sauer et al., 2014). t is the effective boundary traction of Eq. (17) and both να and mτ

are defined in Sec. 2.2. We note that, in the following sections, the inertia term is neglected,i.e. ρ0 v = 0. The corresponding tangent matrices can be found in Appendix A.

4.4 Edge rotation conditions

This section presents a general approach to describe different rotation conditions. These condi-tions are required due to the fact that the presented formulation has only displacement degreesof freedom as unknowns. As shown in Fig. 2, the G1-continuity constraint is required for multi-patch NURBS in order to transfer moments. Additionally, other rotation conditions may beneeded such as fixed surface fold constraints, symmetry (or clamping) constraints, symmetryconstraints at a kink and rotational Dirichlet boundary conditions.

There are various methods to enforce the continuity between patches, such as using T-Splines(Schillinger et al., 2012), the bending strip method (Kiendl et al., 2010) and the Mortar methodfor non-conforming patches (Dornisch, 2015). Alternatively, Nitsche’s method as described inNguyen et al. (2014); Guo and Ruess (2015a) can also be used. Recently, Lei et al. (2015b)applied static condensation and a penalty method to enforce G1-continuity of adjacent patchesbased on subdivision algorithms.

20

Here, we introduce a general approach that can systematically enforce different edge rotationconditions, including G1-continuity of adjacent patches, within the framework of a curvilinearcoordinate system. The presented approach has new features compared to existing formula-tions. For instance, our approach is conceptually and technically different from the methodof Lei et al. (2015b). First and foremost, the presented method is not only restricted to G1-continuity of adjacent patches of NURBS-based models but it also includes other constraints asmentioned above. Second, here the constraint is enforced by systematically adding a potentialto the weak form (Eq. (21)). In Lei et al. (2015b), the G1-continuity is enforced by constrainingthe position of control points in the vicinity of the shared edge so that the adjacent NURBSsurfaces have the same tangent plane at the intersection points. This leads to a completelydifferent constraint equation.Like the bending strip method, we focus on conforming meshes, where the control points of adja-cent patches coincide at their interface. For nonconforming meshes, the proposed method shouldbe modified or alternative methods, like Nitsche’s or Mortar methods, can be used. Comparedto the bending strip method (Kiendl et al., 2010), the proposed method requires only line inte-gration instead of surface integration. For nonlinear problems, the implementation of Nitsche’smethod (e.g. (Guo and Ruess, 2015a)) becomes more complex as it requires the tractions andtheir variations on the interface, which depend on the type of constitutive equations. Our for-mulation is independent of the choice of material model. Further, as shown in Secs. 4.4.1 and4.4.2, our constraint equation, enforced by both the penalty and Lagrange multiplier methods,gives an exact transmission of both traction and moment across the interface.

a. b. c. d. e.

Figure 2: Edge rotation conditions: a. G1-continuity constraint, b. fixed surface folds (e.g. V-shapes and L-shapes), c. symmetry (or clamping) constraints, d. symmetry constraint at akink, e. rotational Dirichlet boundary condition. Surface edge L, shown by a filled circle, isperpendicular to the plane and parallel to the inward pointing unit direction τ .

In order to model all five cases depicted in Fig. 2, we first consider the constraint equation onthe surface edge L

gc := cosα0 − cosα = 0 , ∀x ∈ L , (102)

where

cosα0 := N · N , cosα := n · n . (103)

Here,N is the surface normal of a considered patch and N is the surface normal of a neighboringpatch. They are defined in the initial configuration. Similarly, n and n are the correspondingquantities in the deformed configuration. As shown in Fig. 2a, in the case of G1-continuitybetween two patches, we have N · N = 1. Similarly for fixed surface folds, N · N = cosα0,where α0 is the angle between the normal of the patches (see Fig. 2b). In the case of symmetry,n = N is the (fixed) normal vector of the symmetry plane (see Fig. 2c-d). Furthermore, theconstraint equation (102) can be used to apply rotational Dirichlet boundary conditions, bysetting N · N = 1, and n is then the prescribed normal direction.

The constraint equation (102) can uniquely realize angles within [0, π]. However, Eq. (102)becomes non-unique if α and α0 go beyond this range. Therefore, an additional constraint

21

equation in the form of a sine function is considered,

gs := sinα0 − sinα = 0 , ∀x ∈ L (104)

with

sinα0 :=(N × N

)· τ 0 , sinα :=

(n× n

)· τ , (105)

where τ and τ 0 are the unit directions of interfaces L and L0 (see Sec. 2.2). Here, Eq. (105)implies that α is measured from n to n, and likewise for α0. Together, Eqs. (102) and (104)uniquely define any physical angles of α0 and α ∈ [0, 2π]. One can easily show that they areequivalent to the system of equations

gc := 1− cos(α− α0) = 0 ,

gs := sin(α− α0) = 0 .(106)

It is also worth noting that an initial NURBS mesh imported from CAD programs is usuallygiven with G1-continuity instead of C1-continuity between patches. In this case, a strict C1-continuity enforcement without a mesh modification will affect the FE solution. Thus, a G1-continuity constraint is more practical in such a case. In the following, we present a penaltyand a Lagrange multiplier method for enforcing constraints (102) and (104).

4.4.1 Penalty method for edge rotations

The system of constraints (102) and (104) can be enforced by the penalty formulation

ΠL =

∫L0

ε

2

(g2

c + g2s

)dS =

∫L0ε(1− c0 cosα− s0 sinα

)dS , (107)

which is a locally convex function w.r.t. α and α0. Thus, for the Newton-Raphson method, theexistence of a unique solution is guaranteed, provided that |α− α0| < π. Here ε is the penaltyparameter and s0 := sinα0 and c0 := cosα0. Taking the variation of the above equation yields

δΠL = −∫L0ε(δτ · θ + δn · d+ δn · d

)dS , (108)

where we have defined

θ := s0n× n , d := c0 n+ s0 ν , d := c0n+ s0 ν (109)

and ν := τ ×n = −τ ×n. The linearization of the above equation can be found in Appendix D.We note that, for some applications, the variation of τ and n in Eq. (108) vanishes since theyare considered fixed.From Eq. (108), the bending moments mτ = mτ = ε sin(α0−α) can be identified. This impliesthat the bending moment is transmitted across the interface exactly.

Remark 11: If α0 = 0 (for the cases in Figs. 2a and 2e), the integrand in Eq. (107) reduces toε (1− cosα) such that ΠL becomes

ΠL =

∫L0

ε

2(n− n) · (n− n) dS , (110)

enforcing the constraintgL := (n− n) = 0 . (111)

22

4.4.2 Lagrange multiplier method for edge rotations

Alternatively, the Lagrange multiplier approach can be used to enforce the system of continuityconstraints (106). In this case, we consider the constraint potential

ΠL =

∫L0q(gc + gs

)dS , (112)

where q is the Lagrange multiplier. This potential guarantees unique solutions of α and α0 ∈[0, 2π] provided that |α−α0| < π/4. Furthermore, within this range, the gradient of gL := gc+gsw.r.t α and α0 is non-zero, which is the necessary condition to have a solution for the Lagrangemultiplier q (see e.g. Bertsekas (1982)).

Taking the variation of Eq. (112) yields

δΠL =

∫L0gL δq dS −

∫L0q(δτ · θ + δn · d+ δn · d

)dS , (113)

where θ, d and d are now defined as

θ := (s0 − c0)n× n ,

d := (s0 + c0)n+ (s0 − c0)ν ,

d := (s0 + c0) n+ (s0 − c0) ν .

(114)

Here, we find the bending moment mτ = mτ = −q, which is also transmitted exactly across theinterface. The linearization of Eq. (113) can be found in Appendix D .

5 Numerical examples

In this section, the performance of the proposed shell formulation is illustrated by severalbenchmark examples, considering both linear and non-linear problems. The computationalresults are verified by available reference solutions.

5.1 Linear problems

For the linear problems discussed below, we will consider two material models: the Koiter shellmaterial model of Eq. (35), and the projected shell material model obtained from the numericalintegration of Eqs. (65) and (78), where Λ = E ν/

[(1 + ν)(1− 2 ν)

]and µ = E/

[2 (1 + ν)

]. For

the sake of simplicity, hereinafter we denote the former as the Koiter model and the latter asthe projected model. Furthermore, all physical quantities are introduced in terms of unit lengthL0 and unit stress E0.

5.1.1 Pinching of a hemisphere

A hemisphere pinched by two pairs of diametrically opposite forces F = 2E0 L20 on the equator

is examined in this example. The model parameters are adopted from Belytschko et al. (1985)as E = 6.825 × 107E0, ν = 0.3, R = 10.0L0 and T = 0.04L0. Due to the symmetry, 1/4 ofthe hemisphere is modeled as shown in Fig. 3a. Here, the symmetry boundary conditions areapplied on the Y = 0 and X = 0 planes by the penalty method of Eq. (107) with the penalty

23

a. b.

Number of elements101 102 103 104

Norm

alize

ddispla

cem

ent

0

0.2

0.4

0.6

0.8

1

1.2

QuadraticCubicQuarticQuintic

c.Number of elements

101 102 103 104

Norm

alize

ddispla

cem

ent

0

0.2

0.4

0.6

0.8

1

1.2

QuadraticCubicQuarticQuintic

d.

Figure 3: Pinching of a hemisphere: a. Undeformed configuration with boundary conditions.Here, the blue curves denote the symmetry lines. b. Deformed configuration (scaled 50 times),colored by radial displacement. Radial displacement at the point load vs. mesh refinement forc. the Koiter and d. the projected shell model, considering various NURBS orders.

parameter ε = 800E L20 and the rigid body motions are restricted by fixing the top control

point.

The benchmark reference solution for the radial displacement under the point loads is 0.0924L0

(Belytschko et al., 1985; Macneal and Harder, 1985; Morley and Morris, 1978). As shown inFig. 3, the numerical results converge to the reference solution as the mesh is refined and theNURBS order is increased. Here, the radial displacement of the force is normalized by thereference solution. As can be observed in Figs. 3c-d, for linear elastic deformations, both theKoiter and the projected model are identical. This is also the case for several other problemsexamined here. In the following, only the results of one of the models are reported if thedifferences are negligible.

5.1.2 Simply supported plate under sinusoidal pressure loading

As a second example, we analyze a plate with size L× L = 12× 12L20, thickness T = 0.375L0,

Young’s modulus E = 4.8 × 105E0, Poisson’s ratio ν = 0.38, subjected to sinusoidal pres-sure p(x, y) = p0 sin(π x/L) sin(π y/L)E0. According to Navier ’s solution (Ugural, 2009), the

24

Number ofelements101 102 103

Norm

alize

ddispla

cem

ent

0.95

1

1.05

1.1

1.15QuadraticCubicQuarticQuintic

Number of elements101 102 103

Relative

displa

cem

enter

ror

10!10

10!8

10!6

10!4

QuadraticCubicQuarticQuintic

a. b.

c. d.

Figure 4: Simply supported plate under sinusoidal pressure: a. Initial configuration with bound-ary conditions, b. deformed configuration (scaled 104 times) colored by the vertical displacement,c. displacement of the plate center normalized w.r.t. the analytical solution and d. relative errorof the displacement.

maximum deflection is at the middle point and given by

wmax =p0 L

2

4π4D, (115)

where D := E T 3/12 (1−ν2) is the flexural rigidity of the plate. The setup of the computationalmodel is shown in Fig 4a. Only 1/4 of the plate is modeled using symmetry boundary conditionsenforced by the penalty method of Sec. 4.4.1 with the penalty parameter ε = 10−2 np−1E L2

0,where p is the NURBS order and n is the number of elements in each direction. Fig. 4bshows the deformed plate with the Koiter model and Fig. 4c shows the convergence of thecomputational solution to the analytical one as the mesh is refined. For the comparison, thevertical displacement at the center of the plate is normalized by the analytical solution givenin Eq. (115). The corresponding relative error is shown in Fig. 4d. As expected, more accuracyis gained by increasing the NURBS order.

5.1.3 Pinching of a cylinder

Next, we consider the pinched cylinder test with rigid diaphragms at its ends. It is designed toexamine the performance of shell elements in inextensional bending modes and complex mem-

25

brane states (Belytschko et al., 1985). The analytical solution for this problem was introducedby Flugge (1962) based on a double Fourier series, see Appendix E.

Number of elements101 102 103 104

Norm

alize

ddispla

cem

ent

0

0.2

0.4

0.6

0.8

1

QuadraticCubicQuarticQuintic

Number of elements101 102 103 104

Relative

displa

cem

enter

ror

10!5

10!4

10!3

10!2

10!1

100

QuadraticCubicQuarticQuintic

a. b.

c. d.

Figure 5: Pinching of the cylinder with rigid end diaphragms: a. Setup of the computationmodel, b. deformed shell (scaled 106 times) colored by the radial displacement, c. normalizedradial displacement at the point load and d. error w.r.t. the reference solution as the mesh isrefined.

The parameters are adopted from Belytschko et al. (1985) as E = 3 × 106E0, ν = 0.3, R =300L0, L = 600L0, T = 3L0, where E is Young’s modulus, ν is Poisson’s ratio; R, L and Tare radius, length and thickness of the cylinder, respectively. For the FE computation, 1/8 ofthe cylinder is modeled due to its symmetry as is shown in Fig. 5a. The symmetry boundaryconditions are enforced by the penalty method discussed in Sec. 4.4.1. The penalty parametersused are ε = 2 × 102 np−1

l E L20 for the axial symmetry, and ε = 2 × 102 np−1

t E L20 for the

circumferential symmetry. Here, nt and nl are the number of elements in circumferential andaxial directions, respectively.

Two opposing pinching forces F = 1E0 L20 are applied at the middle of cylinder. For the

symmetric model, a quarter of the force is applied. The deflection due to the loads is measuredfor comparison with the reference solution. For 80 × 80 Fourier terms, the deflection underthe point load is 1.82488× 10−5, which is the reference value commonly used in the literature.However, for 8192 × 8192 Fourier terms the converged solution is 1.82715781 × 10−5 and for214 × 214 Fourier terms, we obtain the solution of 1.82715797× 10−5.

Fig. 5c shows the convergence behavior of the computed displacement at the loading pointw.r.t. the analytical solution as the mesh is refined. From Fig. 5d, we observe that the relative

26

error only converges up to a certain point and then seems to get stuck. This is due to severalnumerical and theoretical reasons. Firstly, although the NURBS order can be increased forthe patches, the patch boundaries are only G1-continuous according to Eq. (107). This affectshigher order NURBS, especially at the boundaries around the singular point load. Secondly,the analytical solution represents the concentrated load by a pressure distribution in the form ofa Dirac delta function. Numerically, this is represented by a Fourier series and thus the Gibbseffect can lead to a loss in numerical accuracy for high order terms (see Appendix E). Therefore,the Fourier solution is truncated at some point and we cannot expect the error to go down tomachine precision as the mesh is refined. We note that, although both the results of the Koitershell material model of Eq. (35), and the projected shell material model of Eqs. (65) and (78) areidentical, the computational time of the former is roughly half of the latter (considering threequadrature points through the thickness). Thus, the presented Koiter shell material model ismore efficient for the same accuracy.

5.2 Nonlinear problems

In the following, several nonlinear test cases are presented to illustrate the robustness andaccuracy of the proposed shell formulation.

5.2.1 Pure bending of a flat strip

We first examine the pure bending of a flat strip with the dimensions S × L. The strip is fixedas is shown in Figs. 6a-d in order to apply the moment M at the two ends of the strip. TheCanham material model, Eq. (32), is used. For this problem, the analytical solution is given bySauer and Duong (2015). Accordingly, the strip is deformed into a curved sheet with dimensionss× l = λ1 S × λ2 L with constant curvature κ1 that is linearly related to M as

M = c κ1 . (116)

Since all the boundaries are free to move, we expect that the in-plane stress components vanish.This condition leads to the stretches λ1 and λ2 in terms of the applied moment M as

λ2 = λ1/a0 , a0 :=M2

2µc+

√(M2

2µc

)2

+ 1 , (117)

and

λ1 =

√−µ (a2

0 + 1) +õ2(a2

0 + 1)2 + a20 (4 µ+ 1) , µ :=

µ

2Λ. (118)

To assess the FE computation, four different mesh schemes using quadratic NURBS are consid-ered as is shown in Figs. 6a-d: 1. A single patch with regular mesh; 2. A single patch with skewmesh with distortion ratio r = 1/5; 3. A double patch with regular mesh; and 4. A double patchwith skew mesh with the same distortion ratio. Here, the mesh refinement is carried out by theknot insertion algorithm (see e.g Hughes et al. (2005)). In particular for the second scheme, theskew mesh is obtained by distortion of the knot vectors as explained in detail in Appendix F.

For all mesh schemes, the material parameters µ = 10L2/c, Λ = 5L2/c and the bendingmoment M = 1 c/L are applied (setting mτ = M in Eq. (101.4)). For mesh schemes 3 and4, the G1-continuity constraint is used between the two patches (see Sec. 4.4.1) with ε =10 000 (nS × nL)L/c, where nS and nL are the number of elements along the length and widthof the strip, respectively. Fig. 6e shows the deformed mesh and the mean curvature error for

27

Number of elements100 101 102 103 104 105

L2

erro

rof

U

10!8

10!6

10!4

10!2

100

Single patch: regular meshSingle patch: skew meshTwo patches: regular meshTwo patches: skew mesh

a.

b.

c.

d.

e.

f.

Figure 6: Pure bending of a flat strip considering: a. Single patch with regular mesh, b. singlepatch with skew mesh, c. two patches with regular mesh, and d. two patches with skew mesh.e. Deformed configuration for the mesh in d. The color here shows the relative error of meancurvature H. f. L2 error of u (Eq. 119) w.r.t. mesh refinement.

mesh 4 × 8 of scheme 4. Fig. 6f shows the L2 error of the displacement field for the fourconsidered mesh schemes. Here, the L2 error is defined as

‖uh − u‖L2 :=

√1

S L

∫S0

(uh − u

)2dA, (119)

where uh is the displacement obtained from the FE analysis and u is the corresponding analyt-ical quantity calculated at each point on S. Here, u can be computed for any applied momentby using the parametrization described in Sauer and Duong (2015).

The convergence observed in Fig. 6f verifies the presented finite element formulation. It alsoshows, that the accuracy of the double patch meshes is of the same order as the single patchresults. This indicates the effectiveness of the penalty constraint presented in Sec. 4.4.

5.2.2 Pure bending of a folded strip

Next, we demonstrate the robustness of the edge rotation constraint presented in Sec 4.4 formultiple patch interfaces with kinks. For this purpose, we reconsider the pure bending test ofthe previous section, but here the mesh consists of 8 patches and has a kink at 3/4π L from the

28

left end, as shown in Fig. 7a. The material of the strip is the same as the example in Sec. 5.2.1.The deformed configuration is shown in Fig. 7b.

102

103

104

10−7

10−6

10−5

10−4

10−3

10−2

10−1

Number of elements nel

L 2 err

or o

f U

LM: N2Q1cLM: N2Q0

PM: ε = 1/64 nel2

PM: ε = 5/64 nel2

PM: ε = 10/64 nel2

a. b.

c.

Figure 7: Pure bending of a folded strip: a. Initial FE configuration and boundary conditions(for S1 + S2 = πL, opening angle β0 = π/6, discretized with quadratic NURBS elements);b. current FE configuration and relative error of H w.r.t Hexact = 0.8 (1/L) for an imposedbending moment M = 1.6 c/L; c. L2 error of u (Eq. (119)) w.r.t. mesh refinement, consideringthe penalty method (PM) and the Lagrange multiplier method (LM).

Here, the continuity between patch interfaces including both G1-continuity and the kink con-straints are enforced by the penalty method (PM) (107) and the Lagrange multiplier method(LM) (112). Further, for LM, we consider a constant and a linear interpolation scheme of theLagrange multiplier q denoted as N2Q0 and N2Q1c, respectively. We note that at patch junc-tions, q can jump across interfaces as there is a change of patch pairings. In this case, N2Q0can automatically capture the jump since q is interpolated discontinuously over the elements,whereas for N2Q1c, the continuous interpolation of q would result in over-constraining, i.e. isLBB-unstable. However, this problem can be simply solved by repeating the pressure degreesof freedom at patch junctions for each interface so that the jump can be captured.

In order to assess the FE results, the L2 error defined by Eq. (119) is computed. As theanalytical solution, given by Eqs. (116) and (117), is also valid for this problem, we can use thesame procedure as in Sec 5.2.1 to compute u and uh. It only needs to be adapted to includethe kink.

Fig. 7c shows the convergence of the computed L2 error as the mesh is refined. With anappropriate choice of the penalty parameter ε, the rate of convergence of PM can be achievedat the same order as for LM. Furthermore, the rate of convergence here is also the same orderas for bending of the flat strip (see Fig. 6). Besides, the accuracy of PM approaches that ofLM as ε is increased. However, note that the stiffness matrix becomes ill-conditioned if ε is toohigh. For LM, we observe that both interpolation schemes N2Q0 and N2Q1c are robust andstable. It can be seen that N2Q0 is as accurate as N2Q1c here.

5.2.3 Cantilever subjected to end shear forces

The large deflection of a cantilever beam, with dimensions L×W ×T = 10× 1× 0.1L30, due to

shear traction t = F /W applied to its free end is computed. The forces at the nodes locatedon the free end are derived from Eq. (101.3). The fixed end is clamped by the penalty method

29

Displacement [L0]0 2 4 6 8

Force[E

I=L2]

0

0.5

1

1.5

2

2.5

3

3.5

4

wA (Quadratic)wA (Cubic)wA (Ref.)! vA (Quadratic)! vA (Cubic)! vA (Ref.)

Displacement [L0]0 2 4 6 8

Force[E

I=L2]

0

0.5

1

1.5

2

2.5

3

3.5

4

wA (Quadratic)wA (Cubic)wA (Ref.)! vA (Quadratic)! vA (Cubic)! vA (Ref.)

a. b.

c. d.

Figure 8: Cantilever subjected to end shear force: a. Undeformed configuration, b. deformedconfiguration colored by the vertical displacement. Horizontal (−vA) and vertical (wA) dis-placement at tip A for c. a regular mesh with uniform element length and d. a skew mesh asshown in b. The results are compared with Sze et al. (2004).

(Sec. 4.4.1) with ε = 1000E L20. The material parameters are E = 1.2 × 106E0 and ν = 0.0

(Sze et al., 2004). The beam is modeled with the Koiter shell material and is discretized by1×10 NURBS elements, considering both a regular mesh and a skew mesh with distortion ratior = 0.2 in y-direction. The applied maximum shear force is Fmax = 4F0 with F0 = EI/L2

and I = 1/12W T 3. As shown in Fig. 8, considering 10 elements, cubic NURBS give very goodresults for both regular and skew meshes.

To examine the effects of both bending and large membrane strains, the same problem is studiedwith an extra Dirichlet boundary condition applied at the beam tip imposing zero displacementsin the y-direction. Here, since the beam is considerably stretched, the in-plane membrane strainsare significant. Instead of a shear force, a vertical displacement is applied at the tip and thereaction forces are measured. In addition to the Koiter model and the projected shell model,a mixed model that combines the bending part of a Koiter shell material with a Neo-Hookeanmembrane formulation is considered (see Eq. (41)). As shown in Fig. 9, all three models predictsimilar results as long as the bending effects are dominant. By increasing the membrane strains,the mixed formulation is very close to the projected model while the full Koiter model has adifferent trend. This shows that the simple, more efficient mixed Koiter model can accuratelycapture the full 3D model behavior.

30

Vertical displacement [L]0 0.2 0.4 0.6 0.8 1

Fz[E

I=L2]

#104

0

1

2

3

4

5

6wA (Koiter + Neo-Hooke)wA (Full Koiter)wA (Projected model)

Vertical displacement [L]0 0.2 0.4 0.6 0.8 1

Fy[E

I=L2]

#104

0

1

2

3

4

5

6wA (Koiter + Neo-Hooke)wA (Full Koiter)wA (Projected model)

a. b.

Figure 9: Cantilever with prescribed tip displacement: a. Vertical reaction force Fz vs. verticaldisplacement and b. horizontal reaction force Fy vs. vertical displacement.

5.2.4 Pinching of a hemisphere with a hole

In this example, we compute the large deformation of a hemisphere with a hole at the top,under two pairs of equally large but opposing forces that are applied on the equator of thehemisphere as shown in Fig. 10. The parameters are extracted from Sze et al. (2004): R = 10L0,

a. b.

c. Displacement [L0]0 2 4 6 8 10 12

Force[E0L2 0]

0

50

100

150

200

250

300

350

400

vA (Quadratic)vA (Cubic)vA (Quartic)vA (Ref.)!uB (Quadratic)!uB (Cubic)!uB (Quartic)!uB (Ref.)

d.

Figure 10: Pinching of a hemisphere with a hole at 18: a. Undeformed configuration withboundary conditions, b-c. deformed configuration colored by the radial displacement and d.force vs. displacement of points A and B compared with the reference solution of Sze et al.(2004).

31

T = 0.04L0, E = 6.825× 107E0, ν = 0.3 and the point load Fmax = 400E0 L20. The symmetry

of the hemisphere is modeled by the penalty method with ε = 6× 103E L20 (see Fig. 10a). The

Koiter shell model is used and the surface is meshed with 20× 20 quadratic, cubic and quarticNURBS based finite elements. As observed in Fig. 10d, the computed results approach thereference solution as the NURBS order is increased.

5.2.5 Pinching of a cylinder with end rigid diaphragms

We reconsider the pinched cylinder test of Sec. 5.1.3, but here with large deformations as shownin Fig. 11. Here, the length, thickness and radius of the cylinder are L = 200L0, T = 1.0L0

and R = 100L0, respectively. Both the Koiter and the projected shell material models areused with E = 30× 103E0 and ν = 0.3 and they give indistinguishable results. The point loadFmax = 12×103E0 L

20 is applied. Due to the symmetry, only 1/8 of the cylinder is modeled. The

symmetry boundary conditions are enforced by the Lagrange multiplier method (see Sec. 4.4).The cylinder is discretized by 50 × 50 quadratic NURBS finite elements. Fig. 11f shows goodagreement with the reference result of Sze et al. (2004).

Displacement [L0]0 20 40 60 80

Forc

e[E

0L

2 0]

0

2000

4000

6000

8000

10000

12000 !vA (Koiter)!vA (Proj.)!vA (Ref.)uB (Koiter)uB (Proj.)uB (Ref.)

a.

b.

c. d.

e.

Figure 11: Pinching of a cylinder with rigid end diaphragm: a. Undeformed configuration withboundary conditions. Deformed configurations in b. 3D view, c. y-axis view, d. z-axis view.The color here denotes the radial displacement. f. Force vs. displacement at points A and Bcompared to the results of Sze et al. (2004).

32

5.2.6 Spreading of a cylinder with free ends

In this test, a cylinder, with dimensions L×R× T = 10.35× 4.953× 0.094L30, with open ends

is pulled apart by a pair of opposite forces up to Fmax = 40 × 103E0 L20, which are applied at

the middle of the cylinder (see Fig. 12). The Koiter material is used with E = 10.5 × 106E0

and ν = 0.3125. Here, the results of Sze et al. (2004) are used as a reference for comparison.Due to the symmetry, only 1/8 of the cylinder is modeled as shown in Fig. 12a. The cylinderis discretized by 20 × 20 NURBS finite elements. The symmetry boundary conditions areenforced by the penalty method (see Sec. 4.4) with ε = 20E L3

0/L along the axial edge andε = 20E L3

0/π R along the circumferential edge. A good agreement with the reference results isalso observed in Fig. 12c.

Displacement [L0]0 1 2 3 4 5

Forc

e[E

0L

2 0]

#104

0

1

2

3

4vA (Quadratic)vA (Cubic)vA (Quartic)vA (Ref.)!uB (Quadratic)!uB (Cubic)!uB (Quartic)!uB (Ref.)!uC (Quadratic)!uC (Cubic)!uC (Quartic)!uC (Ref.)

a.

b.

c.

Figure 12: Spreading of a cylinder with free ends: a. Undeformed configuration with boundaryconditions, b. deformed configuration colored with radial displacement, c. force vs. displacementof points A, B and C compared to the results of Sze et al. (2004).

6 Conclusion

A new unified FE formulation for rotation-free thin shells is presented. The formulation usesisogeometric analysis to benefit from its high order continuity. We have also introduced apenalty and a Lagrange multiplier approach to enforce continuity at patch boundaries. The ap-proach can be used to model G1-continuity required for multi-patch NURBS, fixed surface folds,symmetry (or clamping) constraints, symmetry constraints at a kink and rotational Dirichletboundary conditions. It is required in order to transfer moments at patch boundaries. Weobserved that the penalty method provides a simple and very efficient implementation, yet stillmaintains sufficient accuracy. The proposed formulation is tested by several numerical examplesconsidering both linear as well as non-linear regimes. The numerical results are verified eitherby available analytical solutions or reliable reference solutions. The results demonstrate therobustness as well as good performance of the formulation.

Further, we have presented a detailed and systematic procedure to obtain shell material modelsbased on through-thickness integration of existing 3D constitutive laws. Besides that, our

33

formulation is also designed to accept material models given directly in surface energy form.Since there is no need for numerical integration over the thickness, such material models aremuch more efficient. For the considered numerical examples, it turns out that these models,particularly the mixed Koiter formulation, are equally accurate as the expensive integrationmodels, even for large deformations and stretches. Further, our formulation is presented fullyin curvilinear coordinates. This allows for a direct and efficient implementation, since no localcoordinate transformation is needed. The formulation also allows for a straightforward andefficient inclusion of material anisotropy (Roohbakhshan et al., 2016) and stabilization schemesfor liquid shells (Sauer et al., 2016).

AcknowledgmentFinancial support from the German Research Foundation (DFG) through grant GSC 111, isgratefully acknowledged. The authors also wish to thank Callum J. Corbett and Yannick A.D.Omar for their help.

A FE tangent matrices

Using Eqs. (93) and (94), the terms in Eq. (25) become

cαβγδ 12δaαβ

12∆aγδ = cαβγδ δxT

e NT,α (aβ ⊗ aγ) N,δ ∆xe ,

dαβγδ 12δaαβ ∆bγδ = dαβγδ δxT

e NT,α (aβ ⊗ n) N;γδ ∆xe ,

eαβγδ δbαβ12∆aγδ = eαβγδ δxT

e NT;αβ (n⊗ aγ) N,δ ∆xe ,

fαβγδ δbαβ ∆bγδ = fαβγδ δxTe NT

;αβ (n⊗ n) N;γδ ∆xe ,

(120)

andταβ 1

2∆δaαβ = δxTe NT

,α ταβ N,β ∆xe , (121)

∆δbαβ = − δxTe

[NT,γ (n⊗ aγ) N;αβ + NT

;αβ (aγ ⊗ n) N,γ + NT,γ a

γδ bαβ (n⊗ n) N,δ

]∆xe .

(122)Thus, the linearization of Geint yields

∆Geint = δxTe

(keττ + keτM + keMτ + keMM + keτ + keM

)∆xe , (123)

with the material stiffness matrices

keττ :=

∫Ωe0

cαβγδ NT,α (aβ ⊗ aγ) N,δ dA ,

keτM :=

∫Ωe0

dαβγδ NT,α (aβ ⊗ n) N;γδ dA ,

keMτ :=

∫Ωe0

eαβγδ NT;αβ(n⊗ aγ) N,δ dA ,

keMM :=

∫Ωe0

fαβγδ NT;αβ (n⊗ n) N;γδ dA

(124)

and the geometric stiffness matrices

keτ :=

∫Ωe0

NT,α τ

αβ N,β dA ,

keM = keM1 + keM2 + (keM2)T ,

(125)

34

with

keM1 := −∫

Ωe0

bαβMαβ0 aγδ NT

,γ (n⊗ n) N,δ dA ,

keM2 := −∫

Ωe0

Mαβ0 NT

,γ (n⊗ aγ) N;αβ dA .(126)

For an efficient implementation of these matrices see Appendix B. Similarly, the linearizationof Geext yields

∆Geext = δxTe

(keextp + keextm

)∆xe , (127)

with the external FE tangent matrix7

keextm =

∫∂mΩe

mτ NT,α

(νβ n⊗ aα + να aβ ⊗ n

)N,β ds −

∫∂mΩe

mτ να NT

,α (n⊗ aξ) N,ξ ds ,

(128)which corresponds to f eextm, in Eq. (101.4). Here, ξ denotes the convective coordinate of thecurve ∂mΩe. For the application of surface pressure through Eq. (101.2), the correspondingkeextp can be found e.g. in Sauer et al. (2014).

B Efficient FE implementation

This section presents an efficient implementation of the above equations. First a general algo-rithm is suggested, which can be used for a variety of models like for example the Koiter model(35) and the projected shell model (71). Then a special implementation for models like theCanham model (32) is presented.

B.1 For general cases

For an efficient implementation, we arrange the tangent components cαβγδ as

C :=

c1111 c1122 c1112

c2211 c2222 c2212

c1211 c1222 c1212

, (129)

and likewise dαβγδ, eαβγδ and fαβγδ are arranged as D, E and F. Note that for model (35),C = CT, E = DT and F = FT, leading to keMτ = keTτM . The stress, moment, and curvaturetensors are written in Voigt notation as

τ := [τ11 , τ22 , τ12]T ,

M0 := [M110 , M22

0 , M120 ]T ,

b := [b11 , b22 , 2 b12]T .

(130)

Taking n as the number of control points per element, we define the (3n× 1) arrays

Laαβ := NT,α aβ ,

Lnα := NT,αn ,

Gnαβ := NT

;αβ n

(131)

7The unsymmetric rear term in Eq. (128) disappears if mτds is constant, which corresponds to dead loading.

35

and organize them intoLa = [La11 , La22 , La12 + La21] ,

Gn = [Gn11 , Gn

22 , Gn12 + Gn

21] .(132)

We thus obtain

f eintτ =

∫Ωe0

La τ dA ,

f eintM =

∫Ωe0

Gn M0 dA ,(133)

for the FE forces in Eq. (98) and (99) introduced in Sec. 4.3, and

keττ =

∫Ωe0

La C LTa dA ,

keτM =

∫Ωe0

La D GTn dA ,

keMτ =

∫Ωe0

Gn E LTa dA ,

keMM =

∫Ωe0

Gn F GTn dA ,

(134)

for the material stiffness matrices in Eq. (124). Similarly, for the geometric stiffness matricesin Eq. (126), we get

keM1 = −∫

Ωe0

bM

(a11 Ln1 LnT

1 + a22 Ln2 LnT2 + 2 a12 Ln1 LnT

2

)dA , bM := b

TM0 ,

keM2 = −∫

Ωe0

(Ln1 a

1T + Ln2 a2T)(

M110 N;11 +M22

0 N;22 + 2M120 N;12

)dA .

(135)

B.2 For particular cases

For some material models, the material tangents fit into the format

cαβγδ = caa aαβ aγδ + ca a

αβγδ + cab aαβ bγδ + cba b

αβ aγδ + cbb bαβ bγδ , c = c, d, e, f , (136)

with suitable definitions of coefficients caa, ca, cab, cba and cbb. We can then obtain a moreefficient implementation of the possible contractions within Eqs. (124) and (125). Examplesfor this case are the Canham model (Sec. 2.6.1) and Helfrich model (Sauer et al., 2016). Thesequential computation is as follows.

Precompute the (3× 3n) arrays

Nαβ := aαγ N;βγ ,

Na := Nαα ,

Nb := bαβ N;αβ

(137)

and the (3n× 1) vectors

Lαβ := NT,β a

α , Lαβ := NαTβ n ,

La := Lαα , La := NTa n ,

Lb := NT,α b

α , Lb := NTb n ,

Lαa := NTa a

α , Lαb := NTb a

α ,

Lα := NT,αn , LA := NT

,α Aα

(∗) ,

(138)

36

with bα := bαβ aβ and Aα

:= Aαβaβ. The equation marked by (*) is only needed for thestabilization schemes for liquid shells (Sauer et al., 2016) and can be skipped for solid shells.

(Depending on the type of stabilization scheme used, either Aαβ := Aαβ or Aαβ := aαβpre, where

aαβpre is the corresponding quantity at the previous load step.)

We then compute the (3n× 3n) arrays

N2a := aαβ NT

,α N,β ,

N2A := Aαβ NT

,α N,β ,

N2b := bαβ NT

,α N,β ,

L2 := aαβ Lα LTβ .

(139)

Withταβ = τa a

αβ + τb bαβ + τA A

αβ ,

Mαβ0 = M0

a aαβ +M0

b bαβ ,

(140)

andcαβγδ = caa a

αβ aγδ + ca aαβγδ + cab

(aαβ bγδ + bαβ aγδ

)+ cbb b

αβ bγδ ,

dαβγδ = daa aαβ aγδ + da a

αβγδ + dab aαβ bγδ + dba b

αβ aγδ + dbb bαβ bγδ ,

fαβγδ = faa aαβ aγδ + fa a

αβγδ ,

(141)

we thus get

f eintτ =

∫Ωe0

(τa La + τb Lb + τA La

)dA ,

f eintM =

∫Ωe0

(M0

a La +M0b Lb

)dA

(142)

and

keττ =

∫Ωe0

c dA , keτM =

∫Ωe0

d dA = (keMτ )T , keMM =

∫Ωe0

f dA , (143)

with

c := caa La LTa + cab

(La LT

b + Lb LTa

)+ cbb Lb LT

b −ca

2

(Lβα Lαβ

T + N2a − L2

)d := daa La LT

a + dab La LTb + dba Lb LT

a + dbb Lb LTb − da Lβα LαT

β ,

f := faa La LTa − fa Lβα LαT

β ,

(144)

and

keτ =

∫Ωe0

t dA , keM1 =

∫Ωe0

m1 dA , keM2 =

∫Ωe0

m2 dA (145)

witht := τa N2

a + τA N2A + τb N2

b ,

m1 := −(2HM0

a + (4H2 − 2κ)M0b

)L2 ,

m2 := −M0a Lα LαT

a −M0b Lα LαT

b .

(146)

For further efficiency, the symmetric matrices c, f , t and m1 should only be computed in uppertriangular form, integrated and then reflected after integration (i.e. after the quadrature loop).Likewise keMτ and keM should be determined from keτM , keM1 and keM2 after quadrature.

37

C Linearization of the projected shell model

In this section, the material tangents cαβγδ, dαβγδ, eαβγδ and fαβγδ that are needed for thelinearization of the projected shell model, i.e. Eqs. (65) and (78), are derived considering ξ ≈ ξ0.

C.1 Linearization of gαβ

The linearization of gαβ gives∆gαβ = gαβγδ ∆gγδ , (147)

where (see Sauer and Duong (2015))

gαβγδ := −1

2

(gαγ gβδ + gαδ gβγ

). (148)

Further, from Eq. (51) we have

∆gγδ = aεηγδ ∆aεη + bεηγδ ∆bεη , (149)

withaεηγδ :=

ga2

(δεγ δ

ηδ + δηγ δ

εδ

)+ κ ξ2 aγδ a

εη − ξ2 bγδ bεη ,

bεηγδ :=gb2

(δεγ δ

ηδ + δηγ δ

εδ

)− ξ2 aγδ b

εη + ξ2 bγδ aεη .

(150)

We thus obtain

∆gαβ = gαβγδa ∆aγδ + gαβγδb ∆bγδ , (151)

with

gαβγδa := gαβεη aγδεη , gαβγδb := gαβεη bγδεη . (152)

C.2 Linearization of ταβ

Considering that ταβ has the form (see e.g. Eq. (78))

ταβ = µ Gαβ + f(J∗) gαβ, (153)

we have

∆ταβ =

(f ′J∗

2gαβ gγδ + f gαβγδ

)∆gγδ . (154)

Taking into account Eq. (149), we obtain

∆ταβ =1

2cαβγδ ∆aγδ + dαβγδ ∆bγδ , (155)

with

cαβγδ := 2∂ταβ

∂aγδ= J∗f ′ gαβ gεη aγδεη + 2 f gαβγδa ,

dαβγδ :=∂ταβ

∂bγδ=J∗f ′

2gαβ gεη bγδεη + f gαβγδb .

(156)

38

C.3 Linearization of ταβ and Mαβ0

From Eq. (65), we have8

cαβγδ := 2∂ταβ

∂aγδ=

∫ T2

−T2

s0

[2κ ξ2 ταβ aγδ + (1− ξ2 κ) cαβγδ

]dξ ,

dαβγδ :=∂ταβ

∂bγδ=

∫ T2

−T2

s0

[−ξ2 ταβ bγδ + (1− ξ2 κ) dαβγδ

]dξ ,

eαβγδ := 2∂Mαβ

0

∂aγδ=

∫ T2

−T2

s0

[−ξ2 ταβ bγδ − (ξ −H ξ2) cαβγδ

]dξ ,

fαβγδ :=∂Mαβ

0

∂bγδ=

∫ T2

−T2

s0

[1

2ξ2 ταβ aγδ − (ξ −H ξ2) dαβγδ

]dξ .

(157)

D Linearization of edge rotation conditions

In this section, the linearization of the constraint equations, introduced in Sec. 4.4 for theedge rotation conditions, is presented for both the penalty method and the Lagrange multipliermethod. Simplifications are provided in Remarks 13 and 14.

D.1 Linearization for the penalty method

For the penalty method Eq. (108), one has (e.g. see Sauer and Duong (2015); Sauer (2014))

δn = Rα δaα, δn = Rαδaα , (158)

where

Rα := −aα ⊗ n , Rα

:= −aα ⊗ n . (159)

Along the patch interface, one can define the tangents τ and τ 0 as

τ :=aξ‖aξ‖

, τ 0 :=Aξ

‖Aξ‖, (160)

where aξ := N,ξ xe, Aξ := N,ξ Xe, and ξ denotes the convective coordinate along the interface.From this we find

δτ = M ξ δaξ , (161)

where

M ξ :=1

‖aξ‖(1− τ ⊗ τ

). (162)

We thus have from Eq. (108)

δΠeL = δxT

e f en + δxTe f en ,

∆δΠeL = δxT

e kenn ∆xe + δxTe kenn ∆xe + δxT

e (kenn)T ∆xe + δxTe kenn ∆xe ,

(163)

8Note that due to the approximation assumed in Eq. (65), the tangents in Eq. (157) lose major symmetry,but they still have minor symmetry. Therefore C 6= CT , E 6= DT and F 6= FT in Appendix B.1. If Eq. (66) isused the symmetry is retained.

39

with

f en := −∫

Γe0

ε(NT,ξM

ξ θ −NT,α d

αa n

)dS , f en :=

∫Γe0

ε NT,α d

αa ndS , (164)

and

kenn :=

∫Γe0

ε[NT,αQ

αβ N,β + NT,ξQ

ξξ N,ξ − 2 s0 sym(NT,αn⊗ nαa M ξ N,ξ

) ]dS ,

kenn :=

∫Γe0

ε NT,α Q

αβN,β dS ,

kenn :=

∫Γe0

ε(s0 NT

,ξMξ nαa ⊗ n N,β −NT

,α aαβ n⊗ n N,β

)dS .

(165)

Here, we have defined

Qξξ :=[(τ · θ) (1− 3 τ ⊗ τ ) + θ ⊗ τ + τ ⊗ θ

]/‖aξ‖2 ,

Qαβ := dn aαβ (n⊗ n) + dαa R

β + (dαa Rβ)T ,

Qαβ

:= dn aαβ (n⊗ n) + dαa R

β + (dαa Rβ)T ,

(166)

with dαa := d · aα, dn := d · n, dαa := d · aα, dn = d · n, nαa := n × aα, nαa := n × aα andaαβ := aα · (c0 a

β − s0 τ × aβ).

Remark 12: Note that here N and N denote the shape function arrays and should not beconfused with N and N . In Eqs. (164) and (165), the shape functions and their derivativeswith the indices α and β affect all control points of the elements. Here the shape function arrayN,ξ is understood to have the same dimension as N,α – i.e. appropriate zeros have to be addedto N,ξ.

Remark 13: In case of α0 = 0, for Figs. 2a and 2e, Eqs. (164) and (165) are further reducedto

f en :=

∫Γe0

εNT,α (n⊗ n)aα dS , f en :=

∫Γe0

ε NT,α (n⊗ n) aα dS , (167)

and

kenn :=

∫Γe0

εNT,α

[dn a

αβ (n⊗ n)− dαa (aβ ⊗ n)− dβa (n⊗ aα)]

N,β dS ,

kenn :=

∫Γe0

ε NT,α

[dn a

αβ (n⊗ n)− dαa (aβ ⊗ n)− dβa (n⊗ aα)]

N,β dS ,

kenn := −∫

Γe0

εNT,α (aα · aβ) (n⊗ n) N,β dS ,

(168)

with dn = dn := n ·n, dαa := n · aα and dαa := n · aα. In the case of Fig. 2.b, f en and the relatedtangents are not required.

D.2 Linearization for the Lagrange multiplier method

For the Lagrange multiplier method, using the interpolation q = Nq qe, from Eq. (113) we find

δΠeL = δxT

e f en + δxTe f en + δqT

e f eq ,

∆δΠeL = δxT

e kenn ∆xe + δxTe kenn ∆xe + δxT

e kenn ∆xe + δxTe (kenn)T ∆xe

+ δxTe kenq ∆qe + δqT

e (kenq)T ∆xe + δxTe kenq ∆qe + δqT

e (kenq)T ∆xe .

(169)

40

As seen in Eq. (113), the last integral has the same form as for the penalty method, Eq. (108).Therefore, the quantities f en , f en , kenn, kenn, and kenn are already given in Eqs. (164) and (165)with the substitution ε = 1, c0 = q (s0 + c0), s0 = q (s0 − c0); and d, d, θ are now given inEq. (114). Additionally, we find

f eq :=

∫Γe0

NTq gL dS , (170)

and

kenq := −∫

Γe0

[NT,α (Rα)T d+ NT

,ξMξ θ]

Nq dS , kenq := −∫

Γe0

NT,α (R

α)T d Nq dS .

(171)

Remark 14: Note that in the cases of symmetry constraints and rotational Dirichlet boundaryconditions, n is given and therefore Eqs. (164.2), (165.2-165.3) and (171.2) vanish.

E Analytical solution of the pinched cylinder with rigid enddiaphragms

This appendix provides the analytical solution following the approach of Flugge (1962) for thepinched cylinder with rigid diaphragms. It is based on the double Fourier representation of thedisplacement field and the applied loading.

Accordingly, if the shell deformation is described by u, v and w in the axial, circumferentialand radial directions, respectively, the equilibrium of the system satisfies the following systemof partial differential equations (Flugge, 1962)

u′′ +1− ν

2u∗∗ +

1 + ν

2v′∗ + v w′ + k

[1− ν

2u∗∗ − w′′′ + 1− ν

2w′∗∗

]= −pxR

2

d,

1 + ν

2u′∗ + v∗∗ +

1− ν2

v′′ + w∗ + k

[3

2(1− ν) v′′ − 3− ν

2w′′∗

]= −

pφR2

d,

ν u′ + v∗ + w + k

[1− ν

2u′∗∗ − u′′′ − 3− ν

2v′′∗ + w

′′′′+ 2w′′∗∗

+ w∗∗∗∗ + 2w∗∗ + w]

=pr R

2

d,

(172)

with k := T 2/(12R2), d := (E T )/(1− ν2), where R and T are the radius and thickness of thecylinder, respectively, ν is Poisson’s ratio, E is Young’s modulus and the partial derivatives aredenoted by (•)′ := R∂(•)/∂x and (•)∗ := ∂(•)/∂φ. Further, as shown by Bijlaard (1954), if Nf

concentrated radial loads are applied at the middle of the cylinder and they are equally spacedalong the circumferential direction, they can be expressed in terms of the double Fourier seriesas

px = 0 ,

pφ = 0 ,

pr =∞∑m=0

∞∑n=1

prmn cos(Nf mφ) sin

(λx

R

),

(173)

41

where λ :=nπ R

L, L is the length of the cylinder,

prmn = (−1)(n−1)/2NfP

πRL, if m = 0 and n = 1, 3, 5, · · · ,

prmn = (−1)(n−1)/2Nf2P

πRL, if m 6= 0 and n = 1, 3, 5, · · · ,

prmn = 0 , otherwise ,

(174)

and P is the magnitude of the loads.

It should be noted that generally the representation of concentrated loads by a series such asEq. (174) has numerical deficiencies. The prmn terms alternate in sign and do not decay asm→∞. Although the series still converges in theory, it does not converge numerically due toinaccuracies caused by the Gibbs effect.

In general, Eq. (173) can represent any symmetric pressure distribution. If the cylinder ispinched by two opposing forces applied at the middle, the Fourier components can be found bysetting Nf = 2, φ = 0 or φ = π and x = L/2. The corresponding solution for Eq. (172) can befound by assuming the following ansatz for the displacement field

u =∞∑m=0

∞∑n=0

umn cos(Nf mφ) cos

(λx

R

),

v =

∞∑m=1

∞∑n=1

umn sin(Nf mφ) sin

(λx

R

),

w =∞∑m=0

∞∑n=1

umn cos(Nf mφ) sin

(λx

R

).

(175)

As seen, Eq. (175) satisfies the boundary conditions of the problem with rigid diaphragms orsimple supports, i.e. v = w = 0 if x = 0 or x = L. Plugging Eq. (175) into Eq. (172), thefollowing system of equations is obtained

[λ2 +

1− ν2

N2f m

2 (1 + k)

]umn +

[−1 + ν

2λNf m

]vmn +

+

[−ν λ− k

(λ3 − 1− ν

2λN2

f m2)]

wmn = 0 ,[−1 + ν

2λNf m

]umn +

[N2

f m2 +

1− ν2

λ2 (1 + 3 k)

]vmn +

+

[Nf m+

3− ν2

k λ2Nf m

]wmn = 0 ,

[−ν λ− k

(λ3 − 1− ν

2λN2

f m2)]

umn +

[Nf m+

3− ν2

kλ2Nf m

]vmn +

+[1 + k (λ4 + 2λ2N2

f m2 +N4

f m4 − 2N2

f m2 + 1)

]wmn =

prmnR2

d,

(176)

which can be solved numerically for the unknown coefficients of the Fourier series (umn, vmnand wmn) in Eq. (175).

42

F Skew surface meshes

This appendix explains the procedure to create skew meshes out of regular meshes for a singlepatch, which is used in Secs. 5.2.3 and 5.2.1. Fig. 13a shows the knot grid of a regular rectangularmesh on a single patch. Assuming the knot space has the dimensions L × S in general. Theskew mesh is obtained by two steps:

Step 1. Distorting the regular knot grid by the mappingYd(s, η) = Y (ξ, η) ,

Xd(ξ, η) = X(ξ, η)− 2∆S

Lη

(1− 4

S2ξ2

),

(177)

where (X,Y ) and (Xd, Yd) are the coordinates of the regular and distorted knot grids, respec-tively; ∆S = L/2 tan θ, where θ is the angle of the distorted mid-line w.r.t. the η-axis (seeFig. 13b). The distortion or skewness ratio, r, is defined by

r :=S1 − S2

S1 + S2=

2∆S

S=L

Stan θ , (178)

where r ∈ [0, 1].

Step 2. The positions of control points are recomputed with the corresponding distorted knotgrid using the knot insertion algorithm of Hughes et al. (2005). Here, the Bezier extractionoperator is computed approximately at the center point of the knot grid.

A E B

C F D

S/2

L/2

a.

A BE

C F D

b.

Figure 13: Distortion of a regular knot grid. a: Regular knot grid and b: skew knot grid. Thecontrol points are then recomputed using the knot insertion algorithm (see e.g. Hughes et al.(2005))

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