J

J. Siruel. Meek, I (/), 113-131 (/972)

Mixed Conjugate Finite-Element

Approximations of Linear Operators

J. T. ODEN and J. N. REDDYRESEARCH INSTITUTE

TilE UNIVERSITY OF ALABAMA

HUNTSVILLE, ALABAMA

ABSTRACT

The concept of conjugate approximalion functions is generalized to mixedformulations of linear boundary value problems. Biorthogonal basis functionsare generated using two distinct finite-element models of the domain of a givenfunction. Ylirious specilll cases arc considered. It is shown that four distinctmodels can be developed corresponding to a single linear operator. A com-parison of the concepts of the hypercirc1e and conjugate approximation isgiven.

I. INTRODUCTION

In earlier investigations of the theory of conjugate finite-element ap-proximations [1-6] it has been assumed that the global approximationfunctions 'Pa(X) and their corresponding conjugate approximation functionstp'(X) belong to the same space. This has led to the concept of "consistentstress approximations" for finite-element models of displacement fields inlinear elasticity [3-6].

113

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114 J. T. Odell alld J. N. Reddy

For example, suppose that the approximation ii of a function II is a linearcombination of a set of G linearly independent functions cp,(X):

),

(I)

(therc IX is summcd from I to G). Then, if (II. v) = (v. II) is an inner producton %(ii E £'), a system of biorthogonal basis functions (P'(X), having theproperty

(2)

can be constructed using the relations

(3)

whercin cPa is the inverse of the 1/ x 1/ fundamental matrix

(4)

Then. for a given II. the proper coefficients a' in (I) are immediately given by

Alternately,

a' = (II. cp') (5)

(6)

wherein aa = (CPa. II). We can then proceed tc use (2)-(6) to derive ap-proximations of linear operators. solutions of various boundary-valueproblems, derivatives, integrals. stresses, etc. (e.g .. [3]).

By introducing (3), however, we have limited the choice of conjugateapproximation functions to the G-dimensional subspace spanned by CPa(X),In the present paper. we investigate the consequences of removing thisrestriction. In particular, we develop herein a generalization of the theory ofconjugate approximations in which (2) holds, but cpP are not determined by agiven set of functions CPa' We show that this approach leads to "mixcd"finite-element approximations and that, depending upon the properties of thefunctions CPa and cpP, at least four dilferent finite-element models of a givenlinear operator can be derived. The latter observation leads to the question ofstability of the finite-elemcnt operators; however, we shall postpone consid-eration of this question until a later paper. Finally, we discuss the relationshipbetween the concept of conjugate approximations and the method of thehypercircle [7, 8].

..A'fixecl COlljllgate Finite-Element Approximations

II. DUAL AND CONJUGATE SPACES

115

To fix ideas, we shall review briefly certain definitions and concepts relatedto dual and conjugate spaccs.

Consider two linear vector spaces. '11 and f, and suppose that there isdefined on ql ® f a linear mapping that associates with ordered pairs ofelements II, v (II E 151I, tJ E f) rcal numbcrs. If (II, v) denotes the real number sodefined, corresponding to the pair II and v. then f is the dual space of iill(denotcd f = .:lit') if the following hold:

(u, IXVt + fJ(2) = rx(lI, L't) + P(u. (2)

(:wt + {JU2' v) = rx(lIt. v) + P(U2' v)

(II, t') = 0 for fixed II and all v =;> II = 0

(II, v) = 0 for fixed tJ and all II =;> tJ = 0 (7)

Clearly (II. v) is a linear functional on ql. It follows that f (or il71') is equiva-lent to (isomorphic to) the spacc :171* of all linear functionals on iili. The spaceql* is called the conjugate space of <1« and it suffices our purposes here totreat JU' as synonomous with '111*.

Now suppose that the linearly independent elements cp" CP2"'" CPnprovide a basis for the spacc v71 and that the elements Xt, X2

, .•. , t' provide abasis foro/I' = f. Then every element II E °Il and v E f is of the form

(8)

To determine the coefficients 0', ba (c.c = I, 2 ..... n) for given u and t', weassume that {CPa} and {;(} constitute dual or biorthogonal bases: i.e.,

(9)

where D~ is the Kronecker delta. Then it follows from (8) that

(10)

It is clear that if X' == cp', where cp' is given by (3), then (9) and (10) aresatisfied. Then the theory prcscnted in [1-5] follows. However, (10) is ob-

116 J. T. Odell and J. N. Reddy

tained from (9) without assuming that (3) holds; that is, Xa need not belongto JII.

III. GENERALIZED CONJUGATE FINITE-ELEMENTINTERPOLATION FUNCTIONS

We now wish to examine the application of the notion of dual and conju-gate spaces discussed previously to finite-element approximations of functionsand of solutions to linear boundary-value problems. The present developmentseeks to generalize the theory of conjugate approximations presented carlier[1-5). Consider a region rJi of k-dimensional Euclidean space tek which is thedomain of functions II(X), X = (XI' X2, ... , Xk) E rJi, of a class to bespecified. In general, the values of II(X) at a point X E ,tJt may be scalars,vectors, or tensors of any order; but, for simplicity, it is convcnient totemporarily regard II(X) as scalar-valued.

To conslruct finite-clement approximations of u(X), we procecd as de-scribed in [3] and replace 9i by a finite-element model fj which consists of acollection of E subregions t., called finite elements, connected continuouslytogcther at nodal points and along interelemcnt boundaries. We idcntify Gnodal points xa (a. = I, 2, ... , G) in the connccted model iJi. Other prop-erties then follow the plan discussed in [3].

In the present study, however, it is essential to first eOllsider the possibility oftwo different finite-element models of the same region gf-one model !Jtrepresenting the domain of approximations to functions belonging to afunction space OU and a second model ~ representing the domain of functionsbelonging to the dual or conjugate space "Y of ijll.

Let '11 and "Y denote linear vector spaces, the elements of which arefunctions u(X), and veX), respectively. The domain of the elements II(X) E OUand veX) E -y is the same closed region rJi c t1k• Now let cW and .y denote finitedimensional subspaces of lI/I and -y, respectively. which are spanned by thesets of G linearly independent basis functions C{Jl(X), CP2(X), ...• CPc,(X) E cWand i (X), leX), ... , xG(X) E.y. Then the projection nil = ii of an elementII E iff into the subspace Jfj is of the form

ii = d'cp,,(X) (II)

where a. is summed from I to G, whereas the projection fiv = jj of an elementt) E "Y into the subspace .y is of the form

ii = b.r(X) (12)

Mixeil Conjugate Fillile-Elemelll Approximatiolls 117

We assume further that there is defined on Oft ® l' (and u7i ® ji") a bilinearmapping (u, v) of ordered pairs of elements into real numbers that satisfies(7). Finally, and most important, we assume that the functions cp,(X) andx'(X) are biorthogonal basis of qj and ji"; i.e., they satisfy (9).

If (II) and (12) are to describe finite-element approximations on ;11, thebasis functions C{J,(X) and X'(X) must be endowed with the usual finite-element properties.

To obtain CPa(X):

I. ;11 is replaced by a finite-elementmodel !it consisting of E con-nected subregions re' called finiteelements, joined continuouslytogether at inter-element bound-aries

To obtain r(X):

1. t1l is replaced by a finite-elementmodel f1 consisting of E con-nected finite elements Te' joinedcontinuously together at inter-element boundaries

(13)

2. A finitc number G of points in iJiarc identified called nodes: theseare laheled xa

ex = I, 2, .... G

3. A typical element re is isolatedfrom the collection iJt, and localcoordinate system Xe is estab-lished. A finite number Ne oflocal nodal points N are identi-fied re and are labeled x~,N = I. 2, ... , Ne.

4. The connectivity and decompo-sition, respectively. of the modelfj is established by the mappings

2. A finite number G of nodalpoints in .~ are identified andlabeled Xa

a. = I, 2.... , G

3. A typical element Te is isolatedfrom the collection 92, and alocal coordinate system x" isestablished. A finite number liJ e

of local nodal points N areidentified in Te and are labelediN' lit = I. 2, .... N e'

4. The connectivIty and decom-position, respectively, of themodel ;j is established by themappIllgs

(e)

X' = A~xl'e) (14)

118

where

J. T. Odell and J. N. Reddy

where

(e>

II! =.V

if node N of element Ie

is coincident with nodeIX of the connectedmodel° if otherwise

if node IX of ;j iscoincident with nodeN of Ie in the con-nected model° if otherwise

(f)7'\i _HR -

(f)

A~ =

if node N of elementIJ is coincident withnode a. of the con-nected model° if otherwise

if node ii of 91 iscoincident with nodeN of fJ in the con-nected model° if otherwise

(15)lel

and AN is the transpose ofleIn:.

5. If u(X) E Wand lIe(X) is therestriction of ii(X) to Ie. we ap-proximate II.(X) locally accordingto

/I_(X) = /I~t/1~(x)

where u~ is the value of u_(x) atnode N of element re' and N issummed from I to Ne, and thelocal interpolation functionshave the properties

t/1~(X) = 0, x rt Ie

Ie>and A: is the transpose ofIe>n~.

5. If veX) E j7' and ve(x) is therestriction of vex) to fe' weapproximate v,,(x) locally ac-cording to

( 16)

where vN is the value of ve(x) atnode N of element fe' and Iii isslimmed from I to N.. and thelocal interpolation functionshave the properties

t/1:(x) = 0, x rt re(17)

Ne

I t/1J.(x) = 1N=I

6. Globally (except on a set ofmeasure zero), we have

ii(X) = u'cpiX)

Ne

I l/1:(x) = I.R=t

6. Globally (except on a set ofmeasure zero), we have

veX) = v..x'(X)£

= IlIe(x)e

E= L Ve(X)

e(18)

Mixed Conjugate Finite-Elemelll Approximations 119

.. Examples of such models are shown in Fig. I. Note that at this point werequire only that the models of /leX) and v(X) have the same number of"degrees of freedom." For first-order representations of the type described in(18), this property can be insured by endowing rh with the same number ofnodes as ~: however, the location of these nodal points in !Jt need notcoincide with those of rho We shall add further restrictions on the topology of:i subsequently.

Filt. 1 Finite-element models of a two-dimensional region ~.

The basic requirement of the basis functions cp,(X) and X'(X) is that theysatisfy (9). Later, we prove that such functions do indeed exist by citingspecific examples.

IV. SOME PROPERTIES OF GENERALIZED CONJUGATEFINITE-ELEMENT APPROXIMATIONS

A. General

Without further restrictions on the character of the models grand ~, we

120 J. T. Odell and J. N. Reddy

can only list rather general properties of the finite-element approximation iiand v of (17). Consider, for example, the form of the scalar product (7) on thesubspaces rfi and j?

E (p) E (f)

<- -) (a P) _ a (~n.N,/,(e) ~ riP ,/,N )1/,11 = Ii (Pa' VpX - Ii Vp £.... Ua'l'N , £.... UN'I'(f)

e= I 1= I

E E (e) (f)= /I'V ~ ~ O,Vnp(./,(e) ./,N )/l £.... £.... /I. N 'I'N , 'I'(f)e 1

( 19)

wherein ex,p = I, 2, ... , G; N = I, 2, ... , Ne. Due to the localized char-acter of the functions ¢<;) and ¢(~) described in (17), it is clear that

Otherwise, the scalar product (20) defines an Ne x N 1 matrix

(ef'lf),,/it /,/.(e) ./,N )UN = \'I'N' '1'(1)

(20)

(21 )

and (19) reduces to ..(22)

B. Restricted Models

With the discrete model of the scalar product now given by (22). we canproceed to generate a variety of conjugate finite-element approximations.However, we shall postpone a discussion of such general models to a laterpaper and, henceforth. we confine our attention to the important special casein which

(23)

that is, essentially the same model ~ is used to model domain of functions in(el

17' as is used to model the domain 91 of clements of rfi. Then n~is simply(e)

the transpose of O~ and we may write

(e) (e)

ON = A~ (24)

Mixed Conjugate Finite-Element Approximations 121

(e)

where AN is the connectivity matrix described in [3]. It follows that (22)rcduces to

F. (e) (e) (e)

(ii, v) = 112Vp L n~A~r.(\~e

where

(e).(\M = (",(e) ,/,.\, > = 0 if X ~ rN 'I" N ' 'I' (e) 'F e

In view of the biorthogonality condition (9),

so that

(ii, v) = u'va

(25)

(26)

(27)

(28)

If we design the functions t/J<;)(x) and t/J~)(x) so that (9) is satisfied locally,i.e ..

then (27) reduces to the composition

f: (e) (e)

c5~ = L n~A~e~1

(29)

(30)

Moreover, if (29) holds. we have (except possibly on a set of measure zero)

(31)

whcre II(~) and v~) are the valucs of lI(e) and vee) at node N [sec (17)1·

C. Other Properties

We outline briefly a number of intrinsic properties of the generalizedconjugate approximation functions.

122 J. T. Odell and J. N. Reddy

1. Distributions. Suppose we take II = t5(X - A), A E at where t5(X - A) isthe Dirac delta function defined by

roo, x = At5(X - A) = /0. x ~ A

and

tf(X) t5(X - A) dX = f(A) (32)

where A is a specified point in the domain !]f. Then the projection of c5(X - A)on the space Jij is

1t()(X - A) = ~acp,(X)

1t*C5(X - A) = ~aX'(X)

wherein. according to (10)

~' = (t5(X - A), xa)

~, = (cp" J(X - A»If we definc the scalar product of two functions u and v by

(II. v) = [ IIV d:Jl.~

thcn it follows from (34) that

(33)

(34)

(35)

~' = ,-'(A) (36)

that is, the valucs of thc conjugate basis functions t' and CPa at the nodal poil/ts:C arc the coefficients of thc projections of thc dclta functions t5(X - X') injl' and JiJ, respectively.

2. Nodal Values, Suppose we define, as in Eq. (17),

cp,(Xp) = c5~

x'(XP) = 0'11 (37)

AlL'red Conjugate Fillite-Element Approximatiolls

Then

II(X) = u'~o,(X)

II(XP) = 1I'~'a(XIl) = IIIl

and

t'(X) = vat(X)

v(Xp) = t1a1.'(XIl) = vaaaP

Then

if and only if a'P = J'p.

123

(38)

(39)

3. Moments and Volumes. To demonstrate another property of the conjugateapproximation functions, we introduce moments M, and Ma defined by

11-/, == (ffJ., I)(e)

= L n~(lfr~'), I)•

(d

= ~n~III~)

•where

Similarly, we may define

M' == (I. t>(~ (M

= L AN( I. cp(~» = L ANmi~)e e

where

Then. if (. , .) is given by (35),• • • G

\d9f. = J I· d3f = \ L ffJa(X) dJP

.~ ~ .~,=I

(40)

(41)

(42)

(43)

124 J. T. Odell alld J. N. Reddy

because the functions CPa(X) satisfy the condition (17). Now observe that

(44)

where V is the volume of !fl.. Similarly,

(45)

V. LINEAR OPERATORS

We now arrive at the important problem of conjugate finite-elementapproximations of linear operators. Consider the linear operator !i' encoun-tered in boundary-value problcms characterized by equations of thc form

9'(/1) = f (46)

in which, for the moment, /I is to satisfy homogeneous boundary conditions,We seek approximate solutions of (46) in the subspaces rfi and j? of the

forms

(47)

whcre the repeated index a. is summed from I to G.It is natural to examine projections of !i'(II) into fjj and j? in which ii

appears as the argument of !i'(II) Tather than /I; that is,

or

wherein

n[!i'(u) - f] = n[!i'(ii)] - f'cp,(X) ~ 0

n*[!i'(u) - f] = n*[!i'(ii)] - fa.X:(X) ~ 0

f' = (I, X:) and fa. = (cp" I)

(48)

(49)

(50)

Following [2], we assume (at least approximately) the commutativity of the

Mixed Conjugate Finite-Element Approximations

.. operators 7l and !R and n:* and !R:

125

Then

(51)

and

where

L! = (!l'(P,. 'l)

N,p = (CPa' !l' cpp)

Map = (!l'X~, l)

P,/ = (cpp. !i'r) (54)

Therefore, if tfi alld j? are distinct. there are fO/lr distinct approximations of theoperator!i' COl/sistent with (47) and (51).

Introducing (52) and (53) into (48) and (49), and equating like coefficientsof the basis functions cp,(X) and, separately, X'(X), we obtain the followingdiscrete analogues of (46):

L!a' - jP = 0

M'Pba - jP = 0

Np,a' - /p = 0

P/b, - fp = 0

It is clear from (55)-(58) that

L!a' = M'Pba

Np,a' = P/ba

Define

[L!rl == L;I·[I

[MaP]-1 = M- .p

[Naprt == NaP

[ppar1 == Pi'"

(55)

(56)

(57)

(58)

(59)

(60)

(61 )

(62)

(63)

(64)

126

Hence, from Eqs. (59) and (60) it follows that

where NJJP is given by Eq. (63).

A. Local Approximations

J. T. Ode" and J. N. Reddy

(65)

Wc assume that the elcmcnt re coincidcs with re. Then, locally, wc candefine

(e)

"

M _ «(.f),I,(e) ./,M>N - .L 'I' N ' 'I'(e)

(e)_ (/,(e) (D.I,(e»

lIMN- II'M,.L'I'N

Then. from (54). it follows that

(e)MS «(D./,M ./,N)

11/ =.L 'I' (e)' 'I' (c)

(l')p'.\! = (./,(e) (0./,.\1.)

N 'I' N • .L'I'(,') (66)

£ (,,) I: (e)

L/' = (2'tP.' XP) = (2' L n~IjJ'N), L Q~IjJ~f),'= I 1= I

I: I: (,,) (I)

= L L n~Q~(2'IjJ<;), IjJfJ»e= I Ie I

and

(67)

" E (I) (f)

M'P = (2',/:. l) = L L Qf.,rI~{(2'1jJl'f)' ljJi~}) (68)l=tf=1

£ F. (e) (el

N,p = (CPa' 2'CPp) = L L n~n~((ljJc;', !t"/JW) (69)e= 1 c= I

In the special case in which the conditions (23) and (24) hold, we have theimportant results

(e) (el

L! = L n~A~fl,~Ue

(e) I.,) (e)

M'P = L ANA~tlll.\!Ne

(7\ )

(72)

Mixed Conjugate Finite-Element Approximations

(e) (e) (e)

Nap = r.n:n;tnMNe

( ..) (e) (e)

P'% ~ nNA' 'Mp = '- H, .\tPNe

127

(73)

(74)

(e)

where, for example. fIr represcnts the local component of L/ relative to theelement (e).

B. Specific Linear Operators

Here we give some specific examples of the linear operators. One of themost familiar linear operators is the partial differential operator.

(75)

We assume that the partial derivatives of the basis functions cp,(X) andX'(X) exist and belong to the spaces J{j and of', respectively. (This requirementusually holds only approximately.) Then, according to Eq. (54),

Also, if

we have

a a!l' == ax; ax}

(76)

(77)

128

and

J. T. Odell and J. N. Reddy

Likewise, if

a2u D'fJV ' ).(X) AafJV ).(X)axjax, ~ al ).pjll X = U, ).PjU,X

~ Vapj.11).uarp).(X) = t5/;.1~).uaCP).(X)

2'u = rK(s, X)u(s) ds

(78)

(79)

then

Lit = (2'Ip" l) = ( rK(s, X)CPa(s) ds, xp)

M'fJ = (2'1..', XfJ) = ( rK(s, X)t'(s) ds. l)NaP = «(Pa' 2'CPp) = (CPa, rK(s, X)cpp(s) dS)

Pit = (cpp, 2'X') = (cpp, r K(s, x)xa(s) dS) (80)

Obviously, many other examples could be cited. Since the procedure forconstructing such approximations is amply demonstrated by the aboveexamples, we shall not elaborate further here.

VI. RELATIONSHIP OF CONJUGATE APPROXIMATIONSTO THE HYPERCIRCLE CONCEPT

We shall conclude this investigation with a discussion of the relationshipbetween the concept of generalized conjugate approximations and the conceptof the hypercircle of Prager and Synge [7] and Synge [8]. Consider theboundary-value problem

9'(g(X) 2'w(X» = I(X) in .~

&!J(w) = heX) on iJJP 1 ~(w) = j(X) on iJJP2 (81)

Mixed COl/iI/gate Fillite-Elemelll Approximatiol/s 129

where .!f!(g.!f!) is a linear clliptic partial differential operator of order 2p, &I(w)is a linear differential operator of order p- I describing the principal boundaryconditions, and rc(w) is a linear differential operator of ordcr 2p- I describingthe natural boundary conditions. Here a:"1t = 0.111 + C.1t 2'

Thc problem (81) can be "split" by defining two linear manifolds CJ71 and "Y.the elements in v71 being those functions .!f!1I(X) where the II(X) are such that

~(II) = II on oBi I

Likewise, if I' E "Y, then

g.!f!(v) = fin Bi and lC(v) = jon o3i2

(82)

(83)

It is assumed that thcre exists a unique solution 1\'* of (81) which lies at theintersection ofJ71 and "Y.

The scalar product defined on JIf ® "II" is defined by

[II, v] = (II, .!f!(g.!f!I'» = (2(11), g.!f!(v» + k(lI, v) (84)

where k(lt. v) is a bilinear functional representing certain boundary values of1/ and v and their derivatives. Then the "energy" norm is III/II = [1/, 1t]1/2 andthe squared distances from arbitrary elements 1/ E '11 and v E "Y from the exactsolution w· are

d~(I/, I~'*) = 111/ - 1\'*11 (85)

To obtain approximate solutions of (81). we consider two finite-dimen-sional subspaces u7/ and 17 of 11 and "Y (actually. in finite-element approxima-tions the spaces (lfj and j? are selected so that (82) and (83) are only ap-proximately satisfied: hence. C/7/ and 17 need not actually be subsets of 11 and"Y). In the hypercircle method. elcments in 1i are assumed to be of theform ii = iio + Ii. where iio satisfies (at least approximately) (82) and Iisatisfies homogeneous principal boundary conditions on o3t I: i.e .. Ii is anelcment of a linear spacc qi' and iio describes a translation of Ii'. Likewise.v = Vo + V. where Vo satisfies (83) and v satisfies homogeneous conditions ofthe form (83). Then v belongs to a linear vector space ..p' and Vo describes atranslation. It can then be shown that Jii' is orthogonal to j?'.

In contrast. the method of conjugate projections takes ii and v to be of theform (11) and (12) and the coefficients 0', b,. and the basis functions cpa(X),X'(X) can he adjusted so that (82) and (83) are satisfied.

Turning now to (48), the approximate solutions ii. v can be determined bychoosing the coefficients d', ba such that the following distances are mini-

130

mized:

d~(ii,II'*) = llii - 11'*11 d~(v. 11'*) = Ilv - 11'*11

J. T. Odell and J. N. Reddy

d~,v(ii, v) = II ii - vII (86)

It is easily shown that minimization of cach member of (86) leads directly tothe approximations (55}-(58). For example. introducing (11) into the firstequation of (86), we havc

d~(ti, 11'*) = (a'cpa - 11'*, aP !l'(g!l'cpp) - £'(g£'II'*»

= a'aP(cp" £'(g£'«1p» - 2a"'(CPaJ) + 1111'*112 (87)

Or. using the notation of (50) and (54),

(88)

Hence, d;(ii, \\'*) is a minimum if a2d,,/caP = 0; or

(89)

which is idcntical to (57). Similarly, minimization of d,7(v. 11'*) of (86) withrespect to ba gives (56). Equations (55) and (58) can be obtaincd by seckingthc minimum of thc second equation of (86), first holding a' fixed and varyingba and then holding ha fixed and varying (l'.

Wc remark that it is also possible 10 establish bounds on 1111'*11 in o'li and "f'as described in [8], and that a generally better approximation to 11'* can oftenbe obtained by averaging the approximations ii and ii which minimize thefirst and second equations of (86).

ACKNOWLEDGMENT

The support of this work by the U.S. Air Force Office of Scientific Researchunder Contract F44620-69-C-0124 is gratefully acknowledged.

REFERENCES

1. J. T. Oden, A general theory of finite elements. I: Topological considcrations, Int.J. Numer. Met". Eng. 1: 205-221 (1969).

2. H. J. Brauchli and J. T. Odcn. Conjugate approximation functions in finite clementanalysis. Q. Appl. Math. 29 (3): 65-90 (1971).

3. J. T. Oden, Finite Elements of Nonlinear Continua, McGraw Hill, New York. 1971.

Mixed COl/jugate Fil/ife-Elemellt Approximatiol/s 131

4. J. T. Odcn, Theory of conjugatc projections in finitc clement analysis. NATO Adl'al/ct'llSI/ICb'll/stifllle 01/ Fil/ite Elemelll A1ethods ill COl/lil/lilimMechtll/ics. Lisbon. Sept. 1971.

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Receired August 19. 1971