ICCAD

International CCnlre for Computer Aided Design

secondINTERNA TIONAL SYMPOSIUM

on

FINITE ELEMENT METHODSin

FLOW PROBLEMS

..'.

Santa Margherita Ligure (!taly) June 14· June 18. 1976

rv~IXED HYBRID METHODS FOR THE ANALYSISOF POTENTIAL FLOW

J.T. Oden) 1.K. Lee(U.S.A.)

SESSION 6: CONVERGENCE AND STABILITY PROBLEMS

<"0 •. P"I<'lOfJ ...,..ere, £4 3_*,fifr1Jfr'8$¥t-$4Mf'A_!;ffW#.iJfjO Ek' .¥8i!P.,./. .•..•• M~WEblW" '''''''11*''*''''''"''""" "".".•.•.. ' ,_ ._ . 'I -.-,1. ~ .. I!,,"'" - I ,:--, . :.-; •.. '" :"~~," "'~ /~' __ '"

. . .~". ~.,\,'

499

MIXED-HYBRID METHODS FOR THE

ANALYSIS OF POTENTIAL FLOW

J. T. Oden and J. K. Lee

.Texas Institute for Computational MechanicsThe University of Texas at Austin

Austin, Texas, U.S.A.

SUllllllarJ

We describe in this paper a class of mixed-hybrid finite elementmethods for the analysis of flow problems. Emphasis is given to ques-tions of accuracy, numerical stability, and to criteria for selectingappropriate basis functions which guarantee convergence.

Introduction

Hybrid fin~te'element methods are known to have rather delicateconvergence properties, despite the fact that they also have severaladvantages over conventional methods in terms of weaker continuityrequirements on base functions, In this paper, we describe some re-sults whicn provide a basis for selecting shape functions for two im-portant classes of mixed-hybrid finite elements described below. Whileour general method of attack is applicable to a wide class of physicalproblems, we describe them in connection with the problem of determin-ing stream lines, Specifically, we consider the model problem,

wherein

- M • f in II ]I It- on an2(1)

, = 81 on all and an - g2

a2 a2 26 • - + - .. Laplacian in~h2 ay2

• - stream function

2n ..open, bounded domain in~ with boundary

an • anI U an~, anl () an2 _ ,.

f, gi' 82 = data

If anl• li\ we require that f fdx + ¢82ds ..O. Since we also want toconsider mixed finite element methods, we shall actually be concernedwith the equivalent system,

...~ ;p,: j40Ef4 ; ·*"8"'~·;'''SA;i*. - ....... "'. - ...., .... " .. ,"~." ,•• p.y.'-s+§4ilfMA''''''.'" ...,'I.' .• 1" <. <" .• ' .' . •. :.... :

..J

V~ = v } in fl }-V'y=f (2)

1 and 2~ ..gl on all y • ~ " g2 on an

where! is the velocity vector (V~ = grad ~).

500

.' ):....;_".i.~~·.~~·",. .~_..

Mixed-Hybrid Finite Element Methods

I

(3)

(6)

(5)

(4 )

(qe~e + qe~e)ds}

v (x,y)-e -

• V. + (V~ - v ) "v )dxdye e -e -e

fljleF(~)

• ::t (x,y)e e ,

where ~ .. (.,v,q) and, for simplicity in notation, we have denoted,with E being the number of elements

E E

A(~,1).. [Ae(~JI) • [ {r (Yee-1 e"l )ne

To obtain the element matrices corresponding to this choice ofunknowns, we use the three-field variational principle associated withthe functional

Here ~N' w , and ui contain polynomials of degree k, 1', and t respec--Q i

tively, and an denotes a boundary segml!nt I>(~tweenvertices of thee

finite element n, We further require2that Q (s) satisfies the boun-dary condition, i,e" Q (s) = g2 on an n an ~ "An example of such anelement is shown in Fig~ lao e

I. Model I. Here we approximate ~ and v independently on the interiorof 8 finite element II and the flux - v :n = q on the boundary an :

e - - e

We shall consider two types of mixed-hybrid methods associatedwith (2):

.."

':

'.'iifKf·~·r.ltc

..,.".

501

(2)

:iated

I interiorall :e

X,y)

(3)

respec-the

e boun-lIuch an

ce oC~ed with

(4)

Pted,

x

x

Q • v· n (poly. deg. < t)

t (poly. deg ~ k)

v • grad ~(poly. deg,~ r)

(a) Type I Element

t (poly. deg. ~ k)

v • grad ~(poly. deg.~ 1')

w •• (poly. deg. ~ t)

(b) Type II Element

Figure 1. Two types of mixed-hybrid finite elements.

!CdyThe functional J of (4) assumes a stationary value whenever, and vsatisfy (2), or, variationally, whenever

A(~,~) - F(~) 'lIEU (7)

where U is the space of admissible variations (to be defined later).Thus q behaves as a Lagrange multiplier defined on the total interele-

e -ment boundary

;1

(12)

(11)

(9)

(8)

fSUU.'D'JI42. F~_""""",*,",,·,·_M - 1. =.:7 "'.<' " =~''''~?'''';:''

..l'

J

"' [ wjBj (5)

j-l

Ian (1 ane' In this case, the perti-

-.- .-..

E

Ue=l

1 ~i ~H

r •

.,

w - +\ : We alii ee

I. .....,\',.

~ - (~,y,w),E E

B(~,~) - [ Be (~,~) = [ <j (-$ V . v - ~ V • V - v . v )dxdye~ ~e e- -e -e ee"'l e-l n

e+ ~n (v . nw + v • n; )ds} (13)-e - e -e ~ e

e·

THa - Fb = 0; H b + Lc = r

where We is such that We a gl onnent equations are as follows:

11. Model II. Model II differs from I in that the stream function onthe boundary is approximated rather than the flux. We thus use inter-ior approximations of ~ and y as in (3) (polynomials of degree k andir) and an independent approximation of ~ on the boundary segments aneusing polynomials of degree t:

Here !, ~,and £ are vectors of coefficients of the approximan~s ~,y, and Q, respectively, and

"Na -1 Vlj/N'~a dxdYi FaB ..f ~a· ~B dxdYi LNi - [{ /N IIidsn. II j an j

e e e (10)

PN - fa felj/Ndxdy i G~ - j( gllJi(s) dsanln all

e e

and

Introduction of (3) into (7) leads to the following linear sys-tems for each element:

where M k J for triangles, 4 for quadrilaterals, etc.

502

Jfa', ... db7e,iir:«fflS;'id'ri¢;i;;:f'cle'dBw, fflsftd&tlwh;iaixt:1ili;i'#M- ..·1>it!l';4e'5S firi t' 1'£+kftit·SiYrH*fl'i'tS: .....&iI'1m w oft(\'<~.

. 503

(8) 1 Il..W ds}.dxdy + 2-, e (14)an " alle

Variationally.

B(w,w) v wE V (15)

r sys-

(9)

where V is now a new space of admissible variations (defined preciselybelow).

Mathematical Properties

Problems (7) and (l5) fall into the general framework of abstractvariational boundary-value problems of the following type: find an

.element u in a Hilbert space H such that

where a(·,·) is a bilinear form from H x H intom and f is a continu-ous linear functional on H.

~8 t,a(u,v) '" f(v) vvE.H (16)

~i ds

(10)

It is known that (16) possesses a unique solution u that dependscontinuously on the data f whenever the following conditions hold

Here M and y arc positive constants and (17)1 holds V U,v E. H. More-over, when similar conditions hold in a closed subspace Hh of H, the

, Galerkin approximation U of u exists which satisfies:tion onJ inter-k andi\ts alle

(11)

a(U,V) .. fey)

and the error e • u - U satisfies the inequality

H -IleliH ~ (1 + :y)llu - uIIHh

(18)

(20)

perti-

(l2)

V )dxdye

where Yh is the approximation of y in Hh' Proofs of these assertionsare due to Babuska [I]; for a detailed account, see the recent book ofOden and Reddy (5),

Now it can be verified that the forms in the variational problems(7) and (15) do indeed satisfy conditions (l7) so that they both pos-sess unique solutions which are equal, in fact, to the solution of theoriginal problem (2). These equivalences have been proved by. Babuska,Oden. and Lee [2J and Lee [4], where it is shown that the spaces ofadmissible variations are the following product spaces: i,

I

(lJ)Hodel I (21)

where the component spaces are defined on the partition Por the intere1ement boundary r of (8):

~-i 'f :......

(22)

E(= Une)

e

•• " i' ,_..~',•. ,

Model II

1' •. _.

504

...~~..... ,

1 -~ iHere H (n ) and H (all) are.the ~ell-known Sobo1ev spaces defined overe e

the indicated domains.

(23)

(24)

(25)

(26)

(27)

1- H (P) modulo constants

{4>: 4>ln (HI(lle>' 1 ~e ~E}e

L2(P) - {Ij>: 4>111 ~ L2(ne), 1 ~ e ~ E}e

~2(P) ..L2(P) x L2(P); ~l(p) = HI(P) x H1(p)

L iM{r) .. {q: ql i ( H--'(alle), 1 ~ e ~ E, 1 ~ i ~M}

alle

Norms on U and V are directly related to the energies J of (4)and K of (12):

II ~ II ~ -t (\I 174> 112

cal 12(lle)

',..s;,

',~

mcO,l with the nota-

Finite Element Approximations

The key question in the formulation of mixed-hybrid finite ele-ment methods is whether or not the existence conditions (17) hold forthe approximate problem for all partitions P of n. In finite elementmethods, we identify a particular partition by means of the mesh para-meter h ..max{dia(lle)}, and any two partitions differ by only a quasi-

euniform mesh refinement (see, e,g, [5]). We consider questions ofexistence and convergence of each model separately.

Hodel 1: Let Pk(n ) and P (II ) be the spaces of polynomials of degreeere i

< k and < r, respectively, defined on element nand P (an) the poly-- - i e t enomia1s of degree ~ t on aile' Then model I involves a constructionof elements in the spaces

I1

I

'~~{1

:t,,' .' tN· ,;tit"bW"'ii ;~nte?tme-stZ&i+i)!·'d'Sfjjs14$ ld'iw..wi;ithiz"tN +".;.:,aw 4iti' 6':1$4 ;;bhtt f g dmSt't.'/t,lch";'WO:lM •. We.

. 505

(22)

Ifie) S (r) - (Q:t .

~ EPk(n), l<e<E}; S(P)-S(P)')(S(P)e, e - - -1' I' 1',

QI i E: p (alii), 1 < e < E, i- l,2, ..·,Ml (30)all t e --

e(23)

(24)

(25)

(26)

(27)

which are endowediwith the usu~~/~nterpolation properties:• { as(1l ), v ~ H (n ). q ~ Hm (an ), w ( HP(all ), there

e - e e et E 5.., V ~ 5 , Q E St' W ~ S such that

... - -I' t

II,- ill = O(h v)Hn(n ) e

e

IIq~QII -Ii. H (aile)

Ifexist

(31)

1 e e 1~h - liS min{yh) , Yh c min{~ --Y

1

15 e e 2 e' v - -~ } (34)e 2 e

ele- fan Qn~z ds

\d for - -

lement~ . inf

ee QE Stefl IIQI12

_",(35)

I para- -quaai-

H a(an )e

of IIn~?~\IL (n )v . inf

-2 e

degree

et~Sk(P) Ily~11

I(36)

I poly-!-2(ne)

:10n

where

Clearly, the approximation lies in the space

II

I

II

'II

I

II!..

(33)

(32)

which i8 a subspace of U of (2l). Thus, condition (l7)1 is satisfied.The central problem, then, is ~hether or not (17)2 holds in Uh. Itcan be shown (see (2) that for our choice of subspaces,

where h c dia(all ) ande e.

3n-O,1; ~-min(k+l-n,s-n); v-min(r+l-~,t-n); aamin(t+2,m-l); a-min(t+l,p)

(29)

(4)

led over

} (28)I

, nota-

(37)

, (38)

Pk(lle) with respect

(36) and (37), nO-1'

IInOvz - ~z IlL (II )-r- -2 e

IIYzll~2(lle)

V~ dxdy • ~ Q~ ds

~Ile

Vz •f.e

Y ..e

506

IIn (35), z E L2(Oe) is such that

.'--,:~i.~,I',i

·.yL'rHtt''iNL;,tc ." Ms5!!ikw-s.,!, " ... ·4ii);.;Jh ...6M'Wtitfi4'i ··:'i§>',t.t,Sfl";i&ib :jia••~

I 1and nk is the orthogonal projection of L2(ne) onto

to the inner product (u,v) 1 • 1Vu· Vv dxdy. InL2 ne

is the L2-projection of Pk len ) onto P (II).• - e _r e

In view of (34), conditions for Yh > 0 depend on the behavior of.the parameters u , v , and y. Physically, the parameter U is a mea-- e e e Iesure of consistency between the flux data Q E St(r) and ~z where zis a potential function. We also note that the condition u > 0,e1 < e < E, is critical, which can be achieved if and only if the fol-lOWing-rank condition is satisfied: for a Q E s (r)

t

~ Q~ ds • 0 V ~ E Sk(P) ~ Q ..0laoe

,..:",-.~......~.

,-< ,

:;1

;i i"1 j

" I,t. l(I'

Ii:'II,l~

~

For a triangular element, it can be sho~~ that (see (7) the rank condi-tion is satisfied if and only if k > t+1 when t is even, k > t+2 when

't is even. We further remark that Yh > 0 1s only a sufficient condition

for existence of a unique finite element solution of (7) and yet ue > 0(eq~lvalently, satisfying the rank condition (38) is a necessary condi-tio~, that is to say that if ue ..0 the resulting matrix equation of

(9) will be singular. For this reason (38) is called the rank condi-tion.

As for the parameter y defined by (37), it 15 easy to see that ye eis a measure of distance between the spaces of velocity and streamfunction approximations. For this reason, we would like to have Ye • 0

which is the case when k-l < r. Note that v s 1 for this case. Hence,if k-l ~ 1', - e

(39)

·1

in place of (34) and ue > 0 is now a nec~ssary and sufficient conditionfor existence of a unique finite element solution of (7).

Model II: Similar considerations apply to Model II. We begin by

.fI

...~.

507

setting Vh = Uh C V and introducing parameters

(40)

I. y. ~W dsran

einf 112

YE.~r(P) Ilw L2(arl

e)

1II -e

(37)

(44)

(43)

(41)

(42)

. (45)ell 1Bh = min(ll tV ,y }e e e

f (£4> - y) . ~'i ~y dxdyfle

infV E s (P)~ -1'

1v =e

Iy =e

As previously, physical interpretations can be given for the para-meters and one can show that (see (4] for details)

where

o 1 1where "k and" are LZ and L2 projections onto Pk(rl ) and P (II),

l' 1 ere·respectively. In (42), y ~ L2(rle) is such that

I Vy· Vv dxdy = L W dxdy V v E L~ (rle)

n IIe e

. From the definition (40), it is not difficult to see that \11 > 0 if":and only if the following rank condition holds. e

t.. V .nW ds = 0 V V ( P (n ) ~ W = 0 (46)~ - - -r e

e

The following results are shown in [4]:

(1) The rank condition is satisfied, i.e" ul

> 0, ifeI by

condition

(39)

(38)

e that yereamve 'Ye - 0e. Hence,

.nk condi-~2 whencondition

:t \Ie > 0ry condi-Jon of .

condl-

respect

') n°, -r

,ior ofI a mea-

Ire z0,

Ie f01-

-,.p.":".- ~.~~._ ..__.;,.. \-*

(50)

(49)

(47)

Hr-k+1

.. ... ~'~H/~ .• ~:

for a quadrilateral

for a triangle

8 "'min{r+1,t+l,sl

5n - min{k+1,t +'2' s)

_ 3T - min{k,r+1,t+'2' s-l}

r>l iftcO

r>2 ift~O

if r > k + 1

ifr<k+l

II._~II _O(hn)L2(P)

Discussion

Let (48) hold and if r = k+l~

IIv+-vll c 0(h8) ,

~ ~ ~2(P)

l' > t > 1

r > t + 1

Let (47) hold and if k = rtl,

(11)

(iii)

"

Model II:

~todel I:

Model I: If the rank condition (38) holds,

Model II: If the rank condition (46) holds and if r ~ k+l,

Ilell • o (hE;) , £."' minCk+I,r,t,s-l) (48)~ V

Variational principles for both models are based on modifiedReissner's functional and first utilized by Wolf [8].

For certain special cases, the following L2-estimates can also beobtained (sec (4] for proof):

Error Estimates

As shown in Section 4, satisfaction of rank conditions is essen-tial to guarantee proper rank of resulting matrix equations, Similarconditions for different types of hybrid models are discussed earlierby Pian and Tong (e,g" see (6]), by Raviart [7], Brezz i and Harini(3] .

Thus, if l' > k + 1, ~l > 0 is the necessary and sufficient condition- e

for existence of a unique solution.

For both cases, complete polynomials of order r for V in nand t forW on each side of an requiring continuity at vertices areeassumed.

e

Error estimates follow immediately from (20) and (3l) fors• E. H (P), s ~ I,

508

,.•..., .

... ~~. . ., ,if ••

d$, d4IPi ¥;,;,*',i&WJi;..:....~ri.%6'*iL. d4xl:'(trti6Isf~ ..W,,,siNnz ""ffi*l· ..it r .=.¥if';";" .web ~=M • .qiit.,.·wes?i6tt: ...;,.,"ii¥)';,ar~

509

Nodel I:d t forlimed.

dition

(47)

(48)

be

Finally, reviewing the error es timates, the bes t possible choicesof polynomial orders are:

{= t+1 if t is even }

k • 1'+1 for a triangle= t+2 if t is odd

Hodel II: (a) k .,r-l=t-l, t > l, for a triangle wi th nodal pointson vertices

(b) k - 1'-1= t, t > l, for a quadrilateral with nodalpoints on vertices,

When a smooth velocity profile is desired, Model II may be more attrac-tive than I because it gives continuous velocity at nodal points dueto the nature of the variational principle.

Acknowledgement

The support of this work by the U, S. Air Force Office ofScientific Research through Grant 74-2660 is gratefully acknowledged.

References

1. Babuska, I,,' "Error Bounds for Finite Element ~tethod," Mum. Math.,Vol. 16, pp, 322-333 (1971),

2. Babuska, 1., Oden, J, T" and Lee, J. K .. "Mixed-Hybrid FiniteElement Approximations of Second-Order Elliptic Boundary-ValueProblems," TICOM Report 75-7, The Texas Institute for Computa-tional Hechanics, The University of Texas at Austin, 1975.

3. Brezzi, F. and Marini, L. D., "Or. the Numerical Solutionof PlateBending Problems by Hybrid Methods," RAIRO Report, 1975.

5. Oden, J, T., and Reddy, J, Ne, An Introduction to the MathematicalTheory of Finite Elements, Wiley Interscience, New York (in press),

4. Lee, J. K" "Convergence of Mixe'-Hybrid Finite Element Methods,"Ph.D, Dissertation, Div. of Engr Mech., University of Texas atAustin, 1976.(49)

(50) 6.

7.

essen-Similarearlierarini

\8.

Jed

Pian, T. It, II. and P. Tong, "Basis of Finite Element Methods forSolid Continua," lnternational Journal 'of Numerical Methods inEngineering, Vol. 1, pp. 3-85, 1969.

Raviart, P. A" "Hybrid Finite Element Hethods for Solving 2ndOrder Elliptic Equations," Topics in Numerical Analysis II, Proc,of the Royal Irish Conf. on Num, Analysis, 1974, ed. by J. J. H.Miller, Academic Press, 1976,

Wolf, J. P., "Generalized Hybrid Stress Finite Element Models,"AlAA Journal, Vol. II, No.3, pp. 386-388, 1973.

IIIIfIIi