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Pergamon 00457949(95)00017-8Compulers & Swuclures Vol. 56, No. 2/3, pp. 225-237, 1995

Copyright 0 1995 Elsevier Science LtdPrinted m Great Britain. All rights reserved

004s7949/95 19.50 + 0.00

A MIXED DISPLACEMENT-BASEDFINITE ELEMENT FORMULATIONFOR ACOUSTIC FLUID-STRUCTURE INTERACTION

K. J. Bathe, C. Nitikitpaiboon and X. WangMassachusetts Institute of Technology, Cambridge, MA 02139, U.S.A

Abstract-The solutions of fluid-structure interaction problems, using displacement-based finite elementformulations for acoustic fluids, may contain spurious non-zero frequencies. To remove this deficiency,we present here a new formulation based on a three-field discretization using displacements, pressure anda vorticity moment as variables with an appropriate treatment of the boundary conditions. We proposespecific finite element discretizations and give the numerical results of various example problems.

I. INTRODUCTIONThe interaction between fluids and structures can, inmany practical engineering analyses, significantlyaffect the response of the structure and hence needsto be taken into account properly. As a result, mucheffort has gone into the development of general finiteelement methods for fluid-structure systems.

A number of finite element formulations have beenproposed for acoustic fluids in the analysis offluid-structure interaction problems, namely, thedisplacement formulation (see Hamdi et al . [I],Belytschko and Kennedy [2], Belytschko [3], Olsonand Bathe [4]), the displacement potential andpressure formulation (Morand and Ohayon [5]), andthe velocity potential formulation (Everstine [6],Olson and Bathe [7]). The displacement formulationhas received considerable attention, because it doesnot require any special interface conditions or newsolution strategies (for example, in frequency calcu-lations and response spectrum analyses) and becauseof the potential applicability to the solution of abroad range of problems.

It has been widely recognized that the displace-ment-based fluid elements employed in frequency ordynamic analyses exhibit spurious non-zero circula-tion modes. Various approaches have been intro-duced to obtain improved formulations. The penaltymethod has been applied by Hamdi et al. [l] and hasbeen shown to give good solutions for the casesconsidered in that reference. Subsequently, Olson andBathe [4] demonstrated that the method locks up inthe frequency analysis of a solid vibrating in a fluidcavity and that reduced integration performed on thepenalty formulation yields some improvement inresults but does not assure solution convergence in ageneral case. More recently, Chen and Taylor pro-posed a four-node element based on a reduced inte-gration technique together with an element massmatrix projection [8]. The element is used to solve

some example problems, but an analysis or numericalresults on whether the element formulation is stableand reliable are not presented.

Therefore, although much research has been ex-pended on the topic, we believe that the currentlyavailable displacement-based formulations of fluidsand fluid-structure interactions are not yet satisfac-tory.

Our objective in this study was to develop aformulation to overcome the aforementioneddifficulties. Since mixed methods are now used withconsiderable success for other types of constraintproblems, we endeavored to develop a displacement-based mixed finite element formulation for the fluid.As reported in this paper, we arrived at a three-fieldmixed variational principle and new fluid elementswhich yield very promising results.

In the following sections we first summarize theassumptions and governing equations used in ourmathematical model. Next, we discuss the difficultiespresent in the ordinary displacement formulation.Finally, the new three-field displacement-pressure-vorticity moment (u-p-A) formulation is presented indetail with some solution results.

2. GOVERNING EQUATIONSOur objective is to evaluate the lowest frequencies

of fluid-structure systems. We assume an inviscidisentropic fluid undergoing small vibrations for whichthe governing equations in the fluid domain Vf are

pe+vp =o (1)/w.v+g =o (2)

where v is the velocity vector, p is the pressure, p isthe mass density and the bulk modulus, /?, can be

225

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226 K. J. Bathe rf alwritten in terms of the sound wave speed, c, as

fl = pc=. (3)The boundary conditions are that D is the pre-

scribed pressure on the surface S, and U, is theprescribed normal displacement on the surface S,,with the unit normal vector n pointing outward fromthe fluid domain. Also, we have &US/= S ands,ns,= @.

We note that eqn (2) implies

v.+where II is the displacement vector. Therefore, whenalmost incompressible conditions are considered,(that is, when /l is large while p is of the order of thestress and hence relatively small) we have that V uis close to zero. Also, eqn (1) implies

$ V x v) = 0and hence the motion is always circulation preserv-ing, i.e. it is a motion in which the vorticity does notchange with time.

In terms of the displacements only, we have bysubstituting from eqn (4) into eqn (1)

/lV(V II) + fB = 0 (6)where fB = -pii. Hence fB is a vector of body forcesper unit volume which we assume to contain only theinertia forces, but other contributions could directlybe included.

In the frequency solution, the governing equationis

c2V(V 0) + &J = 0 (7)where 6 represents the eigenfunction. It then followsthat

gv x (V(V 0)) + pcd(V x 6) = 0. (8)Since V x V4 = 0 for any smooth scalar valued

function C#J,we have thatw=(vxQ=o (9)

It is then apparent that in a frequency analysis thesystem admits two types of analytical solutions;namely, solutions corresponding to the rotationalmotions (V x 0 # 0) for which the frequencies arezero, and solutions corresponding to the irrotationalmotions (V x G = 0). One may view the first typeof solution as the way the system responds to therotational initial boundary conditions.

The variational form of eqn (6) is given bys: {/l(V.u)(V.&)-fB.bu}dlI+su;dS = 0 (10).%where uf is the displacement normal to S,. The

variational indicator is

I-I=

This variational indicator is used for the ordinarydisplacement-based finite element discretization ofacoustic fluids. It has been observed that a finiteelement solution based on eqn (11) may yield, otherthan the zero frequency circulation modes, also circula-tion modes with non-zero frequencies. While the circu-lation modes with zero frequencies can easily berecognized and discarded in a dynamic solution, thatis, a mode superposition solution, spurious non-zerofrequencies are difficult to identify in practice andshould not be present in a reliable finite elementanalysis.

In order to eliminate the spurious non-zero fre-quencies, we impose the constraint

vxu=Axwhere A is a vorticity moment. The magnitude ofA shall be small while tl is a constant of large value.

In addition, when the fluid acts in almost incom-pressible conditions, we have the usual constraint ofnear incompressibility, see eqn (4). As demonstratedby Olson and Bathe, a pure displacement formulationwith the two constraints of irrotationality and(almost) incompressibility yields much too stiff re-sponse predictions, typical of the locking behav-ior [4].

However, we now recall our experiences withthe displacement/pressure formulation for solids(or velocity/pressure formulation for viscous fluids)which, with the proper choice of elements, is veryeffective in almost incompressible situations [9].These experiences lead us to propose a displacement-based mixed formulation which properly incorpor-ates the above two constraints and provides excellentpromise for general applications.

3. VARI ATIONAL FOR MULATION

Based on our experience with the displacement(velocity)/pressure formulation for (almost) incom-pressible media, we propose the following variationalindicator for the finite element formulation of the

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Finite element formulation for acoustic fluid-structure interaction 227fluid considered here:

+su, dS (13)s,where the variables are the pressure p, displacementII, vorticity moment A, and the Lagrange multipliersA, and 1,. The constant fi is the bulk modulus and theconstant tl have large values. We note that the firsttwo terms correspond to the usual strain energy(given in terms of the pressure) and the potential ofthe externally applied body forces. The third termimplies the constraint in eqn (4), the fourth term isincluded to be able to statically condense out thedegrees of freedom of the vorticity moment and thefifth term imposes the constraint in eqn (12). For thefourth and fifth terms we require that the constant c(is large, and we use c( = 1000/J. However, from ournumerical tests, we find that TVan be any numericallyreasonable value larger than /I, say 10/I < tl < 104p.

The last term is the potential due to any appliedboundary pressure on the surface S,.

Invoking the stationarity of II, we identify theLagrange multipliers iP and 1, to be the pressure pand vorticity moment A, respectively, and we obtainthe field equations

vp-P+VxA=Ov.u+E =oP

vxu-A=0c1

and the boundary conditions

(14)

(15)

(16)

u.n=Ls, on S,PP on S,A=0 on S. (17)

The pressure j is commonly assigned to be zero onthe free surface. Also, in the variational indicator,ignoring gravity effects and other body forces, wehave fB= --pii.

4. FINITE ELEMENT DISCRETIZATTONWe use the standard procedure of finite element

discretization [9] and hence for a typical element, we

have

p=H,pA=H,,A

V.u=(V.H)fi=BfiVxu=(VxH)fi=Dti

where H, HP and H, are the interpolation matrices,and 6, p and A are the vectors of solution variables.

The matrix equations of our formulation are

where

M= s pHTHdI/L=- sTHP d V/

Q= s DH, d Vfr 1

A=- f H,TH,dI

G=- J H;H,dV R=- s H;Tp dS./ a %(19)

The key to the success of the finite element dis-cretization is to choose the appropriate interpolationsfor the displacements, pressure and vorticity moment.Without restricting the essence of our exposition, letus consider two-dimensional solutions. The finiteelement formulation for three-dimensional solutionsis then directly obtained [9].

Considering (almost) incompressible analysis ofsolids and viscous fluids, it is well-established thatfinite elements which satisfy the inf-sup conditionare effective and reliable. Appropriate interpolationsare summarized in Refs [9], [IO] and [ 111. The u/pelements give continuous displacements (or velocities)and discontinuous pressures whereas the u/p-c el-ements yield continuous displacements (or velocities)and pressures across the element boundaries.

For the inviscid fluid element formulation con-sidered here, we use the displacement/pressure inter-polations that satisfy the inf-sup condition in theanalysis of solids (and viscid fluids) and use the sameinterpolation for the vorticity moment as for thepressure. Thus, two proposed elements are the 9-3-3and 9-4c-4c elements schematically depicted in Fig. 1.

Hence, for the 9-3-3 element, we interpolate thepressure and vorticity moment linearly as

P =Pl +p,r+p,s cw

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228 K. J. Bathe et al

l nodal point displacementx pressure and vorticity moment

9-3-3 elementcontinuous displacement, discontinuous pressureand vorticity moment

l nodal point displacement@ pressure and vorticity moment

9-4c-4c element

continuous displacement, pressure and vorticity momentFig. 1. Two elements for u-p-A formulation. Full numeri-

cal integration is used (i.e. 3 x 3 Gauss integration).

A = A, f A,r + A,s (21)and for the 9-4c-4c element we use the bilinearinterpolations

(22)A= A, + A,r + A,s + A,rs. (23)

Of course, additional elements could be proposedusing this approach. In each case, the vorticity mo-ment would be interpolated with functions that whenused for the pressure interpolation give a stableelement for (almost) incompressible conditions. Forinstance, a reasonable element is also the 9-3-l el-ement, which however does not impose the zerovorticity as strongly as the 9-3-3 element. On theother hand, the 4-l-l element cannot be rec-ommended (see Section 6 for some numerical resultsobtained with this element). For all elements fullnumerical integration is used [9].

While we do not have a mathematical analysisto prove that the proposed elements are indeedstable, based on our experience we can expect agood element behavior. Namely, the constraintsin eqn (4) and (12) are very similar in nature andare not coupled in the formulation, see thevariational indicator in eqn (13). Hence, we canreasonably assume that the two constraints shouldbe imposed with the same interpolations in theelement discretization.

5. BOUNARY CONDITIONSThe above finite element formulation appears to be

rather natural, when considering the literature avail-able on elements that perform well in incompressibleanalysis of solids (or viscous fluids). However, somesubtle points arise in imposing the boundary con-ditions.

Let us consider typical boundary lines composed oftwo adjacent three or four-node elements, see Fig. 2,and two nine-node elements, see Fig. 3. Since the fluidis inviscid, the displacement component tangential tothe solid is not restrained (while the displacementcomponent normal to the solid is restrained to beequal to the displacement of the solid). The choice ofdirections of nodal tangential displacements (fromwhich also follow the directions for the normaldisplacements) is critical.

Considering the solution of actual fluid flows andfluid flows with structural interactions (in which thefluid is modeled using the Navier-Stokes equationsincluding wall turbulence effects or the Eulerequations), we are accustomed to choosing thesedirections such that in the finite element discretizationthere is no transport of fluid across the fluid bound-ary. It is important that we employ the same conceptalso in the definition of the tangential directions atboundary nodes of the acoustic fluid considered here,although the acoustic fluid does not flow but onlyundergoes (infinitesimally) small displacements.While we do not recommend using the 4-l-lelement of our u-p-A formulation (see Table 4 andFigs 15 and 16) lower-order elements (three-nodetriangular two-dimensional and four-node tetrahe-dral three-dimensional elements with bubble func-tions) are effectively employed in fluid flow analysisand could be used in our u-p-A formulation [9, 121.

1

UAAa2. .inflow outflow

Fig. 2. Tangential direction given by y at node A for 3 or4-node elements.

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Finite element formulation for acoustic fluid-structure interactionboundary of element b,length Lb \ ub= (-s l-S2---2 2 >U.4.

boundary of element a.length La

CPYL xFig. 3. Tangential directions given by PA and 8, at nodes Aand B for 9-node elements.

Hence consider first the boundary representationin Fig. 2. This case was already discussed by Doneaet al. [13]. For the displacement U,, we have thespurious fluxes AV, and AVb across the elementlengths L, and L,. Mass conservation requires thatA V, + A Vb = 0 and hence we must choose y given by

tan y = L, sin ci, + L, sin 5(*L, cos CC, L, cos a* (24)

We note that y is actually given by the directionof the line from node one to node two. In general,the direction of U,d is not given by the mean of theangles CC, nd CQ, but only in the specific case whenL, = Lh

y = f(a, + a*).We employ the same concept to establish the

appropriate tangential directions at the typical nodesA and B of our nine-node elements in Fig. 3. Fornode A, we need to have using PA

where -n, dl = -$ds,2 ds >nb dI = -gds,%ds >

In the above equations, x,, y,, x, and y, are theinterpolated coordinates on the boundaries of el-ements a and 6, while II: and II; are the interpolateddisplacements corresponding to the displacement U,at node A,

For element a, mass conservation requires

s, . n, dl = 0L229

(28)

where II: is the interpolated displacement on theboundary of element n based on the nodal displace-ment U,

u,=(l -s*pJp (30)The relation (29) shows that the appropriate

tangential direction ps at node B is given by

ay axtanpB=- -1 Is as B (31)Our numerical experiments have shown that it is

important to allocate the appropriate tangential di-rections at all boundary nodes. Otherwise spuriousnon-zero energy modes are obtained in the finiteelement solution.

6. NUMERICAL SOLUTIONSTo demonstrate the capability of the proposed

formulation, we give the results obtained in thesolution of three problems. We consider these analy-sis cases to be good tests because, while they pertainto simple problems, the degree of complexity seems tobe sufficient to exhibit deficiencies in formulations.Until the development of the u-p-A formulationdescribed here, we were not able to obtain accuratesolutions to all these problems with a displacement-based formulation (or a formulation derived there-from).

In all test problems, we want to evaluate the lowestfrequencies of the coupled fluid-structure system.The problems were already considered by Olson andBathe [4, 71.

Figure 4 describes the tilted piston-container prob-lem. The massless elastic piston moves horizontally.Figure 5 describes the problem of a rigid cylindervibrating in an acoustic cavity. The cylinder is sus-

l- 8m lm

I 12 m lmFig. 4. Problem one: tilted uiston-container system

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230 K. J. Bathe ef al

Springk=l.Ox IO5 N/m

p=2.1 x IO9 N/ m2p=l.O x 10 kg/m3

Em HRigid ellipse 0.4 mMass=2.0 x IO3 kg Fluidj&2.1 x lo9 Ni m2p=l.O x lo3 kg/m3

8m 8mFig. 5. Problem two: a rigid cylinder vibrating in an Fig. 6. Problem three: a rigid ellipse vibrating in an acousticacoustic cavity. cavity.

pended from a spring and vibrates vertically in the obtained with the velocity potential (that is u-4)fluid. Figure 6 shows a rigid ellipse on a spring in the formulation which can be considered a very reliablesame acoustic cavity. Figures 7-9 show typical nine- procedure [7]. The meshes used in these analyses havenode element discretizations used in the analyses of been derived by starting with coarse meshes andthe problems. Table 1 lists the results obtained using subdividing in each refinement each element into twothe 9-3-3 element, and Table 2 gives the results or four elements. We also give in Table 1 the number

i,

Fig. 7. Typical mesh for problem one.

Table 1. Analysis of test problems using 9-3-3 elements

Test caseTiltedpiston-containerproblemRigidcylinderproblemRigidellipseproblem

Mesh,no. of

elements4

163264283264

2832

64

Number of zerofrequencies

Theory Result>7 I

231 31263 63

a105 10521 1213 13261 61

2125 125>l 1

>13 13361 612 125 125

FirstI.895I.8671.8621.8593.8464.2284.2814.2885.5116.6126.8276.829

Frequencies (rad SC)Second Third Fourth6.054 9.256 10.325.696 9.236 9.7925.603 9.189 9.3915.589 8.617 9.175628.2 928.6 1341609.5 1191 1326589.4 1138 1252587.4 1131 1234629.9 932.6 1246589.1 1220 1232512.1 1155 1176571.2 1149 1167

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Finite element formulation for acoustic fluid-structure interaction 231

D

D

D

D

E

1%B /-c /cD -fE --F -cFig. 8. Typical mesh for problem two.

of the zero frequencies k present in the analyses usingthe u-p-A formulation. The theoretical result isobtained from the formula

k>nd-nctl (32)where nd is the number of unconstrained displace-ment degrees of freedom in the model and nc is thesum of the pressure and vorticity moment degrees offreedom. Of course, we do not calculate the modeshapes corresponding to the zero frequencies, butsimply shift to the non-zero frequencies sought [9].

The results in Tables 1and 2 show that the u-p-Aformulation gives results close to those calculated

Ul2a /-c ,cD -/r

with the u-4 formulation, and that the number ofzero frequencies matches the mathematical predic-tion. Figures 10-12 show pressure band plots of the

Fig. 9. Typical mesh for problem three.

modes considered in Table 1. The bands are quitesmooth, indicating accurate solutions.To illustrate the importance of assigning the appro-priate tangential displacement directions at the fluidnodal points on the fluid-structure interfaces, wepresent the results of two test cases.

In the first test case, the very coarse finite elementmodel shown in Fig. 13 for the response analysis ofthe rigid cylinder is considered. The assignment of thecorrect tangential directions at nodes 10 and 11 iscritical. If we use the actual cylinder geometry, the

Table 2. Results obtained using IJ~ formulation for analysis of three test problems

Test caseMesh,no. of elements

Frequencies (rad s- )First Second Third Fourth

Tilted piston problem 32 1.858 5.569 9.116 9.299Rigid cylinder problem 32 4.269 581.8 1124 1224Rigid ellitxe oroblem 32 7.071 563.2 1138 1158

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232 K. J. Bathe et ulY Y

MODE 64F = 0.2963 cycles/set L x z0:Eof;;917 cycles/set L xPRESSURE PRESSUREMODE 64 MODE 65

//I 31.1.5 t I- 433.388.1_ -37.9 - 94.5- -77.4 - -99.0- -116.9 - -292.6

Y YMODE 66 L MODE 61F = 1.462 cycles/set x F = 1.495 cycles/set L x

PRESSURE PRESSUREMODE 66 MODE 67t 1031.11. t 1047.96.,_ 283. ,_ 228.- -144. _ -239.- -572. - -707.

Fig. 10. Pressure bands of the first four modes of the tilted piston-container problem(mesh of 32 9-3-3 elements).

tangential directions are 60 and 120 degrees from thex-axis. However, using eqn (31) we obtain 45 and135 degrees, respectively. Table 3 lists some solutionresults and shows that a spurious non-zero frequencyappears when the incorrect tangential directions areassigned.

In the second test case, the finite element modelshown in Fig. 14 is used for the frequency solution ofthe rigid ellipse vibrating in the fluid-filled cavity. Ifwe use the tangential directions given by the actualgeometry of the ellipse, a non-zero spurious fre-quency is calculated, whereas if the tangential direc-tions are assigned using eqns (26) and (29) goodsolution results are obtained, see Table 3. The com-parison of the angles (y) from the x-axis obtainedusing eqns (26) and (29) with the angles using theactual geometry is as follows:

node 10: angle ylmode,= 86.93.angle Y geometrY 87.04

node 7: angle y Imode, 87.40angle Y geomctrj 83.41

node 11: angle y Imode, 70.89angle 7 Igcomc,ry 71.22.

It is interesting to note that based on the masstransport requirement the angle y at node 7 is largerthan at node 10, which is an unexpected result basedon the actual geometry of the ellipse.

Finally, we also show results obtained with the4- I- 1 element. Since the 411 displacement/pressureelement (that is, the element with four corner nodesfor the displacement interpolation and a constantpressure assumption) for incompressible analysisdoes not satisfy the inf-sup condition, we expect thatthe 4-l-l element will not provide a stable discretiza-tion. Table 4 and Figs 15 and 16 show some solutionresults. These clearly indicate that the 4-l-l elementis not a reliable element; the predicted lowest frequen-

Table 3. Spurious modes due to assignment of incorrect tangential displacement directions at thefluid-structure interfaces

Number of zerofrequencies Frequencies (rad s-l)

Tangent ~~~Test case direction Theory Result First Second Third FourthRigid cylinder incorrect 25 4 4.649 86.17 693.7 1154model in Fig. 13 correct >5 5 3.861 688.2 I169 1324Rigid ellipse incorrect 213 12 6.719 34.65 591.4 1213model in Fig. 14 correct 213 13 6.612 589.1 1220 1232

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Finite element formulation for acoustic fluid-structure interaction 233Table 4. Solution results using 4-l-l elements

Test caseRigid cylinder problemRigid ellipse problem

Mesh,no. of

elements128128

Frequencies (rad s-)First Second Third Fourth11.91 581.8 1135 1234

551.5 988.2 1157 1180

ties are not accurate and the checkerboard pressure and vorticity moment are associated with elementbands obtained here are typical of those observed internal variables and are statically condensed out atwith the 4/l element in incompressible analysis. the element level. Then the only nodal point variables

used in the assemblage of elements are those corre-7. CONCLUSIONS sponding to the displacements. This enables the direct

A new effective three-field mixed finite element coupling of the fluid elements with the structuralformulation for analysis of acoustic fluids and their elements. Of course, in principle, we could alsointeractions with structures has been presented. The employ element nodal pressures and vorticity mo-fluid formulation has the following attributes: ments that provide for continuity in these variables

(and hence cannot be statically condensed out at the(1) The displacement, pressure and vorticity mo- element level).ment are used as independent variables. For the (2) The key point is to choose the appropriateelement interpolations it is effective that the pressure interpolations for the displacements, pressure and

NODE 62= 0.6814 cycles/srcPRESSURE

COK)DE 4: = 181.1 cyc1es,sec F = 199.2 cycles/setPRESSURE

L -1.057E+C

Fig. 11. Pressure bands of the first four modes of the rigid cylinder problem (mesh of 32 9-3-3elements).

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4ODE62

:

1.087cycles,src

PRESSURE

--1336706.

4ODE64

7

183.9cyclss/sec

L3.596E+OG

3.902E+06

-l.l13E+O7

-1.186E+07

Fg1P

b

oh

om

ohgdeppoem(m

o3933

eem

D-E-C

Fg1R

ay

ogdcn

uow93eem

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i "1%BY-c/c

Fg1R

ay

ogdepuoeg933eem

Fg1C

bdp

b

opoemtwwh4

eem

K21

--heckerboardode

L,'F

2.59cycles,'sec

IODE

F1.896cycles/set

PRESSURE

MODE 1 6 3

0

--3

--67197

-1

PRESSURE

MODE 5 4

1

2 1356481 1

: 'ip-1356480

-2

--4069440

rH)DE3

F

80.6cycles/set

LMODE

xF

96.4cycles/set

PRESSURE

MOD3

PRESSURE

MODE

t1.406E+07

!l.O5:E+07

*3.515E+06

-5.200EcOl

:1F i-3.516E+06

--7.031E+06

--l.O55E+07

l.l35E+07

7.715E+06

4.07&+06

4.363E+05

-3.203E+06

-6.843E+06

--l.O48E+07

--1.412Et07

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236 K. J. Bathe et al.

PRESSUREMODE 1

5505413.

4129072.

2752730.

1376388.

46.

-1376296.

- -2752637.

- -4128979.

9.662EtO6

i- 6.484E+06

3.307E106

- -3.048E+"6

- -b.226E+O6

- ~9.403E106

Y10DE 2 -- checkerboard mode L.1 = 157.3 cycles/set

PRESSUREMODE 2

1.956Et07

6.521E+06

4.980E+OZ

-6.52OEt06

1.430E+07

1.065Et07

6.993EcOfi

3.341EtO6

-3 964E+06

- -7.617E+06- -1,127E+07

Fig. 16. Checkerboard pressure band of problem three with 4-l-l elements

vorticity moment. We employ the same interpolationsas used for stable discretizations in the displacement(or velocity)/pressure formulations of almost incom-pressible media. We have no mathematical proof thatthe inf-sup condition is satisfied using these interpola-tions, but believe that our interpolations are reason-able because the constraints of pressure and vorticitymoment are imposed independently in the formu-lation.

(3) Specific attention need be given to the evalu-ation of the tangential (slip) directions at the fluidsolid interfaces. These directions need to satisfy re-lations of mass conservation.

(4) We have tested the formulation on some simplebut well-chosen analysis problems, which heretoforewe could not solve with displacement-based finiteelement formulations (or formulations derived there-from). The new formulation has performed well in allanalysis cases. The results of some numerical exper-iments on the use of a four-node element which is notrecommended for general use and inappropriate

boundary tangential directions have been included inorder to illustrate the importance of our recommen-dations.

While the formulation presented here shows muchpromise, we are continuing our work on the topic ofthis paper because we expect that an even simplerand more effective mixed displacement-based finiteelement formulation can be reached.Acknowledgement-We would like to thank F. Brezzi, Uni-versity of Pavia, Italy, for our discussions on this research.

REFERENCES1. M. A. Hamdi, Y. Ousset, and G. Verchery, A displace-ment method for the analysis of vibrations of coupledflu&structure systems. Int. J. numer. Meth. Engng 13,139-150 (1978).2. T. B. Belytschko and J. M. Kennedy, A fluid-structurefinite element method for the analysis of reactor safetyproblems. Nucl. Engng Des. 38, 71-81 (1976).3. T. Belytschko, Fluid-structure interaction. Cornput.Struct. 12, 459-469 (1980).

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