Shell Coupled FSI _example

8/13/2019 Shell Coupled FSI _example

1/14

Compulers & Structures Vol.8, No. 5/6, pp. 515-528, 1991Printedn Great Britain. oLl45-7949191 3.00 + 0.000 1991 Pergamon Press plc

COUPLED SHELL-FLUID INTERACTION PROBLEMSWITH DEGENERATE SHELL AND THREE-DIMENSIONALFLUID ELEMENTS

R. K. SINGH,t T. KANTI and A. KAKODKAR~tReactor Engineering Division, Bhabha Atomic Research Centre, Trombay, Bombay 400 085, India

JDepartment of Civil Engineering, Indian Institute of Technology, Powai, Bombay 400 076, India(Recei ved 3 Januar y 1990)

Abstract-Discrete methods for practical coupled three-dimensional fluid-structure interaction problemsare developed. A Co explicitly integrated two-dimensional degenerate shell element and a three-dimen-sional fluid element are coupled to study shell dynamics, fluid transient and coupled shell-fluid interactionproblems. The method of partitioning is used to integrate the fluid and shell meshes in a staggered fashionin an optimum manner. Effective explicit-implicit partitioning is shown to achieve high computationalefficiency for this type of problem.

1. INTRODUCTIONFluid-structure interaction problems have attracted agreat deal of attention from the nuclear, space andoffshore industries. The important areas of appli-cations are the analysis of liquid containers subjectedto earthquake ground motion, resulting in impulsiveand sloshing motion of the contained liquid, analysisof submerged structures and components subjected topressure pulses generated due to accidents or other-wise, and analysis of fluid reservoirs supposed tocontain accident-induced high-pressure fluid. Most ofthe attempts for the transient analysis of such prob-lems have been limited to simplified two-dimenbionalstudies. The three-dimensional transient analysisposes problems in terms of size of the problem andlarge storage and CPU time required by the com-puter. The analysis is prone to further difficulties interms of storage space/CPU time if the structureshows nonlinear behaviour and/or if the fluid under-goes large motions or if the fluid behaviour is gov-erned by some additional field variables such astemperature, phase, etc. Fluid-structure interactionstudies are characterized by the oscillations inducedin fluid and its effect on the surrounding struc-tures. Moreover, the structure displacements alsoalter the fluid response. Thus this phenomenon hasto be studied in a coupled manner. Some simplifiedapproaches are also available in which the fluid-structure interaction problem is studied in a decou-pled manner. The fluid response is first obtainedassuming the structure to be rigid and the resultingpressure field is imposed on the structure to obtainthe structural response. This approach overestimatesthe structural response in many cases. Moreover, ifcoupled modes are excited, this approach gives un-conservative results [141. Another simplified methodof design has been to use the added mass approach,

treating the fluid as incompressible. This approachwill be suitable for a case when fluid oscillationfrequencies are sufficiently separated from a struc-tures predominant frequencies. Thus there is a needto study the fluid-structure interaction problems in acoupled manner.

Earlier studies on fluid-structure interaction prob-lems were made by Bathe and Hahn [l], Liu [4],Akkas et a l 5], Wilson and Khalvati [6], Shantaramet al. [7] and Deshpande et al. [8]. The basic method-ology in these studies was to modify an existingstructure dynamics code to account for the effect offluid motion. In this method the shear modulus isset to zero in the fluid domain. The fluid-structureinterface is constrained to have normal displacementcontinuity. The eigenvalue analysis indicates the pres-ence of zero energy modes as described by Akkaset al. [5]. Wilson and Khalvati [6] have suggested amethod to select an optimum penalty parameter sothat these spurious modes can be suppressed. In thisapproach an irrotational flow condition is enforced.Deshpande et al. [8] have also come up with anoptimum penalty parameter to couple the structureand fluid meshes which is a function of the density offluid and the acoustic speed in the fluid medium.Au-Yang and Galford [9] define this method as struc-ture mechanics priority or continuum mechanicsapproach. This approach is advantageous only for alimited class of problems, such as the linear behaviourof a fluid. Moreover, it does not prove to be economi-cal for a three-dimensional transient analysis, even inthe case of a linear problem, due to the large size ofthe fluid domain; the number of discrete equationsbecomes very large.

Another approach with which to tackle such acoupled problem is to couple a structure dynamicscode with a fluid transient code as described in [g-14].Historically, this second approach is older than the

515


2/14

516 R. K. SINGH et al.first approach. However it has not gained popularityuntil recently. The major limitation of this method isthat the resulting coupled equations from the twofields make the system of equations unsymmetric.The bandwidth becomes very large due to fluid-structure coupling. However, the recent developmentof a partitioning method with a staggered solutionapproach as suggested by Park and Felippa [lo],Paul [1 ] and Felippa and Geers [12] makes thisapproach very attractive. The major advantage ofthis method is that the coupled field problems canbe tackled in a sequential manner. The analysis iscarried out for each field and interaction effect isaccounted for by updating the field variables of boththe fields in the respective coupling terms. Though themethod is iterative the number of variables in thefluid is one at each node for a linear fluid behaviour.For three-dimensional fluid-structure interactionstudies this is quite beneficial. Another advantage ofthis approach is that it is modular in nature. Variousfluid transient codes can be coupled to a structuredynamics code to study the interaction effects. Fur-ther, the modular adaptation makes this approachvery attractive in a parallel processing environment.

The present paper demonstrates the capability toanalyse three-dimensional fluid-structure interactionproblems in an optimum manner by the method ofpartitioning. The structure response is obtainedthrough the use of Co degenerate shell elements. Thiselement, with associated through the thickness nu-merical integration, was introduced by Ahmad etal. [15] for linear static analysis. A number of defi-ciencies have been found in this element in the lastdecade. Studies with regard to locking behaviour andzero energy modes have helped to improve the per-formance of this element. Further, this element hasnot been used extensively to tackle problems of shelldynamics and coupled problems due to its cumber-some formulation requiring large computationaltime. Efficient two-dimensional degenerate shellelements based on explicit through thickness inte-gration due to Belytschko [16] and Milford andSchnobrich [171 have recently been developed. Thepresent paper demonstrates the capability of its effec-tive implementation for coupled dynamic problems.The fluid transient behaviour is studied with three-dimensional brick elements. The presence variable isthe only unknown at nodal points, to keep thenumber of degrees of freedom to a minimum in thefluid domain. The shell and fluid meshes are coupledin an optimum manner by assigning a separateequation numbering system for coupling terms toovercome the bandwidth problem described earlier.In the present work the nine-noded degenerate shellelement has been used for shell dynamics, which givesthe best combination of economy and accuracy asdemonstrated by Pugh et al [18] after comparing theperformance of four-, eight-, nine-, 12- and 16-nodedquadrilateral elements. The fluid transient behaviouris studied by an eight-noded trilinear brick element.

This element is sufficient to obtain dynamic loadingon the shell surface in an accurate and economicalway.Another important aspect of transient analysis isthe selection of an optimum integrator for second-order ordinary differential equations resulting fromsemidiscretization. There are two well knownmethods reported in the literature [l 1, 13, 19-241,implicit method and explicit method. Implicitmethods are unconditionally stable and larger timesteps can be used based on the order of accuracyrequired. However, the computational effort requiredis more in this case as the resulting simultaneousequations have to be solved at each time step. On theother hand, explicit methods are conditionally stable,thus time step size is limited by the minimum periodof the mesh (or more simply by the period of thesmallest element). However, in this method the CPUtime and storage space required are small as it doesnot call for solution of a system of simultaneousequations. As described by Belytschko [131, ransientproblems are of two categories, wave propagationproblems and inertial problems. For wave propa-gation problems the wave front has to be tracedaccurately, thus the number of modes which have tobe integrated properly is very large. In the case ofinertial problems, the response lies in a few lowermodes and only these modes have to be integratedproperly. Another important requirement of the inte-grator is that higher modes which are not importantand which are associated with large errors must befiltered out. In the case of fluid-structure interactionproblems, all the pressure modes in the fluid domainmust be traced with accuracy if it is likely to affect theprominant structure modes. In the present work amixed implicit-explicit scheme due to Hughes [19] hasbeen used. Some aspects of mesh partitioning andselection of a proper integrator for fluid and shelldomains are discussed.

The organization of the paper is as follows. InSec. 2, we describe the formulation of the coupledshell fluid dynamics problem where shell dynamicsand the fluid transient problem are discussed in detail.Section 3 describes the coupled equation solutionstrategy for the present problem and some guidelinesare given to achieve efficient partitioning. In Sec. 4 anumber of fluid-structure interaction problems aresolved to demonstrate the capability of the presentapproach. The paper ends with conclusions for theeffective use of the partitioning method for coupledproblems.

2. COUPLED SHELL-FLUID DYNAMICSIn the present work coupled shell-fluid-dynamics

problems have been analysed by solution of the shelldynamics equation and acoustic wave equation byfinite element discretization. The resulting coupledsecond-order ordinary differential equations for twofields are numerically integrated by Newmarks


3/14

Coupled 3D fluid-structure interaction 517method. The fluid-structure interaction effect is stud-ied by transferring the shell normal acceleration tothe three-dimensional fluid domain and the fluidpressure to the shell surface at the shell-fluid interfacefor each time step in an iterative manner. Using thisapproach, single-field problems of shell dynamics orfluid transients (fluid with rigid boundaries) alongwith coupled shell-fluid dynamic problems can beanalysed. The coupling at the shell-fluid interface isaccounted for in a special manner by assigning adifferent equation numbering system to couplingmatrices in order to achieve optimum computerstorage space.2.1. Shell dynami cs w it h tw o dimensional degenerat eshel l elements

Figure 1 shows a three-dimensional brick elementwith a quadratic displacement field description. Thiselement can be degenerated to form a quadratic shellelement. The shell geometry is either defined by topand bottom surface nodes of the shell element or bythe middle surface nodes along with the thickness ateach node point. Here a first-order shear deformation

shell theory is used, in which it is assumed that thenormal to the shell middle surface remains straightduring deformation and strain energy correspondingto the shell normal stress is neglected.In order to develop a two-dimensional degenerateshell element eight strains are defined at any pointon the shell middle surface, viz. cxo, Q, y,@, K,, K,,,K*,,, yXro nd Y,,~. Here an explicit integration in thethickness direction is used [16, 171 n the strain energyexpression to obtain mid-surface stress resultantscorresponding to these strains. This is called a two-di-mensional model of a three-dimensional degenerateshell element. This method is different from theconventional approach [15,20,25] of considering fivestrains L,, cy, yXv, yXzand yvr at any point in theshell body and then either carrying out two-pointGaussian integration of element matrices (stiffness,mass or load) by taking linear variation of strainacross the thickness or by a layered approach wherea one-point Gaussian integration is applied for eachlayer. The present approach of a two-dimensionaldegenerate shell element with exact representation ofinplane stress/strain across the thickness and explicit

3-D Element

Degenerate shell elementmidsurface nodes

tkgenerote shell elementwith nodes on top andbottom surface

with 2-D Model of degenerateshell elementY

(X,Y,Z) GlobalCX,,Y.,Z,) Nodal(Xa,Yo ,Z,) Lacal -x0Coordinate systems m

DisplacementsFig. 1. Development of two-dimensional degenerate shell element.

as 3*,5,6--c


4/14

518 R. K. SINGH et al.integration results in computational ease and betteraccuracy than the conventional approach. Thismethod is particularly suitable for coupled shell-fluidinteraction problems, as shown in the latter part ofthe paper.

2.1.1. Coordinat e systems for tw o-dimensional de-generat e shell element . As shown in Fig. 1, three setsof coordinate systems, global (X, Y, Z), nodal(x,, y,,, z,) and local (x,, y,, z,), are defined. Theglobal coordinate defines the shell geometry. Thenodal coordinate defines the nodal displacements andglobal stiffness, mass and load matrices. The localcoordinate system is used for numerical integration ofelement matrices. Nodal and local coordinates aredefined by a shell normal vector V, at a node n andany point c( within the element by

v,=v


5/14

Coupled 3D fluid-structure interaction 5192.2. Fluid transients with three-dimensional fluid The above boundaries define the fluid boundary r,elements completely.

In order to study the effect of shell-fluid inter-action, three-dimensional modelling of the fluiddomain is required. A number of finite elementformulations are available, namely pressure formu-lation, displacement formulation, displacementpotential and velocity potential formulation. Thesemethods have been reviewed in the work of Paul [111.The pressure formulation was chosen for the pres-ent work. The advantage of this method is that thereis only one variable at each node (pressure). Thisresults in significant computational economy interms of computer time and storage space comparedto other formulations, particularly for three-dimen-sional problems. Another advantage of this approachis that the pressure field obtained from the analysis ofthe fluid field can be directly transferred to the shellstructure at the shell-fluid interface, unlike othermethods where pressure has to be calculated fromvelocity or displacement or its potential. In thepresent work an eight-noded trilinear brick elementhas been developed to study the fluid transientbehaviour. Since the shell response is of prime con-cern in most of the problems the trilinear fluidelement is considered to be adequate for an optimumcombination of economy and accuracy.

r,=r, rR rF rp. (19)The governing equation (14) along with bound-ary conditions (15)-(18) can be shown toresult in an equation of the following form aftersemidiscretization.

M, + C/P + K/P = p,Q,@ + fig>+ r,, (20)where Mf, C, and K, are the global fluid mass,damping and stiffness matrices, respectively. Q,couples the fluid equations to structure and groundacceleration vectors il and iig, respectively, and fj isthe fluid force vector.The above global matrices can be assembled fromthe fluid element matrices, which can be described asfollows. Pressure in the element is given by the fluidelement shape function N, and nodal pressure p as

P = N/P. (21)The fluid element mass matrix is obtained byimpulsive and free surface sloshing terms as

The governing acoustic wave equation in terms ofdynamic pressure variable p for an inviscid, com-pressible fluid with small displacement is (M;)i, = l/c2 s N;NJdhl/01

V2p = --$jj in fluid domain Sz,, (14)

where c is the acoustic speed. The three types ofpressure gradients at the fluid boundary with outwardnormal nf are as follows.

+ l/g s N;NB dr,. (22)r,=The fluid element radiation damping matrix isconsidered for a nonreflecting boundary rR as

(qij = 1 c N;Nfi drR. (23)(i) Interaction boundary r, with

p, n = -& .

(ii) Free boundary rF with(15) Finally, the fluid element stiffness matrix is

P, n = -@lg.

(iii) Radiation boundary rR with16) + N,G, 9 z) da,. (24)

The coupling matrix Q, for the fluid domain isgiven byp, n = -p/c. (17)

p,, ii, and g are the fluid density, shell normalacceleration at the shell-fluid interface and accelera-tion due to gravity, respectively. It may be noted thateqn 15) becomes homogeneous for a fluid interactingwith a rigid boundary. Another type of boundary isfor prescribed pressure p

where n, gives the unit normal component at the shellsurface on the shell-fluid interface.

p =p on r,. (18)

2.3. Coupled shell --ui d systemsThe semidiscrete shell dynamics equations (12)for a shell coupled with an acoustic medium of


6/14

520 R. K. SINGHet althe above description is modified with coupling termQ, as

M,iI + C,I + K,u = f, - MdiIg + QSp. 26)It may be easily recognised that the coupling matriceshold the relation Q, = -QT. The system of equations(20) and (26) are coupled second-order ordinarydifferential equations which define the coupledshell-fluid system completely. These sets of coupledequations are solved on two different meshes of fluidand shell in a staggered fashion-a scheme proposedby Park and Felippa [lo], Paul [l l] and Felippa andGeers [12]. The advantage of the two-dimensionaldegenerate shell element presented in Sect. 2.1 may benoted here. The nodal coordinate system in the shellformulation gives the normal shell acceleration,which can be directly used with the coupling matrixQf in eqn (20). Similarly, the fluid pressure at theshell-fluid interface can be directly introduced withthe coupling matrix Q, of eqn (26). The pressure atmidside nodes of shell element edges and at theelement centre (r = 0.0 and q = 0.0) are linearlyinterpolated from the corner node values of a three-dimensional fluid element at the shell-fluid interface.A separate equation numbering system is evolved atthe shell-fluid interface in the skyline solution pro-cedure by keeping only active degrees of freedom(one at each node) at the interface. Finally, thecoupling terms in the load vectors are transferred tothe original equations of shell and fluid domains. Thisscheme requires minimum computer storage space forcoupling matrices and is very powerful for three-dimensional shell-fluid interaction problems, particu-larly on small computers with inherent limitations onstorage space.

3 TIME INTEGRATION OF COUPLED FIELDEQUATIONS

The set of coupled second-order ordinary differen-tial equations (20) and (26) are solved by method ofpartitioning. The coupling terms make the set ofequations unsymmetric, so the equations of each fieldshould be solved in sequence. The interaction effectbetween the two fields is studied by transferring fieldvariables from one field to another field at each timestep in a staggered manner. This approach requiresminimum storage space, as bandedness of equationsis maintained. Another advantage of this approach isthat the same time integrator can be applied to boththe fields. In the present work Newmarks predic-tor-multi-corrector algorithm due to Hughes [19] hasbeen used, which has been proven to be very power-ful for such coupled problems [ll]. This methodrecognises each element of either field as explicit orimplicit. Thus a purely explicit or implicit integrationmethod along with mixed explicit-implicit methodcan be used for one or both of the fields. The criticaltime steps for the fluid and shell elements are of

verying magnitude, thus this integration schemeallows significant flexibility in selecting an optimumintegration scheme from a stability, accuracy andconvergence point of view.

As a general guideline, in most of the practicalcases the fluid field integrator may be fully explicit,since the critical time period for the fluid domain isnormally orders of magnitude larger than the criticaltime period of the shell domain for the same meshsize. The shell equations could be treated by implicitmethod as the critical time period in this case isnormally so low that an explicit integration methodwill require very large computational time, over-shadowing the advantage of the lesser computationaleffort and time required at each time step. The explicitintegration for the fluid field is particularly advan-tageous in the present case compared to the implicitintegration. The latter will require large computerstorage space. In addition, it results in the burdenof solving a huge system of simultaneous equa-tions, obtained from three-dimensional discretiza-tion, at each time step. The present formulation withquadratic displacement field description for the shelldomain and linear pressure field description for thefluid domain is very powerful. Here the nodal spacingin the fluid domain is twice that of the shell domain,which allows the above type of explicit-implicit par-titioning for fluid and shell meshes more con-veniently. In the next paragraph we present thepredictor-multi-corrector algorithm [11, 13, 191 forthe semidiscrete coupled second-order equations (20)and (26) used in the present work.

The governing second-order equation at time stepn + 1 is given by

Mii,,+,+Ci,+,+Ku,+, =F,+,. (27)Here the subscript has been dropped as it may beused for any field. The force term augments appliedforce, specified boundary conditions and interactionterms from the other field. In the predictor phase thefield variables are expressed as

_&+,=,+IIi+,= 3,,iib+ I =o,

where i is the iteration count and

(28)(29)(30)

Tu,+, = 8, + Ar(1 - y)ii, (31)_II,, , = u, + At L, + l/2 At2(1 - Zj)iI,,. (32)

Here y and fl are Newmarks parameters and At is thetime step.


7/14

Coupled 3D fluid-structure interaction 521In the solution phase the following equation is

formed and solved.

whereK*Au=~:,, (33)

andK* = M/p At + yC/~ At f K (34)

~~,=F;+,-M~~~+,-CI$+,--KU;+,. (35)Once the increment in the field variable is obtained,field variables are updated as follows:

i+1_ in+I -II,,+, +Au (36)**r+f_Q,l - (II$~, - ti, +1 /@At2 (37)bi t. 1 =cintl I I 1 y ArW;;, . (38)

Finally, a convergence check is made on the norm ofthe increment in the field variable compared to thenorm of the total field variable as follows.Is I/AuII/IIu+ I/ 6 e (the specified tolerance)? (39)

If NO, i-G + I, and go to eqn (33) for nextiteration.

If YES, n +n + 1, and go to eqn (27) for next timestep.

Element-wise mesh ~rtitioning is done byrecognizing elements as explicit or as implicit,

M=M$ME (40)C=C+CE (41)K=K+KE (42)F=F+FE, (43)

and modifying the governing equation (27) asMii,+,+MEii,+, +Cti,+,+Ku,+,

=F:+,+F:+,. (44)

PC??-t

The stability criterion of Newmarks integrator fora single field with a single pass is well estab-lished 113, 19,211. y 3 l/2 and /? = (7 + l/2)z/4 leadsto unconditional stability. In the case of coupledproblems the stability criterion is a function of meshintegrator, predictor formula and computationalpath [lO-121. Analytical evaluation of the critical timestep is difficult in the present case as the order ofthe characteristic polynomial for stability analysis islarge. However, in the present formulation the inter-action term appears as a force term, which may beregarded as similar to the pseudo force term of anonlinear dynamics problem. Therefore, a smallreduction in the time step (approx. lO-20% of thecritical time step) is desirable, as suggested byBelytschko [3] for nonlinear problems, in order toachieve rapid convergence.

4 NUMERICAL EXAMPLES

A computer code FLUSHEL [ZS] has been devel-oped by the present authors to study the performanceof the present formulation. This code can be used tosolve single-field problems of shell dynamics or fluidtransients along with the coupled shell-fluid inter-action problems. A number of problems have beensolved with Newmarks parameter /l = 0.25 andy = 0.5 and a tolerence e = 1 O E - 05 for conver-gence criteria by explicit and implicit methods. Theresults reported here have been obtained on aNORSK DATA machine ND 560 in double pre-cision. The problems presented here have been takenfrom the paper of Akkas et al . S], where the resultswere reported for a continuum mechanics approach.4. I. Pressure w ave propagat i on p robl em

Figure 2a shows the problem of pressure wavepropagation in a rigid pipe. A 20-in.-long pipe witha cross-section of 4 x 4 in. containing fluid has beenanalysed for a pressure pulse of 1psi applied at oneend. The other end of the pipe is closed with a steelcap. The fluid domain has been modelled with20 eight-noded brick elements. Solution has beenobtained for the fluid domain with a homogeneousboundary, i.e. the fluid boundary is assumed to besurrounded by a rigid pipe. A time step of 0.028 msechas been used, which is the critical time step for this

K = 120,000psipt = 0.935 x iO~4ib-secVi n~

Pressurepulse

Fig. 2a. Pressure wave propagation in a pipe.


8/14

522 R. K. SINGH et al.0 - ExactI Ref r512 -.-.- Jmplicit consistent3 ---- Explicit4 --- Implicit lumpedAt = 0.028msec II

I I I I0.3 0.6 09 1.2

Time msec)Fig. 2b. Pressure history in element 1.

II 5

mesh obtained from eigenvalue analysis. This isof same order as h/c, where h is the mesh sizealong the axis of the fluid domain. Results areshown in Figs 2b-2e at the centre of elements 1, 5,15 and 20 for the case of implicit integration witha consistent mass matrix of fluid, explicit inte-gration with a lumped mass and implicit inte-gration with a lumped mass. In the same figures theexact results and results of [S] are also shown. Thenumerical results of Akkas et al. [5] and the pres-ent results show dispersion or oscillatory behaviourat the shock front due to discontinuity, and spatialand temporal discretization errors as discussed by

2.50 -

2.00 -

=:: I50-

ih I .oo -

0.50 -

Schreyer [29]. It may be inferred from the natureof the results that the matched method of timeintegration, i.e. explicit method with lumped massand implicit method with consistent mass, rep-resents the stock wave more accurately. This hasbeen discussed in detail by Hughes [19], where thetheoretical argument given is that spatial and tem-poral discretization errors tend to cancel each otherin matched method. It is realized from the pres-ent example that the explicit method with a lumpedmass matrix can be used effectively to trace the shockwave in an economic way for three-dimensionalproblems.

:li0 I---I/I 0 _ Exact:v ----- Ref 1512 -.-.- Implici t consistent3 ---- Explicit4 -- ImpLicit lumped

At = o.o28msec

I I0.6 0.9Time msec)

I I1.2 I 5

Fig. 2c. Pressure history in element 5.


9/14

Coupled 3D fluidstructure interaction 523250

200

51: 150

eilf 1.00

0.50

0

0 - ExactI -- Ref C512 -.--- Implicit consistent3 --- Explicit4 -- Implicit lumpedAt = 0.028 m set

L I I I I03 06 0.9 12 15

4.2. Plat e-Juid i nteracti on problemIn Fig. 3a the problem of a fluid reservoir

(150 cm x 100 cm x 50 cm) supported by a steel plate(5 cm x 100 cm x 50 cm) at one end is shown. Atthe other end of the reservoir a pressure pulse of100 kg/cm* is applied. The pressure history is tracedat point A (x = 142.5 cm, y = 75 cm) in the fluiddomain. The analysis has been carried with a15 x 10 x 1 mesh of the fluid domain with eight-noded brick elements. The steel plate is modelled with10 nine-noded shell elements. The critical time stepsfor fluid and shell meshes are 0.035 and 0.0039 msec,respectively. Here an explicit-implicit (E-I) partition-ing for fluid and shell meshes can be used, since the

2 50 r

Time fm set)Fig. 2d. Pressure history in element 15.

200

I 50

time step for the shell is more critical. Transientanalysis has been carried with a time step of0.032msec. Figure 3b shows the results, and theresults due to Akkas et al. [5] are also shown for therigid plate case and the flexible plate case. Again thepressure history shows oscillation near the shockfront for the rigid plate case compared to the exactsolution. However, it is less than that reported in [5]due to proper selection of the integrating schemeand mass matrix. In the case where the solution isobtained by taking into account the flexibility of thesteel plate, the pressure history is lower than thatreported in [5]. The results of [5] are based on bilineartwo-dimensional plane strain elements, where onlyone element has been taken across the thickness of

0 - Exact

2 --------- Implicitef [51 consistent3 ----- Explic it4 --- Implicit lumpedAt = 0.028msec

0.3 06 09 1.2Time (msecl

Fig. 2e. Pressure history in element 20.


10/14

524 R. K. SINGH t al.

Pressurep&e--I

P 5 = 0.0 ReservoirPoint A I = 0.0 c = I 43333 x 105cm/sect pt 1.019x 10e6 kg-sec2/cm4g 991 all/s&

= 9.002 x Is4 kg- sec2/cm4

15Qcm

Fig. 3a. Plate-fluid interaction problem.the plate. This element is known to be very stiff inbending, so the reported pressure is high. In orderto check this, the results have been compared withthe two-dimensional results of Singh et al. [30] withcode FLUSOL, which is a two-dimensional code forfluid-structure interaction. In two-dimensional anla-ysis the fluid has been modelled with 5 x 5 nine-noded plane fluid elements and the plate has beenmodelled with a 5 x 1 mesh of nine-noded planestress elements. As shown in Fig. 3b, the two-dimensional results match well with the presentthree-dimensional fluid and shell element results.4.3, Reservoir with base motion

Problems of water reservoirs and fluid tankssubjected to base excitation are one of the import-ant areas where considerable attention has been

created[4,31-361. We now present the problemof a water reservoir of 300 ft x 300 ft size which issubjected to a constant acceleration of 1.0 g at thebase. If the fluid boundary is assumed to be sur-rounded by a rigid wall, comparison can be madewith the exact pressure history available due toChopra et al. [31] and two-dimensional finite elementresults due to Akkas et al. [S]. Figure 4 shows theimpulsive pressure history at x = 30 ft and y = 30 ftfrom the base for a three-dimensional fluid modelwith 6 x 5 x 1 mesh by implicit and explicit inte-gration methods, with a time step of 2.5 msec. Inthe same figure, the two-dimensional finite elementresults of Akkas et al. [5] are also shown by directintegration (DI) and mode superposition (M-S)methods. It may be noted here that the presentresults trace the impulsive pressure response more

200 -%zs 150-

pL??

n loo-

50 -

7,,.- O-Rigid exactI _-_--Rigid pf [512--- Flexi Ref t533 ----Rigid 304 Flexi

---3D Fresent

5 -- Flexi 2 D 1(30)

At n 0.032 msec for 3DAt =0.05msec far 2D

I I0 05 IO 1.5 2.0 25Time (msec)

Fig. 3b. Plate4luid interaction.


11/14

Coupled 3D fluid-structure interaction 525

iig I.Ogt

o- Exact 311I ----- DI [512 -.--- M- s [ 53 --_ Implicit4 _-_ Expl icit PresentIAt = 2.5msecc = 4720ftkcg = 32.2ft/s&p, 1.94 lb-secVft4

0 60 120 I60 240 300lime (msec 1

Fig. 4. Reservoir under base motion (three-dimensional result).

accurately than that given in [5]. The phase shiftand amplitude change are a minimum for the pres-ent case. This is due to the fact that pressure isobtained directly in the present case, while in thecontinuum mechanics approach the pressure isobtained from the gradient of the displacementfield. Some spurious modes are also present in theseresults due to circulation and presence of pseudofrequency, which is typical of continuum mechanics-based fluid elements, unless these are suppressednumerically.4.4. Exteri or shell -f l uid int eracti on problem

One of the classical problems of shell-fluid inter-action is that of a spherical shell submerged in an

L=05in. (Exact)L i 9.5 in. (Exact 10751 i in. (Exact)

infinite acoustic medium. A spherical shell withinside radius of 75 in. and thickness of 1 in. has beentaken for analysis. The shell is subjected to animpulsive internal pressure of 1.0186 psi and thepressure wave is traced at distances of 0.5, 9.5, 19.5and 29.5 in. from the shell outer surface. Due tosymmetry 2 sectors in CircumferentiaLand longitudi-nal directions have been taken on the shell surface,which is modelled with a 2 x 2 element mesh. Thesurrounding fluid domain has been modelled witha 2 x 2 x 15 mesh of brick elements in the presentcase. The fluid mesh is taken up to a radial distanceof 29.5 in. from the shell outer surface. A radi-ation boundary condition is applied at this end.The critical time steps for the fluid and shell meshes

I --. L = 0.5 i n.2---L = 9.5 in.3---L*l9.5in.J 4 -- L =-. 29.5 in.At = 0.0225 msecImplicit fluid,implicit shell

0 PFiIR * 75 in. t--_ 0 a4 0.6 1.2 1.6 2.0

Time ( m set )Fig. 5a. Pressure wave propagation from submerged sphere.


12/14

526 R. K. SINCW t al.

r =0.5in.(Exact)L= 9.5in.fExaatf075

9 rL=1g.5in.tExactI I----L= 0.5 in.

e-C in.(Exact) 2-- La 9.5 in.3---L= 19.5in.4---t = 29.5 in

At = 0.0225msecExplicit fluid,imrAicit shell

Kw3.33 x 105psiThickness 8 lin. pf = x9582 x lo-4lb - se&in4

I I I0 04 08 I2 I6 2.0Time (msec)

Fig. Sb. Pressure wave pro~gation from submerged sphere.

are 0.0233nd 0.00286sec, respectively. Figure 5ashows the pressure wave propagation obtainedby implicit-implicit (I-I) partitioning for fluid andshell meshes, respectively, with a time step of0.0225 msec. Figure 5b shows the results withexplicit-implicit (E-I) partitioning with same timestep. In this case the fluid radiation dampingmatrix C, is lumped in the same manner as Mby a special lumping technique 1261. The radialdeflection of the shell is shown in Fig. 5c andboth types of partitionings give responses whichoverlap each other. The present results in allthese cases match very well with the exactresults [S].

4.5. Int eri or shell -f l uid int eraction problemAnother shell-fluid interaction problem of interest

is that of a spherical shell containing fluid inside it.This problem has been solved by Akkas et al. [5] andShugar and Katona [37]. The shell is subjected to animpact pressure of 100 kg/cm* applied at the shell toppole, as shown in Fig. 6. A sector of 30 in thecircumferential direction and 180 in the longitudinaldirection of the shell is modelled with proper sym-metry conditions. The fluid mesh is made consistentwith the outer shell cover. The analysis has beencarried out for E-I and I-I partitionings of fluid andshell meshes for time steps of 1.25 E - 03 msec and

100

80

O-Exact, ___- _

At 8 0.0225 msec

Thickness = I in. E= 29x IO6 psiY f 0.3pr = 7.345 x ld41b- secz/in

I I I I 10 0 4 0.8 1.2 16 20

Time (msec 1Fig. 5c. Radial deflection of submerged sphere.


13/14

Coupled 3D fluid-structure interactionP

521lOOkg/cm2

PI t10000 r R=7cm 0-Ref t51Thickn ess = 0.5cm I ----I-I (3D) 1

At =2.5 x 10m3 msec (20)At = 2.5x Id3 msec (3DJ-I) 3--I-I (20)At= 1.25 x iO~3msec (3D,E-I)

~~105kg/cmzVO.3

p = 2x 16 kg-secVcm4

K- 2. I x 104kg/cm2p, = I x I G6 g - se&cm 4

Time (msec)Fig. 6. Interior shell-fluid interaction.

2.5 E - 03 msec, respectively. The top pole pressure(at a radius of 6.5 cm and an angle of 5 from thevertical axis) in the fluid is compared with the resultsof [5] and [37]. The present results with the shellelement show a higher peak pressure. Moreover, inthe latter portion of the curve (approx. 0.2msec)pressure again rises. This behaviour is not shown inthe results of [5] or [37]. The second pressure peak isdue to the superposition of the pressure wavereflected from the lower pole after 0.2 msec, which isthe time required for the wave to reach the top poleafter reflection from the lower pole. In order to checkthis again comparison has been made with results dueto Singh et al. [30], where the analysis is carried outwith two-dimensional axisymmetric elements avail-able in code FLUSOL. The two-dimensional resultwith I-I partitioning with a time step of 2.5 E -03 msec is also shown in Fig. 6. This result also showsthe effect of the reflected wave after 0.2 msec.

5. CONCLUSIONSTransient analysis of three-dimensional fluid-

structure interaction problems has been demon-strated through an effective partitioning scheme. Thepresent approach is economical in terms of thestorage space and CPU time required for such prob-lems. The economical implementation has beenachieved through a displacement-based two-dimen-sional degenerate Co shell element and a three-dimen-sional fluid element with only pressure variables asunknowns at nodal points. An explicit integrationscheme for the fluid domain (which is normally oflarge size) and an implicit scheme for the shellstructure makes this approach very attractive. By

assigning separate equation numbering to the coup-ling matrices and using a nodal coordinate system forthe shell structure the storage requirement for thecoupling matrices reduces to a minimum. A numberof case studies presented in the present paper indicatethat this formulation has potential for nonlinearfiuid-structure interaction problems as an extensionto the present approach.

1.

2.

3.4.

5.

6.

REFERENCESK. J. Bathe and W. F. Hahn, On transient analysis offluid-structure systems. Compur. Struct. 10, 383-391(1979).W. K. Liu and H. G. Chang, A method of computationfor fluid structure interaction. Compuf. Struct. 20,31 l-320 (1985).T. Belytschko, Fluid-structure interaction. Comput.Struct. 12, 459469 (1980).W. K. Liu, Development of finite element proceduresfor fluid structures interaction. Ph.D. thesis, EERL80-86, Californian Institute of Technology, Universityof California at Berkeley, CA (1981).N. Akkas, H. U. Akay and C. Yilmaz, Applicability ofgeneral purpose finite element programes in solid-fluidinteraction problems. Compuf. Struct. 10, 773-783(1979).E. L. Wilson and M. Khalvati, Finite elements for thedynamic analysis of fluid-solid systems. Znf. J. Numer.M efh. Engng 19, 1657-1668 (1983).D. Shantaram, D. R. J. Owen and 0. C. Zienkiewicz,Dynamic transient behaviour of two or three dimen-sional structures, including plasticity, large deformationeffects and fluid interaction. Earthquake Engng. strucr.Dynam. 4, 561-576 (1976).S. S. Deshpande, R. M. Belkune and C. K. Ramesh.,Dynamic analysis of coupled fluid-structure interactionproblems. In Numerical M ethods for Coupled Problems(Edited by E. Hinton, P. Bettes and R. W. Lewis),pp. 367-378. Pineridge Press, Swansea, U. K. (1981).


14/14

528 R. K. SINGH et l .9

10.

11.

12.

13

14.

15.

16.

17.

18.

19.20.

21.22.

23.

M. K. Au-Yang and J. E. Galford, Fluid-structureinteraction-a survey with emphasis on its applicationto nuclear steam system design. Nucl. Engng Des. 70,387-399 (1982).K. C. Park and C. A. Felippa, Partitioned analysis ofcoupled systems. In Computati onal M ethods for Transi-ent -Anal ysis (Edited by-T. Belytschko and T. J. R.Hushes). Ch. 4. North Holland. Amsterdam (1983).D. -K. Paul, Single and coupled multifield problems.Ph.D. thesis c/ph/64/82, University College of Swansea,University of Wales, U.K. (1982).C. A. Felippa and T. L. Geers, Partitioned analysis forcoupled mechanical systems. Engng Comput. 5,123-133(1988).T. Belytschko and T. J. R. Hughes (Eds), Compu-tat ional M ethods for Transient Anal ysis. North Holland,Amsterdam (1983).T. L. Geers and C. Ruzicka, Finite element/boundaryelement analysis of multiple structures excited bytransient acoustic waves. In Numerical M etho& forTransient and Coupled Problems (Edited by R. W.Lewis, P. Bettes and E. Hinton), pp. 150-162. PineridgePress, Swansea, U.K. (1984).S. Ahmad, B. M. Irons and 0. C. Zienkiewicz, Analysisof thick and thin shell structures by curved elements.Inl. J. Numer. Meth . Engng 2, 419-451 (1970).T. Belytschko, Stress projection for membrane andshear locking in shell finite element, Compur. Meth.appl . M ech. & gng 51, 221-258 (1985).R. V. Milford and W. C. Schnobrich. Deneneratedisoparametric finite elements using explicit integration.Int . J. Numer. M eth . Ertgng 23, 133-154 (1986).E. D. L. Pugh, E. Hinton and 0. C. Zienkiewicz, Astudy of quadrilateral plate bending elements withreduced integration. Int . J. Numer. M eth. Engng 12,1059-1079 (1978).T. J. R. Hughes, The Fini te Element M ethod.Prentice-Hall, Englewood Cliffs. NJ (1987).D. R. J. Owen and E. Hinton, Finite Elements inPl asticit y: Theory and Practi ce. Pineridge Press,Swansea, U.K. (1980).0. C. Zienkiewicz, The Fini te El ement M ethod, 3rd Edn.McGraw-Hill, London (1976).T. Belytschko, A survey of numerical methods andcomputer programs for dynamic structural analysis.Nucl. Engng Des. 37, 23-24 (1976).T. Belytschko and R. Mullen, Stability and explicit-implicit mesh partitions in time integration. Int. J.Numer. Mefh. Engng 12, 1575-1586 (1978).

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.34.

35.

36.

37.

T. Belytschko, H. J. Yen and R. Mullen, Mixedmethods for time integration. Comput . M efh. appl .Mech. Engng 17/U, 259-275 (1979).E. Hinton and D. R. J. Owen, Fini te Element Softw arefor Plates and Shells. Pineridge Press, Swansea, U. K.(1984).E. Hinton, T. Rock. and 0. C. Zienkiewicz, A note onmass lumping and realted processes in the finite elementmethod. Eart hquake Engng struct. Dynam. 4, 245-249(1976).K. S. Surana, Lumped mass matrices with non-zeroinertia for general shell and axi-symmetric shell elements,Int . J. Numer. M eth. Engng 12, 1635-1650 (1978).R. K. Singh, T. Kant and A. Kakodkar, Studies ofshell fluid interaction problems. In Engineering Soft-w are (Edited by C. V. Ramakrishnan, A. Varadarajanand C. S. Desai) (Proc., International Conference,ICENSOFT-89, 4-7 December 1989, New Delhi),pp. 865-872. Narosa, New Delhi (1989).H. L. Schreyer, Dispersion of semidiscretized and fullydiscretized systems. In Computational Methods forTransient Anal ysis (Edited by T. Belytschko and T. J. R.Hughes), Ch. 6. North-Holland, Amsterdam (1983).R. K. Singh, A. Kakodkar and T. Kant, Some studieson fluid structure interaction problems. Report No.BARC-1424, BARC, Trombay, Bombay (1988).A. K., Chorpa, E. L. Wilson and L. Farhoomand,Earthquake analysis of reservoir dam systems.Proc. 4th World Conference on Earthquake Engin-eering, Santiago, Chile (1969).P. Chakrabarti and A. K. Chopra, Earthquake responseof gravity dams including reservoir interaction effects.EERC 72-6, University of California, Berkeley, CA(1972).N. M. Newmark and E. Resenblueth, Fundamentals ofEarthquake Engineering. Prentice-Hall, EnglewoodCliffs, NJ (1971).J. R. Roberts, E. R. Basurto and P. Y. Chen, Sloshdesign handbook 1. NAS 8-1111 I, Northrop SpaceLaboratories, Huntsville, AL (1966).D. D. Kana, Status and research needs for prediction ofseismic response in liquid containers. Nucl. Engng Des.69, 205-221 (1982).D. C. Ma, J. Gvildys and Y. W. Chang, Seismicbehaviour of liquid-filled shells. Nucl. Engng Des. 70,437455 (1982).T. A. Shugar and M. G. Katona, Development of finiteelement head injury model. J. Engng Mech. D i u, AX E101, 223-239 (1975).

Documents

Shell Coupled FSI _example