PARTITIONED ANALYSIS OF COUPLED SYSTEMS

COMPUTATIONAL MECHANICSNew Trends and Applications

E. Onate and S.R. Idelsohn (Eds.)c©CIMNE, Barcelona, Spain 1998

PARTITIONED ANALYSIS OF COUPLED SYSTEMS

Carlos A. Felippa,∗ K.-C. Park∗and Charbel Farhat∗

∗Department of Aerospace Engineering Sciencesand Center for Aerospace StructuresUniversity of Colorado at Boulder

Boulder, Colorado 80309-0429, USAWeb page: http://caswww.colorado.edu

Key words: Coupled systems, multiphysics, partitioned analysis, staggered solution,parallel computation

Abstract. This article reviews the use of partitioned analysis procedures for the analy-sis of coupled dynamical systems. Attention is focused on the computational simulationof coupled mechanical systems in which a structure is a major component. Importantapplications in that class are provided by thermomechanics, fluid-structure interactionand control-structure interaction. In the partitioned solution approach, systems arebroken down into partitions chosen in accordance with physical or computational char-acteristics. The solution is separately advanced in time over each partition. Interactioneffects are accounted for by periodical transmission and synchronization of coupled statevariables. Recent developments in the use of this approach for multilevel decompositionaimed at massively parallel computation are discussed.

C. A. Felippa, K. C. Park and C. Farhat

1 INTRODUCTION

The American Heritage Dictionary lists eight meanings for system. By itself theterm is used here in the sense of “a functionally related group of components formingor regarded as a collective entity.” This definition uses “component” as a generic termthat embodies “element” or “part,” which connotes simplicity, as well as “subsystem,”which connotes complexity.

We further restrict attention to mechanical systems, and in particularly those of im-portance in Aerospace, Civil and Mechanical Engineering. The downplaying of systemsof interest to Physics, Chemistry and Electrical Engineering does not imply restrictionin methodology extension. It is simply that less experience is available as regards theuse of the methods described here.

Systems are analyzed by decomposition or breakdown. Complex systems are amenableto many kinds of decompositions chosen according to specific analysis or design objec-tives. We focus here on decompositions suitable for computer simulation. This kindof simulation usually aims at describing or predicting the state of the system underspecified conditions viewed as external to the system. A set of states ordered accordingto some parameter such as time or load level is called a response.

System designers are not usually interested in detailed response computations per se,but on project goals such as cost, performance, lifetime, fabricability, inspection require-ments and satisfaction of mission objectives. The recovery of those overall quantitiesfrom simulations is presently an open problem in computer system modeling and onethat is not addressed here.

1.1 Coupled System Terminology

Because coupled systems can be studied by many people from many angles, termi-nology is far from standard. The following summary is one that has evolved for thecomputational simulation, and does reflect personal choices. Many of the definitionsfollowed usage introduced in a 1983 review article.1 [Casual readers may want to skimthe following material and return only for definitions.]

A coupled system is one in which physically or computationally heterogeneous me-chanical components interact dynamically. The interaction is multiway in the sense thatthe response has to be obtained by solving simultaneously the coupled equations whichmodel the system. “Heterogeneity” is used in the sense that component simulationbenefits from custom treatment.

The computational decomposition of a complex coupled system is always hierarchical.The hierarchy must be constructed top-down. It usually spans several levels, 2 or 3 beingthe most common. At the first level one encounters two types of subsystems, embodyin the generic term field:Physical Fields. Subsystems are called physical fields when their mathematical modelis described by field equations. Examples are mechanical and non-mechanical objectstreated by continuum theories: solids, fluids, heat, electromagnetics. Occasionally those


Structure

Fluid

3D SPACExi

TIME t

TIMESTEP h

End time t

Start time tn

n+1

Splitting in time(fluid field only)

Partitioningin space

Solve fluid in

this direction

Then along

this one,

and so on

Figure 1. Decomposition of an aeroelastic FSI coupled system: partitioning in space andsplitting in time. 3D space is shown as ”flat” for visualization convenience. Spatialdiscretization (omitted for clarity) may predate or follow partitioning. Splitting (herefor fluid only) is repeated over each time step and must follow time discretization.

components may be intrinsically discrete, as in actuation control systems or rigid-bodymechanisms. In such a case the term “physical field” is used for conveniency only, withthe understanding that no spatial discretization process is involved.Computational Fields. Sometimes artificial subsystems are incorporated for computa-tional conveniency. Two examples: dynamic fluid meshes to effect volume mappingof Lagrangian to Eulerian descriptions in interaction of structures with fluid flow, andfictitious interface fields that facilitate information transfer between two subsystems.

A coupled system is said to be two-field, three-field, etc., according to the number offields that appear in the first-level decomposition.

For computational treatment of a dynamical coupled system, fields are discretized inspace and time. A partition is a field-by-field decomposition of the space discretization.A splitting is a decomposition of the time discretization of a field within its time stepinterval. See Figure 1. In the case of static or quasi-static analysis, actual time isreplaced by pseudo-time or some kind of control parameter.

Partitioning may be algebraic or differential. In algebraic partitioning the completecoupled system is spatially discretized first, and then decomposed. In differential parti-tioning the decomposition is done first and each field then discretized separately.

Splitting may be additive or multiplicative. As the present article focus on partitions,these aspects will be only briefly covered below.

A property is said to be interfield or intrafield if it pertains collectively to all parti-


tions, or to individual partitions, respectively. A common use of this qualifier concernsparallel computation. Interfield parallelism refers to the implementation of a parallelcomputation scheme in which all partitions can be concurrently processed for time-stepping. Intrafield parallelism refers to the implementation of parallelism for a specificpartition using a second-level decomposition.

1.2 Examples

Experiences reported in this survey comes from systems where a structure is one ofthe fields. Accordingly, all of the following examples list structures as one of the fieldcomponents. The number of interacting fields is given in parenthesis.• Fluid-structure interaction (2)• Thermal-structure interaction (2)• Control-structure interaction (2)• Control-fluid-structure interaction (3)• Fluid-structure-combustion-thermal interaction (4)

When a fluid is one of the interacting fields a wide range of computational modelingpossibilities opens up as compared to, say, structures or thermal fields. For the latterfinite element methods can be viewed as universal in scope. On the other hand, therange of fluid phenomena is controlled by several major physical effects such as viscosity,compressibility, transport, gravity and capillarity. Incorporation or neglect of theseeffects gives rise to widely different field equations. For example, the interaction of anacoustic fluid with a structure in aeroacoustics or underwater shock is computationallyunlike that of high-speed gas dynamics with an aircraft, or of flow through porousmedia. Even more variability can be expected if chemistry and combustion effects areconsidered. Control systems also exhibit modeling variabilities that tend not to be sopronounced, however, as in the case of fluids.

The partitioned treatment of some of these examples is discussed further in subse-quent sections.

1.3 Splitting and Fractional Step Methods

Splitting methods for the equations of mathematical physics predate partitionedanalysis by two decades. In the American literature they can be originally traced tothe development by 1955 of alternating direction methods by Peaceman, Douglas andRachford.2,3 A similar development was independently undertaken in the early 1960s bythe Russian school led by Bagrinovskii, Godunov and Yanenko, and eventually unifiedin the method of fractional steps.4 The main computational applications of these meth-ods have been the equations of gas dynamics in steady or transient forms. They areparticularly suitable for problems with layers or stratification, for example atmosphericdynamics or astrophysics, in which different directions are treated by different methods.

The basic idea is elegantly outlined in the monograph by Richtmyer and Morton5:suppose the governing equations in divergence form are ∂W/∂t = AW , where the oper-


ator A is split into A = A1 + A2 + . . .+ Aq. Chose a time difference scheme and thenreplace A successively in it by qA1, qA2, . . ., each for a fraction h/q of the temporalstepsize h. Then a multidimensional problem can be replaced by a succession of simpler(e.g., one-dimensional) problems.

Comparison of such methods with partitioned analysis makes clear that little overlapoccurs. Splitting methods are appropriate for the treatment of an individual partitionsuch as a fluid, if the physical structure of such partition display strong directionalitydependencies. Thus, splitting methods are seen to belong to a lower level of a top-downhierarchical decomposition.

1.4 Computational Multiphysics

Recently the term multiphysics has gained prominence within computational physicsand mechanics. This term can be viewed as being more general than coupled mechanicalsystems in two senses:

• It embraces problems of importance in the physical sciences, which may go beyondengineering systems. For example, electromagnetic and chemistry phenomena figuremore prominently.

• Wider ranges of interacting length and time scales are addressed. This may leads tomultiple models of the same field, which coexist in space and time. Examples of thisclass of problems occur in computational micromechanics and in fluid turbulence.

Given this additional generality, one must be cautious in claiming that a certainpartitioned approach, proven useful for a class of coupled systems, is a panacea formultiphysics. Merits and disadvantages must be assessed on a case by case basis.

2 COUPLED SYSTEM SIMULATION

2.1 Scenarios

Because of their variety and combinatorial nature, the simulation of coupled problemsrarely occurs as a predictable long-term goal. Some more likely scenarios are as follows.

Research Project. Two or more persons in a research group develop expertise in modelingand simulation of two or more isolated problems. They are then called upon to poolthat disciplinary expertise into a coupled problem.

Product Development. The design and verification of a product requires the concurrentconsideration of interaction effects in service or emergency conditions. The team doesnot have access, however, to software that accommodates those requirements.

Software House. A company develops commercial application software targeted to singlefields as isolated entities. For example: a CFD gas-dynamics code, a structure FEMprogram, or a thermal conduction analyzer. As the customer base expands, requests arereceived for allowing interaction effects targeted to specific applications. For example


the CFD user may want to account for moving rigid boundaries, then interaction witha flexible structure, and finally to include a control system.

The following subsection discusses three approaches to the problem.

2.2 Approaches

To fix the ideas we assume that the simulation calls for dynamic analysis that in-volves following the time evolution of the system. [Static or quasistatic cases can beincluded by introducing a pseudo-time control parameter.] Furthermore the modelingand simulation of isolated components is well understood. The three approaches to thesimulation of the coupled system are:Field Elimination. One or more fields are eliminated by techniques such as integraltransforms or model reduction, and the remaining field(s) treated by a simultaneoustime stepping scheme.Monolithic or Simultaneous Treatment. The whole problem is treated as a monolithicentity, and all components advanced simultaneously in time.Partitioned Treatment. The field models are computationally treated as isolated entitiesthat are separately stepped in time. Interaction effects are viewed as forcing effects thatare communicated between the individual components using prediction, substitution andsynchronization techniques.

Elimination is restricted to special linear problems that permit efficient decoupling.It often leads to higher order differential systems in time, which can be the source ofnumerical difficulties.

Both the simultaneous and partitioned treatment are general in nature. No technicalargument can be made for the overall superiority of either. Their relative merits are notonly problem dependent, but are interwined with human factors as discussed below.

2.3 Good or Bad?

The key words that favor the partitioned approach are: customization, independentmodeling, software reuse and modularity.Customization. This means that each field can be treated by discretization techniquesand solution algorithms that are known to perform well for the isolated system. Thehope is that a partitioned algorithm can maintain that efficiency for the coupled problemif (and that is a big if) the interaction effects can be also efficiently treated. As discussedin Section 5, in the original problem that motivated the partitioned approach that happycircumstance was realized.Independent Modeling. The partitioned approach facilitates the use of non-matchingmodels. For example in a fluid-structure interaction problem the structural and fluidmeshes need not coincide at their interface. This translates into project breakdownadvantages in analysis of complex systems such as aircraft. Separate models can beprepared by different teams or subcontractors, which can be geographically distributed.


Software Reuse. Along with customized discretization and solution algorithms, cus-tomized software (private, public or commercial) may be available. Furthermore, thereis often a gamut of customized peripheral tools such as mesh generators and visualiza-tion programs. The partitioned approach facilitates taking advantage of existing code.This is particularly suitable to academic environments, in which software developmenttends to be cyclical and loosely connected from one project to another.Modularity. New methods and models may be introduced in a modular fashion accordingto project needs. For example, it may be necessary to include local nonlinear effects inan individual field while keeping everything else the same. Implementation, testing andvalidation of incremental changes can be conducted in a modular fashion.

These advantages are not cost free. The partitioned approach requires careful formu-lation and implementation to avoid serious degradation in stability and accuracy. Par-allel implementations are particularly delicate. Gains in computational efficiency over amonolithic approach are not guaranteed, particularly if interactions occur throughout avolume as is the case for thermal and electromagnetic fields. Finally, the software mod-ularity and modeling flexibility advantages, while desirable in academic and researchcircles, may lead to anarchy in software houses.

In summary, circumstances that favor the partitioned approach for tackling a newcoupled problem are: a research environment with few delivery constraints, access to ex-isting software, localized interaction effects (e.g. surface versus volume), and widespreadspatial/temporal component characteristics. The opposite circumstances: commercialenvironment, rigid deliverable timetable, massive software resources, extensive interac-tion effects, and comparable length/time scales, favor a monolithic approach.

The remaining of this article covers the partitioned approach. It presents the essen-tials for a simple problem, and gradually generalizes the methodology to more complexcases.

3 THE KEY IDEA

Consider the two-way interaction of two scalar fields, X and Y , color sketched inFigure 2. Each field has only one state variable, identified as x(t) and y(t), which areassumed to be governed by the semidiscrete first-order differential equations

3x+ 4x− y = f(t)y + 6y − 2x = g(t)

(1)

in which f(t) and g(t) are the applied forces. Treat this by Backward Euler integrationin each component:

xn+1 = xn + hxn+1, yn+1 = yn + hyn+1 (2)

where xn ≡ x(tn), yn ≡ y(tn), etc. At each time step n = 0, 1, 2, . . . one gets[3 + 4h −h−2h 1 + 6h

] [xn+1yn+1

]=

[fn+1 + 3hxngn+1 + hyn

](3)


g(t)

f(t) x(t)

y(t)Y

X

Figure 2. The red and the black.

1. (P) Predict:

2. (Ax) Advance x:

3. (S) Substitute:

4. (Ay) Advance y:

BASIC STEPS

xn+1 =1

3 + 4h( fn+ 1 + 3hxn + hy P

n+ 1)

xn+1 = xn+1

(for example)

(trivial here)

y Pn+1 = yn + hyn

yn+1 =1

1 + 6h(gn+1 + hyn + 2hxn+ 1)

.

Figure 3. Basic steps of a red/black staggered solution.

in which x0, y0 are given from initial conditions. In the monolithic or simultaneoussolution approach, (3) is solved at each timestep, and that is the end of the story.

3.1 Staggering

A simple partitioned solution procedure is obtained by treating (3) with the followingstaggered partition with prediction on y:

[3 + 4h 0−2h 1 + 6h

] [xn+1yn+1

]=

[fn+1 + 3hxn + hyPn+1

gn+1 + hyn

](4)

where yPn+1 is a predicted value; two choices being yPn+1 = yn or yPn+1 = yn + hyn. Thebasic solution steps are displayed in Figure 3. The main achievement is that systems Xand Y can be solved in tandem. A state-time diagram of these steps, with time alongthe horizontal axis, is shown in Figure 4.

Suppose that X and Y are handled by two communicating programs called the SingleField Analyzers, or SFAs. If the intrafield advancing arrows are suppressed, we obtain azigzagged picture of interfield data transfers between the X program and the Y program,as depicted in Figure 5(a). This graphical interpretation motivated the name staggeredsolution procedure.6


Step 2: Ax

Step 4: Ay

Step 3: SStep 1: P

Timexn xn+1

yn+1yn

Figure 4. Interfield+intrafield time-stepping diagram ofthe staggered solution steps listed in Figure 3.

Y program:

X program:

Y program:

X program:

TimeTimestep h

I I

SP

P P

P P

S P S

(a)

(b)

Figure 5. Interfield time stepping diagrams: (a) sequential staggered solution ofexample problem, (b) modification suitable for parallel processing.

3.2 Concerns

The immediate concern with partitioning should be degradation of time-steppingstability caused by predictions. For the example system this is not significant. If anA-stable method such as Backward Euler or Trapezoidal Rule is used on each field,it is well known that the simultaneous solution (3) guarantees unconditional stabilityfor the coupled problem of this nature. A simple von Neumann amplification analysisof (4) shows that the staggered solution maintains unconditional stability if predictorsyPn+1 = yn + σhyn, with σ ∈ [0, 1], are used. But for more complex problems thereduction of stability can become serious or even catastrophic.

Prediction could be done on the y field, again leading to a zigzagged diagram withsubstitution on y. The stability of both choices can be made identical by appropriatelyadjusting predictors.

If satisfactory stability is achieved, the next concern is accuracy. This is usuallydegraded with respect to that attainable by the monolithic scheme. In principle this


P SC

A

A

ScA

A A+MC

h

Prediction

Lockstep advancing

Midpointcorrection

Full stepcorrection

Subcycling Augmentation

Substitution Interfield Iteration

I

Time

Figure 6. Devices of partitioned analysis time stepping.

can be recovered by iterating the state between the fields. Iteration is done by cyclingsubstitutions at the same time station. However, intrafield iteration generally costs morethan cutting the timestep to attain the same accuracy level. If the monolithic solutionis more expensive than the staggered solution for the same timestep, we note here theemergence of a tradeoff.

An examination of Figure 5 clearly shows than this staggered scheme is unsuitablefor interfield parallelization because SFAs must execute in a strictly sequential fashion:first program X, then Y, then X, etc. This was of little concern when the method wasformulated in the mid 1970s, because all computers — with the exception of one exoticmachine known as the ILLIAC IV — were sequential.

The modification sketched in Figure 5(b) permits both programs to advance theirinternal state concurrently and thus fits a parallel computer better. Unfortunately itrequires predictions on both fields, which can have a serious impact on stability. Even ifthis is overcome, interfield iteration — denoted by I in Figure 5(b) — may be required toregain accuracy, which increases data communication overhead. More practical parallelschemes are discussed in Section 6.

3.3 Devices of Partitioned Analysis

As this simple example makes clear, partitioned analysis requires the examination ofmany possibilities, and the study of tradeoffs. Figure 6 displays the main “tools of thetrade” in field time stepping diagrams. Some devices such as prediction, substitutionand iteration, have been discussed along with the foregoing example. Others will emergein the application problems discussed in Sections 5 and 6.


4 HISTORICAL BACKGROUND

The partitioned treatment of coupled systems involving structures emerged indepen-dently in the mid 1970s at three locations: Northwestern University by T. Belytschkoand R. Mullen, Cal Tech by T. J. R. Hughes and W. K. Liu, and Lockheed Palo AltoResearch Laboratories (LPARL) by J. A. DeRuntz, C. A. Felippa, T. L. Geers andK.-C. Park. These three groups targeted different applications and pursued differentproblem-decomposition techniques. For example, Belytschko and Mullen7–9 studiednode-by-node partitions whereas Hughes and coworkers developed element-by-elementimplicit-explicit partitions,10–12 a work that later evolved at Stanford into element-by-element iterative solvers.13 The work of both groups focused on structure-structure andfluid-structure interaction treated by all-FEM discretizations.

Work in Coupled Problems at LPARL originated in the simulation of the elastoa-coustic underwater shock problem for ONR. In this work a finite element computationalmodel of the submerged structure was coupled to Geers’ Doubly Asymptotic Approxi-mation (DAA) boundary-element model of the exterior acoustic fluid.14–17 In 1975 Parkdeveloped a staggered solution procedure for this coupling, which was presented in an1977 article by Park, Felippa and DeRuntz6 and later extended by Park and Felippato more general applications18,19 The staggered solution scheme was eventually to besubsumed in the more general class of partitioned methods.20,21 These were surveyed in1983 and 1984 by Park and Felippa1,22 and in 1988 by Felippa and Geers.23

In 1985-86 Geers, Park and Felippa moved from LPARL to the University of Coloradoat Boulder. Park and Felippa started the Center for Aerospace Structures or CAS(originally named the Center for Space Structures and Control). Research work inCoupled Problems continued in CAS but along individual interests. Park began workin computational control-structure interaction,24,25 whereas Felippa began studies insuperconducting electrothermomagnetics.26 Charbel Farhat, who joined CAS in 1987,began investigations in computational thermoelasticity27 and aeroelasticity.28 The lattereffort prospered as it eventually acquired a parallel-computing flavor and was combinedwith advances in the FETI structure-solver method .29,30

Team research in Coupled Problems at CAS was given a boost in 1992 when theNational Science Foundation announced awards for the first round of Grand ChallengeApplications (GCA) Awards competition. This competition was part of the U. S. HighPerformance Computing and Communications (HPCC) Initiative established by an actof Congress in 1991. An application problem is labeled a GCA if the computationaldemands for realistic simulations go far beyond the capacity of present sequential orparallel supercomputers. The GCA coupled problems addressed by the grant were:aeroelasticity of a complete aircraft, distributed control-structure interaction, and elec-trothermomechanics with phase changes.

A renewal project to address multiphysics problems was awarded in 1997. Thisgrant addresses four application problems involving fluid-thermal-structural interac-tion in high temperature turbine blades, turbulence in ducts, piezoelectric and control-


structuralnodesDAA1 BE model

(shown offset fromwet surface for clarity)

shockwave

FE model (internal structure omitted)

"DAA-1 membrane'' Linear ornonlinear structure

Acoustic fluid

Submerged structure

Typical cross section

BE controlpoints

Submerged Structure hit by Shock Wave

model

Figure 7. Submerged structure hit by shock wave: (a) physics of the coupled system,(b) acoustic fluid modeled as “DAA1 membrane” (c) typical cross-section of coupledFEM-BEM discretization on the wet surface with the interior structure omitted. In(b) and (c) the BEM fluid model is shown offset from the structure for clarity.

surface control of aeroelastic systems, and design-optimization as a coupled system.These focus applications are interweaved with research in computer sciences, appliedmathematics and computational mechanics.

5 ACOUSTIC FLUID STRUCTURE INTERACTION

5.1 The Source Problem

The original coupled problem that motivated the development of partitioned analysisprocedures for coupled systems is depicted in Figure 7(a). A linear or nonlinear three-dimensional structure, externally fabricated as a stiffened shell, is submerged in aninfinite fluid. A pressure shock wave propagates through the fluid and impinges on thestructure. Because of the fast nature of the transient response, the fluid can be modeledas an acoustic medium.

As indicated in Figure 7(b) the structure is discretized with the finite element method(FEM). For the fluid, a boundary-element method (BEM) discretization is convenientas only a surface mesh needs to be created. These modeling decisions are appropriatebecause the structure response (especially as regards its survivability) is of primaryconcern whereas what happens in the fluid is of little interest. The fluid was modeledby the first-order Doubly Asymptotic Approximation (DAA1) of Geers.14,15 This modelallowed us to replace the exterior fluid by its wet surface. The “DAA1 membrane” isasymptotically exact in the limits of high and low frequencies, hence its name.

Figure 7(c) reinforces the message that FEM and BEM meshes on the “wet surface”do not generally coincide: one fluid element generally overlaps several structural ele-


ments. This happens because stress computation requirements generally demand a finerstructural discretization, whereas the main function of the fluid elements is to transmitappropriate scattered-pressure forces.

The semi-discrete governing equations for a linear structure interacting with an DAA1exterior-fluid model — after some computational rearrangements for the latter — are

Msu + Dsu + Ku = fs −TA(pI + p)

Af q + ρcAfM−1f Afq = ρcAf (T

T u− vI)(5)

For the structure model: u = u(t) is the structural displacement vector, Ms, Ds andKs are the structural mass, damping and stiffness matrices, respectively, and fs collectforces directly applied to the structure. For the fluid model: Mf = MT

f is the full,

symmetric fluid mass matrix, Af = ATf is the diagonal matrix of fluid-element areas,

vI is the vector of incident normal-fluid-particle velocities p are pI are the scatteredand incident components, respectively, of the nodal pressure vector, and vector q = pis defined for convenience. All fluid matrices and vectors are referred to the controlpoints of the wet-surface BEM mesh. The two meshes are related by T, which isthe transformation matrix that relates node forces at wet-surface structural nodes andfluid-control-point node forces. This matrix is completed with zero rows for all interiorstructural freedoms.

5.2 Staggered Solution: a Free Lunch?

Initial studies using the coupled system (5) dealt with its validation against experi-ments conducted by the Navy in 1973 on a submerged stiffened cylindrical shell, laterknown as “the ONR shell” in the underwater shock community. (Test results havebeen recently declassified.) The simple models initially used for this testbed permittedexperimentation with field elimination and monolithic solution techniques.

Experimental validation showed that both DAA1 and the interaction model wereadequate. The next step was challenging: to incorporate this analysis capability into a3D production code. The development of a new large-scale structural program was ruledout because Navy contractors were already committed to existing FEM codes such asNASTRAN and GENSAM. (Over the next two decades, ADINA, STAGS and DYNA3Dwere added to the list.) Implementing a monolithic solution with a commercial codesuch as NASTRAN, however, rans into serious logistic problems. First, access to thesource code is difficult if not impossible. Second, even if the vendor could be persuadedto create a custom version to be used in classified work, upgrading the custom version tokeep up with changes in the mainstream product can become a contractual nightmare.

The staggered solution approach, constructed by Park in 1975, solved this problem.6

A three-dimensional BEM fluid analysis program called USA (for Underwater ShockAnalysis) was written and data coupled to several existing structural analysis codes


Explicit

Implicit, A-stable

Implicit, A-stable

Acoustic Fluid (BEM)

Cavitating Fluid (FEM)

USA

DYNA3DGENTRANNASTRAN

STAGS

Structure (FEM)

Structure (FEM)

Acoustic Fluid (BEM)

DYNA3DSTAGS

USA

CFA

Implicit, A-stable Implicit, A-stable

(a) Two-field FSI problem: Structure + acoustic fluid

(b) Three-field FSI problem: Structure, acoustic fluid + cavitating fluid

Figure 8. Implementation of partitioned analysis treatment of underwater shockanalysis: (a) as two-field problem coupling acoustic fluid andsubmerged structure models, (b) as three-field problem couplingacoustic fluid, cavitating fluid and submerged structure models.

over the years, as sketched in Figure 8(a). For example, the marriage of USA andNASTRAN is called USA-NASTRAN. This plug-in modularity has important advan-tages. It simplifies upgrade and maintenance of the more complex part, which in thisproblem is the structural analyzer. Furthermore the structural analyzer can be “plugreplaced” to either fit existing structural models or the problem at hand. For example,if the structure experiences strong nonlinear material behavior, the well tested materiallibrary, contact algorithms, and highly efficient explicit time integration capabilities ofDYNA3D may be exploited.

An additional advantage is computational efficiency per time step. Let us assumedthat implicit time integration is used on both systems. The structural system is large butsparse whereas the fluid system is dense (because Mf is full) but small. A monolithicmarriage produces a large system where sparseness is severely hindered by the fluidcoupling. In the staggered approach the cost per step is roughly the same as that ofprocessing the FE and BE models as separate entities. For realistic 3D structures thecost is dominated by that of solving the structural problem. The overhead introduced by


DAA1 Fluid Model

u

(a) Original Equations

Augmented DAA1 Model

u

(b) Structure-to-Fluid Augmented Equations

p = q.

p = q.

f f f f ff.

A q + ρc(A M A + A M A ) q = ρc A v ρc A M A p + residual term

.. -g

If ff

I. __ -g

-1

.

..

Structural FEM Model

M u + K u = f TA (p + p )_f

Iss s

..

Structural FEM Model

M u + K u = f TA (p + p )_f I

ss s..

f f f

f

fI

.. _

A q + ρc A M A q = ρc A (T u v )

-1

T

TTT-g -1sM = C (T M T) C, C = T T

(structural damping ignored)

(identical to above)

Figure 9. Reformulation of two-field semidiscrete FSI equations: (a) original (after someDAA1 rearrangements), (b) upon augmentation of fluid model by structural terms.Augmentation permitted A-stability to be retained in staggered solution.

the flow of information is insignificant because it consists exclusively of computationalvectors of dimension equal to the number of BEM control points on the wet surface.

If this sound too good to be true, it is. The high computational efficiency per timestep is counteracted by the fact that satisfactory numerical stability is hard to achieve.In fact the practical feasibility of the partitioned approach hinges entirely on the stabilityanalysis. In the original publication it is shown that maintaining A-stability for thisproblem requires a reformulation of the governing equations. This is done by a procedurecalled augmentation,1,6,19 which involves transferring selected properties of the structureto the fluid model. The result is that the semidiscrete DAA1 model is modified asindicated in Figure 9. This strategy was considered preferable to augmentation of thestructural model, because the structural program, for the reasons noted above, wasdeemed to be untouchable.

5.3 Three-Field FSI: Cavitation

A BEM treatment of the acoustic fluid works well as long as the fluid is linear. Butif the shock wave is strong and the ambient hydrostatic pressure small, bulk cavitationmay occur. If the structure is sufficiently flexible, hull cavitation may occur. In eithercase the fluid must be modeled as a bilinear acoustic fluid, and a BEM treatment forthe volume affected by cavitation is not possible. It is then convenient to partition thecoupled system into 3-fields, as sketched in Figure 8(b). The near-field, where cavitationmay occur, is modeled by acoustic fluid-volume elements. This region is truncated bya DAA1 boundary. This is an example of a computationally driven partition.

Accordingly, cavitation analysis was implemented as the three-field coupled systemdiagramed in Figure 8(b). A second fluid program, called CFA (for Cavitating FluidAnalyzer) was written and data coupled to the other two. Thus USA-CFA-STAGS, for


example, denotes the linkage of USA and CFA with the nonlinear structural analysisprogram STAGS. The structure, near-field fluid volume and DAA boundary were pro-cessed by implicit, explicit and implicit time stepping, respectively. It is important tonote that the state of the explicit partition must be evaluated and advanced at halftime stations tn+1/2, whereas implicit partitions are processed at full time stations tnand tn+1. With this and other precautions the stable time step, which is controlled bythe CFL condition in the cavitating fluid mesh, was not degraded.19

6 BEYOND THE GREEN DOOR

Since the source problem described above, partitioned methods have been applied byus and other research groups to the simulation of coupled problems in structure-structureinteraction, thermomechanics, control-structure interaction, and various kinds of fluid-structure interaction involving fluid flow for aeroelasticity and flow in porous media.Because of space limitations the following outline is necessarily sketchy and admittedlyfocuses on problems treated by the authors.

6.1 Aeroelasticity

The application of partitioned methods to exterior elasticity has been pioneered byFarhat and coworkers since 1990.28,31–35 The long term ambitious goal of this project isto fly and maneuver a flexible airplane on a massively parallel computer. Essentially theproblem involves the interaction of an external gas flow modeled by the Navier Stokesequations, with a flexible aircraft, as illustrated in Figure 10(a). The aircraft structureis modeled by standard shell and beam finite elements whereas the fluid is modeledby fluid volume elements that form an unstructured mesh of tetrahedra. Additionalalgorithmic and modeling details are provided in a WCCM IV keynote presentation byLesoinne and Farhat.36

Three complicating factors appear in this application: parallel computation, the ALEproblem, and different response scales. The influence of parallelism is discussed belowin Subsection 6.5. The second complication arises because the structure motions aredescribed in a Lagrangian system, in which the structural mesh follows its motion,whereas flow is described in a Eulerian system, in which the gas passes through thefluid mesh. Hence the fluid mesh, or at least a near-field portion of it, must move inlockstep with the structural motions.

There are many proposed solutions to the ALE problem which work well in twodimensions, but which often lose robustness in three dimensions because of mesh cellcollapse. The approach selected by Farhat and his team exploits an idea originallyproposed by Batina.37 A fictitious network of linear springs, which may be augmentedby dampers, masses and torsional springs, is laid down along the edges of the fluidvolume elements, as sketched in Figure 10(b). This network may be viewed as a coupledcomputational field embedded within the fluid partition. The springs are fixed at theouter edges of the region where ALE effects are deemed important. They are driven


Fluid (Eulerian)

Structure (Lagrangian)

��Structure

Dynamic fluid mesh

Fluid (near field)

Fluid (far field)

2

1

3

1

Computational Domain

V

STRUCTURE PARTITION FLUID PARTITION

Structure Near-field fluid

Dynamic fluid mesh

Far-field fluid

Figure 10. The exterior aeroelastic problem as a three-field, two partition system.

by the motion of the aircraft surface, and operate as transducers that feed this motion,appropriately decaying with distance, to the fluid mesh nodes. The resulting three-fieldinteraction is diagramed in Figure 10(c). Although the diagram is topologically similarto the three-field cavitating FSI of Figure 8(b), the nature of flow computations makesthis a far more difficult coupled problem.

In commercial aircraft aeroelasticity, structural motions are typically dominated bylow frequency vibration modes. On the other hand, the fluid response must be capturedin a smaller time scale because of effects of shocks, vortices, and turbulence. Thus theuse of a much smaller timestep for the fluid is natural. This device is called subcycling.The ratio of structural to fluid timestep may range from 10:1 through as high as 1000:1,depending on the particular application.

6.2 Control-Structure Interaction

Partitioned solution methods have been also applied to the problem of interactionof an active control system with a “dry” structure (that is, a structure not coupled


Actuatordynamics

Sensordynamics

STRUCTURE PARTITION

CONTROL PARTITION

Control law Observer

Structure Actuatordynamics

Sensordynamics

STRUCTURE PARTITION

CONTROL PARTITION

Control law Observer

FLUID PARTITION


Dynamic fluid mesh

Far-field fluid

(a) Active control of dry structure (b) Active control of wet structure

Figure 11. Active control of dry and wet structures as two-field and three-field coupled systems.

State

Control ResponseSystem

Environment

Control Design state

Response

EnvironmentResponseequations

Designoptimizer

State synthesis

Design evaluator

System model

Modelupdater

Merit &sensitivities

(b) Practical computer implementation

(a) Abstract interpretation as optimal control problem (a) Structural design partition as component of FSI application

Designoptimizer

Designevaluator

DESIGN PARTITION



Dynamic fluid mesh

Far-field fluid

Statesynthesis

Modelupdater

Figure 12. Computer driven design and optimization of a wetstructure as component of a coupled system.

with a fluid), by Park and coworkers.24,25,38,39 The interaction diagram for this two-fieldproblem is shown in Figure 11(a). One novelty is the modeling of the control partitionas a second-order system, which permits stabilization techniques previously studied forstructure-structure interaction20,21 to be used as starting point.

The interaction fo a control system with a “wet” structure (a structure which interactswith a fluid flow, is a more formidable problem which is the presently the subject ofactive research. Figure 11(b) shows the diagram of this three-field coupled system.


STRUCTURE PARTITION

THERMOSOLID PARTITION POWER PARTITION

FLUID PARTITION

Thermal state instructure

Dynamicfluid mesh

FixedStructure

MovingStructure

Enclosed fluid

Combustion + turbulent mixing & transport

Figure 13. Simulation of a high-temperature gas turbine engineas four-field coupled system.

Important applications include flutter and vibration suppression in aircraft wings, andstall flutter reduction in helicopter blades.

6.3 Design and Optimization as Coupled System

Computer-driven optimization may be embedded in a partitioned framework it isconvenient to view it as a control problem. The block-diagram of this interpretationis sketched in Figure 12(a), and a practical realization shown in Figure 12(b). A designdriver program outputs control variables that modify the system model. This model,subject to environmental interactions (for example, with a surrounding fluid) producesa response from which state variables are synthesized and fed to the design evaluator,which supplies design merit and sensitivities to the optimizer to close the loop.

The block diagram of Figure 12(b) looks like the control partition of Figure 12(a). Itmay be — at least in principle – made into a “design partition” of a coupled system, asillustrated in Figure 12(c). This interpretation suggest a parallel implementation andpoints the way to future research into computer-driven design by simulation.

6.4 Gas Turbine Simulation: a Four Field Coupled System

The simulation of a high-temperature gas turbine (for aircraft, powerplants or mi-cromotor applications) involves the interaction of four partitions shown in Figure 14:structure (moving and fixed), enclosed fluid flow, power (combustion and turbulenttransport) and heat conduction. This is an ambitious application that presently liesbeyond our modeling and computer power. It points the way, however, to the kindof systems may be treated in the next century as petaflop parallel computers becomeavailable.



Dyn fluid mesh (spring network)

Fluid mesh (FVM) Structuremesh (FEM)

...... ...... ......F1 N1 N2 N25F2 F64S2 S16S1

Fluid

Dynamic fluid mesh

Structure

Fixed fluid mesh in fluid partitionMoving fluid mesh in fluid partitionDynamic mesh in fluid partitionStructure partitionMesh interface transfers

Processor assignment:

Figure 14. (a) The first two decomposition levels in parallel analysis of a coupled system:partitions and subdomains; illustrated for the exterior aeroelastic problem.(b) Mapping of subdomains to processors of massively parallel computer. Note:picture assumes that each subdomain in mapped to one processor for simplicity;in recent work several subdomains may be mapped to one processor onshared-memory architectures such as SGI’s.

6.5 Parallelization

The gradual evolution and acceptance of massively parallel computers over the pastdecade has brought new opportunities to partitioned analysis methods, as well as toughchallenges.

The opportunities are obvious. Massive parallelization relies on divide and conquer:breaking down the simulation into concurrent tasks. Since partitioned analysis relies onspatial decomposition, it provides an appropriate start. One can envision, for example,that in a fluid structure interaction problem a parallel computer can advance the fluidand structure state simultaneously.

This picture, however, is a gross oversimplification. Existing parallel computers havemore than 2 processors. In practice 16 through 512 processors are common, and thou-sands will be available in the fine-grained parallel machines being installed or underprocurement at Department of Energy laboratories. To take advantage of such finegranularities, it is necessary to introduce a second decomposition level: subdomains,


Fluid timestep

Structure timestep

Time

A

S

S

Sy

P

Fluid

DynMesh

Structure

Figure 16. Field and subdomain time stepping on a parallel computer.An idealized diagram; actual implementation is more complex.

which are carried out by programs called domain decomposers or mesh decomposers.Unlike partitions, this subdivision is not related to physics. Subdomains, or groups ofsubdomains, are then mapped to processors.

The two-level decomposition and CPU-mapping procedure is illustrated in Figure 13.One result of the partitioned approach is that CPUs are “colored” or tagged according tothe program they execute. The diagram assumes that for an exterior aeroelastic problemthe structure, fluid and dynamic mesh are decomposed into 16, 64 and 25 subdomains,respectively (the restriction to perfect squares is only to get nice group pictures). Theseare mapped to 16+64+25 = 105 processors. Of these, 16 execute the structure code, 64the fluid code and 25 the ALE code. Each of those processors runs its program copy ondifferent data structures predefined by the domain decomposition. Information transferbetween programs is of two types. Intrapartition transfers (e.g. structure to structureor fluid to fluid) are handled by standard message passing techniques based on theMPI protocol. Interpartition transfers (e.g. structure to fluid or control to structure)are also handled by messages but may require a mesh interpolation procedure becausemesh nodes do not necessarily match at physical interfaces.

The time integration is carried out with partitioned schemes suitably improved andrefined for subcycling and computational load balancing. Figure 14 provides an idealizedview of the time stepping. The state of the art on these algorithms, which includebackward corrections as essential ingredient, are discussed by Lesoinne and Farhat in aWCCM IV keynote presentation.36

As previously noted, the ultimate goal of the aeroelastic project, combined withcontrol and maneuvers, is to fly a full airplane through rough and emergency conditionson the computer; see Figure 17.


Figure 17. The poster of the Domain Decomposition 10 (DD10) conference held atBoulder on August 1997; organized by C. Farhat, X.-C. Cai and J. Mandel.

6.6 Geotechnical Applications

Staggered procedures have been applied by Schrefler and coworkers to consolidationproblems with mixed success. These applications are described in the forthcoming edi-tion of the Lewis-Schrefler book.40. Experiences with staggered schemes by the Swanseagroup are surveyed in the second volume of the Zienkiewicz-Taylor monograph.41

7 RESEARCH AREAS

Some areas that deserve further study in conjunction with partitioned analysis ofcoupled systems are listed below.Stability Analysis. This is the first and foremost concern. The ideal goal is: a partitionedtreatment should not degrade stability of the disconnected subsystems. That is:1. If all subsystems are treated by unconditionally stable time-stepping methods, the

coupled system should also be unconditionally stable.2. If one or more partitions are treated explicitly and the overall stable time step is

hmax, the partitioned system should be integrable with up to that stepsize.These goals are generally difficult or impossible to achieve without a reformulation

of the original system. Stability analysis by the von Neumann (Fourier) method isgenerally impossible since the modes of individual subsystems are not modes of thecoupled problem. For some linear problems it is possible to use test systems containingas many modes as partitions. But that is not possible in general, and the only recourseappears to be the use of energy methods. A general theory of stability of discrete andsemidiscrete nonlinear coupled systems remains to be developed.Accuracy Analysis. Accuracy is generally checked only after a stable algorithm is devel-oped. Similar degradation concern may appar. A common one is: second-order accuratealgorithms are used in each subsystem, but the accuracy of the partitioned integration


is only first order. A related area is study of tradeoff between interfield iteration versustime step reduction.

Interface Modeling. This topic has received recent impetus with the development ofFETI and mortar method over the past decade. These techniques address informationtransfer between matching or nonmatching intrafield meshes. For example: structure tostructure, and fluid to fluid. Much remains to be done, however, for interfield meshes.

Nonsmooth Problems. Applications such as contact and impact using partitioned analy-sis procedures deserve study to assess whether partitioned analysis methods can providebreakthroughs in modeling and computational power. Related to this topic are problemsinvolving sliding solid and fluid meshes, as in turbomachinery.

Treatment of Volume and Nonlocal Couplings. Problems of fluid turbulence, transportand mixing fall into this category, as do many applications in forming processes andChemistry. Element crossover effects of this nature can incur substantial overheadpenalties in domain decomposition methods for parallel processing.

8 CONCLUSION

It is hoped that the present article give the reader a feel for the largely unexploredsubject of computer-analysis of coupled systems. Even with the restriction to mechani-cal system of importance in Engineering, there remains a great wealth of opportunity fortheory, physical modeling, algorithms and simulation. Even greater multidisciplinaryrichness results if multiphysics effects are considered. The availability of high perfor-mance parallel computers of teraflop power (and beyond) promises to be a source ofexciting theoretical, modeling and algorithmic advances. But any such developmentsmust necessarily be subjected to the ultimate test of experimental validation.

ACKNOWLEDGEMENTS

The preparation of this article was supported by the National Science Foundation under GrantECS-9725504 and by Sandia National Laboratories under ASCI Contract AS-9991.

REFERENCES

[1] K.-C. Park and C. A. Felippa, “Partitioned analysis of coupled systems,” in: Chapter 3 ofComputational Methods for Transient Analysis, ed. by T. Belytschko and T. J. R. Hughes,North-Holland, Amsterdam, 157–219 (1983)

[2] D. W. Peaceman and H. H. Rachford Jr., “The numerical solution of parabolic and ellipticdifferential equations,” SIAM J., 3, 28–41 (1955)

[3] J. Douglas and H. H. Rachford Jr., “On the numerical solution of the heat equation in twoand three space variables,” Trans. Amer. Math. Soc., 82, 421–439 (1956)

[4] N. N. Yanenko, The Method of Fractional Steps, Springer, Berlin (1991)[5] R. L. Richtmyer and K. W. Morton, Difference Methods for Initial Value Problems, 2nd ed.,

Interscience Pubs., New York (1967)


[6] K.-C. Park, C. A. Felippa C. A. and J. A. DeRuntz, “Stabilization of staggered solution proce-dures for fluid-structure interaction analysis,” in: Computational Methods for Fluid-StructureInteraction Problems, ed. by T. Belytschko and T. L. Geers, AMD Vol. 26, American Societyof Mechanical Engineers, New York, 95–124 (1977)

[7] T. Belytschko and R. Mullen, “Mesh partitions of explicit-implicit time integration,” in:Formulations and Computational Algorithms in Finite Element Analysis, ed. by K.-J. Bathe,J. T. Oden and W. Wunderlich, MIT Press, Cambridge, 673–690 (1976)

[8] T. Belytschko and R. Mullen, “Stability of explicit-implicit mesh partitions in time integra-tion,” Int. J. Numer. Meth. Engrg., 12, 1575–1586 (1978)

[9] T. Belytschko, T. Yen and R. Mullen, “Mixed methods for time integration,” Comp. Meths.Appl. Mech. Engrg., 17/18, 259–275 (1979)

[10] T. J. R. Hughes and W.-K. Liu, “Implicit-explicit finite elements in transient analysis: I.Stability theory; II. Implementation and numerical examples,” J. Appl. Mech., 45, 371–378(1978)

[11] T. J. R. Hughes, K. S. Pister and R. L. Taylor, “Implicit-explicit finite elements in nonlineartransient analysis,” Comp. Meths. Appl. Mech. Engrg., 17/18, 159–182 (1979)

[12] T. J. R. Hughes and R. S. Stephenson,“Stability of implicit-explicit finite elements in non-linear transient analysis,” Int. J. Engrg. Sci., 19, 295–302 (1981)

[13] T. J. R. Hughes, The Finite Element Method – Linear Static and Dynamic Finite ElementAnalysis, Prentice-Hall, Englewood Cliffs, N.J. (1987)

[14] T. L. Geers, “Residual potential and approximate methods for three-dimensional fluid-structureinteraction,” J. Acoust. Soc. Am., 45, 1505–1510 (1971)

[15] T. L. Geers, “Doubly asymptotic approximations for transient motions of general structures,”J. Acoust. Soc. Am., 45, 1500-1508 (1980)

[16] T. L. Geers and C. A. Felippa, “Doubly asymptotic approximations for vibration analysis ofsubmerged structures,” J. Acoust. Soc. Am., 73, 1152–1159 (1980)

[17] T. L. Geers, “Boundary element methods for transient response analysis,” in: Chapter 4 ofComputational Methods for Transient Analysis, ed. by T. Belytschko and T. J. R. Hughes,North-Holland, Amsterdam, 221–244, (1983)

[18] C. A. Felippa and K.-C. Park, “Staggered transient analysis procedures for coupled-fieldmechanical systems: formulation,” Comp. Meths. Appl. Mech. Engrg., 24, 61–111 (1980)

[19] C. A. Felippa and J. A. DeRuntz, “Finite element analysis of shock-induced hull cavitation,”Comp. Meths. Appl. Mech. Engrg., 44, 297–337 (1984)

[20] K.-C. Park, “Partitioned transient analysis procedures for coupled-field problems: stabilityanalysis,” J. Appl. Mech., 47, 370–376 (1980)

[21] K.-C. Park and C. A. Felippa, “Partitioned transient analysis procedures for coupled-fieldproblems: accuracy analysis,” J. Appl. Mech., 47, 919–926 (1980)

[22] K.-C. Park and C. A. Felippa, “Recent advances in partitioned analysis procedures,” in:Chapter 11 of Numerical Methods in Coupled Problems, ed. by R. Lewis, P. Bettess and E.Hinton, Wiley, London, 327–352 (1984)

[23] C. A. Felippa and T. L. Geers, “Partitioned analysis of coupled mechanical systems,” Eng.Comput., 5, 123–133 (1988)

[24] W. K. Belvin and K. C. Park, “Structural tailoring and feedback control synthesis: an inter-disciplinary approach,” J. Guidance, Control & Dynamics, 13(3), 424–429 (1990)


[25] K.-C. Park and W. K. Belvin, “ A partitioned solution procedure for control-structure inter-action simulations,” J. Guidance, Control and Dynamics, 14, 59–67 (1991)

[26] J. J. Schuler and C. A. Felippa, “Superconducting finite elements based on a gauged potentialvariational principle, I. Formulation, II. Computational results,” J. Comput. Syst. Engrg, 5,pp. 215–237 (1994)

[27] C. Farhat, K.-C. Park and Y. D. Pelerin, “An unconditionally stable staggered algorithmfor transient finite element analysis of coupled thermoelastic problems,” Comp. Meths. Appl.Mech. Engrg., 85, 349–365 (1991)

[28] C. Farhat and T. Y. Lin, “Transient aeroelastic computations using multiple moving frames ofreference,” AIAA Paper No. 90-3053, AIAA 8th Applied Aerodynamics Conference, Portland,Oregon (August 1990)

[29] C. Farhat and F.-X. Roux, “Implicit parallel processing in structural mechanics,” Comput.Mech. Adv., 2, 1–124 (1994)

[30] C. Farhat, P. S. Chen and J. Mandel, “A scalable Lagrange multiplier based domain decom-position method for implicit time-dependent problems,” Int. J. Numer. Meth. Engrg., 38,3831–3854 (1995)

[31] M. Lesoinne and C. Farhat, “Stability analysis of dynamic meshes for transient aeroelas-tic computations,” AIAA Paper No. 93-3325, 11th AIAA Computational Fluid DynamicsConference, Orlando, Florida (July 1993)

[32] C. Farhat and S. Lanteri, “Simulation of compressible viscous flows on a variety of MPPs:computational algorithms for unstructured dynamic meshes and performance results,” Comp.Meths. Appl. Mech. Engrg., 119, 35–60 (1994)

[33] N. Maman and C. Farhat,“ Matching fluid and structure meshes for aeroelastic computations:a parallel approach,” Computers & Structures, 54, 779–785 (1995)

[34] S. Piperno, C. Farhat and B. Larrouturou, “Partitioned procedures for the transient solutionof coupled aeroelastic problems,” Comp. Meths. Appl. Mech. Engrg., 124, 79–11 (1995)

[35] M. Lesoinne and C. Farhat, “Geometric conservation laws for flow problems with movingboundaries and deformable meshes, and their impact on aeroelastic computations,” Comp.Meths. Appl. Mech. Engrg., 134, 71–90 (1996)

[36] M. Lesoinne and C. Farhat, “Improved staggered algorithms for the serial and parallel solutionof three-dimensional nonlinear transient aeroelastic analysis,” in: Computational Mechanics,Proc. WCCM IV Conf., ed. by E. Onate and S. Idelsohn, CIMNE, Barcelona (1998)

[37] J. T. Batina, “Unsteady Euler airfoil solutions using unstructured dynamic meshes, AIAAPaper No. 89-0115, AIAA 27th Aerospace Sciences Meeting, Reno, Nevada (January 1989)

[38] R. G. Menon and K.-C. Park, “Parallel solution of control structure interaction for massivelyactuated structures,” Proceedings Third World Congress on Computational Mechanics, ed.T. Kawai et al, Chiba, Japan, 562–563, (August 1994)

[39] K. F. Alvin and K.-C. Park, “A Second-Order Structural Identification Procedure via SystemTheory-Based Realization,” AIAA J., 32, 397–406 (1994)

[40] R. W. Lewis and B. Schrefler, The Finite Element Method in the Deformation and Consoli-dation of Porous Media, Wiley, Chichester, 1st ed. (1987), 2nd ed. to appear.

[41] O. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, 4th ed., Vol. II, McGraw-Hill, London (1991)

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