COMPUTATIONAL MECHANICS
New Trends and Applications
E. Oñate 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 Sciences
and Center for Aerospace Structures
University of Colorado at Boulder
Boulder, Colorado 80309-0429, USA
Web 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 analysis of coupled dynamical systems. Attention is focused on the computational simulation
of coupled mechanical systems in which a structure is a major component. Important
applications in that class are provided by thermomechanics, fluid-structure interaction
and control-structure interaction. In the partitioned solution approach, systems are
broken down into partitions chosen in accordance with physical or computational characteristics. The solution is separately advanced in time over each partition. Interaction
effects are accounted for by periodical transmission and synchronization of coupled state
variables. Recent developments in the use of this approach for multilevel decomposition
aimed 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 the
term is used here in the sense of “a functionally related group of components forming
or regarded as a collective entity.” This definition uses “component” as a generic term
that 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 importance in Aerospace, Civil and Mechanical Engineering. The downplaying of systems
of interest to Physics, Chemistry and Electrical Engineering does not imply restriction
in methodology extension. It is simply that less experience is available as regards the
use of the methods described here.
Systems are analyzed by decomposition or breakdown. Complex systems are amenable
to many kinds of decompositions chosen according to specific analysis or design objectives. We focus here on decompositions suitable for computer simulation. This kind
of simulation usually aims at describing or predicting the state of the system under
specified conditions viewed as external to the system. A set of states ordered according
to 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 requirements and satisfaction of mission objectives. The recovery of those overall quantities
from simulations is presently an open problem in computer system modeling and one
that is not addressed here.
1.1 Coupled System Terminology
Because coupled systems can be studied by many people from many angles, terminology is far from standard. The following summary is one that has evolved for the
computational simulation, and does reflect personal choices. Many of the definitions
followed usage introduced in a 1983 review article.1 [Casual readers may want to skim
the following material and return only for definitions.]
A coupled system is one in which physically or computationally heterogeneous mechanical components interact dynamically. The interaction is multiway in the sense that
the response has to be obtained by solving simultaneously the coupled equations which
model the system. “Heterogeneity” is used in the sense that component simulation
benefits 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 being
the most common. At the first level one encounters two types of subsystems, embody
in the generic term field:
Physical Fields. Subsystems are called physical fields when their mathematical model
is described by field equations. Examples are mechanical and non-mechanical objects
treated by continuum theories: solids, fluids, heat, electromagnetics. Occasionally those
C. A. Felippa, K. C. Park and C. Farhat
TIME
STEP
h
Start time tn
CE
PA
DS
3
id in
e flu
Solv irection
d
this
g
alon
Then ne,
o
this
n
o
so
and
xi
End time tn+1
e
ctur
Stru
TIME
id
Flu
Splitting in time
(fluid field only)
t
Partitioning
in space
Figure 1. Decomposition of an aeroelastic FSI coupled system: partitioning in space and
splitting in time. 3D space is shown as ”flat” for visualization convenience. Spatial
discretization (omitted for clarity) may predate or follow partitioning. Splitting (here
for 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-body
mechanisms. In such a case the term “physical field” is used for conveniency only, with
the understanding that no spatial discretization process is involved.
Computational Fields. Sometimes artificial subsystems are incorporated for computational conveniency. Two examples: dynamic fluid meshes to effect volume mapping
of Lagrangian to Eulerian descriptions in interaction of structures with fluid flow, and
fictitious 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 of
fields that appear in the first-level decomposition.
For computational treatment of a dynamical coupled system, fields are discretized in
space 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 step
interval. See Figure 1. In the case of static or quasi-static analysis, actual time is
replaced by pseudo-time or some kind of control parameter.
Partitioning may be algebraic or differential. In algebraic partitioning the complete
coupled system is spatially discretized first, and then decomposed. In differential partitioning 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-
C. A. Felippa, K. C. Park and C. Farhat
tions, or to individual partitions, respectively. A common use of this qualifier concerns
parallel computation. Interfield parallelism refers to the implementation of a parallel
computation scheme in which all partitions can be concurrently processed for timestepping. Intrafield parallelism refers to the implementation of parallelism for a specific
partition using a second-level decomposition.
1.2 Examples
Experiences reported in this survey comes from systems where a structure is one of
the fields. Accordingly, all of the following examples list structures as one of the field
components. 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 modeling
possibilities opens up as compared to, say, structures or thermal fields. For the latter
finite element methods can be viewed as universal in scope. On the other hand, the
range of fluid phenomena is controlled by several major physical effects such as viscosity,
compressibility, transport, gravity and capillarity. Incorporation or neglect of these
effects gives rise to widely different field equations. For example, the interaction of an
acoustic fluid with a structure in aeroacoustics or underwater shock is computationally
unlike that of high-speed gas dynamics with an aircraft, or of flow through porous
media. Even more variability can be expected if chemistry and combustion effects are
considered. Control systems also exhibit modeling variabilities that tend not to be so
pronounced, however, as in the case of fluids.
The partitioned treatment of some of these examples is discussed further in subsequent sections.
1.3 Splitting and Fractional Step Methods
Splitting methods for the equations of mathematical physics predate partitioned
analysis by two decades. In the American literature they can be originally traced to
the development by 1955 of alternating direction methods by Peaceman, Douglas and
Rachford.2,3 A similar development was independently undertaken in the early 1960s by
the Russian school led by Bagrinovskii, Godunov and Yanenko, and eventually unified
in the method of fractional steps.4 The main computational applications of these methods have been the equations of gas dynamics in steady or transient forms. They are
particularly suitable for problems with layers or stratification, for example atmospheric
dynamics 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-
C. A. Felippa, K. C. Park and C. Farhat
ator A is split into A = A1 + A2 + . . . + Aq . Chose a time difference scheme and then
replace A successively in it by qA1 , qA2 , . . ., each for a fraction h/q of the temporal
stepsize 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 overlap
occurs. Splitting methods are appropriate for the treatment of an individual partition
such as a fluid, if the physical structure of such partition display strong directionality
dependencies. Thus, splitting methods are seen to belong to a lower level of a top-down
hierarchical decomposition.
1.4 Computational Multiphysics
Recently the term multiphysics has gained prominence within computational physics
and mechanics. This term can be viewed as being more general than coupled mechanical
systems in two senses:
•
It embraces problems of importance in the physical sciences, which may go beyond
engineering systems. For example, electromagnetic and chemistry phenomena figure
more prominently.
•
Wider ranges of interacting length and time scales are addressed. This may leads to
multiple models of the same field, which coexist in space and time. Examples of this
class of problems occur in computational micromechanics and in fluid turbulence.
Given this additional generality, one must be cautious in claiming that a certain
partitioned approach, proven useful for a class of coupled systems, is a panacea for
multiphysics. 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 problems
rarely 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 modeling
and simulation of two or more isolated problems. They are then called upon to pool
that disciplinary expertise into a coupled problem.
Product Development. The design and verification of a product requires the concurrent
consideration of interaction effects in service or emergency conditions. The team does
not have access, however, to software that accommodates those requirements.
Software House. A company develops commercial application software targeted to single
fields as isolated entities. For example: a CFD gas-dynamics code, a structure FEM
program, or a thermal conduction analyzer. As the customer base expands, requests are
received for allowing interaction effects targeted to specific applications. For example
C. A. Felippa, K. C. Park and C. Farhat
the CFD user may want to account for moving rigid boundaries, then interaction with
a 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 involves following the time evolution of the system. [Static or quasistatic cases can be
included by introducing a pseudo-time control parameter.] Furthermore the modeling
and simulation of isolated components is well understood. The three approaches to the
simulation of the coupled system are:
Field Elimination. One or more fields are eliminated by techniques such as integral
transforms or model reduction, and the remaining field(s) treated by a simultaneous
time stepping scheme.
Monolithic or Simultaneous Treatment. The whole problem is treated as a monolithic
entity, and all components advanced simultaneously in time.
Partitioned Treatment. The field models are computationally treated as isolated entities
that are separately stepped in time. Interaction effects are viewed as forcing effects that
are communicated between the individual components using prediction, substitution and
synchronization 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 of
numerical difficulties.
Both the simultaneous and partitioned treatment are general in nature. No technical
argument can be made for the overall superiority of either. Their relative merits are not
only 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, independent
modeling, software reuse and modularity.
Customization. This means that each field can be treated by discretization techniques
and solution algorithms that are known to perform well for the isolated system. The
hope is that a partitioned algorithm can maintain that efficiency for the coupled problem
if (and that is a big if) the interaction effects can be also efficiently treated. As discussed
in Section 5, in the original problem that motivated the partitioned approach that happy
circumstance was realized.
Independent Modeling. The partitioned approach facilitates the use of non-matching
models. For example in a fluid-structure interaction problem the structural and fluid
meshes need not coincide at their interface. This translates into project breakdown
advantages in analysis of complex systems such as aircraft. Separate models can be
prepared by different teams or subcontractors, which can be geographically distributed.
C. A. Felippa, K. C. Park and C. Farhat
Software Reuse. Along with customized discretization and solution algorithms, customized software (private, public or commercial) may be available. Furthermore, there
is often a gamut of customized peripheral tools such as mesh generators and visualization programs. The partitioned approach facilitates taking advantage of existing code.
This is particularly suitable to academic environments, in which software development
tends to be cyclical and loosely connected from one project to another.
Modularity. New methods and models may be introduced in a modular fashion according
to project needs. For example, it may be necessary to include local nonlinear effects in
an individual field while keeping everything else the same. Implementation, testing and
validation of incremental changes can be conducted in a modular fashion.
These advantages are not cost free. The partitioned approach requires careful formulation and implementation to avoid serious degradation in stability and accuracy. Parallel implementations are particularly delicate. Gains in computational efficiency over a
monolithic approach are not guaranteed, particularly if interactions occur throughout a
volume as is the case for thermal and electromagnetic fields. Finally, the software modularity and modeling flexibility advantages, while desirable in academic and research
circles, may lead to anarchy in software houses.
In summary, circumstances that favor the partitioned approach for tackling a new
coupled problem are: a research environment with few delivery constraints, access to existing software, localized interaction effects (e.g. surface versus volume), and widespread
spatial/temporal component characteristics. The opposite circumstances: commercial
environment, rigid deliverable timetable, massive software resources, extensive interaction effects, and comparable length/time scales, favor a monolithic approach.
The remaining of this article covers the partitioned approach. It presents the essentials for a simple problem, and gradually generalizes the methodology to more complex
cases.
3 THE KEY IDEA
Consider the two-way interaction of two scalar fields, X and Y , color sketched in
Figure 2. Each field has only one state variable, identified as x(t) and y(t), which are
assumed to be governed by the semidiscrete first-order differential equations
3ẋ + 4x − y = f (t)
ẏ + 6y − 2x = g(t)
(1)
in which f (t) and g(t) are the applied forces. Treat this by Backward Euler integration
in each component:
xn+1 = xn + hẋn+1 ,
yn+1 = yn + hẏn+1
(2)
where xn ≡ x(tn ), yn ≡ y(tn ), etc. At each time step n = 0, 1, 2, . . . one gets
3 + 4h
−2h
−h
1 + 6h
xn+1
f
+ 3hxn
= n+1
yn+1
gn+1 + hyn
(3)
C. A. Felippa, K. C. Park and C. Farhat
f(t)
X
x(t)
g(t)
Y
y(t)
Figure 2. The red and the black.
BASIC STEPS
.
P
yn+
(for example)
1 = yn + h yn
1
P
2. (Ax) Advance x: x n+ 1 =
( f n+ 1 + 3hxn + hyn+
1)
3 + 4h
(trivial here)
3. (S) Substitute: x n+ 1 = x n+ 1
1
4. (Ay) Advance y: yn+ 1 =
(gn+ 1 + hyn + 2hxn+ 1 )
1 + 6h
1. (P)
Predict:
Figure 3. Basic steps of a red/black staggered solution.
in which x0 , y0 are given from initial conditions. In the monolithic or simultaneous
solution 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 following
staggered partition with prediction on y:
3 + 4h
−2h
0
1 + 6h
P
xn+1
f
+ 3hxn + hyn+1
= n+1
yn+1
gn+1 + hyn
(4)
P
P
P
where yn+1
is a predicted value; two choices being yn+1
= yn or yn+1
= yn + hẏn . The
basic solution steps are displayed in Figure 3. The main achievement is that systems X
and Y can be solved in tandem. A state-time diagram of these steps, with time along
the horizontal axis, is shown in Figure 4.
Suppose that X and Y are handled by two communicating programs called the Single
Field Analyzers, or SFAs. If the intrafield advancing arrows are suppressed, we obtain a
zigzagged 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 staggered
solution procedure.6
C. A. Felippa, K. C. Park and C. Farhat
xn
Step 2: Ax
:P
p1
Ste
Time
xn+ 1
Step 3: S
yn
yn+ 1
Step 4: Ay
Figure 4. Interfield+intrafield time-stepping diagram of
the staggered solution steps listed in Figure 3.
(a)
X program:
S
S
P
P
S
Y program:
Timestep h
Time
(b)
X program:
P
P
I
Y program:
I
P
P
Figure 5. Interfield time stepping diagrams: (a) sequential staggered solution of
example problem, (b) modification suitable for parallel processing.
3.2 Concerns
The immediate concern with partitioning should be degradation of time-stepping
stability caused by predictions. For the example system this is not significant. If an
A-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 stability
for the coupled problem of this nature. A simple von Neumann amplification analysis
of (4) shows that the staggered solution maintains unconditional stability if predictors
P
yn+1
= yn + σhẏn , with σ ∈ [0, 1], are used. But for more complex problems the
reduction of stability can become serious or even catastrophic.
Prediction could be done on the y field, again leading to a zigzagged diagram with
substitution on y. The stability of both choices can be made identical by appropriately
adjusting predictors.
If satisfactory stability is achieved, the next concern is accuracy. This is usually
degraded with respect to that attainable by the monolithic scheme. In principle this
C. A. Felippa, K. C. Park and C. Farhat
h
P
Prediction
S
Substitution
C
I
Full step
correction
Time
Interfield
Iteration
ScA
A
MC
A
A
Lockstep
advancing
Midpoint
correction
Subcycling
A+
Augmentation
Figure 6. Devices of partitioned analysis time stepping.
can be recovered by iterating the state between the fields. Iteration is done by cycling
substitutions at the same time station. However, intrafield iteration generally costs more
than cutting the timestep to attain the same accuracy level. If the monolithic solution
is more expensive than the staggered solution for the same timestep, we note here the
emergence of a tradeoff.
An examination of Figure 5 clearly shows than this staggered scheme is unsuitable
for 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 was
formulated in the mid 1970s, because all computers — with the exception of one exotic
machine known as the ILLIAC IV — were sequential.
The modification sketched in Figure 5(b) permits both programs to advance their
internal state concurrently and thus fits a parallel computer better. Unfortunately it
requires predictions on both fields, which can have a serious impact on stability. Even if
this is overcome, interfield iteration — denoted by I in Figure 5(b) — may be required to
regain accuracy, which increases data communication overhead. More practical parallel
schemes are discussed in Section 6.
3.3 Devices of Partitioned Analysis
As this simple example makes clear, partitioned analysis requires the examination of
many possibilities, and the study of tradeoffs. Figure 6 displays the main “tools of the
trade” in field time stepping diagrams. Some devices such as prediction, substitution
and iteration, have been discussed along with the foregoing example. Others will emerge
in the application problems discussed in Sections 5 and 6.
C. A. Felippa, K. C. Park and C. Farhat
4 HISTORICAL BACKGROUND
The partitioned treatment of coupled systems involving structures emerged independently in the mid 1970s at three locations: Northwestern University by T. Belytschko
and R. Mullen, Cal Tech by T. J. R. Hughes and W. K. Liu, and Lockheed Palo Alto
Research Laboratories (LPARL) by J. A. DeRuntz, C. A. Felippa, T. L. Geers and
K.-C. Park. These three groups targeted different applications and pursued different
problem-decomposition techniques. For example, Belytschko and Mullen7–9 studied
node-by-node partitions whereas Hughes and coworkers developed element-by-element
implicit-explicit partitions,10–12 a work that later evolved at Stanford into element-byelement iterative solvers.13 The work of both groups focused on structure-structure and
fluid-structure interaction treated by all-FEM discretizations.
Work in Coupled Problems at LPARL originated in the simulation of the elastoacoustic underwater shock problem for ONR. In this work a finite element computational
model of the submerged structure was coupled to Geers’ Doubly Asymptotic Approximation (DAA) boundary-element model of the exterior acoustic fluid.14–17 In 1975 Park
developed a staggered solution procedure for this coupling, which was presented in an
1977 article by Park, Felippa and DeRuntz6 and later extended by Park and Felippa
to more general applications18,19 The staggered solution scheme was eventually to be
subsumed in the more general class of partitioned methods.20,21 These were surveyed in
1983 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 Colorado
at Boulder. Park and Felippa started the Center for Aerospace Structures or CAS
(originally named the Center for Space Structures and Control). Research work in
Coupled Problems continued in CAS but along individual interests. Park began work
in computational control-structure interaction,24,25 whereas Felippa began studies in
superconducting electrothermomagnetics.26 Charbel Farhat, who joined CAS in 1987,
began investigations in computational thermoelasticity27 and aeroelasticity.28 The latter
effort prospered as it eventually acquired a parallel-computing flavor and was combined
with advances in the FETI structure-solver method .29,30
Team research in Coupled Problems at CAS was given a boost in 1992 when the
National Science Foundation announced awards for the first round of Grand Challenge
Applications (GCA) Awards competition. This competition was part of the U. S. High
Performance Computing and Communications (HPCC) Initiative established by an act
of Congress in 1991. An application problem is labeled a GCA if the computational
demands for realistic simulations go far beyond the capacity of present sequential or
parallel supercomputers. The GCA coupled problems addressed by the grant were:
aeroelasticity of a complete aircraft, distributed control-structure interaction, and electrothermomechanics with phase changes.
A renewal project to address multiphysics problems was awarded in 1997. This
grant addresses four application problems involving fluid-thermal-structural interaction in high temperature turbine blades, turbulence in ducts, piezoelectric and control-
C. A. Felippa, K. C. Park and C. Farhat
Submerged Structure hit by Shock Wave
Submerged structure
"DAA-1 membrane''
Linear or
nonlinear structure
model
shock
wave
Typical cross section
BE control
points
FE model (internal
structure omitted)
Acoustic fluid
DAA1 BE model
(shown offset from
wet surface for clarity)
structural
nodes
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 coupled
FEM-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, applied
mathematics and computational mechanics.
5 ACOUSTIC FLUID STRUCTURE INTERACTION
5.1 The Source Problem
The original coupled problem that motivated the development of partitioned analysis
procedures for coupled systems is depicted in Figure 7(a). A linear or nonlinear threedimensional structure, externally fabricated as a stiffened shell, is submerged in an
infinite fluid. A pressure shock wave propagates through the fluid and impinges on the
structure. Because of the fast nature of the transient response, the fluid can be modeled
as 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 convenient
as only a surface mesh needs to be created. These modeling decisions are appropriate
because the structure response (especially as regards its survivability) is of primary
concern whereas what happens in the fluid is of little interest. The fluid was modeled
by the first-order Doubly Asymptotic Approximation (DAA1 ) of Geers.14,15 This model
allowed us to replace the exterior fluid by its wet surface. The “DAA1 membrane” is
asymptotically 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-
C. A. Felippa, K. C. Park and C. Farhat
ments. This happens because stress computation requirements generally demand a finer
structural discretization, whereas the main function of the fluid elements is to transmit
appropriate scattered-pressure forces.
The semi-discrete governing equations for a linear structure interacting with an DAA1
exterior-fluid model — after some computational rearrangements for the latter — are
Ms ü + Ds u̇ + Ku = fs − TA(pI + p)
T
I
Af q̇ + ρcAf M−1
f Af q = ρcAf (T u̇ − v )
(5)
For the structure model: u = u(t) is the structural displacement vector, Ms , Ds and
Ks are the structural mass, damping and stiffness matrices, respectively, and fs collect
forces directly applied to the structure. For the fluid model: Mf = MTf 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 scattered
and incident components, respectively, of the nodal pressure vector, and vector q = ṗ
is defined for convenience. All fluid matrices and vectors are referred to the control
points of the wet-surface BEM mesh. The two meshes are related by T, which is
the transformation matrix that relates node forces at wet-surface structural nodes and
fluid-control-point node forces. This matrix is completed with zero rows for all interior
structural freedoms.
5.2 Staggered Solution: a Free Lunch?
Initial studies using the coupled system (5) dealt with its validation against experiments conducted by the Navy in 1973 on a submerged stiffened cylindrical shell, later
known as “the ONR shell” in the underwater shock community. (Test results have
been recently declassified.) The simple models initially used for this testbed permitted
experimentation with field elimination and monolithic solution techniques.
Experimental validation showed that both DAA1 and the interaction model were
adequate. The next step was challenging: to incorporate this analysis capability into a
3D production code. The development of a new large-scale structural program was ruled
out because Navy contractors were already committed to existing FEM codes such as
NASTRAN and GENSAM. (Over the next two decades, ADINA, STAGS and DYNA3D
were added to the list.) Implementing a monolithic solution with a commercial code
such as NASTRAN, however, rans into serious logistic problems. First, access to the
source code is difficult if not impossible. Second, even if the vendor could be persuaded
to create a custom version to be used in classified work, upgrading the custom version to
keep 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 Shock
Analysis) was written and data coupled to several existing structural analysis codes
C. A. Felippa, K. C. Park and C. Farhat
(a) Two-field FSI problem: Structure +
acoustic fluid
(b) Three-field FSI problem: Structure,
acoustic fluid + cavitating fluid
Implicit, A-stable
Implicit, A-stable
Implicit, A-stable
Explicit
Implicit, A-stable
Acoustic Fluid (BEM)
USA
DYNA3D
GENTRAN
NASTRAN
STAGS
USA
Acoustic Fluid (BEM)
Cavitating
Fluid (FEM)
CFA
Structure (FEM)
Structure (FEM)
DYNA3D
STAGS
Figure 8. Implementation of partitioned analysis treatment of underwater shock
analysis: (a) as two-field problem coupling acoustic fluid and
submerged structure models, (b) as three-field problem coupling
acoustic fluid, cavitating fluid and submerged structure models.
over the years, as sketched in Figure 8(a). For example, the marriage of USA and
NASTRAN is called USA-NASTRAN. This plug-in modularity has important advantages. It simplifies upgrade and maintenance of the more complex part, which in this
problem is the structural analyzer. Furthermore the structural analyzer can be “plug
replaced” to either fit existing structural models or the problem at hand. For example,
if the structure experiences strong nonlinear material behavior, the well tested material
library, contact algorithms, and highly efficient explicit time integration capabilities of
DYNA3D may be exploited.
An additional advantage is computational efficiency per time step. Let us assumed
that implicit time integration is used on both systems. The structural system is large but
sparse whereas the fluid system is dense (because Mf is full) but small. A monolithic
marriage produces a large system where sparseness is severely hindered by the fluid
coupling. In the staggered approach the cost per step is roughly the same as that of
processing the FE and BE models as separate entities. For realistic 3D structures the
cost is dominated by that of solving the structural problem. The overhead introduced by
C. A. Felippa, K. C. Park and C. Farhat
.
(a) Original Equations
p=q
DAA1 Fluid Model
.
Af q + ρc Af M f-1Af q =
.
ρc Af (TT u _ v I)
Structural FEM Model
..
Ms u + Ks u = fs _TAf (p + pI )
.
(structural damping ignored)
(b) Structure-to-Fluid
Augmented Equations
u
.
p=q
Augmented DAA1 Model
..
.
Af q + ρc(Af Mf-1Af + Af M -gAf ) q =
_ ρc Af v. I _ ρc Af M-gAf p I + residual term
Structural FEM Model
..
Ms u + Ks u = fs _TAf (p + pI )
..
u
(identical to above)
M
-g
-1
= C T(TT Ms T) C,
C = TT T
Figure 9. Reformulation of two-field semidiscrete FSI equations: (a) original (after some
DAA1 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 computational
vectors 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 time
step 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 stability
analysis. In the original publication it is shown that maintaining A-stability for this
problem requires a reformulation of the governing equations. This is done by a procedure
called augmentation,1,6,19 which involves transferring selected properties of the structure
to the fluid model. The result is that the semidiscrete DAA1 model is modified as
indicated in Figure 9. This strategy was considered preferable to augmentation of the
structural model, because the structural program, for the reasons noted above, was
deemed 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. But
if the shock wave is strong and the ambient hydrostatic pressure small, bulk cavitation
may occur. If the structure is sufficiently flexible, hull cavitation may occur. In either
case the fluid must be modeled as a bilinear acoustic fluid, and a BEM treatment for
the volume affected by cavitation is not possible. It is then convenient to partition the
coupled system into 3-fields, as sketched in Figure 8(b). The near-field, where cavitation
may occur, is modeled by acoustic fluid-volume elements. This region is truncated by
a DAA1 boundary. This is an example of a computationally driven partition.
Accordingly, cavitation analysis was implemented as the three-field coupled system
diagramed in Figure 8(b). A second fluid program, called CFA (for Cavitating Fluid
Analyzer) was written and data coupled to the other two. Thus USA-CFA-STAGS, for
C. A. Felippa, K. C. Park and C. Farhat
example, denotes the linkage of USA and CFA with the nonlinear structural analysis
program STAGS. The structure, near-field fluid volume and DAA boundary were processed by implicit, explicit and implicit time stepping, respectively. It is important to
note that the state of the explicit partition must be evaluated and advanced at half
time stations tn+1/2 , whereas implicit partitions are processed at full time stations tn
and tn+1 . With this and other precautions the stable time step, which is controlled by
the 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 by
us and other research groups to the simulation of coupled problems in structure-structure
interaction, thermomechanics, control-structure interaction, and various kinds of fluidstructure interaction involving fluid flow for aeroelasticity and flow in porous media.
Because of space limitations the following outline is necessarily sketchy and admittedly
focuses on problems treated by the authors.
6.1 Aeroelasticity
The application of partitioned methods to exterior elasticity has been pioneered by
Farhat and coworkers since 1990.28,31–35 The long term ambitious goal of this project is
to fly and maneuver a flexible airplane on a massively parallel computer. Essentially the
problem involves the interaction of an external gas flow modeled by the Navier Stokes
equations, with a flexible aircraft, as illustrated in Figure 10(a). The aircraft structure
is modeled by standard shell and beam finite elements whereas the fluid is modeled
by fluid volume elements that form an unstructured mesh of tetrahedra. Additional
algorithmic and modeling details are provided in a WCCM IV keynote presentation by
Lesoinne and Farhat.36
Three complicating factors appear in this application: parallel computation, the ALE
problem, and different response scales. The influence of parallelism is discussed below
in Subsection 6.5. The second complication arises because the structure motions are
described 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 the
fluid mesh. Hence the fluid mesh, or at least a near-field portion of it, must move in
lockstep with the structural motions.
There are many proposed solutions to the ALE problem which work well in two
dimensions, but which often lose robustness in three dimensions because of mesh cell
collapse. The approach selected by Farhat and his team exploits an idea originally
proposed by Batina.37 A fictitious network of linear springs, which may be augmented
by dampers, masses and torsional springs, is laid down along the edges of the fluid
volume elements, as sketched in Figure 10(b). This network may be viewed as a coupled
computational field embedded within the fluid partition. The springs are fixed at the
outer edges of the region where ALE effects are deemed important. They are driven
C. A. Felippa, K. C. Park and C. Farhat
Computational Domain
;;
Fluid (Eulerian)
V
2
Structure
1 Fluid (near field)
3 Dynamic fluid mesh
Structure (Lagrangian)
1 Fluid (far field)
STRUCTURE PARTITION
Structure
FLUID PARTITION
Near-field
fluid
Far-field
fluid
Dynamic
fluid mesh
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-field
interaction is diagramed in Figure 10(c). Although the diagram is topologically similar
to the three-field cavitating FSI of Figure 8(b), the nature of flow computations makes
this a far more difficult coupled problem.
In commercial aircraft aeroelasticity, structural motions are typically dominated by
low frequency vibration modes. On the other hand, the fluid response must be captured
in a smaller time scale because of effects of shocks, vortices, and turbulence. Thus the
use 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 interaction
of an active control system with a “dry” structure (that is, a structure not coupled
C. A. Felippa, K. C. Park and C. Farhat
(a) Active control of dry structure
(b) Active control of wet structure
STRUCTURE PARTITION
STRUCTURE PARTITION
Actuator
dynamics
Control law
Actuator
dynamics
Sensor
dynamics
Structure
Structure
CONTROL PARTITION
Sensor
dynamics
Far-field
fluid
Near-field
fluid
Dynamic
fluid mesh
Observer
Control law
Observer
FLUID PARTITION
CONTROL PARTITION
Figure 11. Active control of dry and wet structures as two-field and three-field coupled systems.
(a) Abstract interpretation as optimal control problem
(a) Structural design partition as component of FSI application
Environment
State
STRUCTURE PARTITION
Control
FLUID PARTITION
Response
System
Model
updater
Structure
State
synthesis
Near-field
fluid
Far-field
fluid
(b) Practical computer implementation
Response
equations
Model
updater
Environment
System
model
Response
Control
Design
optimizer
Merit &
sensitivities
Design
evaluator
State
synthesis
Design
state
Design
optimizer
Design
evaluator
Dynamic
fluid mesh
DESIGN PARTITION
Figure 12. Computer driven design and optimization of a wet
structure as component of a coupled system.
with a fluid), by Park and coworkers.24,25,38,39 The interaction diagram for this two-field
problem is shown in Figure 11(a). One novelty is the modeling of the control partition
as a second-order system, which permits stabilization techniques previously studied for
structure-structure interaction20,21 to be used as starting point.
The interaction fo a control system with a “wet” structure (a structure which interacts
with a fluid flow, is a more formidable problem which is the presently the subject of
active research. Figure 11(b) shows the diagram of this three-field coupled system.
C. A. Felippa, K. C. Park and C. Farhat
STRUCTURE PARTITION
Fixed
Structure
Moving
Structure
Thermal
state in
structure
THERMOSOLID PARTITION
FLUID PARTITION
Dynamic
fluid mesh
Enclosed
fluid
Combustion +
turbulent mixing
& transport
POWER PARTITION
Figure 13. Simulation of a high-temperature gas turbine engine
as four-field coupled system.
Important applications include flutter and vibration suppression in aircraft wings, and
stall flutter reduction in helicopter blades.
6.3 Design and Optimization as Coupled System
Computer-driven optimization may be embedded in a partitioned framework it is
convenient to view it as a control problem. The block-diagram of this interpretation
is sketched in Figure 12(a), and a practical realization shown in Figure 12(b). A design
driver program outputs control variables that modify the system model. This model,
subject to environmental interactions (for example, with a surrounding fluid) produces
a 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). It
may be — at least in principle – made into a “design partition” of a coupled system, as
illustrated in Figure 12(c). This interpretation suggest a parallel implementation and
points 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 micromotor applications) involves the interaction of four partitions shown in Figure 14:
structure (moving and fixed), enclosed fluid flow, power (combustion and turbulent
transport) and heat conduction. This is an ambitious application that presently lies
beyond our modeling and computer power. It points the way, however, to the kind
of systems may be treated in the next century as petaflop parallel computers become
available.
C. A. Felippa, K. C. Park and C. Farhat
STRUCTURE PARTITION
FLUID PARTITION
Structure
mesh (FEM)
S1
......
S2
Dyn fluid mesh
(spring network)
Fluid mesh (FVM)
S16
F1
F2
......
F64
N1
N2
......
N25
Fluid
Structure
Processor assignment:
Fixed fluid mesh in fluid partition
Moving fluid mesh in fluid partition
Dynamic mesh in fluid partition
Structure partition
Mesh interface transfers
Dynamic fluid mesh
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 on
shared-memory architectures such as SGI’s.
6.5 Parallelization
The gradual evolution and acceptance of massively parallel computers over the past
decade has brought new opportunities to partitioned analysis methods, as well as tough
challenges.
The opportunities are obvious. Massive parallelization relies on divide and conquer:
breaking down the simulation into concurrent tasks. Since partitioned analysis relies on
spatial decomposition, it provides an appropriate start. One can envision, for example,
that in a fluid structure interaction problem a parallel computer can advance the fluid
and structure state simultaneously.
This picture, however, is a gross oversimplification. Existing parallel computers have
more than 2 processors. In practice 16 through 512 processors are common, and thousands will be available in the fine-grained parallel machines being installed or under
procurement at Department of Energy laboratories. To take advantage of such fine
granularities, it is necessary to introduce a second decomposition level: subdomains,
C. A. Felippa, K. C. Park and C. Farhat
Fluid timestep
Fluid
A
Sy
S
Dyn
Mesh
S
Time
P
Structure
Structure timestep
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 of
subdomains, 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 to
the program they execute. The diagram assumes that for an exterior aeroelastic problem
the 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). These
are mapped to 16+64+25 = 105 processors. Of these, 16 execute the structure code, 64
the fluid code and 25 the ALE code. Each of those processors runs its program copy on
different data structures predefined by the domain decomposition. Information transfer
between programs is of two types. Intrapartition transfers (e.g. structure to structure
or fluid to fluid) are handled by standard message passing techniques based on the
MPI protocol. Interpartition transfers (e.g. structure to fluid or control to structure)
are also handled by messages but may require a mesh interpolation procedure because
mesh nodes do not necessarily match at physical interfaces.
The time integration is carried out with partitioned schemes suitably improved and
refined for subcycling and computational load balancing. Figure 14 provides an idealized
view of the time stepping. The state of the art on these algorithms, which include
backward corrections as essential ingredient, are discussed by Lesoinne and Farhat in a
WCCM IV keynote presentation.36
As previously noted, the ultimate goal of the aeroelastic project, combined with
control and maneuvers, is to fly a full airplane through rough and emergency conditions
on the computer; see Figure 17.
C. A. Felippa, K. C. Park and C. Farhat
Figure 17. The poster of the Domain Decomposition 10 (DD10) conference held at
Boulder 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 consolidation
problems with mixed success. These applications are described in the forthcoming edition of the Lewis-Schrefler book.40 . Experiences with staggered schemes by the Swansea
group 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 of
coupled systems are listed below.
Stability Analysis. This is the first and foremost concern. The ideal goal is: a partitioned
treatment 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 is
generally impossible since the modes of individual subsystems are not modes of the
coupled problem. For some linear problems it is possible to use test systems containing
as many modes as partitions. But that is not possible in general, and the only recourse
appears to be the use of energy methods. A general theory of stability of discrete and
semidiscrete nonlinear coupled systems remains to be developed.
Accuracy Analysis. Accuracy is generally checked only after a stable algorithm is developed. Similar degradation concern may appar. A common one is: second-order accurate
algorithms are used in each subsystem, but the accuracy of the partitioned integration
C. A. Felippa, K. C. Park and C. Farhat
is only first order. A related area is study of tradeoff between interfield iteration versus
time step reduction.
Interface Modeling. This topic has received recent impetus with the development of
FETI and mortar method over the past decade. These techniques address information
transfer between matching or nonmatching intrafield meshes. For example: structure to
structure, and fluid to fluid. Much remains to be done, however, for interfield meshes.
Nonsmooth Problems. Applications such as contact and impact using partitioned analysis procedures deserve study to assess whether partitioned analysis methods can provide
breakthroughs in modeling and computational power. Related to this topic are problems
involving sliding solid and fluid meshes, as in turbomachinery.
Treatment of Volume and Nonlocal Couplings. Problems of fluid turbulence, transport
and mixing fall into this category, as do many applications in forming processes and
Chemistry. Element crossover effects of this nature can incur substantial overhead
penalties in domain decomposition methods for parallel processing.
8 CONCLUSION
It is hoped that the present article give the reader a feel for the largely unexplored
subject of computer-analysis of coupled systems. Even with the restriction to mechanical system of importance in Engineering, there remains a great wealth of opportunity for
theory, physical modeling, algorithms and simulation. Even greater multidisciplinary
richness results if multiphysics effects are considered. The availability of high performance parallel computers of teraflop power (and beyond) promises to be a source of
exciting theoretical, modeling and algorithmic advances. But any such developments
must necessarily be subjected to the ultimate test of experimental validation.
ACKNOWLEDGEMENTS
The preparation of this article was supported by the National Science Foundation under Grant
ECS-9725504 and by Sandia National Laboratories under ASCI Contract AS-9991.
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