computational aspects of structural acoustics and vibration

CISM COURSES AND LECTURES

Series Editors:

The RectorsGiulio Maier - Milan

Jean Salençon - PalaiseauWilhelm Schneider - Wien

The Secretary GeneralBernhard Schrefler - Padua

Executive EditorPaolo Serafini - Udine

The series presents lecture notes, monographs, edited works andproceedings in the field of Mechanics, Engineering, Computer Science

and Applied Mathematics.Purpose of the series is to make known in the international scientificand technical community results obtained in some of the activities

organized by CISM, the International Centre for Mechanical Sciences.

INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES

COURSES AND LECTURES - No. 505

COMPUTATIONAL ASPECTS OFSTRUCTURAL ACOUSTICS AND

VIBRATION

EDITED BY

GÖRAN SANDBERGLTH, LUND UNIVERSITY, SWEDEN

ROGER OHAYONCNAM, PARIS, FRANCE

This volume contains 113 illustrations

This work is subject to copyright.All rights are reserved,

whether the whole or part of the material is concernedspecifically those of translation, reprinting, re-use of illustrations,

broadcasting, reproduction by photocopying machineor similar means, and storage in data banks.

© 2008 by CISM, UdinePrinted in ItalySPIN 12567607

All contributions have been typeset by the authors.

ISBN 978-3-211-89650-1 SpringerWienNewYork

2 J.-F. Deü and W. Larbi and R. Ohayon

Variational Formulations of Interior Structural-Acoustic… 3

24 G. Sandberg, P.-A. Wernberg and P. Davidsson

Fundamentals of Fluid-Structure Interaction 25


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170 F. Ihlenburg

Model Based Partitioned Simulation of CoupledSystems

Carlos A. Felippa and K. C. ParkDepartment of Aerospace Engineering Sciences and Center for Aerospace Structures,

University of Colorado at Boulder, Boulder, Colorado 80309-0429, USAemail: [email protected]

Abstract. This tutorial paper is extracted from a set of graduate lectureson the time-domain simulation of structural dynamics and coupled systems.This material has also served as a basis for a CISM lecture series on FSI. Forthe treatment of coupled systems, emphasis is placed on partitioned analysisprocedures. Although the subject emerged in the present form over 20 yearsago, the time-consuming study of competing formulations and implementa-tions can be streamlined through the use of various tools such as reductionto model equations, and the help of computer algebra systems.

Keywords: computational structural dynamics, coupled systems, fluid-structure interaction, multiphysics, computer algebra, partitioned analysis,time integration.

1 Introduction

What’s hot in computational mechanics? The three “multis”: multiscale, multi-physics and multiprocessing. Collectively these trends pertain to the formulationand model-based simulation of coupled systems: systems whose behavior is drivenby the interaction of functionally distinct components. The nature of these com-ponents broadly defines the “multi” discipline. Material models that span a rangeof physical scales (for example, molecular through crystal) are the framework ofmultiscale simulations. Multiphysics addresses the interaction of different physi-cal behavior, as in structures and fluids, at similar physical scales. Multiprocessingrefers to computational methods that use system decomposition to achieve concur-rency. Summarizing, system breakdown is governed by: (S) physical scales inmultiscale, (P) physical behavior in multiphysics, and (C) computer implementa-tion considerations in multiprocessing. Plainly a three-level hierarchy: (S)-(P)-(C),can be discerned, but this level of full generality has not been reached in practice.

The “hot areas” share a common feature: explosive complexity. Choicesamong models, algorithms and implementations grow combinatorially in the num-ber of components. Consider for example a fluid-structure interaction problem.

172 C. A. Felippa and K. C. Park

Whereas a FEM model for the structure may be viewed as natural, the choice offluid model can vary across a wide spectrum, depending on what physical effects(flow, turbulence, acoustic shocks, mixing, slosh, cavitation, moving boundaries,bubbles, etc.) are to be captured. Discretization methods vary accordingly. Ifcontrol is added to the picture, for example to simulate maneuvers of a flexibleairplane, further choices emerge. So far this applies to components in isolation.Treating interaction requires additional decisions at interfaces. For example: domeshes match? can meshes slip past each other? how can reduced or spectralmodels be linked to physical models? To make things more difficult, often modelsthat work correctly with isolated components break down when coupled. But thatis not all. Proceeding to the computer implementation and testing levels may bringup further options, in particular if parallel processing issues are important.

How to cope with this combinatorial explosion? Analytical treatment can goso far in weeding out choices. The traditional way to go beyond that frontier is nu-merical experimentation. This has limitations: the most one can hope to do is take“potshots” at the computational application domain. It can only show that a par-ticular numerical model works or doesn’t. A “bridging” tool between human an-alytical thought and numerical testing has gained popularity over the past decade:computer algebra systems (CAS) able to carry out symbolic computations. Thisis due to technical improvements in general-purpose CAS such as Mathematicaand Maple, as well as availability on inexpensive personal computers and laptops.(This migration keeps licensing costs reasonable.) Furthermore, Maple is acces-sible as a toolbox of the widely used Matlab system. A related factor is widerexposure in higher education: many universities now have site licenses, whichfacilitate access and use of CAS for course assignments and projects.

In computational mechanics, CAS tools can be used for a spectrum of tasks:formulation, prototyping, implementation, performance evaluation, and automaticcode generation. Although occasionally advertised as “doing mathematics by com-puter” the phrase is misleading: as of now only humans can do mathematics. But aCAS can provide timely help. Here is a first-hand example: the first author neededfour months to formulate, implement and test the 6-node finite element triangle in1965 as part of thesis work [7]. Using a CAS, a similar process can be completedin less than a week, and demonstrated to students in 20 minutes.

In the present chapter, Mathematica [41] is employed as a CAS “filter tool” inthe design and analysis of time integration methods for structural dynamics andcoupled systems. The main purpose of the filter is to weed out unsuitable methodsby working on model test systems. This initial pass streamlines subsequent stagesof numerical experimentation and computer implementation.

Model Based Partitioned Simulation of Coupled Systems 173

2 Coupled Systems

This section defines coupled systems and introduces pertinent terminology.

2.1 Systems

The American Heritage Dictionary lists eight meanings for system. By itselfthe term is used here in the sense of a functionally related group of componentsforming or regarded as a collective entity. This definition uses “component” as ageneric term that embodies “element” or “part,” which connote simplicity, as wellas “subsystem,” which connotes complexity. In the sequel we restrict attention tomechanical systems, and especially those of importance in Aerospace, Civil andMechanical Engineering.

2.2 System Decomposition

Systems are analyzed by decomposition or breakdown. Complex systems areamenable to many kinds of decomposition chosen according to specific analysis ordesign objectives. This chapter focuses on decompositions called partitions thatare suitable for model-based simulations. Such simulations aim at describing orpredicting the state of the system under specified conditions viewed as external tothe system. A set of states ordered according to some parameter such as time orload level is called a response.

System designers are not usually interested in detailed response computationsper se, but on project goals such as cost, performance, lifetime, fabricability, in-spection requirements and satisfaction of mission objectives. The recovery of thoseoverall quantities from simulations is presently an open problem in computer sys-tem modeling and one that is not addressed here.

The term partitioning identifies the process of spatial separation of a discretemodel into interacting components generically called partitions. The decomposi-tion may be driven by physical, functional, or computational considerations. Forexample, the structure of a complete airplane can be decomposed into substruc-tures such as wings and fuselage according to function. Substructures can be fur-ther decomposed into submeshes or subdomains to accommodate parallel comput-ing requirements. Subdomains are composed of individual elements. Going theother way, if that flexible airplane is part of a flight simulation, a top-level par-tition driven by physics may couple fluid and structure (and perhaps control andpropulsion) models. This kind of multilevel partition hierarchy at common phys-ical scales: coupled system, structure, substructure, subdomain and element, istypical of present practice in modeling and computational technology.


X X YY(a) (b)

Figure 1. Interaction between subsystems and : (a) one way, (b) two way.

2.3 Coupled System Terminology

Coupled systems have been studied by many people from various perspectives,but the field is still advancing rapidly, and terminology is far from standard. Thefollowing summary is one that has evolved for the computational simulation, anddoes reflect personal choices of the authors. Most definitions follow usage estab-lished in a series of articles in the 1980s [10, 12, 24–26]. Casual readers may wantto skim the following material and return only for definitions.

A coupled system is one in which physically or computationally heterogeneousmechanical components interact dynamically.

The interaction is one-way if there is no feedback between subsystems, as pic-tured in Figure 1(a) for two subsystems identified as and . The interaction istwo-way (or generally: multiway) if there is feedback, as illustrated in Figure 1(b)In the latter case, which will be that of primary interest here, the response has tobe obtained by solving simultaneously the coupled equations that model the sys-tem. The “heterogeneous” qualifier is used in the sense that component simulationbenefits from custom treatment. It should be noted that one-way and multiwayinteraction are called weak and strong, respectively, by some authors.

As noted in Section 2.2, the decomposition of a complex coupled system forsimulation is hierarchical with two to four levels being common. At the first levelone encounters two types of subsystems, embodied in the generic term field:

Physical Subsystems. Subsystems are called physical fields when their mathemat-ical model is described by field equations. Examples are mechanical and non-mechanical objects treated by continuum theories: solids, fluids, heat and elec-tromagnetics. Occasionally those components may be intrinsically discrete, asin actuation control systems or rigid-body mechanisms. In such a case the term“physical field” is used for expediency, with the understanding that no spatial dis-cretization process is involved.

Artificial Subsystems. Sometimes artificial subsystems are incorporated for com-putational convenience. Two examples: (i) dynamic fluid meshes to effect volumemapping of Lagrangian to Eulerian descriptions in interaction of structures withfluid flow, and (ii) fictitious interface fields, often called “frames”, that facilitateinformation transfer between two subsystems.


TIME t

TIMESTEP hStart time tn

Splitting in time(fluid field only)

Partitioningin space

3D SPACE

xi

End time tn+1

Figure 2. Decomposition of an aeroelastic FSI coupled system: partitioning inspace and splitting in time. 3D space is pictured as “flat” for visualization con-venience. Spatial discretization (omitted for clarity) may predate or follow parti-tioning. Splitting (here for fluid only) is repeated over each time step and obeystime discretization constraints.

2.4 Partitions

A coupled system is characterized as two-field, three-field, etc., according tothe number of different fields that appear in the first-level decomposition.

For computational treatment of a dynamical coupled system, fields are dis-cretized in space and time. A field partition is a field-by-field decomposition inspace. A splitting is a decomposition of the time (or pseudo time) discretization ofa field within its time step interval. See Figure 2. In the case of static or quasi-staticanalysis, real time is replaced by pseudo-time or some kind of control parameter.

Partitioning is the process of spatially decomposing the system into partitions.This may be algebraic or differential. In algebraic partitioning the complete cou-pled system is spatially discretized first, and then decomposed. In differentialpartitioning the decomposition is done before discretization, and each field is thenseparately discretized.

Algebraic partitioning was originally developed for matched meshes and sub-structuring; see Figure 3(a), but later work has aimed at simplifying the treatmentof nonmatched meshes through frames [28, 29]. Differential partitioning oftenleads to nonmatched meshes, which are typical of fluid-structure interaction asdepicted in Figure 3(b), and handles those naturally.

A property is said to be interfield or intrafield if it pertains collectively to allpartitions, or to individual partitions, respectively. A common use of this qualifier


Fluid

Structure

(b)

(a)

Figure 3. Two partitioning types: (a) algebraic versus (b) differential.

concerns parallel computation. Interfield parallelism refers to the implementa-tion of a parallel computation scheme in which all partitions can be concurrentlyprocessed for time-stepping. Intrafield parallelism refers to the implementationof parallelism for an individual partition using a second-level decomposition; forexample breaking up a structure into substructures or subdomains.

2.5 Coupled Problem Examples

Experiences discussed in this chapter come from systems where a structure isone of the fields. Accordingly, all of the following examples list structures as oneof 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)Electro-thermo-mechanical interaction (3)Fluid-structure-combustion-thermal interaction (4)

When a fluid is one of the interacting fields a wider range of computational model-ing possibilities opens up as compared to, say, structures or thermal fields. For thelatter finite element discretization methods can be viewed as universal in scope.


Liquid

GasInlet

Outlet

Fluid

Structure

(b)(a)

Figure 4. Interior versus exterior coupled problem. In (a) the structure (tank) sur-rounds the non structural fields: liquid and gas. In (b) the structure (submarine) isimmersed in a fluid assumed as having infinite extent.

On the other hand, the range of fluid phenomena is controlled by several majorphysical effects such as viscosity, compressibility, mass transport, gravity and cap-illarity. Incorporation or neglect of these effects gives rise to widely different fieldequations as well as an array of discretization techniques.

For example, the interaction of an acoustic fluid with a structure in aeroacous-tics or underwater shock is computationally unlike that of high-speed gas dynamicswith an aircraft or rocket, a surface ship with ocean waves, or flow through porousmedia. Even more variability can be expected if chemistry and combustion effectsare considered. Control systems also exhibit modeling variabilities that tend not tobe so pronounced, however, as in the case of fluids. The partitioned treatment ofsome of those examples is further discussed in subsequent sections.

2.6 Interior and Exterior Problems

When interacting fields occupy nonoverlapping regions of space, and one ofthem is a structure, the following terminology is commonly used.

Interior Problem. The structure surrounds the nonstructural fields. For example,in the tank problem illustrated in Figure 4(a).

Exterior Problem. The structure is surrounded by nonstructural fields that may beviewed to be unbounded; e.g., in the submarine problem depicted in Figure 4(b).

2.7 Scenarios

Because of its combinatorial nature — as discussed in the Introduction —model based simulation of a coupled system rarely occurs as a predictable long-term goal. Some more likely scenarios are as follows.

Research Project. Members of a research group develop personal expertise inmodeling and simulation of two or more isolated problems. The group is thencalled upon to pool that disciplinary expertise into solving a coupled problem.

Product Development. Design and verification of a product requires concurrentconsideration 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 tosingle fields as isolated entities: a CFD gas-dynamics code, a structure FEM pro-gram and a thermal analyzer. As the customer base expands, requests are receivedto allow interaction effects targeted to more ambitious applications. For examplea CFD user may want to account for moving rigid boundaries, interaction with aflexible structure, and eventually linkage to a control system as in [2, 27].

The following subsection discusses approaches to these scenarios.

2.8 Approaches to Coupled Problem Simulation

To fix the ideas we assume that the simulation calls for dynamic analysis thatinvolves following the time response of the system. [Static or quasistatic analysescan be encompassed by introducing a pseudo-time history parameter.] Further-more, the modeling and simulation of isolated components is assumed to be wellunderstood. Approaches to the simulation of the coupled system include:

Field Elimination Treatment. One or more fields are eliminated by techniques suchas integral transforms or model reduction. The remaining field(s) are treated by asimultaneous time stepping scheme.

Monolithic or Simultaneous Treatment. The whole problem is treated as an entity,and all components advanced simultaneously in time.

Partitioned Treatment. The field models are computationally treated as isolatedentities that are separately stepped in time. Interaction effects are viewed as forc-ing effects that are communicated among 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, or temporal convolutions,which can introduce numerical difficulties. On the other hand the monolithic andpartitioned treatments are general in nature. No technical argument can be madefor the overall superiority of either. Their relative merits are not only problemdependent, but are interwined with human factors as discussed below.

2.9 Monolithic vs. Partitioned

Keywords that favor the partitioned approach are: customization, independentmodeling, software reuse, and modularity.

Customization. This means that each field can be treated by discretization tech-niques and solution algorithms that are known to perform well for the isolatedsystem. The hope is that a partitioned algorithm can maintain that efficiency forthe coupled problem if (and that is a big if) the interaction effects can be alsoeffectively treated.


Independent Modeling. The partitioned approach facilitates the use of non-matching models. For example in a fluid-structure interaction problem the struc-tural and fluid meshes need not coincide at their interface; cf. Figure 3(b). Thistranslates into project breakdown advantages in analysis of complex systems suchas aircraft or ships: separate models can be prepared by different design teams,including subcontractors that may be geographically distributed.

Software Reuse. Along with customized discretization and solution algorithms,custom software (private, public or commercial) may be available. Furthermore,there could be a gamut of service tools such as mesh generators and visualizationprograms. The partitioned approach facilitates taking advantage of existing code.This is particularly suitable to academic environments, in which software develop-ment tends to be cyclical and loosely connected from one project to another.

Modularity. New methods and models may be introduced in a modular fashionaccording to project needs. For example, it may be necessary to include localnonlinear effects in an individual field while keeping everything else the same.Implementation, testing and validation of incremental changes can be conductedin a modular fashion.

These advantages are not cost free. The partitioned approach requires careful for-mulation and implementation to avoid degradation in stability and accuracy. Par-allel implementations are particularly delicate. Gains in computational efficiencyover a monolithic approach are not guaranteed, particularly if interactions occurthroughout a volume as is the case for thermal and electromagnetic fields. Finally,the software modularity and modeling flexibility advantages, while desirable inacademic and research circles, may lead to anarchy in software houses.

In summary, circumstances that favor the partitioned approach for tackling anew coupled problem are: a research environment with few delivery constraints,access to existing and reusable software, localized interaction effects (e.g. surfaceversus volume), and widespread spatial/temporal component characteristics. Theopposite circumstances: commercial environment, rigid deliverables timetable,massive software development resources, global interaction effects, and compa-rable length/time scales, favor a monolithic approach.

The material of Sections 3 through 6 will focus on the partitioned approach.

2.10 Historical Note: Splitting and Fractional Step Methods

Splitting methods for the equations of mathematical physics predate partitioned analysisby two decades. In the American literature they can be originally traced to the mid-50sdevelopment of alternating direction methods by Peaceman, Douglas and Rachford [4, 31].Similar developments were independently undertaken in the early 1960s by the Russianschool led by Bagrinovskii, Godunov and Yanenko, and eventually unified in the methodof fractional steps [42]. The main applications of these methods have been the equations ofgas dynamics in steady or transient forms, discretized by finite difference methods. They


are particularly 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 by Richtmyer and Morton [35]. Suppose the gov-erning equations in divergence form are , where the operator is split into

. Pick a time stepping scheme and replace successively in it by, , , each for a fraction of the temporal stepsize . Then a multidimensional

problem can be replaced by a succession of simpler 1D problems. Splitting may be addi-tive, as in the foregoing example, or multiplicative. Since these lectures focus on partitionedanalysis, the discussion of those variants falls beyond its scope.

Comparison of those methods with partitioned analysis makes clear that little overlapoccurs. Splitting is appropriate for the treatment of a field partition such as a fluid, if thephysical structure of such partition display strong directionality, or if the constitutive equa-tions benefit from a split treatment (e.g., near incompressibility). Consequently splittingmethods are seen to pertain to a lower level of a top-down hierarchical decomposition.

3 Partitioned Analysis

This section provides a quick overview of the partitioned analysis procedures ap-plied to a model problem. Focus will be on the so-called staggered solution proce-dures, which are important on account of extensive use in applications as well assimplicity of implementation.

g(t)

f(t) x(t)

y(t)Y

X

Figure 5. Le rouge et le noir.

3.1 The Basic Idea

Consider the two-way interaction of two scalar fields, and , sketched inFigure 5. Each field has only one state variable identified as and , respec-tively, which are assumed to be governed by the first-order differential equations

(1)

Here and denote applied forces. Treat this by Backward Euler timeintegration in each component:

(2)


1. (P) Predict:

2. (Ax) Advance x:

3. (S) Substitute:

4. (Ay) Advance y:

xn+1 = 13 + 4h

( h fn +1+ 3 xn + y Pn+1 )

xn+1 = xn+1

(for example)

(trivial here)

y Pn+1 = yn + hyn

yn+1 = 11 + 6h

( h g

h

hn+1 + yn + 2 xn+1 )

. Step 2: Ax

Step 4: AyStep 3: SStep 1: P

Time

xn x

yyn n+1

n+1

(a) (b)

Figure 6. Basic steps of a red-black staggered solution.

where , , etc. At each time step we get

(3)

in which are provided by the initial conditions. In the monolithic orsimultaneous solution approach, this equation is solved at each timestep, and thatis the end of the story.

3.2 Staggering

A simple partitioned solution procedure is obtained by treating (3) with thefollowing staggered partition that does prediction on :

(4)

Here is a predicted value or simply the predictor. Two common choicesare (called the last-solution predictor) and . Thebasic solution steps are displayed in Figure 6(a). A state-time diagram of thesesteps, with time along the horizontal axis, is shown in Figure 6(b). The mainachievement is that systems and can be now solved in tandem.

Suppose that fields and are handled by two separate but communicat-ing programs. If intrafield advancing arrows are omitted, we obtain a zigzaggedpicture of interfield data transfers between the X-program and the Y-program, assketched in Figure 7. This interpretation motivated the name staggered solutionprocedure introduced in [23].

3.3 Concerns: Stability and Accuracy

In linear problems the first concern with partitioning should be degradation oftime-stepping stability caused by prediction. In the foregoing example this is notsignificant. The spectral analysis presented in Section 5, which embodies this ex-ample as instance, shows that staggering does not harm stability or even accuracy,


Y program:

X program:

Y program:

X program:

TimeTimestep h

I ISP

P P

P PS P S

(a) (b)

TimeTimestep h

Figure 7. Interfield+intrafield time-stepping diagram of the staggered solution stepslisted in Figure 6. Interfield time stepping diagrams: (a) sequential staggered solu-tion of example problem, (b) naive modification for parallel processing.

if the integrator and predictor are appropriately chosen. In fact, staggered proce-dures are very effective for coupled first-order parabolic systems. But for moregeneral problems, particularly those modeled by oscillatory second order ODEs,the reduction of stability can become serious or even catastrophic.

Instead of , prediction might be done on the field, leading to a zigzaggeddiagram with substitution on . The stability of both choices can be made tocoalesce by adjusting predictors.

Once satisfactory stability is achieved, the next concern is accuracy. This isusually degraded with respect to that attainable by the monolithic scheme. In prin-ciple this can be recovered by iterating the state between the fields. Iteration isdone by cycling substitutions at the same time station. However, interfield iter-ation generally costs more than cutting the timestep to attain the same accuracylevel. If, as often happens, the monolithic solution is more expensive than thestaggered solution for the same timestep, we note the emergence of a tradeoff.

In strongly nonlinear problems, such as gas dynamic flows in the transonicregime, stability and accuracy tend to be interwined (because numerical stabilityis harder to define) and they are usually considered together in method design. Theexpectation is for a method that operates well at a reasonable timestep.

Examination of Figure 7(a) shows than this simple staggered scheme is unsuit-able for interfield parallelization because programs must execute in strictly serialfashion: first X, then Y, etc. This was of little concern when the method was for-mulated in the mid 1970s as computers were then sequential. (With the exceptionof an exotic machine known as the ILLIAC IV.) The variant sketched in Figure 7(b)permits the programs to advance their internal state concurrently, which allows in-terfield parallelization. More effective schemes, which do not require predictionon both fields, have been developed over the past 15 years and are discussed atlength in [5, 6, 32, 33].


P SC

A

A

ScA

A A+MC

timestep h

Prediction

Lockstep advancing

Midpointcorrection

Full stepcorrection

Subcycling Augmentation

Substitution InterfieldIteration

I

Time

Figure 8. Devices of partitioned analysis time stepping.

3.4 Devices of Partitioned Analysis

As the simple example illustrates, partitioned analysis requires the examinationof alternative algorithm and implementation possibilities as well as the study oftradeoffs. Figure 8 displays, using interfield time stepping diagrams, the main“tools of the trade.” Some devices such as prediction, substitution and iterationhave been discussed along with the foregoing example. Others will emerge in theexample application problem discussed in Sections 5 and 6.

4 Preliminary Method Design

As is the norm in complex software projects, the construction of a model-basedmultiphysics simulation program goes through several design-build-test (DBT)stages. Resources available at project start can be more volatile than a typicalbuild-everything-from-scratch software project. Existing software may be at handfor some components of the project. These “initial conditions” may be due to in-dependent decisions, or be stipulated as part of the project contractual obligations.In a long-term project several DBT targets can be identified:

(a) Preliminary or proof-of-concept version;

(b) Development version, accessible only to developers;

(c) Alpha and beta versions distributed to selected external testers;

(d) Production version for general release.

The diagram of Figure 9 pertains to the preliminary design stage. This is theonly one discussed here. Stages (b)–(d) acquire heftier software engineering con-text and thus are largely beyond the scope of this chapter.

Preliminary design stage inputs are the models for the components of the cou-pled problem. If model choices exist, those available in reusable software would


Stabilityanalysis

Select

Select

ConstructMETS

Problemmodeling

To development phase

Method OK onMETS Stability cannot

be fixed: changegoverning equations by

augmentation

Model EquationTest System

GoverningSemidiscrete

Equations

PartitionType

ModifiedEquation

CharacteristicEquation

Integrator,Predictor &

ComputationalPath

Fixablestability

problems,retry

Fixableaccuracyproblems,

retry

Stability OK,do accuracy

analysis

Making progress

Regression

Figure 9. Flowchart for preliminary design of partitioned analysis procedures.

be naturally preferred. If all software is to be developed from scratch, the choicemay be dictated by the familiary of developers with certain models. A completefreedom of choice, however, is comparatively rare outside of academia.

4.1 The Root Morass

At the top of Figure 9 we have the governing semidiscrete equations. Thesecome from the space discretization procedure adopted for each component. For astructure this would be normally be FEM, whereas for a fluid a variety of othermethods (FVM, BEM, ...) might be considered. Whatever the choice, full multi-way interaction is assumed.

At this point the root-morass difficulty emerges. To fix the ideas suppose thatthe target problem is linear structure-structure interaction, and that four substruc-tures with up to 10 million degrees of freedom (DOF) each are allowed. All cou-pled semidiscrete systems are of second order in time. A 3-step integration methodis envisioned. Then the characteristic system will havemillion eigenvalues. To verify numerical stability, the location of each eigenvalueon the complex plane would have to be ascertained. This is further complicated bythe fact that in method design one often carries along free parameters.

The presence of this root morass would pose no problem if a monolithic so-lution scheme, whether implicit or explicit, is used. Under usual assumptions onmodal decoupling, the normal modes of the semidiscrete (time-continuous) systemsurvive in the time-discrete (difference) system. One can therefore look at a one-DOF model problem, and infer from its spectral analysis the numerical stability ofthe whole difference system, whether it has one or ten million equations.

Normal modes do not generally survive, however, in the time-discrete system


if a partitioned integration scheme is used. This has two implications: (i) thefull coupled system has to be considered in the stability evaluation, and (ii) moresophisticated mathematical techniques, such as the theory of positive-real polyno-mials, must be resorted to in order to arrive at useful conclusions.

4.2 The Model Equation Test System (METS)

For preliminary method design a compromise is necessary. A model equationtest system (abbreviation: METS) is still used, but this is no longer a scalar ODE:it has as many equations as partitions. (Sometimes interfaces are treated as ad-ditional partitions, in which case the METS includes additional equations [36].)This makes analysis feasible with a computer-algebra system (CAS) while keep-ing free parameters, and the morass is avoided — or at least alleviated. Proceduresfor constructing METS are illustrated in the examples of Sections 5 and 6.

A METS has two types of free parameters: physical and numerical. The formercome from physical properties of the systems being modeled; the latter from thetime integrator and predictor. To make preliminary design effective a key goal isto reduce the number of free parameters. This can be a delicate balancing act.If the number of parameters is too small it may leave out important physics, orweed out useful time integration schemes. If too large, the analysis work grows upsuperlinearly; for example going from 4 parameters to 5 might may increase thedesign cycle time by orders of magnitude. In general it is better to err on the sideof simplicity. An effective method to cut down on free parameters is to reduce theMETS to dimensionless form, as illustrated in the application examples.

4.3 Additional Design Choices

Once a METS is constructed, the designer (or designer team) picks a parti-tion type. This embodies several decisions: (i) choosing algebraic or differentialpartitioning (cf. Section 2.4), (ii) designating variable(s) to be predicted from apartition to another, and (iii) deciding whether additional fields, such as Lagrangemultipliers, are to be placed between partitions [25, 28, 29, 36]. The type of in-tegration scheme to be used in the subsystems: implicit or explicit, may be alsopicked at this stage. The foregoing choices are dictated by physical and modelingattributes. They are usually kept fixed unless it becomes necessary to modify thegoverning equations and consequently the METS.

Next one selects the time integrator, predictor and computational path, a stepabbreviated to IPCP. The choice of integrator and predictor have been discussed.Typically these may be left parametrized if a CAS is used. “Computational path,”a term originally introduced for structural dynamics [8–10,25], identifies how cer-tain auxiliary quantities (e.g., velocities, accelerations, etc.) are computed in eachtime step. The reader is referred to the foregoing papers for details.


4.4 METS Stability Analysis

This is the centerpiece of the design process. A partitioned analysis procedurethat fails to satisfy target stability conditions is useless. What are those conditions?Informally: stability should not be affected by partitioning. More specifically, thebest outcome that can be hoped for is

1. If all partitions are treated by A-stable integrators, the partitioned methodshould retain A-stability.

2. If one or more partitions are treated by conditionally stable integrators, themaximum stable stepsize should not be degraded by partitioning.

Since the METS is normally linear, its stability analysis may be done by spec-tral techniques. Root loci of its characteristic system are investigated for the ap-propriate free parameter ranges. Three possible outcomes of this analysis are:

Lucky. If stability requirements are met, one proceeds to accuracy analysis. Thisis a long but less crucial subject, and so it will not be covered here.

Fixable. If requirements are missed but look achievable by tweaking integrator,predictor and/or computational path, one may return to the IPCP box, make ad-justments, and try again. Changing the partition type might also be attempted.

Unfixable. If stability targets are missed and can be shown to be unattainable, onemay try changing the governing equations. This is done by a technique calledaugmentation, discussed in the example of Section 6.

4.5 From Idealized METS to Reality

Obtaining a preliminary design that meets stability and accuracy targets doesnot guarantee ultimate success. It only gives a candidate method that happensto work on the METS. This system might have incorporated highly simplifyingassumptions to speed up work. Furthermore the METS cannot reflect features thatmay be crucial to implementation efficiency, such as the matrix sparsity structureof the original semidiscrete equations. In the development stage, implementorsapply reality checks. These may include one or more of the following:

1. Considering physical or computational effects that were ignored or discardedon constructing the METS. For example, structural damping, mesh truncationin unbounded domains, and local nonlinearities.

2. Applying the method to the original discrete equations to verify implementa-tion efficiency and (if necessary) compatibility with existing software.

3. Checking that linearizations, if any, are justified.

4. Accounting for applied forcing effects. While these are ignored in spectralstability analysis, they may influence computational accuracy, as well as long-term stability in nonlinear problems.


The next sections present two preliminary design examples. While the first oneis largely academic, the second highlights aspects of a realistic case study.

5 Coupled Parabolic Equations

The title example is one for which designing an unconditionally stable staggeredsolution procedure (SSP) is smooth sailing. Two coupled models governed bylinear parabolic equations (e.g., an unsteady diffusion process) are defined by statevectors and of order and , respectively, for two partitions. Assume thatboth partitions have been spatially discretized, and that the two-way coupling isgoverned by the semidiscrete differential matrix equations

(5)

Here and are symmetric positive definite while and are sym-metric nonnegative. The supermatrix that combines , , and isalso nonnegative. [In heat conduction problems matrices and represent ther-mal capacitance and conductivity, respectively, of the modeled media.] Note thatinterfield coupling occurs through the state vectors but not their time derivatives.This greatly facilitates the construction of stable partitioned solution procedures.

5.1 METS Construction

Consider the two uncoupled generalized symmetric eigenproblems associatedwith the left side of (5):

(6)

in which denotes the mode index, and in which eigenvectors are normalized as

with = Kronecker delta. (7)

By virtue of the stipulated matrix properties, the eigenvalues and are nonneg-ative real [30]. In accordance with (6) we introduce the transformations ,

, in which matrices and are formed with the eigenvectors and ,respectively, as columns, and where and collect associated mode amplitudesas generalized coordinates. Dropping the force terms in (5) and congruentiallytransforming to modalT coordinates yields

(8)

in which and denote the identity matrices of orders and , respectively,diag , diag and .


Although the left side of (8) is uncoupled, each mode pair, say , will be gen-erally coupled through the right hand side terms and . Assumingfor simplicity that and , we can write , in whichthe dimensionless is called the modal coupling coefficient. On supressing theindices for brevity, the following METS emerges:

(9)

Introducing , , , we may rewrite the METS in a formmore convenient for combining with the stepsize introduced below:

(10)

Now everything is dimensionless but for , which has the dimension (1/time). Incompact form: , with . The determinant of is

. Since by the nonnegativity stipulation, it follows that. The case is called full mode coupling. This occurs if two

modes “resonate” or “antiresonate” in the sense that and .Note that if or the off-diagonal term must also be zero since otherwise

; consequently may be set to zero in that case without loss of generality.For time integration we consider the general one-step linear LMS scheme

(11)

where is the stepsize and is a free parameter. Note that no gains canbe expected in this problem should different s be used for and .

5.2 Stability of Monolithic Integration

Even if partitioned time integration is the end goal, it may be instructive toverify the stability of a monolithic scheme applied to the METS. This may providea valuable check since often the expected result is known and the required programchanges minor. In the present example only one line of code had to be modified.All computations reported in this and the following subsection were carried outwith the Mathematica stability analysis modules presented in Appendices A and B.Evaluating (10) at and gives

(12)Eliminating time derivatives from (11)–(12) yields the state amplification relation

or (13)


in which ,, and

. The amplification polynomial is. Through the mapping this is con-

verted to the Hurwitz polynomial . The coefficients of may be somewhatsimplified by taking , in which . The result provided bythe Mathematica modules is

with

(14)

As is quadratic in , the A-stability conditions are , andwhile , and , with as free parameter. Byinspection this happens if or , which is a well known result. Secondorder accuracy is obtained for , which yields the Trapezoidal Rule.

5.3 Stability of Staggered Integration

A staggered solution procedure (SSP) is obtained by replacing in the firstequation of (12) with a predicted value:

(15)

in which is a free predictor parameter. (The previous-step equation in (12) isassumed to be satisfied exactly; this has implications with respect to computationalpath selection [8–10, 25], but the topic is too elaborate to be discussed here.)

Eliminating time derivatives yields a state amplification relation with the sameform as (13), but now ,

, ,and

. From this one obtains the amplification polynomialand the Hurwitz polynomial upon mapping .

Coefficients of may be again simplified by taking . The resultprovided by the Mathematica modules is

with

(16)

Since is quadratic in , the A-stability conditions can be simply expressedas , and for , and , but now


C-STABLE orUNSTABLE

2nd order accurate SSP

Last-solutionpredictor =0

Linear extrapolationpredictor =1

TrapezoidalRule =1/2

BackwardEuler =1

A-STABLE

Figure 10. A-stable region for staggered solution of coupled parabolic equations.

there are two free parameters: as before, and the predictor coefficient. An analysis of these inequalities show that they are satisfied if

(17)

This A-stable region is plotted on the plane in Figure 10. (The regioncontinues beyond but is truncated at since in practice .] Anaccuracy analysis using the method of modified equations [13, Appendix B] showsthat second order accuracy is obtained if and .

In summary, for this particular case the SSP is competitive with the monolitihictreatment in terms of stability and accuracy, but of course the SSP is significantlycheaper per time step. This optimistic picture changes when considering couplingwith a second order system, as evidenced by the next application example.

6 Fluid Structure Interaction for Underwater Shock

This application problem fits the FSI theme of the lectures. It is actually that whichmotivated the invention of staggered methods, which later evolved into the moregeneral partitioned solution procedures. It illustrates difficulties often encounteredin method design. The exposition largely follows the original 1977 paper [23].

6.1 Background

In the late 1960s the US Navy became concerned about the vulnerability of thestrategic submarine fleet to underwater shock (UWS) attacks. Particularly worri-some were new torpedo devices capable of producing directional shockwaves ableto propagate over long distances with small decay under “waveguide” conditions.Until then submarine shock hardness was estimated with a potpourri of empiri-cal and simple analytical recipes based on gross approximations extrapolated from


World War 2 warfare experiences. Those became obsolete as vulnerability predic-tors for the new scenarios.

The time was nonetheless ripe for taking advantage of ongoing developments inmodel-based simulation. By 1970, the Finite Element Method (FEM) was solidlyestablished in structural and solid mechanics. Time-domain dynamic responsemethods were advancing rapidly. The first government-funded, large-scale FEMcode: NASTRAN, had been released and DoD contractors were evaluating it asbackup for in-house programs. Thus, on the structure side things were in promis-ing shape for a direct simulation approach.

The major obstacle to direct simulation was the fluid-structure coupling. Mod-eling the fluid region from the detonation site through full envelopment and be-yond (to account for reflections and scattering) would entail a fluid volume meshwith millions of freedoms. Even treating the water as a linear acoustic medium,a huge computational effort would be required to propagate the shockwave andtrace its interactions after envelopment. Thus, modeling and computational re-sponses would be largely wasted on the portion of the problem less relevant to themain objective: predicting structural and internal equipment damage.

6.2 Initial Modeling and Validation

T. L. Geers, then at the Applied Mechanics Laboratory (AML) of LockheedPalo Alto Research Laboratory (LPARL) advocated a new coupling approach:model the acoustic fluid by a Boundary Element Method (BEM) called the Dou-bly Asymptotic Approximation or DAA [14–17]. This model was asymptoticallyexact in the low and high frequency limits while effecting a smooth transition inbetween. The volume fluid mesh can thereby be reduced to just a “membrane”surrounding the structure. This drastically simplifies modeling while allowing themajor share of computational resources to be allocated to the focus of interest,namely the structure. Before 1972 the DAA had been used, however, only on alimited set of FSI benchmark problems that possessed exact solutions, such as in-finite cylindrical shells described by in-vacuo vibration modes. It had not beencoupled to a FEM structural model nor been experimentally validated.

A project for validating the coupled DAA-FEM model was awarded by theOffice of Naval Research (ONR) to AML-LPARL in 1973. The benchmark wasan axisymmetric, stiffened steel cylindrical shell with heavy end caps and inter-nal rings, designed to be a very rough scaled representation of a submarine hull.(This came to be known as “the ONR shell” in the underwater shock commu-nity.) The test article was instrumented with velocity meters and nondestructivelytested by Navy personnel as a submerged structure in Chesapeake Bay with ta-pered DSX-1 explosive charges, including both side-on and end-on attacks. In-dependently the LPARL team did numerical simulations using what later became


known as a monolithic method to do FSI. Discrete shell equations generated bythe BOSOR 4 axisymmetric shell code of D. Bushnell [1] were coupled to theDAA-BEM equations, and solved simultaneously with direct time integration overa timespan of about 20 milliseconds. Unvarnished comparisons showed that thecoupled DAA-BOSOR model delivered satisfactory velocity predictions for mostof the attack cases.

6.3 Staggered Solution Procedure

Encouraged by the experimental validation, ONR awarded AML-LPARL aproduction contract in 1975 to support the development of a general-purpose,three-dimensional, UWS code. But a logistic problem soon emerged. Submarine-builder contractors insisted that the UWS code had to be compatible with NAS-TRAN, which by then was in heavy use to model ship and submarine structures.But since its initial release NASTRAN had become a proprietary program. Thissnuffed the idea of tightly fitting a DAA fluid solver because the source code wasoff limits. The use of a monolithic solution method was ruled out.

The second author, also a member of AML and participant in the secondproject, proposed what was then an innovative concept: to solve the fluid and struc-ture in tandem with two different codes. These would separately advance in timewhile exchanging interaction data. The technique was baptized staggered solutionprocedure or SSP. Predictors were essential part of the method. The SSP not onlycircumvented the software accessibility problem, but allowed the structure and thefluid to be treated by different integration schemes. Initial studies, summarized inSection 6.9, showed that any such combination would have an unacceptably smallstable timestep, of the same order as that of an explicit scheme. That would haverendered a promising idea unusable since for a steel shell (such as a submarinehull) that stable timestep is on the order of nanoseconds.

Construction of an unconditionally stable SSP turned out to be a mathematicaltour de force. No theory was readily available because governing equations do notmodally decouple. Artifacts from control theory were morphed to suggest promis-ing study paths. Much of the difficulty is of combinatorial nature, as discussedin the Introduction. Finally, one technique, called augmentation, was successfulbeyond expectations in that it produced a set of unconditionally stable time inte-grators without need for expensive iterations at each timestep. Once a satisfactorySSP was found, a DAA-BEM fluid analyzer called USA was plugged to an arrayof structure analyzers over the next 20 years: NASTRAN, GENTRAN, ADINA,LSDYNA, etc. Analysis capabilities were enhanced to include cavitation [11], aswell as free surface effects for surface ship attack simulations [39].

By 1980, SSPs had been recognized as a special case of a more general frame-work: partitioned analysis procedures for coupled systems in which interacting


structuralnodesDAA-1 BEM mesh

(shown offset fromwet surface for clarity)

shockwave

FE model (internal structure omitted)

"DAA-1 membrane'' Linear ornonlinear structure

Acoustic fluid

(b)

(c)

BEM controlpoints

(a)

modelcross section

Figure 11. Submerged structure hit by shock wave: (a) coupled problem physics,(b) acoustic fluid modeled as “DAA membrane” (c) typical cross-section of cou-pled FEM-BEM discretization on wet surface with interior structure omitted. In(b–c) the BEM fluid model is shown offset from the wet surface for clarity.

subsystems can be structures, flowing fluids, control systems, electromagneticwaves, thermal fields, etc. Applications to a range of coupled problems have beendeveloped as driven by target applications. As noted in Section 1, partitionedanalysis is enjoying renewed attention given increased interest in multiphysics,nonmatching meshes, model reduction and parallel processing.

6.4 The Source Problem

The coupled problem is depicted in Figure 11(a). A 3D structure, externallyfabricated as a stiffened shell, is submerged in an unbounded compressible liq-uid. A pressure shock wave due to an underwater explosion propagates throughthe fluid and impinges on the structure. Because of the fast nature of the transientresponse, which spans only milliseconds, the fluid can be modeled as an acousticmedium since no significant flow develops. The structure is discretized by conven-tional finite element methods. For the fluid, a boundary-element method (BEM)discretization is appropriate since only a “wet surface mesh” has to be created. Asdiscussed in Section 6.2, those decisions were dictated by the fact that the structureresponse — especially as regards vulnerability — is of primary concern whereaswhat happens in the fluid is of little interest. Accordingly, the fluid was modeledby the first-order Doubly Asymptotic Approximation (DAA ) of Geers [14–17].

Figure 11(c) reinforces the message that FEM and BEM meshes on the “wetsurface” do not generally coincide: one fluid element generally overlaps severalstructural elements. This happens because stress computation requirements de-mand a finer structural discretization, whereas the main function of the fluid ele-ments is to transmit hull pressure forces and receive feedback normal velocities.


6.5 Governing Equations

The semidiscrete equations of motion coupled through the “wet surface ”may be written in matrix form as

(18)

The first set of matrix equations expresses dynamic equilibrium in terms of thestructural displacements, whereas the second set expresses the surface interactionapproximation being used. (The number of structural equations is usually muchgreater than the number of fluid equations, because the latter come from a BEMdiscretization.) In (18) , and are the structural mass, damping and linear(or linearized) stiffness matrices, respectively; , and are structure responsedisplacements, dry-structure applied force, and nonlinear residual (pseudo-force)vectors, respectively; is the fluid-induced force vector appropriate to the struc-tural mesh on , is the scattered pressure vector and the incidentpressure vector, respectively, appropriate to the fluid mesh on ; is thecorresponding pressure-integral vector; is a fluid-forcing term that is a functionof time derivatives of the structural motion normal to the wet surface and of theincident fluid particle velocity ; and , , are matrices determined bythe specific DAA being used. Finally, superscripts dot ( ) and asterisk ( ) denotetemporal differentiation and integration, respectively.

For the particular case in which the DAA of Geers [14–17] is used as surfaceinteraction approximation, (18) specializes to

(19)

In this case is a diagonal matrix embodying elemental areas of the fluid mesh on, is the fluid added mass matrix as determined from an analysis of incom-

pressible fluid motion appropriate to a distribution of elementary sources on ,is a generally rectangular transformation matrix that relates structural displace-

ments to the control points of the BEM fluid mesh on , and are fluid densityand speed of sound, respectively, and superscript denotes matrix transposition.The treatment of the coupled system (19) is diagramed in Figure 6.5.

6.6 The Model Equation Test System

For stability analysis only the homogeneous linear portion of (19) is retained:

(20)


u

p = q.

.(structural damping ignored)

Structural FEM ModelM u + K u f T A(p +p) f

Iss s..

DAA-1 Fluid Modelf f f

f

fI

.

.A q + c A M A q

c A (T u v )

1

T

Structure

STRUCTURE PARTITION FLUID PARTITION

AcousticFluid

Pressure

Normal interface velocity

Figure 12. Partitioned analysis treatment of UWS coupled problem.

The standard approach to stability analysis of direct time integration for a ODEsystem such as (20) involves three steps: (a) transform the homogeneous formto normal coordinates, (b) apply the time integration procedure to the resultinguncoupled equations, and (c) examine the time-boundedness of the computed so-lution. In the following study, the structural damping term is dropped sinceits effect on the coupled response is in most cases negligible when compared tothe fluid radiation damping term . An appropriate two-DOF model problemassociated with (20), upon setting , is

(21)

in which , , and are generalized quantities resulting from the projectionof , , and , respectively, on normal coordinates and that simultane-ously diagonalize the symmetric pencils and , respectively. Thederivation of (21) is presented in [23, Appendix A]. The first of (21) representsa pressure excited undamped mechanical oscillator in the normal displacement .The second one represents a velocity-excited generalized pressure-decay equationin the normal pressure-integral variable .

The number of physical parameters in (21) can be cut down from six to threeby reducing it to the non-dimensional form

(22)

which in matrix form is

(23)

This is done by introducing the dimensionless variables

(24)


Table 1. Range of Nondimensional Parameters , , and

Illustrative cases / ParametersCavity limitDry structure mode limit indet.Early time response (shock-excited) 0.001–0.1Late time response (shock-excited) 1–100

In (24), denotes a characteristic length of the problem, for instance the radius of asubmerged cylinder or sphere; is a “buoyancy ratio” (structural mass divided bydisplaced fluid mass); is a reduced vibration frequency, and is a generalized-pressure decay exponent. Note that the dot superscript has been redefined to denotedifferentiation with respect to the reduced or dimensionless time rather thanactual time . Because is the time needed by a fluid sound wave of speed

to travel the characteristic length , that unit of time may also be called theenvelopment time in UWS problems.

6.7 Parameter Range

The model system (23) contains three dimensionless physical parameters: ,and . Time discretization as discussed in Sec 6.8 will introduce a dimensionlessstepsize . For the analysis of staggered procedures it is of interest toexhibit the range that those parameters can cover in envisioned applications.

Concerning the physical parameters, two limit cases: no structure and no fluid,are of interest in designing a robust time integration method. The cavity conditionis the limit of modal motions heavily dominated by the fluid inertia, as if the struc-ture would reduce to a weightless bubble. For those modes the analysis of [23]shows that , and . The dry mode condition is realized bystructural modes that do not interact with the fluid; for example torsional modes ofa submerged cylinder or vibrations of “quite” submarine engines. For these modes

can be an arbitrarily large nonnegative number, and is indeterminate.As regards the dimensionless stepsize , two regimes are of

interest. The early-time response spans the period during which the shock waveenvelops the structure: . It is characterized by high frequency struc-tural motions, high radiation damping and relatively small hydrodynamic forces.Here is typically to . The late-time response is the period well past en-velopment, say , characterized by low frequency structural motions, domi-nant hydrodynamic inertia and low radiation damping. Here may typically riseto 1–100. The intermediate time response, say , is transitional innature. Because the practical range of spans several orders of magnitude, forcomputational robustness we are interested in attaining A-stability.

Table 1 subsumes the foregoing discussion on parameter ranges.


6.8 Time Discretization

The METS defined by (22)–(24) is reduced to first order by introducing thestructure velocity as auxiliary vector. Combining this with

and yields the matrix system

(25)

As usual subscripts and will denote values at the last computed solutionand the next time step, respectively, so , etc. System (25) is time-discretized by the one-step LMS method

(26)

in which , and are coefficients in the range . The idea behind (26) isto allow different integrators to be used for each state variable. The restriction tothe simplest schemes is typical of the preliminary design stage: if no satisfactorystability is achievable with those, it is unlikely that more refined ones will work.

The predictor will be performed on the structural velocity that appears in thelast equation. As predictor formula we pick

(27)

in which is a numerical coefficient. Typically . If one obtainsthe so-called last solution predictor . Taking corresponds tousing forward extrapolation in the prediction.

6.9 Velocity-Predicted Staggered Procedure

The staggered solution procedure (SSP) defined by equations (25)–(27) will becalled the velocity predicted SSP. Introduce the state vectorsand . Eliminating all time derivatives from (25)–(27) yields the state advancing equations . The amplificationmatrix produced by Mathematica is complicated and not shown here. Expand-ing gives the amplification polynomial with generally complexroots . For A-stability we require for any combination of nonnegative

and for .This condition was investigated numerically in [23], with the final conclusion

being that A-stability was unattainable. It is instructive to take the cavity condition


0 2 4 6 8 101

0.5

0

0.5

1

h/

STABLE

UNSTABLE

best

0 2 4 6 8 101

0.5

0

0.5

1

h/

STABLE

UNSTABLE

best

Trapezoidal Rule:

v

Backward Euler:

v

Figure 13. Stability regions of velocity-predicted SSP with

, which can be shown to be the worst case for . The stabilitylimit is then controlled by so we take as parameter. The resultingstate amplification system reduces to

or (28)

This yields the amplification polynomial

(29)

Transforming to a Hurwitz polynomial via and removingdenominators yields

(30)

Since is linear in (although is cubic in , the two roots dropin the mapping, as discussed in Section A.3) the stability conditions are

(31)

Stable and unstable regions are shown on the ) plane in Fig. 6.8 forand , which correspond to the Trapezoidal Rule (TR) and Backward Euler(BE), respectively, for the second one-step integrator in (26). In either case thelargest stable timestep is , which is obtained by selecting forthe TR and for BE. Consequently there is no harm in choosing

, that is, the TR for all equations. A deeper investigation varying


1 0.5 0 0.5 1

2

4

6

8

10

STABLE FOR ALL

h/

Figure 14. Stability region in the versus plane for ,and various values of .

and shows that the stable time step is not affected for if , aspictured in Fig. 14. Taking a nonzero does not change the conclusions.

In summary, the stable region is limited to the grey area marked (for the TR)in Fig. 14. The largest possible stable stepsize is , which is obtained for

. This turns out to be of the same order as that of an explicit scheme, and thusunacceptable for calculation of practical submerged structures. (The presence ofshell elements in the structural model would result on an extremely small explicitstable timestep.) A study of the spectral radius of the iteration matrix for a frozen-time iterative correction procedure shows that this is not guaranteed to converge if

. Again this is of the order of the explicit stability limit.An analysis that abstracted the effect of the time integrator and relied heavily on

methods of control theory showed [23], that conditional stability for any consistenttime integrator was unavoidable. This was done by modeling the predictor as a“dead time” or “delay” device. The destabilizing effect was traced to the delayedvelocity feedback, which periodically feeds fictitious energy from one partition tothe other. Since there is no physical mechanism to absorb this feedback energy,Nyquist’s theorem showed that instability was triggered once exceeded a modestmultiple of . Although disappointing, the diagnostic suggested how to stabilizethe SSP by augmentation.

6.10 Stabilization by Augmentation

As just noted, A-stabilization of a SSP for this system requires addition ofdamping terms in the left-hand side to absorb the fictitious energy periodicallyfed by the predictor. Since addition of artificial damping is ruled out on accuracygrounds, the governing equations must be tailored in such a way that dampingterms appear in the structural equations, the fluid equations or both. This technique


was baptized augmentation in [23]. Here it is worked out at the METS level.To augment the structure, insert the fluid equation into the structural equation:

. Transfer the damping term to the left-hand side toget the structure-augmented model system

(32)

or

(33)

The prediction is to be done on the variable, which physically represents a fluidpressure integral. Accordingly this technique was called the Pressure IntegralExtrapolation, or PIE, in [23].

To augment the fluid, insert the structural equation into the time-differentiatedfluid equation: . Transferring the damping term tothe left we get the fluid-augmented model system

(34)

or

(35)

The prediction is to be done on the variable, which physically represents a struc-tural displacement. Accordingly this technique is called the Displacement Extrap-olation, or DE, in [23]. Additional augmented forms can be constructed by mod-ifying both structure and fluid equations. Such forms were not found to possessany particular advantage over the previous two and are not described here.

The spectral analysis of the PIE and DE forms showed that both may be ren-dered A-stable through appropriate choices of time integrators (these being differ-ent for the fluid and structure) and predictor. The accuracy analysis showed thatthe PIE was somewhat preferable on grounds of accuracy, but software implemen-tation constraints discussed in Section 6.13 ruled it out.

6.11 Stability Of Structure-Augmented METS

To analyze the stability of the structure-augmented METS (33), introduce again, reduce to first order system and pass the coupling terms to the right to get

(36)


As predictor take . Time discretization is effected by

(26). As before introduce the station state vectorsand the amplification matrix linking .

This matrix is quite complicated to show here, so as in the previous section westudy the cavity condition limit by setting and defining . Theamplification matrix reduces to

(37)

The cubic amplification polynomial is

(38)

has two roots that drop out on passing to the Hurwitz polynomial, as discussed in Section A.3, and is linear in . The only A-stability

condition is

for any (39)

This is satified if and any , so A-stability for the limit cavity conditionis achieved. A more complete analysis indicates that this is also the case for anynonnegative and if is in the range .

6.12 Stability Of Fluid-Augmented METS

For the fluid-augmented METS (35), introduce and , reduce to afirst order system and pass the coupling terms to the right-hand side to get

(40)

As predictor take . Time discretization is effected by the one-step integrators (26), to which we append .

As before introduce the station state vectors ,and the amplification matrix that connects

. This matrix is quite complicated to display, so as before we studythe cavity condition limit by setting and defining . The


amplification matrix reduces to

(41)

in which . The quartic amplification polynomial is

(42)

has three roots that drop out on passing to the Hurwitz polynomial, as discussed in Section A.3, and is linear in . The only A-stability

condition is

for any (43)

These are satified if so A-stability for the limit cavity condition is againachieved. A more complete analysis indicates that this is also the case for anynonnegative and if is in the range .

6.13 Implementation Constraints

As emphasized in Section 4.5, having constructed an A-stable and accuratepartitioned solution procedure for the METS is only the first step on the way to aproduction implementation. Two aspects that must be considered include

Look at the semidiscrete matrix forms and study implementation feasibility.

Decide how the fluid and structure programs will be developed (using existingcommercial software, available open source, brand new code, etc) and pickthe configuration that best fits implementation constraints.

In this case study the PIE form, although slightly more desirable than the DEform in terms of accuracy, was ruled out because development of a new structuralprogram was precluded. Submarine building contractors were already committedto existing FEM codes such as NASTRAN and GENSAM. (Over the next twodecades, ADINA, STAGS and DYNA3D were added to the list.) Implementing thePIE form with a commercial code such as NASTRAN would have run into seriouslogistic 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 beused in Navy work, upgrading a distribution-restricted custom version to keep upwith changes in the mainstream product can become a contractual nightmare.

The DE version of the staggered solution approach circumvented this problem.A 3D BEM fluid analysis program called USA (for Underwater Shock Analysis)was written and data coupled to several existing structural analysis codes over


STRUCTURE PARTITION FLUID PARTITION

Acoustic fluid (near field)

Structure

PHYSICAL DOMAIN

Acoustic fluid (far field)

shockwave

COMPUTATIONALDOMAIN

Cavitationregion

SilentDAA

boundary

Bilinearfluid

volume

SilentDAA

Boundary

Structure

Figure 15. Three-field partitioned treatment of UWS with cavitation.

the years. For example, the marriage of USA and NASTRAN is called USA-NASTRAN. This plug-in “toolbox modularity” embodies operational advantages:

It simplifies upgrade and maintenance of the more complex software compo-nent, which in this problem is the structural analyzer.

Allows the structural analyzer to be “plug replaced” to either fit existing struc-tural models or to address other physical effects. For example, should thestructure or internal equipment experience strong nonlinear behavior, the welltested material library, contact algorithms, and highly efficient explicit timeintegration capabilities of DYNA3D may be exploited.

A similar capability expansion can also be modularly addressed on the fluidside. For example, the occurrence of fluid cavitation (hull cavitation in sub-marines or bulk cavitation in surface ships) was addressed with the devel-opment of a nonlinear fluid-volume-based program called CFA. This codeoperates as a third partition that can be plugged-in as a pressure transducerbetween the structure and a DAA boundary sufficiently removed away fromthe wet surface to enclose any cavitating region [11]. See Figure 15.

7 Conclusions

The computer analysis of coupled system is in its infancy. Progress has beenslowed down because of the combinatorial nature of the subject, and the atten-dant complexity explosion of methods and implementations. A major goal ofthis exposition is to call attention to the widespread availability of computer al-gebra systems. These can provide time-saving help in facilitating the synthesis of


partitioned analysis procedures to handle complex coupled problems. Analyticalcalculations that were prohibitively difficult by hand when those methods werecreated in the mid-1970s, can be now done in reasonable time. Another pointemphasized throughout is the development of useful model test systems. Theseshould be neither too simple (thus leaving out important physics) nor too complex(thus obscuring primary behavior in a forest of details). The examples presentedin Sections 5–6 aim to illustrate how this method synthesis stage can be addressed.

This Chapter covers two of four lectures given at the CISM short course. Ma-terial presented in the other two lectures: non-matching meshes and Localized La-grange Multipliers (LLM) has been omitted for space reasons. A recent paper [36]addresses those topics and provides references to related prior work.

A Stability Analysis Tools

This Appendix presents computer algebra tools to help the spectral stability analy-sis of multilevel systems of linear difference equations with free parameters. Thisis done by testing stability polynomials for root cluster location. Such polynomialscome in two flavors: amplification and Hurwitz.

A.1 A Multistep Difference Scheme

To motivate the ensuing analysis as well as establishing notation, suppose thatdiscretization of a linear, time-invariant, real-valued test equation with state vari-ables collected in leads to a system of matrix difference equations such as

(44)

The symbols in (44) have the following meaning., , The state -vector at time stations , and , re-

spectively. Here is the current time station, which is thelast computed solution, whereas is the next timestation with as the stepsize. Previous solutions, such as ,are called historical data. Historical terms such as ap-pear if one uses a multistep integration scheme that spans two ormore timesteps, or a multistep predictor.

, , A set of matrices whose real entries depend on parame-ters of the test equation and on the time integration procedure. If

does not enter in the time discretization (44), . Ad-ditional historical terms, such as mayappear if the time integrator uses more previous solutions.The discretized forcing function at . In linear systemsthis term does not depend on the state .


A.2 Amplification Polynomial

To convert (44) to an amplification polynomial, set .Here is a (generally complex) variable that plays the role of amplification fac-tor. (This substitution is connected to the “discrete transform” of Jury [21].)Factoring out gives

(45)

For stability analysis the applied force term is set to zero. To avoid negativepowers it is convenient to multiply through to get

(46)

This equation has a nontrivial solution if and only if the determinant of the matrixvanishes. Expanding it as a polynomial in yields

(47)

where all are real. This receives the name amplification polynomial. Ithas order . Denote the (generally complex) roots of by ,

. The polynomial is called stable if (I) all roots lie on or insidethe unit circle in the complex plane:

(48)

and (II) any root of exact modulus 1 is simple. Condition (II) is of minor practicalimportance and will be only occasionally mentioned. On the other hand (I) iscrucial. It can be expressed compactly by introducing the spectral radius:

stable if and (II) (49)

Requirement (49) can be tested in terms of polynomial coefficients with the Schur-Cohn criterion [3, 38], which is covered in [22, 34]. We use this criterion infre-quently because computations with free parameters tend to get messy. More oftenwe shall transform to a Hurwitz polynomial to apply well-known, fraction-free stability tests more suitable to the presence of free parameters.

A stable time-marching difference system such as (44) will be called strictlystable [18] if the only root on the unit circle is , and all others are inside. Ifthere are several roots on the unit circle the system is weakly stable.

Example A.1. The general one-step LMS method applied to the scalar, homogeneous,exponential-decay model equation , with , is

, in which the real scalar . The resultant difference equation is

(50)


Setting yields . The only root ofis

(51)

whence . Obviously for any if and only if .

Example A.2. Consider the unforced, undamped linear oscillator model equation, in which real is the circular frequency. Treat this by the Newmark method

(52)

Here and are two real parameters that determine stability and accuracy characteristics.The state vector is . Evaluate at and , andeliminate accelerations from the equation of motion to get the difference equation

(53)

in which is dimensionless. Taking the determinant of yields

(54)

The two roots and of , as well as the spectral radius , are complicatedfunctions of , and . It is convenient to use the Hurwitz polynomial form, as done later.

A.3 Hurwitz Polynomial

A Hurwitz polynomial of order in the complex variable will becalled stable if all roots of lie in the negative complex plane:

(55)

and if all roots with are simple. To go or vice-versaone may use — subject to the caveats below — the bilinear transformations, alsocalled Mobius transformations:

(56)

These conformally map the open unit disk onto the open left-hand plane; see Figure 16. The unit circle , excluding , maps to the

imaginary axis . To go replace ,multiply through by and simplify. To go replace

, multiply through by and simplify as appropriate.


z-planes-plane

Figure 16. The involutory bilinear mapping , .

The transformation has a glitch: roots ofvanish from since they map to infinity, as illustrated in Example A.3, andare not recovered in the inverse map. If it is important to preserve roots, thescaled variant of (56) given in (59) may be useful, as illustrated in that example.Alternatively reverse Hurwitz polynomials, covered in Section A.4, may be used.

Example A.3. Consider . This has rootsand so and is stable as per (49). Mapping to the plane via

gives

(57)

This has the only root so it is also stable as per (55). Root maps to(unsigned), which disappears on multiplying through by . To go back to an

amplification polynomial, denoted here by , replace :

(58)

Note that the root is recovered, but the other is not. In the extreme case

, with a positive integer, the restored is just the constant . For somequestions, notably investigation of strict stability, it may be desirable that be mappedto a finite negative value. This can be done with a scaled variant of (56), which maps theopen disk to the open left-hand plane :

(59)

Here one may take, say, with a tiny . For example ifis used on one gets, in exact arithmetic,

(60)


The roots of are and , and those of are exactlyrecovered. Either the standard bilinear mapping (56) or the variant (59) should not be usedin inexact (floating point) arithmetic, as both are highly sensitive to cancellations.

This example suggest a way to detect single or multiple roots using the Math-ematica modules presented in Sections A.5 and A.6, by testing polynomial degree drops.Another method is described in Example A.4.

A.4 Reversed Hurwitz Polynomial

Given a of order , its reversed polynomial is obtained byreplacing and multiplying through by . The coefficient list is reversedand each root is replaced by its reciprocal. For example,

with roots becomeswith roots . [That is the image of

.] Stability is not affected by reversal because the left-hand planemaps onto itself: . An amplification root maps to

whereas a root maps to . This device provides another “saveplus-ones” tool should roots be deemed worthier of preservation than roots.

To directly go from to and vice-versa one may use the mappings

(61)

which for become and .

Example A.4. For the stable polynomial of Example A.3, thetransformation (59) with arbitrary giveswith roots and . If , collapses to

. The reversed polynomial is .Setting gives , which has finite roots and . This

may be directly obtained by mapping with .

A.5 Amplification-to-Hurwitz Mapping

Module HurwitzPolynomialList, listed in Figure 17, performs themapping. It is especially suitable for polynomials with symbolic coeffi-

cients that appear in method design. The module is invoked as

b=HurwitzPolynomialList[a,r,norm] (62)

The arguments are:a Coefficients of the amplification polynomial.r Usually 1. A non-unit or symbolic value performs the mapping (59).norm Normalization flag. Set to True to request that the coefficient of the

highest power in of the Hurwitz polynomial be scaled to one.


HurwitzPolynomialList[a_,r_,norm_]:=Module[{PA,PH,b,k,rep, n=Length[a],s,z,i,j,modname="HurwitzPolynomialList"}, If [n<=0, Return[{}]]; rep=z->r*(s+1)/(s-1); If [IsInexact[a]||IsInexact[r], Print[modname, " error: float input"]; Return[Null]]; k=FindLastNonzero[a]; If [k==0, Return[{0}]]; PA=a[[1]]+Sum[a[[i+1]]*z^i,{i,1,k-1}]; PH=Simplify[Expand[(s-1)^(k-1)*(PA/.rep)]]; b=CoefficientList[PH,s]; If [norm, j=FindLastNonzero[b]; If [j>0,b=b/b[[j]] ]]; Return[b]];AmplificationPolynomialList[b_,r_,norm_]:=Module[{PA,PH,a,k,rep, n=Length[b],s,z,i,j,modname="AmplificationPolynomialList"}, If [n<=0, Return[{}]]; rep=s->(z+r)/(z-r); If [IsInexact[b]||IsInexact[r], Print[modname, " error: float input"]; Return[Null]]; k=FindLastNonzero[b]; If [k==0, Return[{0}]]; PH=b[[1]]+Sum[b[[i+1]]*s^i,{i,1,k-1}]; PA=Simplify[Expand[(z-r)^(k-1)*(PH/.rep)]]; a=CoefficientList[PA,z]; If [norm, j=FindLastNonzero[a]; If [j>0, a=a/a[[j]] ]]; Return[a]];

FindLastNonzero[a_]:=Module[{i,n=Length[a]}, For [i=n,i>0,i--, If [a[[i]]==0,Continue[],Return[i], Return[i]]]; Return[0]];IsInexact[expr_]:=Precision[expr]=!=Infinity;IsExact[expr_] :=Precision[expr]===Infinity;

Figure 17. Modules to produce from and vice-versa.

The module returnsb A list of the coefficients of the Hurwitz polynomial. If

has roots that map to , .

Both a and r must be exact expressions. If a floating-point value is detected ineither argument, the module aborts with an error message to that effect.

A.6 Hurwitz-to-Amplification Mapping

Module AmplificationPolynomialList, listed in Figure 17, performs thetransformation. The module is invoked as

a=AmplificationPolynomialList[b,r,norm] (63)

The arguments areb Coefficients of the Hurwitz polynomial.r Usually 1. A non-unit or symbolic value performs the mapping (59).norm Normalization flag. Set to True to request that the coefficient of the

highest power in of the Hurwitz polynomial be scaled to one.

The module returnsa Coefficients of the amplification polynomial.


Both b and r must be exact expressions. If a floating-point value is detected ineither argument, the module aborts with an error message to that effect.

Example A.5. For the quarticthe call b=HurwitzPolynomialList[ 1,1,-3,-1,2 ,r,False]

with arbitrary returns , , ,, in b. A subsequent call to the

module: a=AmplificationPolynomialList[b,r,False] returns , ,, , in a, so we get back but for a factor .

If the last two coefficients: and , vanish reducing to .The degree drops by 2 because has two plus-one roots. To get the reverse Hurwitzpolynomial with , say b=Reverse[b]/.r->1. This yields the cubic

; its degree is 3 since the root maps to and drops out.

B The Routh-Hurwitz Criterion

In this Appendix we consider the stability conditions for the generic Hurwitz poly-nomial of degree ( is used below instead of for brevity):

with real (64)

Note that is assumed. If the whole polynomial should be scaled by, which does not change the zeros of .

The name “Routh-Hurwitz stability” used in the sequel acknowledges the fact that Routh’stable-based criterion [37], presented 18 years before that of Hurwitz [20], is equivalentwhen written in fraction-free form, although the coalescence was not proven until 1911.

B.1 The Hurwitz Determinants

To assess stability as per (55), introduce the Hurwitz determinant sequences

(65)

in which index ranges from 1 through . Coefficient indices along each rowchange by two, whereas indices along each column change by one. The term isset to zero if or . Note that and .

Both determinant forms displayed in (65) appear in the literature. Are theyconnected? If we associate with then is the form associated with the


reversed polynomial , and vice-versa. Since and share the samestability properties as regards roots in the open left-hand -plane, sequencesand deliver the same information. Their values, however, differ for .

Some authors write down transposes of the foregoing matrices, which of coursedoes not change the determinant. To further compound the confusion, issometimes written as instead of (64). The end result ofthese combinatorial gyrations is that the reader may find 8 ways of stating the“Hurwitz determinantal criterion” in the literature.

B.2 The Hurwitz Stability Criterion

The stability criterion was stated by Hurwitz [20] as: A necessary and sufficientcondition that the polynomial (64) have only roots with negative real parts is that

(66)

be all positive. Readable proofs may be found in Henrici [19], Jury [22] or Us-penky [40]. Hurwitz made the following observations. First, on expandingby the last column it is easily shown that . The requirement that

and is equivalent to and . Thus the theoremremains valid if is replaced by . Second, , , etc., vanish identi-cally since all last column entries vanish. The theorem can therefore be restatedas: all non-identically vanishing terms of the infinite sequence must bepositive. These remarks are obviouly applicable to by appropriately revertingindices.

A necessary condition for (64) to be stable is that all coefficients throughbe positive. The proof is quite simple: if the real part of all roots is negative, everylinear factor of is of the form with , and every quadratic factor isof the form with and . Since the product of polynomialswith positive coefficients likewise has positive coefficients, it follows that a stable(64) can have only positive coefficients. Thus finding a negative coefficient in

or is sufficient to flag that polynomial as unstable. The positivitycondition is not sufficient if , however, as the examples below make clear.

For the more general case of a with complex coefficients, see Jury [22].

Example B.1. For the quadratic polynomial , with , theconditions given by (66), with replaced by , are

(67)

These are satisfied if the three coefficients: are positive.For the real-coefficient cubic polynomial , with ,

and replaced by , the conditions are

(68)


RouthHurwitzStability[b_]:=Module[{n=Length[b]-1,i,c}, n=FindLastNonzero[b]-1; If [n<3, Return[b]]; c=Table[0,{n+1}]; c[[1]]=b[[n+1]]; For [i=2,i<=n+1,i++, c[[i]]=HurwitzDeterminant[b,i-1]]; Return[Simplify[c]]];

HurwitzDeterminant[b_,k_]:=Module[{n=Length[b]-1,i,j,m,A}, If [k<1||k>n||n<=0, Return[0]]; If [k==n,Return[b[[1]]]]; If [k==1, Return[b[[n]]]]; A=Table[0,{k},{k}]; For [i=1,i<=k,i++, For [j=1,j<=k,j++, m=i-2*j+n+1; If [m>0&&m<=n+1, A[[i,j]]=b[[m]] ]; ]]; Return[Simplify[Det[A]]]];

Figure 18. Modules that return Routh-Hurwitz stability conditions.

These are satisfied if the coefficients: are positive, and . The condi-tions delivered by the sequence are , , and .

For the real-coefficient quartic polynomial ,with , and replaced by , the conditions are

(69)

The conditions delivered by the sequence are , , ,and .

B.3 Routh-Hurwitz Stability Modules

Figure 18 lists two Mathematica modules that facilitate the production of theHurwitz determinant sequence (66). The modules are HurwitzDeterminant andRouthHurwitzStability, with the latter calling the former.

The determinant module is invoked as

=HurwitzDeterminant[b,k]

=HurwitzDeterminant[Reverse[b],k] (70)

The arguments are:b Coefficients of the Hurwitz polynomial.k Determinant index: .

The module returns either or as function value. Which one is dictated by(70). If or , the module returns or as appropriate. If is outsidethe range 1 through , it returns zero.


Module RouthHurwitzStability, listed in Figure 18, uses the foregoingmodule to return the determinant sequence (66) as a list, with or replacedby or as appropriate. It is invoked as

=RouthHurwitzStability[b]

=RouthHurwitzStability[Reverse[b]] (71)

The only argument isb Coefficients of the Hurwitz polynomial. Care must be

given to insuring the positivity condition for either or , as appropri-ate. In symbolic work this may require careful a priori checking.

The module returns a list of Hurwitz determinants or , as per (71).

Example B.2. Determine the stability of the polynomial used in [19, p. 557] as example:

(72)

The call b= , , , , , , , c=RouthHurwitzStability[b] returnsc= , , , , , , whence is stable. Calling with reversed coefficients:c=RouthHurwitzStability[Reverse[b]] gives c= , , , , , , ,which confirms the result.

Example B.3. Take the last coefficient of the foregoing polynomial as variable:

(73)

For which values of is the polynomial stable? Now b= ,c=RouthHurwitzStability[Reverse[b]] returnsc= , , , , , , . A study of the sign of the lastthree entries shows that the polynomial is only stable for

, which includes the value used in the previous example. Callingc=RouthHurwitzStability[b] returns , , , ,

, , , which is more complicated than the previous one.So often it pays to try both determinantal forms.

Example B.4. Another example from [19, p. 558]. Given the polynomial

(74)

For which values of is stable? The call b= , , , , ,c=RouthHurwitzStability[b] returns , , , , inc. By inspection, is stable if . Since remains invariant under reversal,the same list is returned for Reverse[b].

Example B.5. Complete Example A.2 by investigating the A-stability of the Newmarkintegration scheme (52) for the model oscillator equation . The amplificationpolynomial (54) is quadratic in . Place its three coefficients into the list a. The call


b=HurwitzPolynomialList[a,1,True] returns b= ,,1 , whence . [The coefficient of returns

as one by setting the third argument to True.] Since is quadratic, for A-stability itis necessary and sufficient that its three coefficients be nonnegative for any .By inspection this happens if and .

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On Topological Design Optimization of Structures Against Vibration and Noise Emission

Niels Olhoff 1 and Jianbin Du2

1Department of Mechanical Engineering, Aalborg University, DK-9220, Aalborg East, Denmark

2School of Aerospace, Tsinghua University, 100084, Beijing, China

Abstract This paper presents a brief introduction to topological design optimization, and gives an overview of the application of this novel method to problems of design of linearly elastic continuum-type structures (without damping) against vibration and noise subject to given external excitation. The design objective of such problems is often to drive the structural eigenfrequencies of vibration as far away as possible from an external excitation frequency, or a band of excitation frequencies, in order to avoid resonance phenomena with high vibration and noise levels. This objective may be achieved in different ways, e.g., by (i) maximizing the fundamental or a higher order eigenfrequency of the structure, (ii) maximizing the distance (gap) between two consecutive eigenfrequencies, (iii) maximizing the dynamic stiffness of the structure subject to forced vibration, or by (iv) minimizing the sound power radiated from the structural surface into an acoustic medium. The mathematical formulations of these topology optimization problems and several illustrative numerical results are presented.

1 Introduction The method of topology optimization of continuum structures first appeared in the literature in 1988, and was originally developed for determining the distribution of an elastic material within an admissible design domain that yields the stiffest possible structure for a prescribed weight, see Bendsøe and Kikuchi (1988) and Bendsøe (1989). Since usual sizing and shape optimization methods generally cannot change the structural topology, the development of the method

218 N. Olhoff and J. Du

of topology optimization was a remarkable break-through in the field of optimum design, as the choice of the best topology generally has the most decisive impact on the gain that can be achieved by optimization. Topology optimization is therefore an important preprocessing tool for sizing and shape optimization, see Olhoff et al. (1991). During the last decade, the method has been extended to handle several other design objectives and constraints. Topology optimization has therefore become a standard tool for synthesis of parts or whole structures in the automotive and aerospace industries, and it is rapidly spreading into other mechanical design disciplines. The reader is referred to the exhaustive textbook by Bendsøe and Sigmund (2003), the IUTAM Symposium proceedings edited by Bendsøe et al. (2006), and the review article by Eschenauer and Olhoff (2001) for recent developments and publications.

Passive design against vibrations and noise was first undertaken by Olhoff (1976, 1977) in the form of shape optimization with respect to eigenfrequencies of freely, transversely vibrating beams. By maximizing the fundamental eigenfrequency for given beam volume, optimum cost designs against vibration resonance were obtained subject to all external excitation frequencies within the large range from zero and up to the fundamental eigenfrequency. Optimization with respect to a higher order eigenfrequency was found to produce a large gap between the subject eigenfrequency and the adjacent lower eigenfrequency, and offered even more competitive designs for avoidance of resonance in problems where external exitation frequencies are confined within a large interval with finite lower and upper limits. In subsequent papers by Olhoff and Parbery (1984) and Bendsøe and Olhoff (1985), the design objective was directly formulated as maximization of the separation (gap) between two consecutive eigenfrequencies of the beam.

Topology optimization with respect to eigenfrequencies of structural vibration was first considered by Dias and Kikuchi (1992), who dealt with single frequency design of plane disks. Subsequently, Ma et al. (1994), Dias et al. (1994), and Kosaka and Swan (1999) presented different formulations for simultaneous maximization of several frequencies of free vibration of disk and plate structures, defining the objective function as a scalar weighted function of the eigenfrequencies. In contrast to this, Krog and Olhoff (1999) and Jensen and Pedersen (2005) applied a variable bound formulation (see Bendsøe et al., 1983) which facilitates proper treatment of multiple eigenfrequencies that very often result from the optimization. The former of these papers treats optimization of fundamental and higher order eigenfrequencies of disk and plate structures, while the latter deals with maximization of the separation of adjacent eigenfrequencies for bi-material plates. The paper (Pedersen, 2000) deals with maximum fundamental eigenfrequency design of plates, and includes a technique to avoid spurious localized modes.

Topology optimization with the objective of maximizing the dynamic stiffness (minimizing the dynamic compliance) of structures subjected to time-harmonic external loading of given frequency and amplitude are, e.g., studied by Ma et al.

On Topological Design Optimization of Structures… 219

(1995), Min et al. (1999), and Jog (2002). It should be noted that the separation of adjacent eigenfrequencies as considered

by Jensen and Pedersen (2005) and in the present paper, is closely related to the existence of so-called phononic (or acoustic) band gaps, i.e., gaps in the wave band structure for periodic materials implying that elastic waves cannot propagate in certain frequency ranges. Sigmund (2001) applied topology optimization to maximize phononic band gaps in periodic materials (see also Diaz et al., 2005, and Halkjær et al., 2006). Moreover, Sigmund and Jensen (2003), Jensen (2003), and Jensen and Sigmund (2005) performed minimization of the response of band gap structures (wave damping).

The present paper lends itself on the authors’ recent work (Du and Olhoff 2007a,b, and Olhoff and Du 2005, 2006, 2008), and gives an overview illustrated by numerical examples of problems of optimum topology design of single- or bi-material beam and plate-type continuum structures against vibration and noise subject to given external excitation.

The paper is organized as follows. Chapter 2 gives a brief account of basic concepts of topology optimization of continuum structures and the SIMP (Solid Isotropic Microstructure with Penalty) material models used in this paper. In Chapter 3, structural topology optimization subject to prescribed volume of material is first considered for problems of maximizing the fundamental or a higher order eigenfrequency, and then problems of maximizing the distance (gap) between two consecutive eigenfrequencies are studied. Chapter 3 also discusses the difficulty that eigenfrequencies subject to optimization often become multiple eigenvalues, which are not differentiable in the usual sense. Section 4 presents several solutions to the types of problems considered in Chapter 3. Chapters 5-8 deal with problems of topology optimization of structures subjected to time-harmonic external dynamic loading of given frequency and amplitude. Thus, the problem of maximizing the dynamic stiffness of a structure subjected to forced vibration in vacuum is discussed in Chapter 5, and several solutions are presented in Chapter 6. Subsequently, the problem of minimizing the sound power radiated into an acoustic medium from the surface of a vibrating structure, is studied in Chapter 7 and numerical examples are presented in Chapter 8. Finally, Chapter 9 concludes the paper.

2 Topology Optimization and Material Interpolations

Contrary to shape optimization, problems of topology optimization are defined on a fixed domain of space called the admissible design domain (see, e.g., Bendsøe and Sigmund, 2003, and Eschenauer and Olhoff, 2001). The topology problem is basically one of discrete optimization, but this difficulty is avoided by


introducing relationships between stiffness components and the volumetric density of material � over the admissible design domain.

Fig. 1 illustrates some basic concepts for a topology optimization problem for a continuum structure to be made of a single material. Given are the admissible design domain (indicated by grey in Fig. 1a), the boundary conditions, loading, and the volume of solid, elastic material for the structure. As indicated in Fig. 1,

usually a fixed finite element mesh is em-bedded in the entire admissible design do-main. Typically, the mesh is a uniform, rectangular partition of space, and the material density � is assumed to be constant within each finite element. For the initial design, the given amount of material normally is distributed uniformly over the ad-missible design domain as indicated in Fig. 1a.

To determine the op-timum structural topo-logy, the densities �e ofmaterial in each of the finite elements are used as design variables de-fined between limits 0 (corresponding to void

as shown by white in Fig. 1b) and 1 (corresponding to solid elastic material shown in black). The aim of the optimization process is to find out, for each of the finite elements in the admissible design domain, whether it should contain solid material or not. In this process (of successive iterations), each of the design variables tend to attain one of their limiting values as explained below, thereby forming a design with aggregations of finite elements with solid material and void, respectively, see Fig. 1b. The result is a rough description of outer as well as inner boundaries of the design that represents the overall optimum topology. This topological design may subsequently be used as a basis for refined shape optimization, see Olhoff et al. (1991).

(a)

Void

(b)

Optimum Design

Figure 1. Illustration of a topology design process from the initial (a) to the optimum design (b).

Initial Design

Evenly distributed material

Solid material


2.1 SIMP Model for Topology Optimization of Single-material Structures

As mentioned above, it is the aim of the optimization process to determine the optimum zero(void)-one(solid) distribution of a prescribed amount of the given material over the admissible design domain. To achieve this goal, many different material models have been developed (see, e.g., Bendsøe and Sigmund, 2003, and Eschenauer and Olhoff, 2001), among which the SIMP (Solid Isotropic Microstructure with Penalty) model proposed by Bendsøe (1989), Rozvany and Zhou (1991) and Rozvany et al. (1992) is a simple and effective one which is widely used in optimum topology design. The SIMP model is normally applied together with a filtering technique, see Sigmund (1997), as this prevents checkerboard formation and dependency of optimum topology solutions on finite element mesh-refinement. According to the SIMP model, the finite element elasticity matrix Ee is expressed in terms of the element volumetric material density �e, 0 � �e � 1, in a power p, p � 1, as

*)( ep

eee EE �� (1)

where *eE is the elasticity matrix of a corresponding element with the fully solid

elastic material the structure is to be made of. The power p in (1), which is termed the penalization power, is introduced with a view to yield distinctive “0-1” designs, and is normally assigned values increasing from 1 to 3 during the optimization process. Such values of p have the desired effect of penalizing intermediate densities 0 < �e < 1 since the element material volume is proportional to �e while the interpolation (1) implies that the element stiffness is less than proportional. Note also that the interpolation (1) satisfies eE (0) = 0 and

eE (1) = *eE , implying that if a final design has density 0 and 1 in all elements,

this is a design for which the structural response has been evaluated with a correct physical model.

By analogy with (1), for a vibrating structure the finite element mass matrix may be expressed as

*)( eqeee MM �� (2)

where *eM represents the element mass matrix corresponding to fully solid

material, and the power q � 1. Apart from exceptions briefly discussed in the following section, normally q = 1 is chosen.

The global stiffness matrix K and mass matrix M for the finite element based structural response analyses behind the optimization, can now be calculated by

��

��EE N

ee

qe

N

ee

pe

1

*

1

* , MMKK �� (3)


Here, *eK is the stiffness matrix of a finite element with the fully solid material

for the structure, and NE denotes the total number of finite elements in the admissible design domain.

In the problem formulations in Chapters 3, 5 and 7, Ve, e = 1,…,NE, denotes the volumes of the finite elements, V0 is the total volume of the admissible design domain, and for single material design, V* denotes the total volume

��

EN

eeeV

1� of solid elastic material which is available for the structure.

2.2 Localized Eigenmodes With values assigned to p and q as stated above, application of the SIMP model for problems of topology optimization with respect to eigenfrequencies may lead to the occurrence of spurious, localized eigenmodes associated with very low values of corresponding eigenfrequencies. The localized eigenmodes may occur in sub-regions of the design domain with low values of the material density (e.g. �e � 0.1), where the ratio between the stiffness (with, say, p = 3 in the interpolation formula) and the mass (with q = 1) is very small. To eliminate these spurious eigenmodes, we may use the method of Pedersen (2000) of linearizing the element stiffness or the approach of Tcherniak (2002) of setting the element mass to zero in sub-regions with low material density. Thus, following Tcherniak (2002) with a slight modification to avoid numerical singularity, the interpolation formula (2) for the finite element mass matrix was modified as

��

�

�1.0,1.0,

)( *

*

eere

eeeee ��

��

MM

M . (4)

Here, the mass is set very low via a high value of the penalization power r insub-regions with low material density. Thus, r is chosen to be about r = 6, i.e., much larger than the penalization power p for the stiffness, which is kept unchanged at a value about p = 3.

It is noted that Eq. (4) is discontinuous at the low value �e = 0.1 of the material density. Numerically this is not a serious problem since the discontinuity only occurs at a single point. However, we can always improve (4) by generating a continuous interpolation model for the mass with respect to any value of the material density between 0 and 1. For example, to achieve continuity of the interpolation model, we may introduce the following revised form of Eq. (4),

��

�

�1.0,1.0,

)( *60

*

eee

eeeee c ��

��

MM

M . (4a)


where the coefficient 50 10�c enforces the C0 continuity at the value �e = 0.1 of

the material density. In several of the examples presented later in this paper, for comparison, we have applied each of the interpolation models (4) and (4a) in the numerical solution scheme and only found negligible differences in the final results. The reason is that in both models, the region with lower density has a very small contribution to the first several eigenfrequencies of the structure. Furthermore, all intermediate values of the material density will approach zero or one during the design process, which implies that the change of the interpolation model in regions with lower density as shown in (4a) must have very limited influence on the final zero-one design.

2.3 SIMP Model for Topology Optimization of Bi-material Structures The SIMP model for topology optimization of structures made of two different solid elastic materials can be easily obtained by an extension of the SIMP model for single-material design. Following Bendsøe and Sigmund (1999), the finite element elasticity matrix for the bi-material problem can be expressed as

2*1* )1()( ep

eep

eee EEE �� (5)

where 1*eE and 2*

eE denote the element elasticity matrices corresponding to the two given solid, elastic materials *1 and *2. Here, material *1 is assumed to be the stiffer one. The penalization power p in (5) was assigned the value 3 in this paper which resulted in distinctive optimum topology designs in the examples of bi-material design considered. It follows from (5) that for a given element, �e = 1 implies that the element fully consists of the solid material *1, while �e = 0 means that the element fully consists of the solid material *2.

The element mass matrix of the bi-material model may be stated as the simple linear interpolation

2*1* )1()( eeeeee MMM �� (6)

where 1*eM and 2*

eM are the element mass matrices corresponding to the two different, given solid elastic materials *1 and *2.

The SIMP model formulated by (1) and (2) (or (5) and (6)) may be regarded as an interpolation scheme for the structural stiffness and mass with respect to material volume density. Recently, a generalized material model based on a polynomial interpolation was proposed by Jensen and Pedersen (2005), and it was shown how proper polynomials corresponding to different design objectives can be easily obtained.

When bi-material design is treated via the problem formulations in Chapters


3, 5 and 7, then V* denotes the total volume ��

EN

eeeV

1� of the stiffer material *1

available for the structure, while the total volume of material *2 is given by V0 - V*, where V0 is the volume of the admissible design domain. In figures in Chapters 4, 6 and 8 presenting optimum topologies of bi-material structures, material *1 is shown in black and material *2 in grey.

3 Eigenfrequency Optimization Problems A frequent goal of the design of vibrating structures is to avoid resonance of the structure in a given interval for external excitation frequencies. This can be achieved by, e.g., maximizing the fundamental eigenfrequency, an eigenfrequency of higher order, or the gap between two consecutive eigenfrequencies of given order, subject to a given amount of structural material and prescribed boundary conditions. The mathematical formulations of these topology optimization problems are developed for linearly elastic structures without damping in this chapter, and several illustrative results are presented in Chapter 4, see also the recent paper by Du and Olhoff (2007b).

Methods for optimization of simple (unimodal) eigenvalues/eigenfrequencies in shape and sizing design problems are well established and can be implemented directly in topology optimization. The formulation for topology optimization with respect to a simple, fundamental eigenfrequency is presented in Section 3.1 of this chapter, and the sensitivity analysis of a simple eigenfrequency subject to change of a design variable �e is outlined in Section 3.2. However, particularly in topology optimization it is often found that, although an eigenfrequency is simple during the initial stage of the iterative design procedure, later it may become multiple due to coincidence with one or more of its adjacent eigenfrequencies. In order to capture this behaviour, it is necessary to apply a more general solution procedure that allows for multiplicity of the eigenfrequency because a multiple eigenfrequency does not possess usual differentiability properties.

In Section 3.3 of this chapter, the abovementioned eigenfrequency optimization problems are conveniently formulated by a so-called bound formulation (Bendsøe, Olhoff and Taylor 1983; Taylor and Bendsøe 1984; Olhoff 1989). Section 3.4 then presents design sensitivity results for multiple eigenvalues derived by Seyranian, Lund and Olhoff (1994), and by usage of these results, the problems can be solved efficiently by mathematical programming (see, e.g., Overton, 1988, and Olhoff, 1989) or by the MMA method (Svanberg, 1987). Moreover, the procedure of treating the multiple eigenvalues can be greatly simplified by using the increments of the design variables as unknowns (see Krog and Olhoff, 1999, and Du and Olhoff 2007b).


3.1 Maximization of the Fundamental Eigenfrequency Problems of topology design for maximization of fundamental eigenfrequencies of vibrating elastic structures have, e.g., been considered in the papers (Diaz and Kikuchi, 1992, Ma et al. 1994, 1995, Diaz et al., 1994, Kosaka and Swan, 1999, Krog and Olhoff, 1999, Pedersen, 2000). Assuming that damping can be neglected, such a design problem can be formulated as a max-min problem as follows,

.,,1,10

,,0

,,,1,,,

,,,1,:

}}{,1

min{max

0**

1

2

2

,,1

Ee

N

eee

jkkj

jjj

jN

Ne

VVVV

Jjkkj

Jj�toSubject

Jj

E

E

�

�

�

��

��

��

� �

��

�

��

��

��

�

��

M��

M�K�T

(7a)

(7b)

(7c)

(7d)

(7e)

Here �j is the jth eigenfrequency and �j the corresponding eigenvector, and K and M are the symmetric and positive definite stiffness and mass matrices of the finite element based, generalized structural eigenvalue problem in the constraint (7b). The J candidate eigenfrequencies considered will all be real and can be numbered such that

,0 21 J�� (8)

and it will be assumed that the corresponding eigenvectors are M-orthonormalized, cf. (7c) where �jk is Kronecker’s delta. In problem (7a-e), the symbol NE denotes the total number of finite elements in the admissible design domain. The design variables �e, e = 1,…,NE, represent the volumetric material densities of the finite elements, and (7e) specify lower and upper limits � and 1

for �e. To avoid singularity of the stiffness matrix, � is not zero, but taken to be

a small positive value like � = 10-3. In (7d), the symbol � defines the volume

fraction 0* /VV , where V0 is the volume of the admissible design domain, and V *

the given available volume of solid material and of solid material *1, respectively, for a single-material and a bi-material design problem, cf. Sections 2.1 and 2.3.


3.2 Sensitivity Analysis of a Simple Eigenfrequency

If the jth eigenfrequency �j is simple (also called unimodal or distinct), i.e., 11 � �� jjj �� , then the corresponding eigenvector j� will be unique (up to a

sign) and differentiable with respect to the design variables �e, e = 1,…,NE. To determine the sensitivity (derivative)

ej �� )( � of the eigenvalue 2jj �� with

respect to a particular design variable �e, we differentiate the vibration equation (7b) with respect to �e, and get

Ejjjjj Neeeee

,,1,))(()()( �� 0�MMK�MK �� (9)

where ee�� )()( . Pre-multiplying (9) by T� j and using the vibration

equation (7b) and the normalization of j� included in (7c) then gives (see also Wittrick, 1962, Lancaster, 1964, or Haftka, 1990),

Ejjjj Neeee

,,1,)()( �� MK�T�� (10)

The derivatives of the matrices K and M can be calculated explicitly from the material models in Section 2. Considering, e.g., the single-material model in (3), the sensitivity of the eigenvalue 2

jj �� with respect to the design variable e�becomes

Ejeq

ejep

ejj Neqpe

,,1,)()( *)1(*)1( �� MK�T �� (11)

The optimality condition for the maximization of a unimodal eigenvalue 2jj �� of given order j, j = 1, 2, …, now follows from (10) (or (11)) and usage

of the Lagrange multiplier method, and takes the form Eej NeV

e,,1,0)( 0 ��

(12)

where �0 ( 0) is the Lagrange multiplier corresponding to the material volume constraint, and the side constraints for �e have been ignored. With this sensitivity result and optimality condition, the design problem (7a-e) may be solved for a unimodal optimum eigenfrequency by using an OC (Optimality Criterion) based method, e.g., the fixed point method (see Cheng and Olhoff, 1982), or a mathematical programming method, e.g., MMA (Svanberg, 1987).

We may also wish to apply a gradient based method of solution. It is then essential that the jth eigenvalue 2

jj �� is simple and differentiable, and hereby admits linearization with respect to the design variables �e, e = 1, …, NE. Hence, if all the design variables are changed simultaneously, the linear increment j��

of 2jj �� is given by the scalar product


��Tjj �� (13)

where � �T��EN�� ,,1 � is the vector of changes of the design variables, and

� �TTT �MK��MK� jjjjjjj ENEN)(,,)(

11 �� (14)

is the vector of sensitivities (or gradients) of the eigenvalue j� with respect to the design variables �e, e = 1, …, NE.

3.3 Bound Formulations for Maximization of the nth Eigenfrequency or the Distance Between two Consecutive Eigenfrequencies

In this section, we first consider the more general problem of maximizing the nth eigenfrequency n� of given order of a vibrating structure, i.e., the fundamental eigenfrequency (n = 1) or a higher order eigenfrequency (n > 1). Employing a bound formulation (Bendsøe et al., 1983, Taylor and Bendsøe, 1984, and Olhoff, 1989) involving a scalar variable � which plays both the role of an objective function to be maximized and at the same time a variable lower bound for the nthand higher order eigenfrequencies (counted with possible multiplicity), the above problem can be formulated as

,,,1,,0

:

}{,

max

2

,,1

JnnjtoSubject

j

NE

�

�

��

��

Constraints: 7(b-e)

(15a)

(15b) (15c)

Here, as well as in Eqs. (16) below, J is assumed to be larger than the highest order of an eigenfrequency to be considered a candidate to exchange its order with the nth eigenfrequency or to coalesce with this eigenfrequency during the design process.

The problem of maximizing the distance (gap) between two consecutive eigenfrequencies of given orders n and n – 1 (where n > 1) may be written in the following extended bound formulation, where two bound parameters are used (see also Bendsøe and Olhoff, 1985, and Jensen and Pedersen, 2005):


,1,,1,0

,,,1,,0

:

}{,,

max

12

22

12,,121

��

��

nj

JnnjtoSubject

j

j

NE

�

�

�

��

��

��

)16()16(

)16(

cb

a

Constraints: 7(b-e). (16d)

Note that if in (16) we remove the bound variable �1 and the corresponding set of constraints (16c) from the formulation, then the eigenfrequency gap maximization problem (16) reduces to the nth eigenfrequency maximization problem (15), and in particular, for n = 1, to the problem of maximizing the fundamental eigenfrequency in (7).

In problem (15) the eigenfrequency n� , and in problem (16) both the eigenfrequencies n� and 1n� of the optimum solution may very well be multiple, and the bound formulations in (15) and (16) are tailored to facilitate handling of such difficulties.

It is also worth noting that the introduction of the scalar bound variables �in (15) and 1� and 2� in (16) implies that even if multiple eigenfrequencies are present, the optimization problems (15) and (16) are both differentiable if they are considered as problems in all variables, i.e. the bound parameter(s) � (or

21,�� ), design variables Ee Ne ,,1, �� , as well as the eigenfrequencies j�and eigenvectors j� , Jj ,,1�� , (implying that all these variables should have been included under the ‘max’ signs in (15a) and (16a)). This type of problem is referred to as one of “Simultaneous analysis and design” (SAND), and is a very large problem in the present context. Therefore, we refrain from solving the current topology optimization problems in this form in our paper.

In the form written above, where only the design variables Ee Ne ,,1, �� ,and the bound parameters � and 21,�� are included under the ‘max’ signs in (15a) and (16a), the topology optimization problems (15) and (16) are non-differentiable because the eigenfrequencies Jjj ,,1, �� , are considered as functions of the design variables Ee Ne ,,1, �� . This is a ‘nested’ formulation which provides the basis for numerical solution by a scheme of successive iterations where, in each iteration, the eigenfrequencies j� and eigenvectors j� ,

Jj ,,1�� , are established for known design, Ee Ne ,,1, �� , by solution of the generalized eigenvalue problem (7b) and implementation of the orthonormality conditions (7c).


To accommodate for occurrence of multiple eigenfrequencies, we in the subsequent Section 3.4 consider some important sensitivity results for such eigenfrequencies. In Section 3.5, we make use of these results in the development of incremental forms of problems (15) and (16) which provide the basis for construction of a highly efficient scheme for numerical solution of the topology optimization problems under study.

3.4 Sensitivity Analysis of Multiple Eigenfrequencies Multiple eigenfrequencies may manifest themselves in different ways in structural optimization problems. One possibility is that an eigenfrequency subject to optimization is multiple from the beginning of the design process, e.g., due to structural symmetry, but an originally unimodal eigenfrequency may also become multiple during the optimization process due to coalescence with one or more of its adjacent eigenfrequencies. In this case, sensitivities of the multiple eigenfrequency cannot be calculated straightforwardly from (10) (or (11)) due to lack of usual differentiability properties of the sub-space spanned by the eigenvectors associated with the multiple eigenfrequency. Investigations of sensitivity analysis of multiple eigenvalues (like eigenfrequencies or buckling loads) are available in many papers (see, e.g., Bratus and Seyranian, 1983, Masur, 1984, 1985, Haug et al., 1986, Seyranian, 1987, Overton, 1988, Seyranian et al., 1994, and papers cited therein).

Following Seyranian et al. (1994), let us assume that the solution of the generalized eigenvalue problem (7b) included in problems (15) or (16) yields a N-fold multiple eigenvalue �

~ ,1,,,~ 2 �� Nnnjjj �� (17)

associated with the N (N > 1) lowest eigenfrequencies j� appearing in the bound constraints (15b) and (16b)*1. Here we shall assume JNn �� 1 , i.e., that the total number J of eigenfrequencies (counted with multiplicity), that is considered in problems (15) and (16) is chosen such that the Jth eigenfrequency J� is larger

than the multiple eigenfrequency corresponding to �~ in (17). The multiplicity of the eigenvalue �~ in (17) implies that any linear combination of the eigenvectors

*1 Similarly, the eigenvalue problem (7b) contained in problem (16) may yield another R-fold eigenvalue 1,,,ˆ 2 �� nRnjjj �� , which corresponds to the R largest eigenfrequencies j� in (16c). This case (for which we assume that

Rn �1 ), is completely analogous to (17).


1,,, �� Nnnjj �� , corresponding to �~ will satisfy the generalized

eigenvalue problem (7b) in (15) and (16), which implies that the eigenvectors are not unique.

In Seyranian et al. (1994) the sensitivity analysis is based on a mathematical perturbation analysis of the multiple eigenvalue and the correspondingeigenvectors. This analysis involves directional derivatives in the design space and leads to the result that the increments j�� of a multiple eigenvalue

1,,,~ 2 �� Nnnjjj �� , as in (17) are eigenvalues of a N-dimensional algebraic sub-eigenvalue problem of the form

� � 0�det � �� sksk ��f T , s, k = n, …, n+N-1, (18)

where �sk is Kronecker’s delta, and fsk denote generalized gradient vectors of the form

� � ,~,,~11

TTT )�MK(�)�MK(�f kskssk ENEN �� s, k = n, …, n+N-1. (19)

According to the definition in (19), each fsk is a NE-dimensional vector, which means that ��f T

sk in (18) is a scalar product. The label ‘generalized gradient vector’ for fsk becomes apparent when comparing (19) with the expression for the gradient vector j�� of a simple eigenvalue j� in (14). Note also that fsk = fks

due to the symmetry of the matrices K and M, and that the two subscripts s and krefer to the orthonormalized eigenmodes from which fsk is calculated.

Assuming that we know the multiple eigenvalue �~ , the associated sub-set of

orthonormalized eigenmodes, and have computed the derivatives of the matrices K and M, we can construct the generalized gradient vectors fsk, s, k = n, …, n+N-1, from (19). Solving the algebraic sub-eigenvalue problem in (18) for �� then yields the increments 1,,, �� Nnnjj �� , of the multiple eigenvalue

�~ subject to a given vector � �

EN�� ,,1 �� of increments of the design variables.

The N increments ,1,,, �� Nnnjj �� constitute the eigenvalues of the sub-eigenvalue problem (18), and represent the directional derivatives of the multiple eigenvalue 1,,,~ 2 �� Nnnjjj �� , with respect to change

e�� of the design variables Ee Ne ,,1, �� . Attention should be drawn to the fact that the increments 1,,, �� Nnnjj �� of the multiple eigenvalue are generally non-linear functions of the direction of the design increment vector �� .Thus, unlike simple eigenvalues, multiple eigenvalues do not admit a usual


linearization in terms of the design variables. Finally, two important special cases should be observed.

Case of simple eigenfrequency. As is to be expected, for N = 1, i.e., j = s = k = n, (17) and (18) reduce to the case of a simple eigenvalue 2

nn �� . Eq. (18) reduces to the simple equation

0� � nnn ��f T (20)

where, according to (13), (19) and (14), we have

nnn ��f (21)

i.e., nnf is simply the vector of sensitivities of the unimodal eigenvalue n� with respect to the design variables �e, e = 1, …, NE, cf. (10) and (14).

Case of vanishing off-diagonal terms. For the case of multiple eigenvalues, cf. (17) with N > 1, a very important observation can be made. If in (18) all off-diagonal scalar products are zero, i.e. if

,1,,,,,0 �� Nnnkskssk ��f T (22)

then the increment j�� of an eigenvalue 2jj �� becomes determined as

,1,,, �� Nnnjjjj ��f T� (23)

where according to (17) and (19)

� � .1,,,,,11

�� Nnnjjjjjjjjj ENEN��

TTT )�MK(�)�MK(�f �� (24)

Hence, if the design increment vector �� fulfils (22), then jjf has precisely the same form as the gradient vector j�� in (14) for a simple eigenvalue, and the eigenvalue increments j�� in (23) are uniquely determined on the basis of the eigenmodes j� , 1,, �� Nnnj � . The formulas for design sensitivity analysis of multiple eigenvalues then become precisely the same as those for simple eigenvalues.

3.5 Computational Procedure The topology optimization problems (15) and (16) can be efficiently solved by an iterative procedure indicated in Fig. 2, which can be used for solution of problems with multiple as well as simple eigenfrequencies.


Figure 2. Flow chart of iterative solution procedure.

0. Problem initialization. Define value of n and

initialize design variables e�

1. Solution of the generalized eigenvalue problem (7b,c) for eigenfrequencies and -modes by FE-analysis. Detect possible multiplicity N of n� (and R of 1n� )

2. Computation of generalized gradients fsk, if N>1 (and R>1) or usual gradients if N=1 (and R=1)

3. Iterative solution of optimization sub-problem (25) (or (26)) for increments e�� of the design variables

4. Update values of the design variables

eee �� : .

e� converged ?

i.e., �� ?

Stop

Yes

NoIncrements e�� converged ?

No

Yes

Inner loop Main loop


The procedure is based on the results of the sensitivity analysis in the preceding section, and is seen to consist of a main (outer) loop and an inner loop. While steps 1, 2 and 4 of the main loop are pretty straight-forward, the third step (the inner loop) needs to be briefly discussed. (The interested reader is referred to Du and Olhoff (2007b) for more details about the iterative procedure.)

The purpose of the third step (the inner loop) in Fig. 2 is to determine optimum values of the increments Ee Ne ,,1, �� , of the design variables, subject to known values of iterates that have been determined in steps 1 and 2 and are fixed in the third step. To enable this, we rewrite the bound formulations (15) and (16) in terms of the vector �� of increments Ee Ne ,,1, �� , of the design variables and corresponding increments of the squared eigenfrequencies

1,,),( 2 �� Nnnjjj �� , (and 1,, � nRnj � , for problem (16)). Hereby, we obtain the following sets of sub-problems to be solved for optimum increments in the third step of the main loop of the computational procedure for

(a) Maximization of the nth eigenfrequency:

� ��

,,,1,10

,,0)(

,1...,,,,0)(�det

,1...,,,0)(

,for,0

:

}{,

max

0**

1

2

22

2

,,1

Eee

N

eeee

sksk

jj

jjj

N

Ne

VVVV

Nnnks

Nnnj

NnJj

toSubject

E

E

�

�

��

��

��

��

��

��

��

��

��

��

��

��

��

��f

��f

T

T

(25a)

(25b)

(25c)

(25d)

(25e)

(25f)

(b) Maximization of the gap (distance) between the nth and (n-1)st eigenfrequencies:


� ��

� ��

� ��

.,,1,10

,,0)(

,1...,,,,0)(�det

,1...,,,,0)(�det

)2if(,1for,0

)1(,1...,,,0)(

,1...,,,0)(

,for,0

:

}{,,max

0**

1

2

2

12

122

222

22

12,,121

Eee

N

eeee

sksk

sksk

jjj

jj

jj

jjj

N

Ne

VVVV

nRnks

Nnnks

nRRnj

nRnRnj

Nnnj

NnJj

toSubject

E

E

�

�

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��

��f

��f

��f

��f

T

T

T

T

(26a)

(26b)

(26c)

(26d)

(26e)

(26f)

(26g)

(26h)

(26i)

Note that in the sub-optimization problems (25) and (26), the only unknowns are the bound variables � and 1� , 2� and the increments of the design variables Ee Ne ,,1, �� , which play the role as independent variables. The

dependent variables are the increments 1,,),( 2 �� Nnnjj �� , of the N-fold

eigenfrequency 21

2�� Nnn �� (in problems (25) and (26)), together with the

increments 1,,),( 2 �� nRnjj �� , of the R-fold eigenfrequency 2

12

�� nRn �� (in problem (26)). All other iterates in (25) and (26), i.e. the material valume densities e� , the eigenfrequencies j� , the generalized gradient vectors skf and the multiplicities N and R have been determined in step 1 and 2 of the main iteration loop, and are kept fixed in the current step 3 of this loop.

Problems (25) and (26) can be solved using the MMA method (Svanberg 1987) or a linear programming algorithm.

Finally, it is interesting to note that if we introduce the additional constraints 0��f T

sk , for ks � , s, k = n, …, n+N-1, i.e. force the off-diagonal terms in (18) to vanish, then the increments j�� are determined in a linearized form with respect to the increments e�� of the material volume densities for both simple and multiple eigenvalues, and as a result, the sub-optimization problems (25) and (26) both reduce to linear programming problems (see Krog and Olhoff, 1999).


4 Numerical Examples of Eigenfrequency Optimization

4.1 Maximization of the Fundamental Eigenfrequency of Beam-like 2D Structures

(a)

(b)

(c)Figure 3(a-c). Admissible design domains (a = 8, b = 1) of beam-like 2D structures with

three different sets of boundary conditions. (a) Simply supported ends. (b) One end clamped, the other simply supported. (c) Clamped ends. The fundamental

eigenfrequencies of the 3 initial designs (uniform distribution of material with density � = 0.5) are all unimodal with values 7.680

1 �a� , 1.10401 �b� and 1.1460

1 �c� .

As a first example, we consider the topology optimization of a single-material beam-like structure modeled by 2D plane stress elements. The admissible design domain is specified, and three different cases (a), (b) and (c) of boundary conditions as shown in Fig. 3 and defined in the caption, are considered. The design objective is to maximize the fundamental eigenfrequency for a prescribed material volume fraction � = 50%, and in the initial design the available material is uniformly distributed over the admissible design domain. The material is isotropic with Young’s modulus E = 107, Poisson’s ratio � = 0.3 and mass density �m = 1 (SI units are used throughout).

The fundamental eigenfrequencies of the initial designs with the three cases

Admissible design domain

a



b


(a), (b) and (c) of boundary conditions are given in the caption of Fig. 3. The optimized topologies are shown in Figs. 4(a-c), and the corresponding optimum fundamental eigenfrequencies are all found to be bimodal with values given in the caption of the figure. Fig 5 shows the iteration history for the first 3 eigenfrequencies of the optimum bimodal design with simply supported ends in Fig. 4(a). The iteration histories for the optimum designs with the two other cases of boundary conditions in Figs. 4(b,c) are qualitatively similar. Figs. 6(a-c) depict the first 3 eigenmodes of the optimized beam-like structure with simply supported ends in Fig. 4(a), and the results show that the first 2 eigenmodes (corresponding to the bimodal fundamental eigenfrequency) of the structure are typical simply supported beam-type vibration modes, while the 3rd one is a more general 2D vibration mode.

(a)

(b)

(c)

Figure 4(a-c). Optimized single-material topologies (50% volume fraction) for the three different sets of boundary conditions defined in Figs. 3(a-c). The optimum fundamental eigenfrequencies are all found to be bimodal and have the values (a) 7.1741 �opt

a� , (b)

7.2881 �optb� , and (c) 4.4561 �opt

c� , implying that they are increased by (a) 154%, (b) 177% and (c) 212% relative to the initial designs.


0 20 40 60 800

100

200

300

400

500

600

Iteration number

Eig

en

fre

qu

en

cie

s

� 1

� 2

� 3

(Maximized)

Figure 5. Iteration history of the first 3 eigenfrequencies associated with the design process leading to the optimum simply supported beam-like structure in Fig. 4(a). It is

seen that the fundamental eigenfrequency is simple for the initial design, but soon coalesces with the second eigenfrequency, and the maximum fundamental eigenfrequency

is bimodal.

(a) 7.1741 �opta�

(b) 7.17412 �� optaa ��

(c) 9.2843 �a�Figure 6(a-c). The three first eigenmodes of the simply supported beam-like structure in Fig. 4(a) with a bimodal optimum fundamental eigenfrequency. (a) and (b) depict the two

modes associated with the optimum fundamental eigenfrequency, and (c) shows the subsequent mode.


4.2 Maximization of the Second Eigenfrequency of Beam-like 2D Structures

We now present an example of topology optimization of single material beam-like structures for maximum value of the second eigenfrequency. The initial data and the three sets of boundary conditions in this example are the same as for the first example in Section 4.1. The resulting topologies are shown in Figs. 7(a-c).

(a)

(b)

(c)

Figure 7(a-c). Optimized single-material topologies (50% volume fraction) for the three different sets of boundary conditions in Figs. 3(a-c). The optimum second

eigenfrequencies are found to be (a) 3.5982 �opta� , (b) 8.7322 �opt

b� , and (c)

0.8492 �optc� , and are all bimodal.

4.3 Maximization of the Distance (Gap) Between Two Consequtive Eigenfrequencies of Beam-like 2D Structure

In this example, we consider the design objective of maximizing the distance (gap) between two consecutive eigenfrequencies (the 2nd and the 3rd eigenfrequencies) of the clamped beam-like structure in Fig. 8(a). A concentrated mass mc is attached at the mid-point of the lower edge of the beam-like structure


as shown in Fig. 8(a), which has the value mc = 1/2mb (Here mb is the total mass of the initial design). We use the same admissible design domain, materials and volume fractions as in the previous example (see Fig. 3(c)). The optimum topology and the corresponding iteration histories of the eigenfrequencies are given in Figs. 8(b) and 8(c). It can be seen that the 2nd eigenfrequency is decreased and the 3rd eigenfrequency is increased. As a result, the design ends up with a maximized gap between the 2nd and the 3rd eigenfrequencies that is equal to 810, which is 548 % higher than the difference between the corresponding eigenfrequencies of the initial design. Note that in Fig. 8(d) the 3rd, 4th and 5th eigenfrequencies form a tri-modal eigenfrequency of the final optimum design.

(a)

(b)

0 20 40 60 80 1000

200

400

600

800

1000

1200

Iteration number

Eig

en

fre

qu

en

cie

s

(Maximized) gap:

� 1

� 2

� 3� 4

� 5

� � 3 2

(c)Figure 8 (a). A clamped beam-like 2D structure with a concentrated mass attached at the mid-point of the lower edge. (b) Optimized topology of the beam-like structure. The gap between the 2nd and the 3rd eigenfrequencies is maximized. (c) Iteration history for the first five eigenfrequencies associated with the process leading to the optimized topology (b). Notice that the 3rd, 4th and 5th eigenvalues have coalesced to a tri-modal eigenvalue

for the optimized topology.


4.4 Maximization of the Fundamental Eigenfrequency of Single-material Plate Structures

In this example, we consider the topology optimization of a single-material plate-like structure modeled by 8-node 3D brick elements with Wilson incompatible displacement models to improve precision. The admissible design domain is specified, and three different cases (a), (b) and (c) of boundary conditions and attached concentrated, nonstructural masses as shown in Fig. 9 and defined in the caption, are considered. The design objective is to maximize the fundamental eigenfrequency for a prescribed material volume fraction � = 50%, and in the initial design the available material is uniformly distributed over the admissible design domain. The material is isotropic with Young’s modulus E = 1011,Poisson’s ratio � = 0.3 and mass density �m = 7800 (SI units are used throughout). The fundamental eigenfrequencies of the initial designs with the three cases (a), (b) and (c) of boundary/mass conditions are given in the caption of Fig. 9. The optimized plate topologies are shown in Figs. 10(a-c), and the corresponding optimum fundamental eigenfrequencies are all unimodal with values given in the caption of Fig. 10.

(a) (b) (c)

Figure 9. Plate-like 3D structure (a=20, b= 20 and t=1) with three different cases of boundary conditions and attachment of a concentrated nonstructural mass. (a) Simple

supports at four corners and concentrated mass mc at the center of the structure ( 3/0mmc � , m0 the total structural mass of the plate). (b) Four edges clamped and

concentrated mass mc at the center ( 10/0mmc � ). (c) One edge clamped, other edges free, and concentrated mass mc attached at the mid-point of the edge opposite to the

clamped one ( 10/0mmc � ). The first eigenfrequencies for the 3 initial designs (uniform

distribution of material with density � = 0.5) are 13.801 �a� , 07.310

1 �b� , 46.301 �c� .


(a) (b) (c)Figure 10(a-c). Optimized single-material topologies (50% volume fraction) for the three different cases of boundary conditions and mass attachment in Figs. 9(a-c). The optimum fundamental eigenfrequencies are found to be (a) 38.161 �opt

a� , (b) 42.651 �optb� , and

(c) 66.91 �optc� , implying that they are increased by (a) 101%, (b) 111% and (c) 179%

relative to the initial designs.

As a second example, single-material topology optimization of an initially quadratic plate-like structure with simple supports at its four corners and center is considered (Fig. 11(a)). The admissible design domain and the material are the same as in the foregoing example. Due to the structural symmetry, the fundamental eigenfrequency of the initial design is bimodal with modes shown in Figs. 11(b-c).

(a) (b) (c)Figure 11. Plate-like 3D structure (a=20, b= 20 and t=1) with simple supports at its four corners and center. (a) Admissible design domain. (b-c) The eigenmodes of the initial design associated with the bimodal fundamental eigenfrequency 56.240

201 �� .

The optimized topology is shown in Fig. 12(a) (50% volume fraction), and the corresponding optimum fundamental eigenfrequency is also bimodal.


0 10 20 30 40 5020

30

40

50

60

70

80

90

Iteration number E

ige

nfr

eq

ue

ncy

Multiple eigenfrequency

1

2

��

�3

(Maximized)

(a) (b)Figure 12. (a) Optimized topology (50% volume fraction, single-material design) associated with the maximum fundmental eigenfrequency 32.601 �opt� , which is

bimodal. (b) Iteration history for the first three eigenfrequencies.

4.5 Maximization of Higher Order Eigenfrequencies of Single- and Bi-material Plate Structures

Here we first present an example of topology optimization of single-material plate-like structures with respect to the second eigenfrequency. The initial data for the example are the same as for the first example in Section 4.4. Thus, we choose the same volume and type of available material, the same admissible design domain, and again consider the three different cases (a), (b) and (c) of boundary conditions and attached concentrated masses as shown in Fig. 9, but we now maximize the second eigenfrequencies. The resulting optimum topologies and the frequency iteration histories for the three cases of boundary conditions and mass attachment in Fig. 9 are given in Figs. 13 and 14.

(a) (b) (c)Figure 13 (a-c). Optimized single-material topologies (50% volume fraction)

corresponding to the three different cases of boundary conditions and mass attachment in Figs. 9(a-c). The values and multiplicities of the optimum second eigenfrequencies are:

(a) 03.462 �opta� (trimodal), (b) 43.1552 �opt

b� (bimodal), (c) 77.392 �optc� (bimodal).


0 20 40 60 800

10

20

30

40

50

60

Iteration number

Eig

en

fre

que

nci

es

� 1

� 2� 3

(Maximized)

� 4

5�

0 20 40 60 800

50

100

150

200

250

Iteration number

Eig

en

fre

qu

en

cie

s� 1

� 2

� 3

(Maximized)

� 4

0 20 40 60 800

20

40

60

80

100

120

Iteration number

Eig

en

fre

qu

en

cie

s

� 1

� 2

� 3

(Maximized)

� 4

5�

(a) (b) (c)Figure 14. Iteration histories of eigenfrequencies associated with the design process

leading to the results in Figs. 13(a-c). For case (c) it is seen that the second eigenfrequency is simple for the initial design, but soon coalesces with the third

eigenfrequency.

In the next example, we consider topology optimization of bi-material structures with respect to higher order eigenfrequencies. Both of the two materials are isotropic. The stiffer material *1 with elasticity and mass matrices

1*1* , ee ME , see Section 2.3, is indicated by black in Fig. 15, and is the same as that used for optimization with a single-material in the preceding examples. The weaker material *2 is indicated by grey in Fig. 15, and has the properties

1*2* 1.0 ee EE � and 1*2* 1.0 ee MM � . We take the volume fraction of material *1 to be 50%, and present results of optimizing the topologies of a bi-material quadratic plate with the same boundary conditions and attachment of a concentrated mass as shown in Fig. 9(b). Figs. 15(a-c) present the optimized plate topologies associated with maximum values of the 4th, 5th and 6th eigenfrequencies.

(a) (b) (c)

Figure 15. Optimized topologies of bi-material plate with all edges clamped and a concentrated mass attached to the center, cf. Fig. 9(b). The topologies correspond to

maximum values of the (a) 4th, (b) 5th and (c) 6th eigenfrequency. The stiffer and the weaker material are indicated by black and grey, respectively, and the volume fraction of

the stiffer material *1 is 50%.


4.6 Maximization of the Distance (Gap) Between Two Consequtive Eigenfrequencies of Bi-material Plate Structures

This example also concerns topology optimization of bi-material plate structures, and we use the same materials and volume fractions as in the previous example. The design objective considered is to maximize the distance (gap) between the 2nd and 3rd eigenfrequencies of the structure. We use the same admissible design domain as in Fig. 9 for the plate structure, and choose the cases (a) and (c) of boundary conditions and concentrated mass attachment as shown in Fig. 9. The results are given in Fig. 16.

0 10 20 30 40 5010

20

30

40

50

60

70

80

Iteration number

Eig

en

fre

qu

en

cie

s

� 1

� 2

� 3� 4� 5

(Maximized) Gap: � � 3 2

(a) (b) (c)

Figure 16. Optimized topology of the plate-like structure with simple supports at four corners and a concentrated mass at the center, cf. Fig. 9(a). The gap between the 2nd and

the 3rd eigenfrequencies is maximized. (b) Iteration histories for the first five eigenfrequencies associated with the process leading to the design (a). It shows that the

second and the third eigenfrequencies form a double eigenfrequency for the initial design, but that they split as the design process proceeds, and the 3rd and the 4th eigenfrequencies end up being a double eigenfrequency of the final design. (c) Optimized topology of the

plate-like structure with the upper horizontal edge clamped, other edges free, and a concentrated mass attached at the mid-point of the lower horizontal edge, cf. Fig. 9(c).

The gap between the 2nd and the 3rd eigenfrequencies is maximized.

5 Minimization of the Dynamic Compliance of Structures Subjected to Forced Vibration

This and the subsequent Chapter 6 deal with the problem of topological design optimization of elastic, continuum structures without damping that are subjected to time-harmonic, dynamic loading with prescribed frequency and amplitude, and is based on a recent paper by Olhoff and Du (2008). An important objective of such a design problem is often to drive the eigenfrequencies of the structure as


far away as possible from the prescribed loading frequency in order to avoid resonance and reduce the vibration level of the structure. This objective is implemented by minimizing the dynamic compliance of the structure subject to the given loading frequency, using the volumetric densities of material in the finite elements in the admissible design domain as design variables. The total structural volume, the boundary conditions, and the material are given.

Topology optimization for minimum dynamic compliance is equivalent to maximizing the dynamic stiffness of structures subjected to time-harmonic external loading of given frequency and amplitude, and have, e.g., been studied by Ma et al. (1995), Jog (2002), and Olhoff and Du (2008). Topology design subject to transient external loading was studied by Min et al. (1999), where the dynamic compliance is defined relative to a specified time interval. Another related problem discussed by Calvel and Mongeau (2005) concerns topology optimization of continuum structures subject to dynamic constraints (e.g. the amplitude of displacement response) where a range of forcing frequencies is considered.

A problem that may arise in structural topology optimization under time-harmonic dynamic loading is that the static compliance (corresponding to the same loading amplitude, but zero frequency) may increase to a very high level (see e.g. Tcherniak, 2002) during the process where the dynamic compliance of the structure is optimized. In extreme cases the static compliance actually tends to infinity, see Olhoff and Du (2008), which reflects that a disintegration of the structure is being created during the design process. However, it was demonstrated by Olhoff and Du (2008) that the design objective of the dynamic problem can be implemented along different optimization paths, and that it is possible to avoid the problem mentioned above by selection of the proper path. Thus, in the present chapter, an approach is presented in which the static compliance of the structure is constrained or decreased during the process of optimizing the dynamic properties. An algorithm developed for this handles the optimum design problem by a continuation technique where the loading frequency is sequentially increased (or decreased) from a sufficiently low (or high) initial value up to (or down to) its prescribed value. This approach can be applied to both low and high frequency loading cases. Numerical examples are presented in Chapter 6 to demonstrate the validity of the approach.

5.1 Formulation of the Problem of Minimizing the Dynamic Compliance

The problem of optimizing the topology of a continuum structure (without damping) for minimum value of the integral dynamic structural compliance can be formulated in a discrete form as follows:


.),,1(,10

,)*(,0*

,)(

,||

:

}{min

01

2

2

1,

Ee

N

eee

p

d

dN,,e

Ne

VVVV

C

toSubject

CF

E

Ee

�

�

��

��

�

�

�

�

�

��

��

�

�

PUMK

UPT

(27)

In (27), the symbol Cd stands for the dynamic compliance defined as || UPT�dC . Here, P denotes the vector of amplitudes of a given external time-

harmonic mechanical surface loading vector ti pet �Pp �)( with the prescribed excitation frequency p� , and U represents the vector of magnitudes of the

corresponding structural displacement response vector ti pet �Ua �)( . Thus, Uand P satisfy the dynamic equilibrium equation included in (27) for the steady-state vibration at the prescribed frequency p� , with K and M representing the Ndimensional global structural stiffness and mass matrices, where N is the number of DOFs. We note that the above expression for the dynamic compliance Cdrepresents the numerical mean value of the magnitudes of the surface displacements weighted by the values of the amplitudes of the corresponding time-harmonic surface loading. For the case of static loading ( p� = 0), the expression directly reduces to the traditional definition of static compliance, i.e., the work done by the external forces against corresponding displacements at equilibrium.

In (27), NE denotes the total number of finite elements in the admissible design domain for the topology optimization problem. The symbols �e, e = 1,…,NE, play the role of design variables of the problem and represent the volumetric material densities of the finite elements, with lower and upper limits � and 1 specified for �e. To avoid singularity of the stiffness matrix, � is not

zero, but taken to be a small positive value like � = 10-3. In the second but last

constraint in (27), the symbol � defines the volume fraction 0* /VV , where V0 is

the volume of the admissible design domain, and V * is the given available volume, respectively, of the solid material for a single-material design problem and of the solid material *1 for a bi-material design problem, cf. Sections 2.1 and 2.3.


It is noted from (27) that the global dynamic stiffness matrix dK defined as

MKK 2pd �� may be negative definite when the prescribed external excitation

frequency p� has a high value, e.g. higher than the fundamental eigenfrequency of the structure. In this case, the scalar product of the transpose of the vector of amplitudes of the external surface loading and the vector of amplitudes of the structural displacement response may become negative, and in order to include this possibility in our problem formulation, we apply the absolute value of this scalar product as the dynamic compliance Cd, see (27). Moreover, to render the problem differentiable, we choose the objective function F as the square of the dynamic compliance. The dynamic equilibrium equation in (27) is solved in a direct way by Gauss elimination in this paper.

5.2 Sensitivity Analysis The sensitivity of the objective function F in problem (27) with respect to the design variables �e is given by

,)()(2)( 2 UPUPUP TTT �� dCF (28)

where prime denotes partial derivative with respect to �e. The sensitivity P� of the load vector will be zero if it is design-independent, otherwise it can be handled using the method described by Hammer and Olhoff (1999, 2000), and also by Du and Olhoff (2004a,b). The sensitivity U� of the displacement vector is given by

,)()( 22 UMKPfUMK ��!�� pp �� (29)

where the sensitivities of the stiffness and mass matrices can be directly obtained from the SIMP material model, i.e. Eqs. (4). The vector f is known as the pseudo load and is defined by the term on the right-hand side of Eq. (29). Instead of solving Eq. (29), the adjoint method (see e.g. Tortorelli and Michaleris 1994) may be used to calculate the sensitivity of the objective function in a more efficient manner, which gives the following result

� �.)(2)(2 2 UMKUPUUP TTT �� pF �(30)

Accordingly, the optimality condition for problem (27) can be expressed in the following form by means of the method of Lagrange multipliers,

� � ,0)(2)(2 2 �"�� ep VUMKUPUUP TTT �(31)

where " is the Lagrange multiplier corresponding to the material volume constraint, and the side constraints for �e have been ignored. The optimization problem (27) can be solved by using the well-known MMA method (Svanberg 1987) or an


optimality criterion method, e.g. the fixed point method, as devised by Cheng and Olhoff (1982).

5.3 An Approach for Minimum Dynamic Compliance Design The dynamic compliance defined in the first constraint of problem (27) may alternatively be written as follows by using the modal superposition technique (without damping)

,1||11

212

��

�##$

%&&'

(

))*

+

,,-

.#$%

&'(��

I

iii

I

i i

i

i

pd cMC

�� PUP

TT (32)

where �i represents the ith eigenfrequency and �i the corresponding eigenmode of the structure, and

2

##$

%&&'

(�

i

iic

��PT

and 12

1

))*

+

,,-

.#$%

&'(�

i

piM �

� (33)

can be interpreted as the contribution of the ith eigenmode of the structure to the dynamic compliance and the corresponding magnification factor (as defined for a single degree of freedom system). It is noted that the lower eigenfrequency normally gives more contribution (implying a larger value of ci) to the dynamic compliance of the structure if the corresponding eigenmode is not orthogonal to the external loading mode. On the other hand, when the loading frequency is close to an eigenfrequency of the structure, the absolute value of the magnification factor corresponding to this eigenfrequency will increase very quickly, which indicates the occurrence of a resonance.

Not surprisingly, minimization of the integral dynamic compliance normally yields a structure whose eigenfrequencies are far from the prescribed excitation frequency �p of the dynamic load; this structural behaviour implies efficient avoidance of resonance phenomena with large displacement amplitudes and low dynamic stiffness. In the present topology optimization problem, the initial design (cf. D1 in Fig. 17(a)), which is normally chosen to have a uniform distribution of material with intermediate density over the admissible design domain, may have a fundamental eigenfrequency (resonance frequency) / = /1

that is smaller than the given loading frequency �p. In this case, a decrease of the dynamic compliance corresponding to �p normally implies an increase of the static compliance (that corresponds to the same loading amplitude but zero frequency � = 0), due to a decrease of the fundamental eigenfrequency /(thereby avoiding resonance), see Fig. 17(a). As a result of this, in particular in single-material problems, the structure may become very weak at the (local) optimum of the dynamic compliance that is obtained. In order to prevent this,


one may introduce an upper bound constraint on the static compliance. However, this will delimit the gain of the optimization of the dynamic compliance.

In fact, much lower values of the dynamic compliance can be obtained if we can start out the optimization procedure using a value � of the loading frequency that is lower than the value of the fundamental eigenfrequency (resonance frequency) /1 for the initial design, and then sequentially increase � up to its originally prescribed value �p , see Fig. 17(b). This procedure has the desirable effect of generating a series of topologies with increasing values of both the fundamental frequency / and the static and dynamic stiffnesses for the sequence of structures produced (we may call this technique a “continuation technique”). Finally, the procedure delivers the optimum dynamic compliance topology solution subject to the originally prescribed loading frequency, �p. The procedure automatically avoids resonance, and works very well as long as the prescribed loading frequency �p is lower than the maximum obtainable value /opt of the fundamental eigenfrequency, i.e. optp /�� . Moreover, since the fundamental eigenfrequency of the structure maintains a value higher than the loading frequency, the dynamic stiffness matrix MKK 2

pd �� , see problem (27), remains positive definite during the design process. This implies a very good feature embedded in the dynamics design, i.e., the global structural response approaches zero when the dynamic compliance of the structure approaches zero.

(a)

Path 2

Cd1

Cd2

D1

Initial design D1

D2

�p/1

Cd

Cs2

Cs1

� = 0 /2 �


(b)

Figure 17. Principle sketch of the dependence of the dynamic compliance Cd on the loading frequency � for the case where the prescribed loading frequency �p is close to the fundamental eigenfrequency /1 of the initial design D1. Symbols /2 and /3 represent the

fundamental eigenfrequencies of the designs D2 and D3. (a) If the fundamental eigenfrequency /1 of the initial design D1 is less than the prescribed loading frequency

�p, i.e. p��/1 , then the design will proceed along path 2 (see (a)) to decrease the

dynamic compliance, and as a result, the dynamic compliance corresponding to �pbecomes smaller, i.e. 12 dd CC � , but the static compliance Cs (corresponding to � =0) for the same design increases, 12 ss CC . (b) If p�/1 , then the design will proceed along

path 1 (see (b)), and as a result, both the dynamic compliance (corresponding to �p) and the static compliance (corresponding to � =0) for the same design decrease, i.e.

13 dd CC � and 13 ss CC � .

Now, if the prescribed value of �p is such that optp /� (but sufficiently smaller than the second resonance frequency of the design associated with the maximum eigenfrequency /opt ), the minimization of the dynamic compliance will drive the fundamental eigenfrequency / of the design towards zero (for single material design). At the same time, the static displacements of the structure become very large, which means that the static stiffness tends to zero. The physical reason for this behaviour is that, in the limit, a disintegration is created in the structure. In this limit, the zero value of the fundamental

Path 1

D1 D3

/1 /3

Cd1

Cd3

�p

Cs1

Cs3

� = 0 �

Cd Initial design D1


eigenfrequency is associated with a rigid body vibration mode of the structure, and the static displacements of the disintegrated part of the structure become infinite, as the structure cannot sustain the static load.

A straight-forward way of avoiding this unwanted structural behaviour is to include an upper bound constraint on the static compliance in the mathematical formulation of the problem of minimizing the dynamic compliance. We have found that such a constraint is extremely effective and well-chosen when minimizing the dynamic compliance for a value of �p that is somewhat larger than /opt .

It should be mentioned at this point that in Chapters 5 and 6, the given excitation frequency �p is tacitly assumed to be lower than the anti-resonance frequency located between the first and second resonance frequencies of the structures considered. The reader is referred to Olhoff and Du (2008) for a general, systematic approach (based on continuation techniques) for determining topology designs of minimum dynamic compliance subject to excitation frequencies that may be far above the lowest resonance frequencies of the structure.

6 Numerical Examples of Dynamic Compliance Minimization

6.1 Minimum Dynamic Compliance Design of a Plate-like Structure

This example concerns optimum topology design of a single-material 3D plate-like structure with support conditions as shown in Fig. 18(a). A time-harmonic, concentrated transverse external load p(t) = Pcos� t is applied to the center of the plate. The design objective is to minimize the dynamic compliance of the plate for a prescribed loading frequency � = �p = 80 and a volume fraction of 50% for the given solid material, which has the Young’s modulus E = 1011, Poisson’s ratio � = 0.3 and the specific mass �m = 7800. The first eigenfrequency of the plate in the initial design (see Fig. 18(a)) is /1 = 61.6, i.e. less than the given loading frequency. Minimization of the dynamic compliance drives the design away from the resonance point which implies a continual decrease of /1 as shown in Fig. 18(b). As a result, the static compliance of the structure increases very quickly (Fig. 18(c)). Fig. 18(d) shows that at iteration step 30, the plate has become very weak at the two fixed supports. This indicates creation of a rigid body vibration mode in association with the first eigenfrequency, and that the structure cannot effectively sustain the static load associated with � = 0.


0 5 10 15 20 25 300

20

40

60

80

100

Iteration number

Fre

que

nci

es

� � 01 Prescribed loading frequency

/ 1

p

First eigenfrequency

(a) (b)

0 5 10 15 20 25 300

0.5

1

1.5

2x 10

-6

Iteration number

Dyn

am

ic a

nd

sta

tic c

om

plia

nce

Dynamic compliance C for

Static compliance C

� � � � 01

2� � 13

p

s

d

(c) (d)

Figure 18. (a) Admissible design domain (a = 3, b = 2 and c = 0.1) with loading and support conditions. (b) Iteration history for the first eigenfrequency of the plate (/1 < �p =

80). (c) Iteration histories for the dynamic and static structural compliance (the latter corresponds to the same loading amplitude but frequency � = 0). (d) Material distribution

at iteration step 30.

In order to avoid such a statically weak design, a more expedient approach is adopted for solution of the above problem. Thus, we use a continuation approach and start out the design problem with a value � = �0 of the loading frequency that is lower than the first eigenfrequency /1 of the initial design, and we then sequentially increase � up to its originally prescribed value � = �p = 80 (Fig. 19(a)). In the converged result, a structure with minimized dynamic compliance and improved static stiffness is now obtained (see Fig. 19(b,c)).


0 5 10 15 20 25 3050

60

70

80

90

100

110

120

130

Iteration number

First eigenfrequency

� � � � 01

Prescribed loading frequency

Fre

qu

en

cie

s

Initial loading frequency �0

p

/ 1

5 10 15 20 25 300

1

2

3

4

5x 10-7

Iteration number

Dynamic compliance C for loading frequency � � � � 01

Static compliance C 2� � 13

Str

uct

ura

l co

mp

lian

ce

pd

s

(a) (b)

(c) �p = 80 (d) �p =150 ( SS CC � )

Figure 19. (a), (b) Iteration histories for the first eigenfrequency of the plate, the loading frequency, and the dynamic and static compliances. (c), (d) Optimum topologies(50%

volume fraction, single-material design) for �p=80 and �p=150 (with an upper bound on the static compliance, i.e. SS CC � ).

Finally let us consider a case with a prescribed, higher value of the loading frequency, �p=150. We have computed the optimum value of the first eigenfrequency of the plate to be /opt = 127.6, i.e., lower than the given loading frequency in this case. Then, to ensure a reasonable static stiffness of the design, we introduce an upper bound 8105 4�� SS CC for the static compliance SC in the formulation of the problem. The optimum topology result for this problem is shown in Fig. 19(d).

Fig. 20(a) shows the iteration histories of the dynamic compliance of the plate subject to the higher loading frequency (� = �p = 150) and four different


upper bound constraints on the static compliance Cs (associated with the same loading amplitude but zero frequency). The graphs show that the optimum dynamic compliance decreases as the upper bound constraint on Cs is increased. In Fig. 20(b), iteration histories are shown for minimum compliance topology design of the plate subject to a given upper bound constraint on the static compliance ( 7105.0 4�� ss CC ) for four different higher loading frequencies. These graphs show that for the higher loading frequency designs, the dynamic compliance of the structure decreases as the prescribed loading frequency is increased. This feature is opposite to that obtained by minimum compliance topology design subject to prescribed lower or medium loading frequencies. As a conclusion, variations of the minimum dynamic compliance with respect to different loading frequencies are depicted in Fig. 21(a), and Fig. 21(b) presents the static compliances associated with the minimum dynamic compliance designs subject to different prescribed loading frequencies.

0 50 100 150 2000

0.5

1

1.5

2

2.5

3

3.5

4x 10

-7

Iteration number

Dyn

amic

co

mp

lian

ce o

f st

ruct

ure

C = 4x10-7

s

C = 3x10-7 sC = 1x10-7

s

C = 0.5x10-7

s

Dynamic compliances C for� � � � 561and different upper bounds on C

p

d

s

0 50 100 150 2000

1

2

3

4

5

6x 10

-7

Iteration number

Dyn

am

ic c

om

plia

nce

of

stru

ctu

re

Dynamic compliances C for� � � � 571p

� � � � 561� � � � 501� � � � 811

p

p

p

with constraintC <= C = 0.5x10s s -7

d

(a) (b)

Figure 20. (a) Iteration histories of the dynamic compliances of the plate subject to a high loading frequency (� = �p = 150 > /opt = 127.6) and four different upper bound

constraints on the static compliance Cs, i.e. ss CC � . (b) Iteration histories of the dynamic compliances of the plate subject to a given upper bound constraint on Cs

( 7105.0 4�� ss CC ), for four different loading frequencies �p = 130, �p = 150, �p = 180 and �p = 200, all of which are higher than the optimum value of the fundamental

eigenfrequency /opt .


0 50 100 150 2000

0.5

1

1.5x 10-7

Loading frequency �

/ opt

Min

imu

m d

yna

mic

co

mp

lian

ce C

C = 0.5x10s-7

d

C <= Cs s

Dynamic design without constraint on C s

Dynamic Design withconstraint

p0 50 100 150 200

0

0.5

1

1.5x 10-7

Loading frequency �

/ optSta

tic c

ompl

ianc

e C

of

m

inim

um d

ynam

ic c

ompl

ianc

e de

sign

s

C = 0.5x10s-7 s

C <= Cs s

Dynamic design withoutconstraint on C s Dynamic

design with constraint

p

(a) (b)

Figure 21. (a) Minimum dynamic compliances Cd vs. different loading frequencies. (b) Static compliances Cs (correspond to the same loading amplitude but zero frequency)

associated with the designs in Fig. 21(a) vs. different loading frequencies. Note that if the prescribed loading frequency is close to or higher than the optimum value /opt = 127.6 of the fundamental eigenfrequency for the corresponding problem of free vibrations of the

plate, an upper bound constraint ss CC � is prescribed for the static compliance in order to avoid obtaining a statically too weak structure from the dynamic design.

6.2 Topology Design of a 2D Inlet Subjected to Hydrodynamic Pressure Loading

As an extension relative to traditional topology optimization with design-independent loading, we now consider an example where the dynamic loading is design-dependent, i.e., both the locations and directions of the loading change as the structural topology changes. A method developed by Du and Olhoff (2004a, 2004b) (see also Hammer and Olhoff, 1999, 2000) is employed to handle the design problems associated with this type of loading.

The example deals with optimum topology design of a single-material inlet for fluid flow. The fluid flow in the channel of the initial inlet is as shown in Fig. 22(a), and is assumed to exert a uniform hydrodynamic pressure loading of given frequency and amplitude on the inner surface of the inlet. Note that the loading from the hydrodynamic pressure is design-dependent, as it changes with changes of the inner surface of the inlet. The material of the inlet is isotropic with Young’s modulus E = 107, Poisson’s ratio � = 0.3 and the specific mass �m = 1. The design objective is to minimize the dynamic compliance of the inlet. Fig. 22(a) shows the admissible design domain and the initial loading boundary. Figs. 22(b-d) show optimized topologies and the associated loading boundaries for three given loading frequencies 0�p� (static loading), 800p �� and


1000�p� . These loading frequencies are all lower than the maximum fundamental eigenfrequency of free vibrations of the inlet, which was found to be /opt = 1328. The corresponding optimum topology of the inlet (with the same material volume fraction as before) is shown in Fig. 23.

Figure 22. Optimized single-material topologies (40% volume fraction) of 2D inlet for three different loading frequencies. (a) Admissible design domain. (b) �p = 0. (c)

800�p� . (d) 1000�p� .

Figure 23. Optimum topology of the 2D inlet (for 40% volume fraction) obtained by maximizing the fundamental eigenfrequency of free vibrations of the inlet. The maximum

fundamental eigenfrequency is /opt = 1328.


By comparing the optimum topologies in Figs. 22 and 23 we find some interesting features. Thus, when the loading frequency is much lower than the optimum fundamental eigenfrequency of the structure, the resulting topology (see Fig. 22(c)) obtained by the dynamic design of the present paper is similar to the static design that sustains the amplitude of the loading at zero frequency, see Fig. 22(b), which implies that the dynamic design is dominated by the spatial distribution of the amplitude of the external loading vector. However, if the loading frequency is closer to the value of the optimum fundamental eigenfrequency of the structure, the design is dominated by the dynamic requirement, and drives the fundamental eigenfrequency of the structure as far away as possible from the prescribed loading frequency. For an intermediate value of the loading frequency, the optimum topology of the inlet is a kind of compromise between the loading amplitude dominated design and the eigenfrequency dominated design (cf. Figs. 22(b-d) and Fig. 23).

7 Minimization of Sound Radiation from Vibrating Structures

The present and the following chapter lend themselves to Olhoff and Du (2006) and Du and Olhoff (2007a), and are devoted to topological design optimization of vibrating bi-material elastic structures of given volume, domain and boundary conditions, with the objective of directly minimizing the sound power radiated from the structural surfaces into a surrounding acoustic medium. As in the preceding two chapters, the structural vibrations are excited by a time-harmonic mechanical loading with prescribed forcing frequency and amplitude, and structural damping is not considered. It is assumed that air is the acoustic medium and that a feedback coupling between the acoustic medium and the structure can be neglected. Certain conditions are assumed, where the sound power radiated from the structural surface can be estimated by using a simplified approach instead of solving the Helmholz integral equation. This implies that the computational cost of the structural-acoustical analysis can be considerably reduced.

A bi-material model (cf. Section 2.3) is employed in the topology optimization. This implies that the boundary shape of the structure is not changed during the design, and leads to a great simplification of the sensitivity analysis, since the calculation associated with the shape gradients of the acoustic pressure loading is avoided. Numerical results are presented in Chapter 8 for bi-material plate and pipe structures with different sets of boundary conditions and excitation frequencies.


7.1 Formulation for Minimization of Sound Radiation In this section, we consider topological design optimization of a vibrating bi-material elastic structure with the objective of minimizing the total sound power (energy flux) 9 radiated from the structural surface S into a surrounding acoustic medium. The structural vibrations are assumed to be excited by a time-harmonic mechanical surface loading vector ti pet �� Pp )( with prescribed forcing frequency �p and amplitude vector P on S or part thereof. Assuming that damping can be neglected, the corresponding structural displacement response vector can be stated as ti pe �U , and the problem of minimizing the sound power 9 can be formulated as follows,

.),,1(,10

,)(,0

,

,)(

:

)Re(21min

01*1*

1

2

*

Ee

N

eee

ff

fp

Snf

Sn

Ne

VVVV

tosubject

dSvpdSI

E

e

��

��

�

��

:;

:<=

:�

:��

��>

�

??

�

�

��

�

�

�

HPGUPCLPPUMK (34)

Here, the symbols pf and *nv in the expression for 9 represent the acoustic

pressure and the complex conjugate of the normal velocity of the structural surface, and Pf denotes the corresponding vector of amplitudes of the acoustic pressure on the structural surface S. The symbol L represents the fluid-structural coupling matrix and the symbols K and M denote the N dimensional structural stiffness and mass matrices, where N is the number of DOFs. The expression

MK 2p� in (34) represents the dynamic stiffness matrix which we may denote

by KD. The matrices G, H and C� can be generated by the discretized Helmholtz integral and calculation of the spatial angle along the structural surface (see, e.g., Christensen et al., 1998). We consider a bi-material design problem (see Section 2.3) where NE is the total number of finite elements and the symbol �e denotes the volumetric density of the stiffer material in element e and plays the role of the design variable in the problem. The symbol � denotes the fraction of the given volume V *1 of the stiffer material (material *1) and is given by 0

1* /VV ,where V0 is the volume of the admissible design domain. The remaining part of the total volume V0 is occupied by a softer material (material *2) as explained in Section 2.3.


7.2 Calculation of Sound Power Flow from the Surface of a Vibrating Structure

The first two constraint equations in (34) denote the structural-acoustic coupling equations (without incoming acoustic waves) and imply quite complicated computations since these equations must be solved in each iterative step of the solution procedure. For simplification, one may consider a special case where the vibration frequency of the structure has a sufficiently high value. In this case, the radiation impedance at the boundary of the structure is approximately the same as the characteristic impedance of the acoustic medium (see Herrin et al., 2003, and Lax and Feshbach, 1947), which implies that the acoustic pressure pf and normal velocity vn of the structural surface approximately satisfy the following linear relationship

nff cvp �� (35)

where c is the sound speed and � f is the specific mass (mass density) of the acoustic medium. Tests performed by Sorokin (2005) for simple beam and sphere examples show that the accuracy of (35) depends on not only the frequency level but also on the size of the structure and the shape of the vibration mode of the structure. Generally speaking, the accuracy of the approximation increases with increasing values of the frequency, but may decrease with a change of the vibration mode. Nevertheless, the tests also show that even for lower frequencies, (35) may still yield a good approximation of the distribution (up to a multiplying factor) of the sound pressure along the structural surface. This is actually useful for our problem of optimizing the global sound radiation, because even a scaled distribution of the sound pressure along the structural surface can yield a topology design which is close to the optimum one.

If we further assume weak coupling, i.e., ignore the acoustic pressure in the structural equation, the first constraint in (34) will be simplified to the equation of a vibrating structure subjected only to the external mechanical loading P (see the third equation in (27) and the paper Olhoff and Du, 2008),

PUKM �� )( 2p� (36)

With the above simplification, the first two constraint equations in problem (34) may be replaced by Eqs. (35) and (36), and the sound power flow from the structural surface can now be calculated in a very efficient way, which is illustrated briefly as follows.

Substituting pf by (35) and noting that )( tipn

peiv �� @� un , where n is the unit normal and u is the amplitude of the displacement, the sound power flow in (34) may be restated as


? @@�>S

pf dSc ))((21 2 unun�� (37)

Using the finite element interpolation eNUu � , where N is the shape function and Ue is the nodal displacement vector of element e, Eq. (37) may be rewritten in matrix form as

USUTnpf c 2

21 ��> (38)

where � ?�� #

#

$

%

&&

'

(��

E

e

E N

e S

N

enen dS

11NnnNSS TT may be termed the surface normal matrix,

and U is the global nodal displacement vector of the structure as in (34).

7.3 Sensitivity Analysis

The sensitivity of the objective function (i.e. the sound power flow) in problem (34) with respect to the design variables �e is given by

USUT ��>� npf c 2�� (39)

where prime denotes partial derivative with respect to �e. Using Eq. (36) and applying the adjoint method, see Tortorelli and Michaleris (1994), the sensitivity(39) of the sound power flow may be calculated in a more efficient manner, which gives the following result

� �UMKUPU TT )( 22 ��>� psspf c �� (40)

Here Us is the solution to the equation snsp fUSUMK !� )( 2� , where fs may be regarded as a pseudo surface load vector. Specifically, we only consider the case of design-independent mechanical load in the present paper, so the sensitivity P� of the mechanical load in (40) will be zero. The sensitivities of the stiffness and mass matrices, i.e. K� and M� , can be derived by introducing the material models (see Section 2.1 and 2.3).

Based on the above sensitivity results, the optimization problem (34) may be solved by using the well-known MMA method (see Svanberg, 1987) or an optimality criterion method, e.g. the fixed point method (see Cheng and Olhoff, 1982).


8 Numerical Examples of Minimization of Sound Power Radiation

8.1 Minimization of Sound Power Radiation from a Clamped Bi-material Plate Excited by Uniform Harmonic Pressure Loading

(a) (b)

Figure 24. Plate-like structure (a=20, b=20, t=1) subjected to uniformly distributed harmonic pressure loading on its upper surface. All edges of the plate are clamped.

The first example concerns optimum topology design of a bi-material plate-like structure with clamped edges (see Fig. 24). A time-harmonic, uniformly distributed, transverse external load p(t) = Pcos�t (with P equal to unity) is applied to the upper surface of the plate. The design objective is to minimize the total sound power radiated from the surface of the plate to its surrounding acoustic medium, i.e. air, for a prescribed loading frequency � = �p and a volume fraction of up to 50% for the given stiffer material *1, which has the Young’s modulus 111* 10�E , Poisson’s ratio � = 0.3 and the specific mass 1*

m� = 7800 (SI units are used throughout). The soft material *2 has the properties

1*2* 1.0 EE � and 2*m� = 0.1 1*

m� , and � = 0.3. The specific mass of the fluid (i.e. air) is � f = 1.2 and the sound speed c = 343.4.

The plate is modeled by 3D 8-node isoparametric elements in a 4044041mesh. Nine prescribed different loading frequencies, �p = 10, 100, 200, 300, 500 and 1000 are considered, which cover the designs from a lower frequency level to a high frequency level. (The fundamental eigenfrequency of the initial uniform design in Fig. 24(b) is �1 = 95.) The finite element mesh used here ensures that the computational results have sufficient accuracy in the frequency range (10 to 1000) considered.


(a) �p = 10 (b) �p = 100 (c) �p = 200

(d) �p = 300 (e) �p = 400 (f) �p = 500

(g) �p = 600 (h) �p = 700 (i) �p = 1000 Figure 25. Optimum topologies of clamped bi-material plate-like structures obtained by

minimization of the total sound power radiation subject to nine different loading frequencies. (a) �p = 10. (b) �p = 100. (c) �p = 200. (d) �p = 300. (e) �p = 400. (f) �p = 500. (g) �p = 600. (h) �p = 700. (i) �p = 1000. A uniformly distributed time-harmonic

pressure loading is applied to the upper surface of the plate.

The corresponding optimum topologies of the plate are presented in Figs. 25(a-i), where the stiffer material *1 is represented by black and the soft material *2 is represented by grey. It can be seen that, as the loading frequency increases,


the optimum topology of the structure shows a more and more complicated periodicity. In comparison to the initial uniform design, the total sound power flow into the acoustic medium is reduced from 3.78410-8 to 3.61410-9 for �p = 10, from 3.50410-4 to 4.01410-7 for �p = 100, from 1.31410-6 to 6.49410-8 for �p

= 200, from 2.10410-6 to 7.43410-8 for �p = 300, from 1.06410-6 to 2.57410-8 for �p = 400, from 3.68410-7 to 1.84410-8 for �p = 500, from 2.47410-7 to 1.69410-8

for �p = 600, from 5.52410-7 to 1.06410-8 for �p = 700, and from 1.03410-7 to 5.61410-9 for �p = 1000. It is noted that the sound power has a remarkable decrease in the design for the given frequency �p = 100. The reason is that the loading frequency �p = 100 is very close to the first resonance point that corresponds to the value 951 �int� of the fundamental eigenfrequency of the initial design (Fig. 24(b)). For the optimum design (see Fig. 25(b)), we have found that its first two eigenfrequencies are 591 �opt� and 1312 �opt� (see Figs. 26(a)-26(c)), which are far away from the given loading frequency �p = 100. This implies that resonance has been avoided effectively by the optimum design and explains the large decrease of the sound radiation for this design.

(a) 591 �opt� (b) 1312 �opt� (c) 13123 �� optopt ��

Figure 26. The first three eigenmodes of free vibration of the optimum topology design subject to the given loading frequency �p = 100. The second and the third

eigenfrequencies constitute a bi-modal eigenfrequency due to the symmetry of the plate.

8.2 Comparison between Topology Designs of Minimum Sound Radiation and Designs of Minimum Dynamic Compliance

As is evident from (34), the acoustic design objective of minimum sound radiation considered in the present Chapter 8 is a global criterion. Now an interesting question arises, namely, for the same given excitation frequency p�and distribution of the external mechanical loading, how much will the design result change if we instead consider a “comparison problem” where we apply the global design objective of minimizing the dynamic compliance, and assume that the structure is elastic, has no material damping, and is subjected to forced vibration in vacuum?


Optimum topology solutions to this “comparison problem” can be directly obtained by the method developed in Chapter 5 and in the paper by the authors Olhoff and Du (2008). Thus, for the investigation in the present section, we define the dynamic compliance as || UPT�dC for the “comparison problem”. Here, P denotes the amplitude vector of the given time-harmonic mechanical surface loading (cf. the first paragraph of Section 7.1), and U represents the vector of magnitudes of the corresponding structural displacement response vector that satisfies the dynamic equilibrium equation (36) for the steady-state vibration at the frequency p� . The above expression for the dynamic compliance of our “comparison problem” represents the numerical mean value of the magnitudes of the surface displacements weighted by the values of the amplitudes of the corresponding time-harmonic surface loading. For the case of static loading ( p� = 0), the expression directly reduces to the traditional definition of static compliance, namely the work done by the external forces against corresponding displacements at equilibrium.

(a) �p = 10 (b) �p = 100 (c) �p = 200

(d) �p = 300 (e) �p = 500 (f) �p = 1000 Figure 27. Optimum topologies of clamped, quadratic bi-material plate-like structures obtained by minimizing the dynamic compliance in vacuum for six different given

excitation frequencies. (a) �p = 10. (b) �p = 100. (c) �p = 200. (d) �p = 300. (e) �p = 500. (f) �p = 1000. A uniformly distributed harmonic pressure loading of constant amplitude is

applied to the upper surface of the plate.

It is worth mentioning at this point that it is customary to define the dynamic compliance differently, namely as the average input power from the external time-harmonic loading over a load cycle, if the structure possesses material


damping and is subjected to forced vibration in a light acoustic medium without feedback (see, e.g., Koopmann and Fahnline, 1997, and Jog, 2002). By the law of conservation of energy, this input power is equal to the sum of the sound power radiated from the structure to the light acoustic medium and the power dissipated due to the material damping in the structure. This means that if the structure is subjected to forced vibration in vacuum and the material damping is ignored (as is the case for our “comparison problem”), then the dynamic compliance defined as the input power from the external loading, will be equal to zero and hence inapplicable as an objective function for our “comparison problem”.

We now consider the initial structure and the same boundary and loading conditions as in Fig. 24, but instead of minimizing the total sound power 9emitted from the structural surface, we use the method developed in Chapter 5 and (Olhoff and Du, 2008) to minimize the dynamic compliance || UPT�dC ofthe plate-like structure in vacuum (and assuming vanishing structural damping). Optimum topologies for the latter problem corresponding to the six prescribed loading frequencies �p = 10, 100, 200, 300, 500 and 1000 are shown in Figs. 27(a-f).

It is seen that for the low excitation frequencies �p = 10 and �p = 100, the optimum topologies shown in Figs. 27(a) and 27(b) are almost indistinguishable from the corresponding ones in Figs. 25(a) and 25(b). However, as the value of the excitation frequency is increased, the differences between the topologies become more pronounced (see Figs. 27 and 25 for �p = 200, 300, 500 and 1000).

For further comparison we have calculated the values of both the sound power and the dynamic compliance of the initial structure and the optimum structures corresponding to the two different design objectives behind Figs. 25 and 27 (assuming the same loading and boundary conditions), and the results are presented in Table 1.

These results provide numerical evidence that topology optimization with respect to either of the two design objectives has a strongly improving effect on the other objective.

Thus, substantial reductions in sound power radiated from a structure immersed in a light acoustic medium like air can be achieved by using the method of topology optimization developed in Chapter 5 (and in Olhoff and Du, 2008) to minimize the dynamic compliance defined above for a structure in vacuum. This approach may sometimes be attractive because it circumvents the need to solve the acoustical equations, thereby achieving saving in computational cost. On the other hand, the results in Table 1 show that for the higher values of the excitation frequency p� , significant further reductions can be obtained by directly minimizing the sound power radiation, and the expense in computational


cost for this is indeed very limited, if one adopts the simplified approach applied in this paper for calculation of the sound power flow from the surface of the vibrating structure. Table 1. Comparisons between designs of minimum sound power and designs of minimum

dynamic compliance.

Initial uniform design (see Figure 24)

Optimum designs of minimum sound power flow from the structural surface (see Figure 25)

Optimum designs of minimum dynamic compliance in vacuum (see Figure 27)

Designs

Excit- ationFrequency 9 Cd 9opt Cd 9 Cd

opt

�p = 10 3.78410-8 1.39410-5 3.61410-9 4.43410-6 3.68410-9 4.42410-6

�p = 100 3.50410-4 1.29410-4 4.01410-7 4.37410-6 4.02410-7 4.36410-6

�p = 200 1.31410-6 3.17410-6 6.49410-8 3.01410-7 1.28410-7 1.47410-8

�p = 300 2.10410-6 4.76410-7 7.43410-8 3.08410-7 3.56410-7 5.74410-10

�p = 500 3.68410-7 6.84410-7 1.84410-8 8.49410-8 4.46410-8 7.85410-9

�p = 1000 1.03410-7 2.09410-7 5.61410-9 4.18410-8 7.41410-8 9.29410-10

8.3 Minimization of Sound Power Radiation from a Corner-supported Bi-material Plate Excited by Concentrated Harmonic Loading

In order to investigate the influence of the boundary conditions and load distribution on the optimum topologies, we consider again the design problem of the bi-material plate as in Fig. 24, but in combination with another set of boundary and loading conditions. Thus, the plate is now simply-supported at its four corners and subjected to a concentrated time-harmonic load at the center of the upper surface (see Fig. 28). Six prescribed values of the loading frequency, �p = 10, 100, 300, 500, 700 and 1000 are considered separately here to generate the corresponding optimum topologies of the bi-material plate. The optimum topology results subject to the six different frequencies are presented in Figs. 29(a-f).

Figure 28. Corner-supported plate subjected to a concentrated harmonic load at the center of its upper surface.


(a) �p = 10 (b) �p = 100 (c) �p = 300

(d) �p = 500 (e) �p = 700 (f) �p = 1000

Figure 29. Optimum topologies of the corner-supported bi-material plate-like structure (cf. Fig. 28) for six different loading frequencies. (a) �p = 10. (b) �p = 100. (c) �p = 300. (d) �p = 500. (e) �p = 700. (f) �p = 1000. A concentrated harmonic load is applied at the

center of the upper surface of the plate.

It is seen that the optimum topologies associated with the same loading frequencies as in Figs. 25 and 29 are quite different, which implies that the boundary and loading conditions may have large influence on the resulting topology of the plate-like structure.

It is noteworthy that in the optimum topologies subject to the concentrated harmonic load (see Fig. 29), the central part of the plate is always filled out with the stiffer material *1 which also has a higher mass density. This local layout is very efficient in counteracting the concentrated load which has not only a given frequency but also a prescribed amplitude. The mass assembly surrounding the point of action of the force yields a large local inertia, which effectively reduces the displacement amplitude at the point of action of the force, and thereby reduces the vibration amplitudes and the density of sound power emission all over the plate.


Figs. 30 and 31 show comparisons of the power flow distribution between the initial designs (subject to the two different boundary and loading conditions in Figs. 24 and 28) and the corresponding optimum designs for the prescribed excitation frequency �p = 1000. Similar comparisons for corresponding designs subject to the prescribed frequency �p = 500 are given in Figs. 32 and 33. It is seen that the distribution of the sound power in the optimum designs subject to the concentrated load show features that are very different from those in the optimum designs subjected to the uniform load.

(a) (b)Figure 30. Distribution of the power flow from the structural surface of the initial design. (a) and (b) correspond to the two different sets of boundary and loading conditions shown

in Figs. 24 and 28, respectively. The loading frequency has the prescribed value �p = 1000.

(a) (b)Figure 31. Distribution of the power flow from the structural surface of the optimum

design. (a) and (b) correspond to the two different sets of boundary and loading conditions shown in Figs. 24 and 28, respectively. The loading frequency has the prescribed value �p

= 1000.


Thus, we find that for the higher excitation frequencies, the main part of the sound power emitted from the optimum designs subjected to the concentrated load (see Figures 31(b) and 33(b)) is limited within a small annular-like area surrounding the mass assembly in the vicinity of the concentrated load. This implies that the optimum designs (see Figures 29(d) and 29(f)) create an efficient isolation of vibration and sound power radiation that to a large extent terminates the transmission of bending waves to the boundary of the plate-like structures at the prescribed frequencies. The features discussed here reveal very strong similarities between the present optimum designs and so-called band gap structural designs (see, e.g., Halkjær et al., 2006).

(a) (b)

Figure 32. Distribution of the power flow from the structural surface of the initial design. (a) and (b) correspond to the two different sets of boundary and loading conditions shown in Figs. 24 and 28, respectively. The loading frequency has the prescribed value �p = 500.

(a) (b)

Figure 33. Distribution of the power flow from the structural surface of the optimumdesign. (a) and (b) correspond to the two different sets of boundary and loading conditions shown in Figs. 24 and 28, respectively. The loading frequency has the prescribed value �p

= 500.


9 Conclusions for the Topology Optimization Problems Considered

9.1 Optimization of Eigenfrequencies Problems of topology optimization with respect to structural eigenfrequencies of free vibrations were studied and presented in Chapters 3 and 4 of this paper. The design objectives were maximization of specific eigenfrequencies and distances between two consecutive eigenfrequencies of the structures.

It was necessary to develop and apply special iterative numerical procedures to handle topology optimization problems associated with both simple and multiple eigenfrequencies. Thus, particularly in topology optimization, where the design space is very large, it is often found that although an eigenfrequency may be simple during the initial stage of the iterative design procedure, later it may become multiple due to coincidence with one or more of its adjacent eigenfrequencies. In order to capture this behaviour, it is necessary to apply an extended mathematical formulation and solution procedure that allows for multiplicity of the eigenfrequency because a multiple eigenfrequency – in contrast to a simple eigenfrequency – does not possess usual differentiability properties.

Several numerical examples of topology optimization of single- and bi-material beam- and plate-like structures were carried out with the abovementioned design objectives and validated the approaches presented. The results demonstrated that multiplicity of optimum eigenfrequencies is the rule rather than the exception in topology optimization of freely vibrating structures and that this needs careful attention. The results also indicated that the techniques enable us, in a most cost-efficient manner, to move structural resonance frequencies far away from external excitation frequencies with a view to avoid high vibration and noise levels.

9.2 Minimization of Dynamic Compliance Topological design with the objective of minimizing the dynamic compliance (maximizing the integral dynamic stiffness) of continuum structures subjected to time-harmonic forced vibration with prescribed frequency and amplitude of the dynamic loading, was studied in Chapters 5 and 6. The results in Chapter 6 show that the design objective yields structures whose eigenfrequencies of free vibration are generally far from the given excitation frequency of the dynamic loading, which implies efficient avoidance of resonance phenomena and reduction of the vibration level of the structure.

It was demonstrated that the design objective may be implemented along different optimization paths according to different levels of the given external exicitation frequency �p. For cases where the loading has a lower or medium


value of �p, the minimum dynamic compliance design process may be driven by a continuation technique where the loading frequency is sequentially increased from a sufficiently low initial value up to its prescribed value, �p. This procedure delivers the desired result that the optimum structure is associated with minimum dynamic compliance subject to the prescribed loading frequency, and also implies an effective improvement (decrease) of the static compliance of the structure. On the other hand, if the excitation frequency was prescribed to be somewhat larger than the maximum obtainable value /opt of the fundamental eigenfrequency of the corresponding free vibration problem, we found that the increase of the dynamic compliance was accompanied by a drastic decrease of the static compliance of the structure. Thus, especially when minimizing the dynamic compliance for single-material structures, we found it to be expedient to introduce an upper bound constraint on the static compliance in order to maintain a reasonable static stiffness of the design.

It is worth mentioning that the approaches presented are not limited to values of the excitation frequency that are below or just above the fundamental eigenfrequency of free vibrations of the structure. By using the continuation technique, topology designs of minimum dynamic compliance can be determined for excitation frequencies that are far above the lowest eigenfrequencies of the structure, see Olhoff and Du (2008).

9.3 Minimization of Sound Power Radiation

Chapters 7 and 8 dealt with problems of topological design of vibrating bi-material elastic structures with the objective of minimizing the sound power radiated from a structural surface into a light acoustic medium. As in Chapters 5 and 6, the structures were subjected to forced vibration of prescribed excitation frequency and force amplitude distribution.

For sufficiently high excitation frequencies, instead of solving the Helmholz integral equation, the sound pressure at the structural surface is determined approximately by the product of the characteristic impedance of the acoustic medium and the normal velocity of the structural surface. This simplifies the structural-acoustic analysis and substantially reduces the computational cost. Actually, even for relatively low frequencies, the above simplification may yield a good approximation of the distribution of the sound pressure over the structural surface, which is very useful in the present context. An extended SIMP model was used for the topology design of the bi-material structures, where the same penalization was applied for the stiffness and mass of the structural volume elements. Sensitivities of the design objective were derived by an adjoint method, and the optimization problem was solved by using the MMA method.


Numerical results are presented for bi-material plate structures with different loading and boundary conditions, and interesting features of the optimum structural topologies, power flow distributions and sound pressure waves, are revealed and discussed. Main conclusions are:

The sound power radiation from structures subjected to forced vibration can be considerably reduced by topology optimization. Just as was demonstrated in Chapter 5 and in the paper (Olhoff and Du, 2008) on topological design for minimum dynamic compliance of structures in vacuum, this is achieved by creation of an optimum design with a large gap between two adjacent eigenfrequencies of free vibrations, with the given excitation frequency placed approximately in the middle of the gap.

Along these lines, topology optimization of structures with respect to either of the two objectives (i) minimum sound radiation into a surrounding light acoustic medium and (ii) minimum dynamic compliance in vacuum, has a strongly improving effect on the other objective. For structures subjected to the same excitation frequency, spatial distribution of the external mechanical loading, and the same boundary conditions, the optimum topologies associated with the design objectives (i) and (ii) are almost indistinguishable at lower excitation frequencies but become more different at higher frequencies.

Independently of the spatial distribution of the external dynamic loading and the boundary conditions, the optimum topologies of the bi-material plate structures exhibit different types of periodicities when the excitation frequency is increased to values above a number of the lowest resonance frequencies of the structures.

When subjected to a concentrated harmonic load at such a value of the excitation frequency, the optimum structural topology shows a local assembly of the heavier material around the point of action of the load. Due to the given amplitude of the load, the inertia of this local mass assembly does not only reduce the displacement amplitude of the point of load action, but also the general level of vibration and sound power radiation of the structure. Almost all the (reduced) sound power radiation from the optimum structure is limited to take place from within a small ring-shaped area of the weaker material around the aforementioned local mass assembly in the case of plates. This means that almost all of the power input from the external harmonic load is radiated as acoustic power from these locations close to the point of action of the load, whereby only a very small part of the power input is transmitted to the plate boundary.

Overall, the problem studied in this paper has several features in common with problems of topological design optimization of band gap structures.


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