fundamentals of fluid-structure interaction

Fundamentals of Fluid-Structure Interaction

Goran Sandberg*, Per-Anders Wernberg*, Peter Davidsson †

* Department of Construction Sciences, Structural Mechanics, Lund University, Lund,SWEDEN

† A2Acoustics, Helsingborg, SWEDEN

Abstract Acoustic and structure-acoustic analysis is of great importance are foundin a number of applications. Common applications for acoustic and structure-acoustic analysis are the passenger compartments in automobiles and aircraft. Theincreased use of light-weight materials in these vehicles usually makes it even morecomplicated to achieve good passenger comfort in terms of low level of interior noise.When the weight of the structure is reduced, the vibrations could be increased andthat could lead to higher noise levels. Another application where structure acousticanalysis is of interest is in light-weight constructions of buildings, to mention but afew.

This paper describes the basic finite element formulations of coupled fluid-structure systems and an overview of the various formulations possible. Then ascheme for treating unsymmetrical coupled systems is outlined. The discretizationis performed using displacement formulation in the structure and either pressure ordisplacement potential in the fluid. Based on the eigenvalues of each subdomainsome simple steps give a standard eigenvalue problem. It might also be concludedthat the unsymmetrical matrices have real eigenvalues. The strategies for formu-lating the structure-acoustic systems are illustrated using the educational softwareroutines and elements.

1

Contents

Contents 2

1 Introduction 3

1.1 Organization of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Structure-acoustic analysis - Governing equations 6

3 Finite element formulation 7

3.1 Structural domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Acoustic fluid domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 The coupled structure-acoustic system . . . . . . . . . . . . . . . . . . . . 113.4 Two-dimensional structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5 Boundary conditions and coupling . . . . . . . . . . . . . . . . . . . . . . 133.6 Alternative formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.7 Modal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Modal representation of Fluid-Structure Systems 18

4.1 Fluid-structure interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Modal representation, p-formulation . . . . . . . . . . . . . . . . . . . . . 194.3 Modal representation, ψ-formulation . . . . . . . . . . . . . . . . . . . . . 224.4 Summary of eigenproblems and eigenvectors . . . . . . . . . . . . . . . . . 234.5 Computational effort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Acoustic and Structure-Acoustic Implementations 33

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 Element routines in Calfem . . . . . . . . . . . . . . . . . . . . . . . . . 345.3 Acoustic elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.4 Structure-acoustic interface elements . . . . . . . . . . . . . . . . . . . . . 435.5 System functions for structure-acoustic analysis . . . . . . . . . . . . . . . 51

Bibliography 55

A Finite element routines 59

A.1 Routine aco2td.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59A.2 Routine cp2s2f.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

B Calfem Examples 62

B.1 Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63B.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65B.3 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66B.4 The complete system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67B.5 A reduced modal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2

C Calfem input files 71

C.1 Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71C.2 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73C.3 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74C.4 The complete system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74C.5 A reduced modal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3

1 Introduction

Vibrating structures inducing pressure waves in a connecting acoustic fluid and the oppo-site case of acoustic pressure waves inducing structural vibrations constitute a thoroughlyinvestigated field of research (see for example the texts by Cremer and Heckl (14), andFahy (29). In (47; 48; 31; 23; 49; 36; 37), the structure-acoustic problem is studiedusing analytical expressions for the two domains. It is evident that the two connectingdomains, the flexible structure and the enclosed acoustic cavity, can be strongly coupledand in that case the structure-acoustic system must be studied in a coupled system toevaluate the natural frequencies and the response to dynamic excitation.

The systems studied often have complex shapes, leading to the conclusion that an-alytical functions cannot be used for describing the spatial distribution of the primaryvariables. Numerical methods must be employed. A review of different solution strate-gies for structure-acoustic problems is given by Atalla (3), where analytical methods andtwo numerical approaches: the finite element method and the boundary element method,are discussed. The development of structure-acoustic analysis using the finite elementmethod for the study of vehicle interior noise is reviewed by Nefske et al. (42). The basicsof the finite element method are described in, for example, Ottosen and Petersson (46).A more thorough investigation of the finite element method is found in, for example, thecited works of Bathe (5) or Zienkiewicz and Taylor (66).

The formulation of coupled structure-acoustic problems using the finite element meth-od is described, for example, in (54; 13; 43; 25). In the finite element formulation, asystem of equations describing the motion of the system is developed, with the numberof equations equal to the number of degrees of freedom introduced in the finite elementdiscretisation. One important property of the equation system derived is the sparsityof the system matrices, i.e. only a few positions in these matrices are populated. Thisproperty results in that the time for solving the system of equations is much shorter,compared to solving a fully populated system of equations with equal size.

In the structural domain, the primary variable is displacement. For the fluid domain,several different primary variables can be used to describe the motion. Using the fluiddisplacement as the primary variable, both the structural and fluid domains can be de-scribed with the same type of solid elements. The fluid domain has no shear stiffness andnormal modes with pure rotational motion are introduced. All rotational modes shouldhave the eigenvalue equal to zero. However, spurious non-zero, and thereby non-physical,modes are introduced when using full integration of the solid element. Reduced integra-tion can be used to make all eigenvalues of rotational modes equal to zero (2). Thehourglass modes due to the reduced integration can however interact with the correctmodes giving spurious modes with the same frequencies as the correct ones. In (18), theelement mass matrix was modified to account for this and the eigenvalue of all spuriousmodes becomes zero. A mixed displacement based finite element formulation was pre-sented by Bathe (5), also removing the spurious modes. Using displacement to describethe fluid domain can be called an one-field formulation, with only the displacement fieldis used to describe the structure-acoustic system.

In order to remove the problem with non-physical modes and to arrive at a more com-pact system of equations, a potential description of the fluid domain can be used, such

4

as the acoustic pressure or fluid displacement potential. The pressure formulation wasused in (20; 50) to determine normal modes and eigenvalues of complex shaped rigid-wallenclosures and also in (21) to study the transient response of structure-acoustic systems.A two-field formulation, with structural displacements and fluid potential function isachieved with only one degree of freedom per fluid node. The derived system of equa-tions using pressure or displacement potential yields an unsymmetric system of equations.A fluid velocity potential can also be used, where a matrix proportional to velocity isintroduced (24). To solve the structure-acoustic eigenvalue problem using the two fieldformulation, one needs an eigenvalue solver that either can handle unsymmetric matricesor can solve quadratic eigenvalue problems. Solving these problems are more computa-tional intensive compared to solving the generalised eigenvalue problem for symmetricsystems (32).

In order to achieve a symmetric system of equations describing the structure-acousticsystem, a three field formulation with structural displacement, fluid pressure and fluiddisplacement potential can be used (41; 51). By condensation of one of the fluid po-tentials, a symmetric two field system of equations can be achieved (13). However, thesystem matrices then lose the positive property of being sparse.

Different types of methods for model reduction are often employed in structure-acoustic analysis. The most commonly used method is to reduce the system using thenormal modes for the structural and fluid domains, derived in separate eigenvalue anal-ysis of the two subdomains (61; 44). In a paper by Sandberg (53), the un-symmetriceigenvalue problem, achieved when using the structural displacement and fluid pressureas primary variables, is made symmetric using the subdomain modes and matrix scaling.Reduction methods using component mode synthesis were also proposed in, for example,(22; 62). In the thesis by Carlsson (13), the Lanczos procedure was used in investigatingstructure-acoustic problems.

When introducing generalised coordinates, a reduced set of basis vectors, compared tothe original (physical) coordinates, are derived. For example, after solving the eigenvalueproblem, only a few of the calculated normal modes with the lowest natural frequencyare needed to describe the system (5). Types of modes other then normal can also beused. When using Ritz vectors (4; 63; 1; 40), or Lanczos vectors (13; 35; 19), a veryefficient modal reduction of the system can be performed.

In condensation methods, a large number of the degrees of freedom, that are notneeded in describing the dynamic behaviour of the system, is removed. This can be doneby static (Guyan) condensation (33), where the choice of the kept degrees of freedom isvery important (56; 34; 6). The reduction of the problem can also achieved by dynamiccondensation, where the influence of the internal degrees of freedom are accounted for ina simplified manner (38; 39; 7; 8; 9).

A frequently used method for substructuring and modal reduction is the componentmode synthesis method. The research conducted to develop this method was reviewed bySeshu (55) and detailed description of the method can be found in, for example, the bookby Craig (15). The problem domain under study is divided into a number of components,or subdomains, and a set of basis vectors is derived for each component to be includedin the description of the whole system. Using generalised coordinates or condensationmethods can be seen as special cases of the component mode synthesis method.

5

Component mode methods are usually divided into the fixed-interface componentmode method and the free-interface component mode method. The fixed-interface modemethod uses static condensation of each component, only keeping the degrees of freedomat the interface between components (16). These basis vectors, which fulfil the displace-ment and force continuity at the interface boundary, are expanded with internal modescalculated with the interface degrees of freedom fixed. The eigenvalue problem of thestatic-reduced system, only including the interface degrees of freedom, can first be solved.The inclusion of only a number of these modes reduces the size of the reduced systemfurther (12; 11). The convergence of the fixed-interface mode method can be improvedin a certain frequency range using quasi-static constraint modes as basis vectors (57).

Using the free interface mode method, the internal modes are calculated withoutconstraints on the interface degrees of freedom, see (10; 59; 58) for recent developmentsof this method. The continuity over the interface is achieved by including attachmentmodes, which are calculated by applying a unit force on each of the interface degrees offreedom with the other interface degrees of freedom unconstrained.

1.1 Organization of the paper

The paper is organized in the following way. First we introduce the basic formulationsfor structure-acoustic systems based on different independent variables. Then, Section4, we show how the unsymmetric formulations can be reformulated to a symmetric al-ternative, thus showing that the eigenvalues are real.

In order to bring a more solid understanding of the details we introduce the imple-mentations of these in a script language, Matlab. It is our belief that to really understandwhat goes on in the mathematically discrete form, how the formulations actually work, aprogramming code is a good tool that further the operational understanding. In Section5 the educational code Calfem is the vehicle for that discussion. The theoretical back-ground for the element routines is described. Particularly the interface elements clearlyshow how the information is passing between the two physical domains. Finally someexample are presented in Appendix B

6

2 Structure-acoustic analysis - Governing equations

This Section investigates the analysis of structure-acoustic systems, here limited to sys-tems consisting of a flexible structure in contact with an enclosed acoustic cavity, withinthe finite element environment.

For the structure-acoustic system, the structure is (here) described by the differentialequation of motion for a continuum body assuming small deformations and the fluid bythe acoustic wave equation. Coupling conditions at the boundary between the struc-tural and fluid domains ensure the continuity in displacement and pressure between thedomains. The governing equations and boundary conditions can, as for example, wasdescribed in detail by Carlsson (13), be written:

Structure :

∇Tσs + bs = ρs

∂2us

∂t2Ωs

+ Boundary and initial conditions

Fluid :

∂2p

∂2t− c20∇

2p = c20∂qf∂t

Ωf

+ Boundary and initial conditions

Coupling :

us|n = uf |n ∂Ωfs

σs|n = −p ∂Ωfs

(2.1)

The structure-acoustic problem schematically sketched in Figure 1. It consists of afluid domain, Ωf , and a structure domain, Ωs. The boundary between the fluid domainand the structure domain is denoted, Ωsf , the fluid boundaries with prescribed pressure,Ωp, with prescribed velocity, Ωv, and with a prescribed impedance, Ωz.

Figure 1. A principal sketch of the problems discussed in this report

The variables and material parameters are defined in the following sections, wherealso the finite element formulation of this coupled problem derived.

7

The structure of interest in most structure-acoustic problems is two dimensional andis therefore often described by plate or shell theory. For derivation of the system matricesfor these problems, see, for example, (5; 66).

3 Finite element formulation

3.1 Structural domain

The structure is described by the equation of motion for a continuum body. Thefinite element formulation is derived with the assumption of small displacements. Thispresentation follows the matrix notation used by Ottosen and Petersson (46).

For a continuum material the equation of motion can be written

∇Tσs + bs = qs (3.1)

with the displacement, us, the body force, bs, and the inertia force, qs,

us =

us1us2us3

; bs =

bs1bs2bs3

; qs = ρs∂2us

∂t2(3.2)

where ρs is the density of the material. The differential operator ∇ can be written

∇ =

∂

∂x10 0

0∂

∂x20

0 0∂

∂x3∂

∂x2

∂

∂x10

∂

∂x30

∂

∂x1

0∂

∂x3

∂

∂x2

; (3.3)

The Green-Lagrange strain tensor, Es, and the Cauchy stress tensor Ss are defined as

Es =

εS11 εS12 εS13εS22 εS23

sym. εS33

; Ss =

σS11 σS12 σS13σS22 σS23

sym. σS33

(3.4)

and in matrix notations the strains and stresses can be written

εs =

εs11εs22εs33γs12γs13γs23

σs =

σs11σs22σs33σs12σs13σs23

(3.5)

8

where γs12 = 2εs12, γs13 = 2εs13 and γs23 = 2εs23. The kinematic relations, the relations

between the displacements and strains, can be written

εs = ∇us (3.6)

For an isotropic material, the stresses and strains are related by the constitutivematrix Ds given by

σs = Dsεs (3.7)

where

Ds =

λ+ 2µ λ λ 0 0 0λ λ+ 2µ λ 0 0 0λ λ λ+ 2µ 0 0 00 0 0 µ 0 00 0 0 0 µ 00 0 0 0 0 µ

(3.8)

The Lame coefficients, λ and µ, are expressed in the modulus of elasticity, E, the shearmodulus, G, and Poisson’s ratio, ν by

λ =νE

(1 + ν)(1 − 2ν)(3.9)

µ = G =E

2(1 + ν)(3.10)

To arrive at the finite element formulation for the structural domain, the weak formof the differential equation is derived. This can be done by multiplying equation (3.1)with a weight function, vs = [v1 v2 v3]

T , and integrating over the material domain, Ωs,

∫

Ωs

vTs (∇Tσs − ρs

∂2us

∂t2+ bs)dV = 0 (3.11)

Using Green-Gauss theorem on the first term in equation (3.11) gives∫

Ωs

vTs ∇TσsdV =

∫

∂Ωs

(vs)T tsdS −

∫

Ωs

(∇vs)TσsdV (3.12)

The surface traction vector ts related to the Cauchy stress tensor, Ss, by

ts = Ssns (3.13)

where ns is the boundary normal vector pointing outward from the structural domain.The weak form of the problem can be written

∫

Ωs

vTs ρs∂2us

∂t2dV +

∫

Ωs

(∇vs)TσsdV −

∫

∂Ωs

(vs)T tsdS −

∫

Ωs

vTs bsdV = 0 (3.14)

Introducing the finite element approximations of the displacements ds and weight func-tions cs by

us = Nsds; vs = Nscs (3.15)

9

where Ns contains the finite element shape functions for the structural domain, thestrains can be expressed as

εs = ∇Nsds (3.16)

This gives the finite element formulation for the structural domain, when described as acontinuum body

∫

Ωs

NTs ρsNsdV ds +

∫

Ωs

(∇Ns)TDs∇NsdV ds =

∫

∂Ωs

NTs tsdS +

∫

Ωs

NTs bsdV

(3.17)and the governing system of equations can be written

Msds + Ksds = ff + fb (3.18)

where

Ms =

∫

Ωs

NTs ρsNsdV ; Ks =

∫

Ωs

(∇Ns)TDs∇NsdV

ff =

∫

∂Ωs

NTs tsdS fb =

∫

Ωs

NTs bsdV

(3.19)

3.2 Acoustic fluid domain

The governing equations for an acoustic fluid are derived using the following assump-tions for the compressible fluid (13):

• The fluid is inviscid.

• The fluid only undergoes small translations.

• The fluid is irrotational.

Thereby, the governing equations for an acoustic fluid are, the equation of motion,

ρ0∂2uf (t)

∂t2+ ∇p(t) = 0 (3.20)

the continuity equation,∂ρf (t)

∂t+ ρ0∇

∂uf (t)

∂t= qf (t) (3.21)

and the constitutive equation,

p(t) = c20ρf (t) (3.22)

Here uf (t) is the displacement, p(t) is the dynamic pressure, ρf (t) is the dynamic densityand qf (t) is the added fluid mass per unit volume. ρ0 is the static density and c0 is thespeed of sound. ∇ denotes a gradient of a variable, i.e.,

∇ =

[∂

∂x1

∂

∂x2

∂

∂x3

]T

; (3.23)

10

The nonhomogeneous wave equation can be derived from equations (3.20) – (3.22). Dif-ferentiating equation (3.21) with respect to time and using (3.22) gives

1

c20

∂2p

∂2t+ ρ0∇

(

ρ0∂2uf

∂t2

)

=∂qf∂t

(3.24)

Substituting (3.20) into this expression gives the nonhomogeneous wave equation ex-pressed in acoustic pressure p.

∂2p

∂2t− c20∇

2p = c20∂qf∂t

(3.25)

where ∇2 = ∂2/∂x2

1 + ∂2/∂x22 + ∂2/∂x2

3.The finite element formulation of equation (3.25) is derived by multiplying with a test

function, vf , and integrating over a volume Ωf .

∫

Ωf

vf

(∂2p

∂2t− c20∇

2p− c20∂qf∂t

)

dV = 0 (3.26)

and with Green’s theorem the weak formulation is achieved∫

Ωf

vf∂2p

∂2tdV + c20

∫

Ωf

∇vf∇pdV = c20

∫

∂Ωf

vf∇pnfdA+ c20

∫

Ωf

vf∂qf∂t

dV (3.27)

where the boundary normal vector nf points outward from the fluid domain. The finiteelement method approximates the pressure field and the weight function by

p = Nfpf ; vf = Nfcf (3.28)

where pf contains the nodal pressures, cf the nodal weights and Nf contains the finiteelement shape functions for the fluid domain. Inserting this into equation (3.27) andnoting that cf is arbitrary gives

∫

Ωf

NTf NfdV pf + c20

∫

Ωf

(∇Nf)T∇Nf dV pf =

= c20

∫

∂Ωf

NTf ∇pnfdS + c20

∫

Ωf

NTf

∂qf∂t

dV

(3.29)

The system of equations for an acoustic fluid domain becomes

Mf p + Kfp = fq + fs (3.30)

where

Mf =

∫

Ωf

NTf Nf dV ; Kf = c20

∫

Ωf

(∇Nf)T∇Nf dV

fs = c20

∫

∂Ωf

NTf nTf ∇p dS ; fq = c20

∫

Ωf

NTf

∂q

∂tdV

(3.31)

11

Note that nTf ∇p in the squared expression above is a scalar function multiplied on

each position of NTf . If part of the boundary is non-flexible, the condition is imposed by

setting ∇p(r, t) = 0 (natural boundary condition). If the fluid volume has a free surface,p = 0 at the corresponding nodes. Finally, if some part of the boundary, it might be aninterior boundary, is interacting with a flexible structure, that interaction is entered byreformulating nTf ∇p.

3.3 The coupled structure-acoustic system

At the boundary between the structural and fluid domains, denoted ∂Ωsf , the fluidparticles and the structure moves together in the normal direction of the boundary.Introducing the normal vector n = nf = −ns, the displacement boundary condition canbe written

usn|∂Ωsf = ufn|∂Ωsf (3.32)

and the continuity in pressureσs|n = −p (3.33)

where p is the acoustic fluid pressure. The structural stress tensor at the boundary ∂Ωsfthus becomes

Ss = −p

1 0 00 1 00 0 1

(3.34)

and the structural force term providing the coupling to the fluid domain, ff (in equation(3.18)), can be written

ff =

∫

∂Ωsf

NTs (−p)

1 0 00 1 00 0 1

ns dS =

∫

∂Ωsf

NTs np dS =

∫

∂Ωsf

NTs nNf dSpf

(3.35)Note that the structural boundary normal vector ns is replaced with the normal vectorn pointing in the opposite direction. The force acting on the structure is expressed inthe acoustic fluid pressure.

For the fluid partition the coupling is introduced in the force term fs (in equation(3.30)). Using the relation between pressure and acceleration in the fluid domain

∇p = −ρ0∂2uf (t)

∂t2(3.36)

and the boundary condition in equation (3.32), the force acting on the fluid can bedescribed in terms of structural acceleration

nT∇p|∂Ωsf = −ρ0nT ∂

2uf

∂t2|∂Ωsf = −ρ0n

T ∂2us

∂t2|∂Ωsf = −ρ0n

TNsds|∂Ωsf (3.37)

and the boundary force term of the acoustic fluid domain, fs, can be expressed in struc-tural acceleration

fs = −c20∫

∂Ωsf

NTf nT∇pdS = −ρ0c

20

∫

∂Ωsf

NTf nTNsdSds (3.38)

12

The introduction of a spatial coupling matrix

H =

∫

∂Ωsf

NTs nNfdS (3.39)

allows the coupling forces to be written as

ff = Hpf (3.40)

andfs = −ρ0c

20H

T ds (3.41)

The structure-acoustic problem can then be described by an unsymmetrical system ofequations

[Ms 0

ρ0c20H

T Mf

] [

dspf

]

+

[Ks −H

0 Kf

] [dspf

]

=

[fbfq

]

(3.42)

This system is the standard form, the most basic form, of FSI-formulation.

3.4 Two-dimensional structure

It is particularly illustrative to derive the finite element expression for a two-dimen-sional fluid-structure interaction problem, e.g. a fluid-backed beam. This will be furtherdiscussed in detail in 5.

The partial differential equation of motion for elementary beam flexure, in a localcoordinate system where the x-axis is directed along the axis of the beam, is

∂2

∂x2(EI

∂2us∂x2

) +m∂2us∂t2

= p(r, t)

∣∣∣∣∂Ωsf

. (3.43)

u(x, t) is the transverse displacement of the beam, EI is the bending stiffness and m isthe mass per unit length. For simplicity, the only distributed load considered here is thetransverse load due to the acoustic pressure. By definition, the transverse displacementis positive in the normal direction of the beam, corresponding to the local y-axis.

The finite element formulation for an undamped beam with length L, expressed inthe local coordinate system, is

Ms¨d + Ksd = ff + fb, (3.44)

where

Ms =

∫ L

0

NTsmNs dx,

Ks =

∫ L

0

BTs EIBs dx,

ff =

∫ L

0

NTs p dx ,

fb =

[dNT

s

dxM − NT

s V

]L

0

,

(3.45)

13

and the definitions of Ns and Bs restricted to one element read

Nes = [Ne

s1(x) Nes2(x) N

es3(x) N

es4(x)] , and Be

s = d2Nes/dx

2. (3.46)

If the beam is arbitrarily rotated, the local coordinate system must be transformedto the global coordinte system by a matrix G, d = Gd. For a two-dimension beam withtwo translation degrees-of-freedom and one rotation degree-of-freedom at each node, thematrix G is defined as

G =

nxy nyy 0 0 0 00 0 1 0 0 00 0 0 nxy nyy 00 0 0 0 0 1

, (3.47)

where nxx defines the direction cosine between the x axis and the x axis, and so on. Notethat the axial deformation of the beam is not present in the model. Premultiplying Eq.(3.44) by GT and using d = Gd yields

Msd + Ksd = ff + fb, (3.48)

where

Ms = GT MsG,

Ks = GT KsG,

ff = GT ff ,

fb = GT fb.

(3.49)

3.5 Boundary conditions and coupling

In the two previous sections we have put some expressions in boxes. The dynamiccoupling of the different domains, structure and fluid, is described by these expressions.They need to be reformulated using the defined shape functions from either system.Firstly, continuity of the fluid displacements and structural displacements is assumed inthe normal direction to the interface. Thus, the following kinematic boundary conditionapplies, in global coordinates,

uf · n|s = us · n|s , (3.50)

where uf is the fluid displacements and X|s denotes the restriction of X to S. Secondly,for the acoustic fluid the equations of motion give the following relation between acousticpressure p and fluid displacements uf

∇p(r, t) = −ρ ∂2uf (r, t)

∂t2, (3.51)

where ρ is the fluid density. At the structure-acoustic interface we get

∇p(r, t) · n|s = − ρ∂2

uf (r, t)

∂t2· n∣∣∣∣s

= − ρ∂2

us(r, t)

∂t2· n∣∣∣∣s

≈ [nxnynz]Ns d, (3.52)

14

where Eq. (3.15) has been used to approximate us. This expression is substituted in fsto get

fsEq. (3.31)

= −ρc2∫

s

NTf [nxnynz]Ns dS d = −ρc2HT d. (3.53)

The last part defines H. ff is discretized using the nodal quantities of p(r, t)

ffEq. (3.19)

=

∫

Sf

NTs p(r, t) · n dS ≈

∫

Sf

NTs

nxnynz

Nf dS p = Hp. (3.54)

The expressions Eq. (3.53) and Eq. (3.54) are particularly simple for the two-dimensionalcase because

∇p(x, y, t) · n|e = −ρNs¨d = −ρNsGd. (3.55)

Hence

fs =

∫ L

0

c2NTf∇p(x, y, t) · n dx = −ρc2

∫ L

0

NTf Ns dx Gus = −ρc2HT us,

ff = GT

∫ L

0

NTs p(x, y, t) dx = GT

∫ L

0

NTs Nf dx p = Hp,

(3.56)where

H = GT

∫ L

0

NTs Nf dx. (3.57)

Eq. (3.18) alt. Eq. (3.44) can thus be written as

Msd + Ksd = fb + Hp (3.58)

and similarly, Eq. (3.30), as

Mf p + Kfp = −ρc2 HT d + fq (3.59)

and the coupled system of equations established as

[Ms 0

ρc2 HT Mf

] [

d

p

]

+

[Ks −H

0 Kf

] [d

p

]

=

[fbfq

]

. (3.60)

This system of equations is the basis for the element routines described in Section 5.

3.6 Alternative formulations

Introduce a potential function, ψ, associated with the fluid displacement, ∇ψ def= uf .

It can be shown that the acoustic wave equation still holds for this new variable,

∂2ψ(r, t)

∂t2− c2∇2ψ(r, t) = c2Q. (3.61)

15

Consequently, we can recycle the weak form presented in Eq. (3.24) by just changingthe notation p → ψ. The boundary conditions have altered, however. From the fluidequation we get

∇ψ(r, t) · n|Sdef= uf (r, t) · n|S = us(r, t) · n|S ≈ [nxnynz]Ns d (3.62)

and thus

fsEq. (3.31)

=

∫

S

c2NTf (∇ψ(r, t) · n) dS = c2 HT d (3.63)

where H has been defined previously. The load on the structural system from the fluidpressure, ff , is still p(r, t)n but it can be expressed in terms of the fluid displacementpotential as

p(r, t) = −ρ ∂2ψ(r, t)

∂t2, (3.64)

hence

ffEq. (3.19)

=

∫

Sf

NTs (−ρ)∂

2ψ(r, t)

∂t2· n dS = −ρ

∫

Sf

NTs

nxnynz

Nf dS Ψ = −ρHΨ.

(3.65)The coupled system based on the fluid displacement potential can now be established:

[Ms ρH0 Mf

] [d

Ψ

]

+

[Ks 0

−c2HT Kf

] [d

Ψ

]

=

[fbfq

]

. (3.66)

The fluid pressure p and fluid displacement potential ψ can furthermore be combined inthe following manner

∇p+ ρ∇ψ = 0,

1

ρc2p+ ∇2ψ =

1

ρQ.

(3.67)

Expand p and ψ in the same set of shape functions Nf , and use ∇Nf as weight functionsin the first equation in Eq. (3.67) and Nf in the second equation. Actually the shapefunctions need not be the same, but for simplicity I have made that choice here. Theresult is

Ms 0 00 ρc−2Kf 00 0 0

d

Ψp

+

Ks 0 −H

0 0 c−2Kf

−HT c−2Kf −(ρc2)−1Mf

d

Ψp

=

fb0fq

.

(3.68)Note that this is a symmetric system but with some deficiences, like zero entries in thediagonal.

Once the mass matrices and stiffness matrices, Mx, Kx (x = s and f), and thecoupling matrix H have been established, any of these systems, Eq. (3.60), (3.66) or(3.68), can readily be created.

The ambition above has not been to present all details, merely to outline some of themathematics of interacting fluid-structure systems. For a more thorough discussion ofthe systems above and some others, see (54)–(? ) to the references.

16

3.7 Modal decomposition

The symmetry property of the matrix blocks for each subdomain makes it possibleto diagonalize each block in Eq. (3.60) independently. This corresponds to solving thestructural problem in vacuo and the fluid domain with stiff boundaries. Assuming Φsand Φf are matrices consisting of the eigenvectors in the structural domain and the fluiddomain respectively, we have

ΦTs MsΦs = Is and ΦTs KsΦs = Ds (3.69)

in the structural domain and

ΦTf MfΦf = If and ΦTf KfΦf = Df (3.70)

in the fluid domain, where Ds and Df are diagonal matrices. It can be shown, see (? ),that by a similarity transformation Eq. (3.60) can be written as

[ηs

ηf

]

+

Ds −√

ρc2√

DsΦTs HΦf

−√

ρc2ΦTf HTΦs√

Ds Df + ρc2ΦTf HTΦsΦTs HΦf

[ηs

ηf

]

=

√

ρc2√

DsΦTs Fs

ΦTf Ff − ρc2ΦTf HTΦsΦTs Fs

(3.71)

Contrary to Eq. (3.60) the system matrix is symmetric and the corresponding eigen-value problem is in standard form. The similarity transformation is

T right =

[Φs 00 Φf

] [

(√

ρc2√

Ds)−1 0

0 If

]

(3.72)

The eigenvectors extracted from Eq. (3.71) can be used to compute the right eigen-vectors to Eq. (3.60) by simply multipling the extracted eigenvectors by the similaritytransformation given above, thus

[dp

]

right eigenvector= Tright

[ηsηf

]

right eigenvector. (3.73)

If the original system is to be diagonalized, the left eigenvectors of the system also needto be computed. A system similar to Eq. (3.71) can be derived for the left eigenvectors

Ds + ρc2ΦTs HΦfΦTf HTΦs −

√

ρc2ΦTs HΦf√

Df

−√

ρc2√

DfΦTf HTΦs Df

ηs

ηf

= λ

ηs

ηf

(3.74)

The corresponding similarity transformation is written

Tleft =

[Φs 00 Φf

] [Is 0

0 (√

ρc2√

Df )−1

]

(3.75)

17

and [dp

]

left eigenvector= Tleft

[ηsηf

]

left eigenvector. (3.76)

In Calfem there are ready-built functions to generate the system matrices defined inEq. (3.71) and Eq. (3.74), once the eigenvalue problems corresponding to the subsystemshave been solved, and to transform the eigenvectors back to the original system, i.e. Eq.(3.73) and Eq. (3.76).

The procedure outlined here can also be used on the ψ-formulation, Eq. (3.66). Thecorresponding routines are also available in Calfem (17).

18

4 Modal representation of Fluid-Structure Systems

The structural formulation is naturally based on the displacement field as independentvariable. In the fluid there exist multiple choices of independent variable, e.g. fluid dis-placement and different scalar fields such as pressure, velocity potential, and displacementpotential and combinations thereof.

The use of fluid-displacement field needs special attention to assure that the irrota-tional displacement field is maintained. This can be done by a proper introduction ofthe displacement field, see Sandberg (54). The disadvantage, compared to a scalar field,is the introduction of three nodal unknowns for a three dimensional problem.

Scalar fields can be used in various ways with different matrix block structures, al-though they all describe the same physical problem. They all automatically enforceirrotationality of fluid motions, however. A straightforward introduction of the pressureor the fluid-displacement potential is used by e.g. Zienkiewicz and Bettess (64) andZienkiewicz et al (65). By means of the finite element method, the discretized differen-tial equations then yield unsymmetrical system matrices. A short note on how coupled,conservative mechanical systems interacting through conjugate variables produce unsym-metrical systems is included in Felippa, (26).

The solution of the vibration eigenproblem for scalar formulation has been given a lotof attention, with the ambition to overcome the deficiency of the unsymmetric matrices.The standard procedure has been to combine, in various ways, the pressure and the fluid-displacement potential, thus achieving symmetric systems, see Ohayon (45), Sandberg(51) and Felippa and Ohayon (27). It is also possible to statically condense the two-fieldfluid formulations, thus producing a one-field symmetric formulation, (54; 13; 27; 28).The drawback is that the resulting systems yields either a full mass matrix or a fullstiffness matrix. Much of the confusion between the various formulations is removed inFelippa and Ohayon (27).

The present paper returns to the unsymmetric formulations based on either pressureor fluid-displacement potential. It suggests a procedure for handling vibration analysisbased on the eigenvectors of each subdomain, fluid and structure. Starting from theeigenvectors in each domain, the coupled system is formed, and after some simple steps,a standard eigenvalue problem is achieved. If the coupled system matrices need to be putin diagonal form, e.g. if the goal is a transient analysis based on the original system, theaccompanying left eigenvalue problem is turned into a similar simple standard eigenvalueproblem.

One advantage of the proposed system is the possibility to select a limited set ofeigenmodes from the subdomains, thus only forming a limited final coupled system.

4.1 Fluid-structure interaction

The particular class of fluid-structure interaction problems treated here is based onthe linear acoustic approximation in the fluid, i.e. compressible, non-viscid and smallvibration fluid flow. Different formulations of the fluid domain using pressure, p, fluiddisplacement potential, ψ, or combination thereof as independent variables, are possible.See further Sandberg (54) and Carlsson (13) for complete derivations of the coupledproblems utilizing different formulations.

19

The coupled system based on fluid pressure formulation reads

(Ms 0

ρc2BT Mf

)(usp

)

+

(Ks −B0 c2Kf

)(usp

)

=

(FsFf

)

(4.1)

and the fluid displacement potential formulation

(Ms ρB0 Mf

)(usψ

)

+

(Ks 0

−c2BT c2Kf

)(usψ

)

=

(FsFf

)

(4.2)

where, in both cases,Ms,Ks, are the structural mass and stiffness matrices, and (Mf )ij =∫

V NifN

jfdV and (Kf )ij =

∫

V ∇N if · ∇N j

fdV are the fluid system matrices. N jf are

the shape functions in the fluid domain. The coupling matrix B is defined as Bij =∫

S Nis · nN j

fdS. Nis are the shape functions in the structural domain and n is the

unit vector in the normal direction at the ‘wet’ surface. If both p and ψ are used asindependent variables in the fluid domain, symmetric formulations can be derived, seefor instance Sandberg and Goransson (51)

Ms 0 00 ρKf 00 0 0

usψp

+

Ks 0 −B0 0 Kf

−BT Kf −(ρc2)−1Mf

usψp

=

FsFfψFfp

.

(4.3)

One advantage of Eq. 4.3, as compared to Eq. 4.1 and Eq. 4.2, is the possibility of usingsymmetric eigenvalue solvers. The drawback is of course the doubling of fluid degrees offreedom. In the next section, the properties of Eq. 4.1 are further investigated.

4.2 Modal representation, p-formulation

This section deals with some modal properties of the p-formulation stated in Eq. 4.1above. Because the system is unsymmetric we need to distinguish between the rightand left eigenvector sets if at some step the system is to be diagonalized, for instance toperform a transient analysis.

A useful property of unsymmetric matrices used both in this section and in Section4.3 is

Remark: Suppose that C1 and C2 are n× n and m×m matrices respectively, andthat A is a n×m matrix. Then

(C1 A0 C2

)−1

=

(C−1

1 −C−11 AC−1

2

0 C−12

)

(4.4)

This is easily verified by checking that the matrix on the right hand side actually is theleft and right inverse. In particular, it is easy to compute the inverse in the case ofdiagonal matrices, C1 and C2.

20

Right eigenvectors of the p-system The symmetry property of the matrix blocksfor each subdomain makes it possible to diagonalize each block independently. Thiscorresponds to solving the structural problem in vacuo and the fluid domain with stiffboundaries. Assuming Φs and Φf are matrices consisting of the eigenvectors in thestructural domain and the fluid domain respectively, we have

ΦTsMsΦs = Is and ΦTsKsΦs = Ds (4.5)


ΦTfMfΦf = If and ΦTfKfΦf = Df (4.6)

in the fluid domain where Ds and Df are diagonal matrices. Changing to the base ofeigenvectors in each sub domain, i.e.

(usp

)

=

(Φs 00 Φf

)(ξsξf

)

(4.7)

and multiplying Eq. 4.1 from the left by

(ΦTs 00 ΦTf

)

(4.8)

yields

(Is 0

ρc2ΦTf BTΦs If

)(ξsξf

)

+

(Ds −ΦTs BΦf0 Df

)(ξsξf

)

=

(ΦTs FsΦTf Ff

)

. (4.9)

According to the Remark Eq. 4.4 in the previous section, matrix inversion applied tothe transformed mass matrix yields

(ξsξf

)

+

(Is 0

−ρc2ΦTf BTΦs If

)(Ds −ΦTs BΦf0 Df

)(ξsξf

)

=

(Is 0

−ρc2ΦTf BTΦs If

)(ΦTs FsΦTf Ff

)(4.10)

or(ξsξf

)

+

(Ds −ΦTs BΦf

−ρc2ΦTf BTΦsDs Df + c2ρΦTf BTΦsΦ

Ts BΦf

)(ξsξf

)

=

(ΦTs Fs

ΦTf Ff − ρc2ΦTf BTΦsΦ

Ts Fs

)(4.11)

The system matrix is almost symmetric. Scaling the system, using

(ξsξf

)

=

(

(√

ρc2√Ds)

−1 00 If

)(ηsηf

)

(4.12)

21

and multiplying from the left by the inverse of the scaling matrix, gives

(ηsηf

)

+

(

Ds −√

ρc2√DsΦ

Ts BΦf

−√

ρc2ΦTf BTΦs

√Ds Df + ρc2ΦTf B

TΦsΦTs BΦf

)(ηsηf

)

=

( √

ρc2√DsΦ

Ts Fs

ΦTf Ff − ρc2ΦTf BTΦsΦ

Ts Fs

)(4.13)

The transformation going from Eq. 4.1 to Eq. 4.13 is performed through a set ofsimilarity transformation; hence the eigenvalues are not affected by this procedure. Infact, the eigenvalues of the corresponding eigenvalue problem of Eq. 4.2 and Eq. 4.3are the same as those of Eq. 4.13. The diagonalization of the unsymmetrical matrices,however, is achieved by different sets of right and left hand eigenvectors, contrary to thestandard eigenvalue problem corresponding to Eq. 4.13.

Relation between the right eigenvectors of the original form and the derived

standard form. Denote the eigenvectors of the η-system in Eq. 4.13 by vη, the systemmatrix by Kη and the eigenvectors of the ξ-system in Eq. 4.11 by vξ and the systemmatrix by Ks

ξ . Then the following relation applies, S−1KsξS = Kη where S is the scaling

matrix defined in Eq. 4.12. BecauseKηvη = λvη where λ is the corresponding eigenvalue,we get

KsξSvη = λSvη (4.14)

hence, the eigenvectors vξ of Ksξ are Svη. But Ks

ξ is actually M−1ξ Kξ where the two

latter matrices are defined by Eq. 4.9. This gives the simple relation between righteigenvectors, vrp, of the original eigenproblem in Eq. 4.1 and the derived standard form,Eq. 4.13,

vrp =

(Φs 00 Φf

)(

(√

ρc2√Ds)

−1 00 If

)

vη (4.15)

Left eigenvectors of the p-system The previous subsections dealt with the righteigenvectors of the p-system. If the original system is to be diagonalized, the left eigen-vectors of the system need to be known. The following statement is trivial, xTK =λxTM ⇐⇒ KTx = λMTx hence the left eigenvalue problem of Eq. 4.1 reads

(Ks 0−BT c2Kf

)(usp

)

= λ

(Ms ρc2B0 Mf

)(usp

)

(4.16)

Note that the diagonal blocks are symmetric, as used in Eq. 4.16. Diagonalize eachsubdomain following the procedure in Section 4.2 and we get

(Ds 0

−ΦTf BTΦs Df

)(ξsξf

)

= λ

(Is ρc2ΦTs BΦf0 If

)(ξsξf

)

(4.17)

The inversion of the mass matrix yields(Is −ρc2ΦTs BΦf0 If

)(Ds 0

−ΦTf BTΦs Df

)(ξsξf

)

= λ

(ξsξf

)

(4.18)

22

or(Ds + ρc2ΦTs B

TΦfΦTf BΦs −ρc2ΦTs BΦfDf

−ΦTf BTΦs Df

)(ξsξf

)

= λ

(ξsξf

)

(4.19)

and we have an almost symmetric system. Applying a scaling matrix(ηsηf

)

=

(Is 0

0 (√

ρc2√Df)

−1

)(ξsξf

)

(4.20)

to perform a change of base, and multiplying from the left by the inverse of the scalingmatrix, similar to Subsection 4.2, we finally get(

Ds + ρc2ΦTs BΦfΦTf B

TΦs −√

ρc2ΦTs BΦf√Df

−√

ρc2√DfΦ

Tf B

TΦs Df

)(ηsηf

)

= λ

(ηsηf

)

(4.21)

We have the following relation between the eigenvectors of Eq. 4.21 and the left eigen-vectors of the original p-system, vlp

vlp =

(Φs 00 Φf

)(Is 0

0 (√

ρc2√Df )

−1

)

vη (4.22)

(note that the scaling matrix is different from the scaling matrices in Subsection 4.2)

4.3 Modal representation, ψ-formulation

This section deals with the modal properties of Eq. 4.2. Because the procedure issimilar to that of Section 3 only some of the intermediate results are stated.

Right eigenvectors of the ψ-system Using the same notation as in Subsection 4.2,we have Φs and Φf , matrices consisting of the eigenvectors in the structural domain andthe fluid domain respectively, i.e.

ΦTsMsΦs = Is and ΦTsKsΦs = Ds (4.23)


ΦTfMfΦf = If and ΦTfKfΦf = Df (4.24)

in the fluid domain, where Ds and Df are diagonal matrices. Changing to the base ofeigenvectors in each sub domain,

(Is ρΦTs BΦf0 If

)(ξsξf

)

+

(Ds 0

−c2ΦTf BTΦs Df

)(ξsξf

)

=

(ΦTs FsΦTf Ff

)

. (4.25)

Taking the inverse of the mass matrix according to Remark 4.4 and apply a scalingmatrix

(ξsξf

)

=

(Is 0

0√

c2

ρ D−1/2f

)(ηsηf

)

(4.26)

23

gives

(ηsηf

)

+

(


TΦs −√

ρc2ΦTs BΦf√Df

−√

ρc2√DfΦ

Tf B

TΦs Df

)(ηsηf

)

=

(ΦTs Fs − ρΦTs BΦfΦ

Tf Ff√

ρc2

√DfΦ

Tf Ff

)(4.27)

which yields a corresponding eigenvalue problem of standard type .The relation between the eigenvectors of the derived standard form and the right

eigenvectors, vrψ of the original problem reads

vrψ =

(Φs 00 Φf

)( Is 0

0√

c2

ρ (√Df)

−1

)

vη (4.28)

The eigenvalue problem in Eq. 4.27 is the same as the left eigenvalue problem of thep-system derived in Subsection 4.2. The relation to the original form differs, however.

Left eigenvectors of the ψ-system The procedure in Subsection 4.2 applied to theψ-form stated in Eq. 4.2 yields a standard eigenvalue problem of the form(

Ds −√

ρc2√DsΦ

Ts BΦf

−√

ρc2ΦTf BTΦs


TΦsΦTs BΦf

)(ηsηf

)

= λ

(ηsηf

)

(4.29)

This is exactly the same as the right eigenvalue problem of the p-system, derived inSubsection 4.2.

The relation between the eigenvectors of the derived standard form and the left eigen-vectors, vlψ of the original ψ-form is

vlψ =

(Φs 00 Φf

)( √c2

ρ (√Ds)

−1 0

0 If

)

vη (4.30)

4.4 Summary of eigenproblems and eigenvectors

To summarize the result, we have two standard eigenvalue systems, originating fromthe p-system and the ψ-system.

The right p-system and the left ψ-system:

(

Ds −√

ρc2√DsΦ

Ts BΦf

−√

ρc2ΦTf BTΦs


TΦsΦTs BΦf

)(ηsηf

)

= λ

(ηsηf

)

(4.31)

The left p-system and the right ψ-system:

24

(


TΦs −√

ρc2ΦTs BΦf√Df

−√

ρc2√DfΦ

Tf B

TΦs Df

)(ηsηf

)

= λ

(ηsηf

)

(4.32)

We have the following set of eigenvectors where the superscript denotes the corre-sponding left or right eigenvector problem,

vrp =

(Φs 00 Φf

)(

(√

ρc2√Ds)

−1 00 If

)

vη (vη from on Eq. 4.31) (4.33)

vlp =

(Φs 00 Φf

)(Is 0

0 (√

ρc2√Df )

−1

)

vη (vη from on Eq. 4.32) (4.34)

vrψ =

(Φs 00 Φf

)( Is 0

0√

c2

ρ (√Df )

−1

)

vη (vη from on Eq. 4.32) (4.35)

vlψ =

(Φs 00 Φf

)( √c2

ρ (√Ds)

−1 0

0 If

)

vη (vη from on Eq. 4.31) (4.36)

4.5 Computational effort

The numerical labor of going from Eq. 4.1 down to Eq. 4.13 lies in the formationof eigenvectors of the subsystems, Eq. 4.5 and Eq. 4.6. From there on, only cheapmatrix multiplications are performed. In most cases only a limited set of eigenvectorsparticipate in a solution, or only a limited interval in the frequency domain is of interest.Therefore, only a few of the eigenvectors need to be extracted from the structural andfluid subdomains. Although the matrix in Eq. 4.27 is full, the limited size and thesymmetry yield but minor numerical work during eigenvalue extraction.

4.6 Numerical examples

To illustrate the capabilities of the proposed scheme, two simple problems are given.The first is a simple ‘tube’ i.e. a one dimensional problem. The second is a two dimen-sional fluid rectangular domain with only one flexible wall.

One dimensional tube The tube example is a one dimensional fluid with a springmounted piston at one end and a fixed end at the other side. See further Fig. 2.

The resonance frequencies f (Hz) can be found as the solution to F (f) = 0 where thefunction F is defined as

F (f) = sin

(2 fπ l

c

)

+

2 cρ fπ cos

(2 fπ l

c

)

k − 4mf2π2, (4.37)

25

Figure 2. Simple one dimensional problem; tube locked at one end and a spring-masssystem at the other.

k and m are the spring stiffness and the piston mass respectively. ρ is the fluid densityand c the speed of sound. Finally l is the tube length. In the table below ‘analytical’refers to values of f where Eq. 4.37 equates zero. The pressure eigenmodes are

p(x) =

2(k − 4mf2π2

)cos

(2 fπ x

c

)

fπc−1 − 4 ρ f2π2 sin

(2 fπ x

c

)

√

k2 − 8 kmf2π2 + 16m2f4π4 + 4 c2ρ2f2π2(4.38)

For a more elaborate discussion of this example see Carlsson and Sandberg10. The firstfive of these eigenmodes are shown in Fig. 3, using data from case I, see below.

x

32.521.510.500

1

0.5

0

-0.5

-1

Figure 3. Analytical eigenmodes; plots of the pressure p(x) for the first five values ofthe eigenfrequency according to Eq. 7.2. The zero mode is excluded.

The fluid is divided into 50 simple linear elements, adding up to 51 degrees-of-freedom,and of course the structure is just a one degree-of-freedom system. The physical datachosen in the fluid are rho = 1000 kg/m, c = 1500 m/s and l = 3 m. Two sets ofruns were performed with different structural data, case I with k = 4.9348 × 108 N/m,

26

m = 200 kg and case II with k = 4.9348 × 105 N/m, m = 0.2 kg. The fluid resonanceoccurs at even 250 Hz steps, if both ends are regarded as fixed. The structural data arechosen in order to get structure resonance at 250 Hz, if vibrating in vacuo.

The analysis is based on an eigenvalue analysis of the fluid domain based on thep-formulation, Eq. 4.1. Different numbers of eigenmodes from this first non-coupledanalyses are utilized to form a coupled system according to Eq. 4.13 or equivalently Eq.4.31. Each such non-coupled eigenmode will be multiples of half cosin functions becausethe ends are rigid. In Fig. 4 the first five computed non-coupled fluid eigenmodes areshown. Finally the eigenmodes of the original system are computed according to Eq.4.15 or Eq. 4.33.

-1

0

1

0 1 2 3

Figure 4. The first five non-coupled fluid modes, i.e. computed with rigid boundaries.The vertical axis shows the fluid pressure. The horizontal axis shows the distance fromthe piston in meters.

The result for case I is shown in Table 1 and Fig. 5; the zero frequency mode isexcluded in the figure. Using only 5 non-coupled fluid eigenmodes, the behavior of thesystem is very well captured. Using all existing fluid modes, 51 in all, gives exactlythe same result as the result based on the unsymmetric system, Eq. 4.1, using anunsymmetric eigenvalue solver. This is of course to be expected because the steps passingfrom the original unsymmetric system to the derived symmetric standard eigenvalueproblem are similarity transformations.

Table 1. First six eigenfrequencies for case I, using 5, 10 and 15 fluid modes.

analytical no. of fluid modes used5 10 15 51

0.0 0.0 0.0 0.0 0.0144.0 149.0 146.2 145.4 144.0362.4 378.4 369.5 366.8 362.6594.1 622.0 605.4 601.1 594.6830.1 874.6 845.2 839.6 831.6

1069.1 1219.1 1088.0 1081.2 1072.3

27

At x = 0 the fluid pressure is less then 1, due to the flexible end. It is noteworthy howjust a limited set catches the overall behaviour, and especially the pressure maximum.The limited set of eigenmodes cannot completely capture the high frequency content atthe piston end, i.e. the slope at x = 0, compare Fig. 5 a) - c) to Fig. 5 d) and to theanalytical result presented in Fig. 3

a) b)

-1

0

1

0 1 2 3-1

0

1

0 1 2 3c)

d)

-1

0

1

0 1 2 3-1

0

1

0 1 2 3

Figure 5. Coupled fluid eigenmodes for case I. a) using 5, b) using 10, c) using 15,d) using all non-coupled fluid modes. The vertical axis shows the fluid pressure. Thehorizontal axis shows the distance from the piston in meters.

The result for case II is shown in Table 2 and Fig. 6. Data given to the piston yieldspractically a free end. Even for this case the limited set of eigenmodes captures thebehaviour, in particular keeping the pressure at x = 0 to zero.

28

Table 2. First six eigenfrequencies for case II, using 5, 10 and 15 fluid modes.

analytical no. of fluid modes used5 10 15 51

0.0 0.0 0.0 0.0 0.0125.0 130.9 127.7 126.7 125.0375.0 393.1 383.1 380.1 375.1625.0 657.3 639.1 633.9 625.6874.9 926.7 895.9 888.5 876.7

1124.9 2931.1 1154.1 1144.0 1128.7

a) b)

-1

0

1

0 1 2 3-1

0

1

0 1 2 3

c) d)

-1

0

1

0 1 2 3-1

0

1

0 1 2 3

Figure 6. Coupled fluid eigenmodes for case II. a) using 5, b) using 10, c) using 15,d) using all non-coupled fluid modes. The vertical axis shows the fluid pressure. Thehorizontal axis shows the distance from the piston in meters.

29

Fluid backed panel The next example is a two-dimensional box, 1.5× 0.45 m2, withone flexible side, shown in Fig. 7. The finite element model consists of 30 beam elementsand 270 8-node fluid elements. The beam has fixed supports at each end. In total thereare 93 structural degrees-of-freedom and 889 fluid degrees-of-freedom. The following datawere used in the fluid, ρ = 1000 kg/m3 and c = 1500 m/s. The beam were given thefollowing data, E = 2.11× 1011 N/m2, I = 8.33× 10−7 m4, cross section area A = 0.001m2 and mass per unit length ms = 2 kg/m.

0.45×1.5 m2

9×30 fluid elements

Flexibel side, 30 beam elements

Figure 7. A two-dimensional box with one flexible side filled with water, length 1.5 m,height 0.45 m.

Table 3 show the resonance frequencies for the structure vibrating in vacuo and thefluid using rigid boundaries.

One test run has been performed using 30 structural modes and 100 fluid modes.The result indicates a very good agreement between the full unsymmetric system andthe reduced proposed scheme, see Table 4. The mode numbering in Table 4 are givenaccording to the full unsymmetric system. Some modes in the proposed scheme havebeen altered e.g. mode 5 and 6. Even higher modes are correctly represented, e.g. mode24. The first 25 modes are computed according to the proposed scheme with less than5% error compared to the full system. The shape of the eigenmodes from the proposedstandard eigenvalueproblem also agree very well with those from the complete system,see Fig. 8 and 9.

30

Table 3. Resonance frequencies in Hz for structure in vacuo and fluid with rigid bound-aries.

Structure Fluid468 0

1290 5002530 10003417 15004182 16676247 17406844 19448726 20001029 22421162 25001376 26031493 30001727 30051865 3333

Table 4. Coupled resonance frequencies in Hz. Left column represents results usingthe complete unsymmetric system, Eq. 2.1. Right column using Eq. 3.10 based on 30structural modes and 100 fluid modes.

Mode Unsymmetric Proposedno system system1 87 882 228 2333 447 4654 688 7205 900 9416 904 9367 1241 13038 1275 13019 1672 175810 1707 172911 2106 220712 2169 218713 2416 2539

· · ·18 2925 3008

· · ·24 3675 3747

· · ·

31

87 Hz 88 Hz

228 Hz 233 Hz

447 Hz 465 Hz

688 Hz 720 Hz

Figure 8. Fluid mode shapes for the first 4 coupled modes. The surface indicates fluidpressure. Left column shows result based on the complete unsymmetric system. Rightcolumn shows result based on the proposed scheme.

4.7 Concluding remarks

This paper proposes a new scheme for solving a class of unsymmetrical coupled fluid-structure problems. The eigenvalues and eigenvectors of the structural and fluid subsys-tems may be extracted by using symmetric procedures. The coupled system is formedfrom the eigenvalues and eigenvectors of the subsystems, and turned into a standardsymmetric eigenvalue problem by some very simple steps.

The left eigenvalue problem is easily created using the same information that producedthe right eigenvaluproblem; thus transient analysis can readily be performed.

It is also shown how a limited set of eigenvectors from each subdomain give a very

32

900 Hz 941 Hz

1241 Hz 1303 Hz

2416 Hz 2539 Hz

3675 Hz 3747 Hz

Figure 9. Coupled fluid mode shapes no. 5, 7, 13 and 24. The surface indicates fluidpressure. Left column shows result based on the complete unsymmetric system. Rightcolumn shows result based on the proposed scheme.

good representation of the coupled problem.A central role of the coupled problem addressed is played by the coupling matrix, B in

this paper. The information concerning the coupled transformed system is represented byΦTs BΦTf . It seems reasonable that the participating modes of a specific coupled analysis

may be selected on the basis of the content of ΦTs BΦTf . This is a matter for furtherinvestigation.

33

5 Acoustic and Structure-Acoustic Implementations

5.1 Introduction

This Section is an excerpt from the educational software Calfem (17). The acousticroutines has been developed primarily by Goran Sandberg, Per-Anders Wernberg, PeterDavidsson and Hakan Carlsson.

Calfem is an interactive computer program for teaching the finite element method(FEM). The name Calfem is an abbreviation of ”Computer Aided Learning of the FiniteElement Method”. The program can be used for different types of structural mechanicsproblems and field problems.

Calfem, the program and its built-in philosophy have been developed at the Divisionof Structural Mechanics starting in the late 70’s. Many co-workers, former and present,have been engaged in the development at different stages.

The software is distributed under license conditions free of charge.See http://www.byggmek.lth.se/Calfem/index.htm for more information.

Figure 10. Calfem logo

34

5.2 Element routines in Calfem

5.3 Acoustic elements

Acoustic elements

p2

p3

p1

aco2td

p4 p3

p1

p2

aco2i4d

p4

p1

p2

p7

p6p8

p5

p3

aco2i8d

p4 p3

p1 p2

p7

p6

p8

p5

aco3i8d

2D acoustic elements

aco2td Compute element matrices for a triangular elementaco2i4d Compute element matrices, four-node isoparametric elementaco2i8d Compute element matrices, eight-node isoparametric element

3D acoustic elements

aco3i8d Compute element matrices, eight-node isoparametric element

35

Purpose:Compute the element stiffness and consistent mass matrices for a triangular acous-tic element. See also Appendix A.1

(x1,y1 )

(x3,y3 )

(x2,y2 )

x

y

p2

p3

p1

Syntax:[Ke,Me]=aco2td(ex,ey,ep)[Ke,Me,fe]=aco2td(ex,ey,ep,eq)

Description:aco2td provides the element stiffness matrix Ke, the element mass matrix Me, andthe element load vector fe for a triangular acoustic element.The element nodal coordinates x1, y1, x2 etc, are supplied to the function by exand ey. The element thickness t and speed of sound c are supplied by ep.

ex = [ x1 x2 x3 ]ey = [ y1 y2 y3 ]

ep = [ t c ]

If the scalar variable eq is given in the function, the vector fe is computed, using

eq =[∂2Q/∂t2

]

where Q is the the mass inflow per unit volume.

Theory:The element stiffness matrix Ke, the element mass matrix Me, and the elementload vector fel , stored in Ke, Me, and fe, respectively, are computed according to

Ke = c2t (C−1)T∫

A

BT B dA C−1

Me = t (C−1)T∫

A

NT N dA C−1

feq = c2t (C−1)T∫

A

NT Q dA

36

if an evenly distributed mass inflow per unit volume Q is present. The evaluationof the integrals for the triangular element is based on the linear approximationp(x, y) of the acoustic pressure and is expressed in terms of the nodal variables p1,p2 and p3 as

p(x, y) = Neae = N C−1ae

where

N = [ 1 x y ] C =

1 x1 y11 x2 y21 x3 y3

ae =

p1

p2

p3

and hence it follows that

B = ∇N =

[0 1 00 0 1

]

∇ =

∂

∂x

∂

∂y

.

Evaluation of the integral for the triangular element yields

Ke = c2t (C−1)T BT B C−1 A

Me = t (C−1)T NT N C−1 A

feq =c2t QA

3[1 1 1]

T

where the element area A is determined by

A =1

2detC

37

Purpose:

Compute element stiffness matrix and consistent mass matrix for a four-nodeisoparametric acoustic element.

(x4,y4 )

x

y

(x1,y1 )

(x3,y3 )

(x2,y2 )

p4 p3

p1

p2

Syntax:

[Ke,Me]=aco2i4d(ex,ey,ep)[Ke,Me,fe]=aco2i4d(ex,ey,ep,eq)

Description:

aco2i4d provides the element stiffness matrix Ke, the element mass matrix Me, andthe element load vector fe for a four-node isoparametric acoustic element.

The element nodal coordinates x1, y1, x2 etc, are supplied to the function by exand ey. The element thickness t, the speed of sound c and the Gauss point numbern are supplied to the function by ep. The number of integration points availableare (n× n) with n = 1, 2, 3.

ex = [ x1 x2 x3 x4 ]ey = [ y1 y2 y3 y4 ]

ep = [ t c n ]

If the scalar variable eq is given in the function, the element load vector fe iscomputed, using

eq =[∂2Q/∂t2

]


Theory:

The element stiffness matrix Ke, the element mass matrix Me, and the elementload vector fel , stored in Ke, Me, and fe, respectively, are computed according to

38

Ke = c2t

∫

A

BeT Be dA,

Me = t

∫

A

NeT Ne dA,

fel = c2t

∫

A

NeT Q dA.

The evaluation of the integrals for the isoparametric four-node element is based onan approximation p(ξ, η) of the acoustic pressure, expressed in a local coordinatessystem in terms of the nodal variables p1, p2, p3, and p4 as

p(ξ, η) = Neae

where

Ne = [ Ne1 Ne

2 Ne3 Ne

4 ] ae = [ p1 p2 p3 p4 ]T

The element shape functions are given by

Ne1 =

1

4(1 − ξ)(1 − η) Ne

2 =1

4(1 + ξ)(1 − η)

Ne3 =

1

4(1 + ξ)(1 + η) Ne

4 =1

4(1 − ξ)(1 + η)

The Be-matrix is given by

Be = ∇Ne =

∂

∂x

∂

∂y

Ne = (JT )−1

∂

∂ξ

∂

∂η

Ne

where J is the Jacobian matrix

J =

∂x

∂ξ

∂x

∂η

∂y

∂ξ

∂y

∂η

Evaluation of the integrals is done by Gauss integration.

39

Purpose:Compute element stiffness matrix and consistent mass matrix for an eight-nodeisoparametric acoustic element.

x

y

p4

p1

p2

p7

p6p8

p5

p3

Syntax:[Ke,Me]=aco2i8d(ex,ey,ep)[Ke,Me,fe]=aco2i8d(ex,ey,ep,eq)

Description:aco2i8d provides the element stiffness matrix Ke, the element mass matrix Me, andthe element load vector fe for an eight-node isoparametric acoustic element.The element nodal coordinates x1, y1, x2 etc, are supplied to the function by exand ey. The element thickness t, the speed of sound c, and the Gauss point numbern are supplied to the function by ep. The number of integration points availableare (n× n) with n = 1, 2, 3.

ex = [ x1 x2 x3 . . . x8 ]ey = [ y1 y2 y3 . . . y8 ]

ep = [ t c n ]

If the scalar variable eq is given in the function, the vector fe is computed, using

eq =[∂2Q/∂t2

]

where Q is the the mass inflow per unit volume.Theory:


Ke = c2t

∫

A

BeT Be dA,

Me = t

∫

A

NeT Ne dA,

fel = c2t

∫

A

NeT Q dA.

40

The evaluation of the integrals for the 2D isoparametric eight-node element isbased on an approximation p(ξ, η) of the acoustic pressure, expressed in a localcoordinates system in terms of the nodal variables p1 to p8 as

p(ξ, η) = Neae

where

Ne = [ Ne1 Ne

2 Ne3 . . . Ne

8 ] ae = [ p1 p2 p3 . . . p8 ]T


Ne1 = −1

4(1 − ξ)(1 − η)(1 + ξ + η) Ne

5 =1

2(1 − ξ2)(1 − η)

Ne2 = −1

4(1 + ξ)(1 − η)(1 − ξ + η) Ne

6 =1

2(1 + ξ)(1 − η2)

Ne3 = −1

4(1 + ξ)(1 + η)(1 − ξ − η) Ne

7 =1

2(1 − ξ2)(1 + η)

Ne4 = −1

4(1 − ξ)(1 + η)(1 + ξ − η) Ne

8 =1

2(1 − ξ)(1 − η2)


Be = ∇Ne =

∂

∂x

∂

∂y

Ne = (JT )−1

∂

∂ξ

∂

∂η

Ne


J =

∂x

∂ξ

∂x

∂η

∂y

∂ξ

∂y

∂η


41

Purpose:

Compute element stiffness matrix and consistent mass matrix for an eight-nodeisoparametric acoustic element.

zx

y

p4 p3

p1 p2

p7

p6

p8

p5

Syntax:

[Ke,Me]=aco3i8d(ex,ey,ep)[Ke,Me,fe]=aco3i8d(ex,ey,ep,eq)

Description:

aco3i8d provides the element stiffness matrix Ke, the element mass matrix Me, andthe element load vector fe for an eight-node isoparametric acoustic element.

The element nodal coordinates x1, y1, z1 x2 etc, are supplied to the function byex, ey and ez. The speed of sound c and the Gauss point number n are suppliedto the function by ep. The number of integration points available are (n× n× n)with n = 1, 2, 3.

ex = [ x1 x2 x3 . . . x8 ]ey = [ y1 y2 y3 . . . y8 ]ez = [ z1 z2 z3 . . . z8 ]

ep = [ c n ]

If the scalar variable eq is given in the function, the element load vector fe iscomputed, using

eq =[∂2Q/∂t2

]


Theory:


42

Ke = c2∫

V

BeT Be dV

Me =

∫

V

NeT Ne dV

feq = c2∫

V

NeT Q dV

The evaluation of the integrals for the 3D isoparametric eight-node element isbased on an approximation p(ξ, η, ζ) of the acoustic pressure, expressed in a localcoordinates system in terms of the nodal variables p1 to p8 as

p(ξ, η, ζ) = Neae

where Ne = [ Ne1 Ne

2 Ne3 . . . Ne

8 ] and ae = [ p1 p2 p3 . . . p8 ]T. The

element shape functions are given by

Ne1 = 1

8 (1 − ξ)(1 − η)(1 − ζ) Ne2 = 1

8 (1 + ξ)(1 − η)(1 − ζ)

Ne3 = 1

8 (1 + ξ)(1 + η)(1 − ζ) Ne4 = 1

8 (1 − ξ)(1 + η)(1 − ζ)

Ne5 = 1

8 (1 − ξ)(1 − η)(1 + ζ) Ne6 = 1

8 (1 + ξ)(1 − η)(1 + ζ)

Ne7 = 1

8 (1 + ξ)(1 + η)(1 + ζ) Ne8 = 1

8 (1 − ξ)(1 + η)(1 + ζ)


Be = ∇Ne =

∂

∂x

∂

∂y

∂

∂z

Ne = (JT )−1

∂

∂ξ

∂

∂η

∂

∂ζ

Ne


J =

∂x

∂ξ

∂x

∂η

∂x

∂ζ

∂y

∂ξ

∂y

∂η

∂y

∂ζ

∂z

∂ξ

∂z

∂η

∂z

∂ζ


43

5.4 Structure-acoustic interface elements

Structure-acoustic elements

p2

p1

u1

u2

u3

u4

u5

u6

p2

p1

u1

u2

u3

u4

u5

u6

cp2s2f

u4

u5

u6

u1

u2

u3

p1

p2

p3

cp2s3f

p4 p3p1

p2

u1

u3 u2

(u4,u5,u6 )

(u10,u11,u12 ) (u7,u8,u9 )

cp4s4f

2D structure-acoustic interface elementscp2s2f Interface between linear acoustic element and two-node beam elementcp2s3f Interface between quadratic acoustic element and two-node beam ele-

ment

3D structure-acoustic interface elementscp4s4f Interface between linear acoustic element and linear solid element

44

Purpose:Compute element matrix for coupling between a linear acoustic element and atwo-node beam element. See also Appendix A.2

(x2,y2 )

(x1,y1 )

p2

p1

u1

u2

u3

u4

u5

u6

x

y(x2,y2 )

(x1,y1 )n

p2

p1

u1

u2

u3

u4

u5

u6

Syntax:[He]=cp2s2f(ex,ey,ep)

Description:cp2s2f provides the element coupling matrix He between a two-node beam elementand a four-node acoustic element.The element nodal coordinates x1, y1, x2 etc, are supplied to the function by exand ey, and the element thickness t by ep.

ex = [ x1 x2 ]ey = [ y1 y2 ]

ep = [ t ]

Theory:The element coupling matrix He, stored in He, is computed according to

45

He = t GT

∫ L

0

NTs Nf dx

where G is the transformation matrix defined in beam2d, see (? ).The evaluation of the integral is based on the local displacement of the beam u(x, t)and the linear approximation p(x, t) of the acoustic pressure along the interface, inlocal coordinates.The displacement of the beam is expressed in terms of the nodal variablesu1, u2, . . . u6 as

u(x, t) = Nesae

where

Nes =

[Nes,1 Ne

s,2 . . . Nes,6

]ae = [ u1(t) u2(t) . . . u6(t) ]

T


Nes,1(x) = 0

Nes,2(x) = 1 − 3

x2

L2+ 2

x3

L3

Nes,3(x) = x(1 − 2

x

L+x2

L2)

Nes,4(x) = 0

Nes,5(x) =

x2

L2(3 − 2

x

L)

Nes,6(x) =

x2

L(x

L− 1)

Note that no interaction is accounted for in the direction tangential to the beam,hence Ne

s,1(x) = 0 and Nes,4(x) = 0.

The acoustic pressure along the interface is expressed in terms of the nodal variablesp1 and p2 as

p(x, t) = Nefa

e

where

Nf = [Nf,1 Nf,2] ae = [p1(t) p2(t)]T


Nf,1 = 1 − x

L

Nf,2 =x

L

The element coupling matrix He is thus a (6 × 2)-matrix, coupling six structuraldegrees-of-freedom to two fluid degrees-of-freedom. After assembling the completeset of element coupling matrices into the global coupling matrix H, it can be usedto form the coupled structure-fluid system according to

46

[Ms 0

ρc2HT Mf

] [

d

p

]

+

[Ks −H

0 Kf

] [d

p

]

=

[fsfq

]

The usage of He is further discussed in the examples.

See also:assem ns

47

Purpose:Compute element matrix for coupling between a quadratic acoustic element and atwo-node beam element.

x

y(x3,y3 )

(x1,y1 )

(x2,y2 )

n

u4

u5

u6

u1

u2

u3

p1

p2

p3

Syntax:[He]=cp2s3f(ex,ey,ep)

Description:cp2s3f provides the element coupling matrix He between a two-node beam elementand an eight-node acoustic element.The element nodal coordinates x1, y1, x2 etc, are supplied to the function by exand ey, and the element thickness t by ep.

ex = [ x1 x2 x3 ]ey = [ y1 y2 y3 ]

ep = [ t ]


He = t GT

∫ L

0

NTs Nf dx

where G is the transformation matrix defined in beam2d, see (? ).The evaluation of the integral is based on the local displacement of the beam u(x, t)and a quadratic approximation p(x, t) of the acoustic pressure along the interface,in local coordinates.The displacement of the beam is expressed in terms of the nodal variablesu1, u2, . . . u6 as

u(x, t) = Nesae

48

where

Nes =

[Nes,1 Ne

s,2 . . . Nes,6

]ae = [ u1(t) u2(t) . . . u6(t) ]

T


Nes,1(x) = 0

Nes,2(x) = 1 − 3

x2

L2+ 2

x3

L3

Nes,3(x) = x(1 − 2

x

L+x2

L2)

Nes,4(x) = 0

Nes,5(x) =

x2

L2(3 − 2

x

L)

Nes,6(x) =

x2

L(x

L− 1)

Note that no interaction is accounted for in the direction tangential to the beam,hence Ne

s,1(x) = 0 and Nes,4(x) = 0.

The acoustic pressure along the interface is expressed in terms of the nodal variablesp1, p2 and p3 as

p(x, t) = Nefa

e

where

Nf = [Nf,1 Nf,2 Nf,3] ae = [p1(t) p2(t) p3(t)]T


Nf,1 =1

2x(x− 1)

Nf,2 = 1 − x2

Nf,3 =1

2x(x+ 1)

The element coupling matrix He is thus a (6 × 3)-matrix, coupling six structuraldegrees-of-freedom to two fluid degrees-of-freedom. After assembling the completeset of element coupling matrices into the global coupling matrix H, it can be usedto form the coupled structure-fluid system according to

[Ms 0

ρc2HT Mf

] [

d

p

]

+

[Ks −H

0 Kf

] [d

p

]

=

[fsfq

]


See also:assem ns

49

Purpose:Compute element matrix for coupling between a linear acoustic element and alinear solid element.

zx

y

n

(x1,y1 )

p4 p3p1

p2

u1

u3 u2

(u4,u5,u6 )

(u10,u11,u12 ) (u7,u8,u9 )

Syntax:[He]=cp4s4f(ex,ey)

Description:cp4s4f provides the element coupling matrix He between a four-node linear solidelement and a four-node linear acoustic element.The element nodal coordinates x1, y1, x2 etc, are supplied to the function by exand ey, and the Gauss point number n is supplied to the function by ep. Thenumber of integration points available are (n× n) with n = 1, 2, 3.

ex = [ x1 x2 x3 x4 ]ey = [ y1 y2 y3 y4 ]

ep = [ n ]


He = GT

∫

A

NTs Nf dA

where G is the transformation matrix defined in beam2d, see (? ).The evaluation of the integral is based on the local transverse displacement of thesolid surface u(ξ, η, t) and a linear approximation p(ξ, η, t) of the acoustic pressurealong the interface, in local coordinates.The transverse displacement of the solid surface is expressed in terms of the nodalvariables u1, u2, . . . u12 as

u(ξ, η) = Nes u

50

where

Nes =

[Nes,1 Ne

s,2 . . . Nes,12

]ae = [ u1(t) u2(t) . . . u12(t) ]

T


Nes,3 = 1

4 (1 − ξ)(1 − η) Nes,6 = 1

4 (1 + ξ)(1 − η)

Nes,9 = 1

4 (1 + ξ)(1 + η) Nes,12 = 1

4 (1 − ξ)(1 + η)

Because no interaction transverse to the beam is accounted for Nes,1 = Ne

s,2 =Nes,4 = Ne

s,5 = Nes,7 = Ne

s,8 = Nes,10 = Ne

s,11 = 0

The acoustic pressure p(ξ, η) along the interface is expressed in terms of the nodalvariables p1, p2, p3 and p4 as

p(ξ, η) = Nef ae

where

Nef =

[Nef,1 N

ef,2 N

ef,3 N

ef,4

]ae = [p1(t) p2(t) p3(t) p4(t)]

T


Nef,1 = 1

4 (1 − ξ)(1 − η) Nef,2 = 1

4 (1 + ξ)(1 − η)

Nef,3 = 1

4 (1 + ξ)(1 + η) Nef,4 = 1

4 (1 − ξ)(1 + η)

The element coupling matrix He is thus a (12×4)-matrix, coupling twelve structuraldegrees-of-freedom to four fluid degrees-of-freedom. After assembling the completeset of element coupling matrices into the global coupling matrix H, it can be usedto form the coupled structure-fluid system according to

[Ms 0

ρc2HT Mf

] [

d

p

]

+

[Ks −H

0 Kf

] [d

p

]

=

[fsfq

]


See also:assem ns

51

5.5 System functions for structure-acoustic analysis

This section contains a set of general commands and commands for modal analysisperformed on decomposed fluid-structure domains. Before using the modal analysiscommands the eigenvalue analysis on each subdomain, fluid and structure, must havebeen performed.

There are two commands that form the interacting modal structure-fluid system,depending on the desired basic formulation. They form a single symmetric matrix forstandard eigenvalue analysis. After the eigenvalue analysis is performed, the eigenvectorsin the original system can be computed. There are four commands depending on basicformulation and left/right eigenvectors.

System functionsassem ns Non-symmetric assembly of element matrices

fsi mod Modal structure–fluid pressure formulation or modal displacement po-tential formulation

fsi egv Transforms computed coupled eigenvectors from the modal coordinatesback to the FE-coordinate system

52

Purpose:Non-symmetric assembly of element matrices.

Syntax:H=assem ns(edof a,edof b,H,He)

Description:assem ns adds the matrix He, stored in He, to the matrix H, stored in H, accordingto the topology matrices edof a and edof b.The topology matrices edof a and edof b are defined as

edof a = [el dof1 . . . dofned︸︷︷︸

global row index

] edof b = [el dof1 . . . dofned︸︷︷︸

global column index

]

where the first column contains the element number, and the columns 2 to (ned+ 1)contain the corresponding global degrees of freedom (ned = number of elementdegrees of freedom).He is added to H using the row index from edof a and the column index from edof b.In the case where the matrix He is identical for several elements, assembling ofthese can be carried out simultaneously. Each row in edof x, (x=a or b), thenrepresents one element, i.e. nel is the total number of considered elements.

edof x =

el1 dof1 dof2 . . . dofnedel2 dof1 dof2 . . . dofned...

......

...elnel dof1 dof2 . . . dofned

one row for each element

Examples:The following set of commandsH=zeros(5,8); edof_a=[1 1 3]; edof_b=[1 3 5 7]; He=[10 20 30;

40 50 60]

H=assem_ns(edof_a,edof_b,H,He)

yields

H =

0 0 10 0 20 0 30 0

0 0 0 0 0 0 0 0

0 0 40 0 50 0 60 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

53

Purpose:Form the modal coupled system matrices for fluid-structure interaction problems,based on either fluid pressure or fluid displacement potential.

Syntax:Amodc = fsi mod(laS,fiS,listS,laF,fiF,listF,H,fp,string)

Description:The function forms the coupled modal matrices based on fluid pressure p, or fluiddisplacement potential ψ.laS and fiS contain the structural eigenvalues and eigenvectors in vacuo as computedbe the eigen-command. laF and fiF contain the fluid eigenvalues and eigenvectorscomputed with stiff boundaries. The eigenvectors shall be normalized, i.e.

φti ·m · φj = δij φti · k · φj = δijλj

listS and listF are lists of eigenvectors to by used in the formation of the coupledsystem. Normally these lists contain fewer eigenvectors than computed. H is thecoupling matrix in the original coordinate system and fp is a vector that containsthe fluid properties ρ and c. In summary we have

laX = [λ1 λ2 ... λn] fiX = [φ1 φ2 ... φn]listX = [m1 m2 ...mr] (r ≤ n) fp = [ρ c]

The output, Amodc, is a system matrix for standard eigenvalue analysis. Forstring=‘right’ the output is the right modal p-system or the left modal ψ-system:

Ds −√

ρc2√

DsΦTs HΦf

−√

ρc2ΦTf HTΦs√

Ds Df + ρc2ΦTf HTΦsΦTs HΦf

For string=‘left’ the output is the left modal p-system or the right modal ψ-system:

Ds + ρc2ΦTs HΦfΦTf HTΦs −

√

ρc2ΦTs HΦf√

Df

−√

ρc2√

DfΦTf HTΦs Df

54

Purpose:Compute the eigenvectors in the original coordinate system.

Syntax:V = fsi egv(v,laS,fiS,listS,laF,fiF,listF,fp,string1,string2)

Description:The eigenvalue analysis of the standard system generated by fsi mod gives a set ofeigenvectors in modal coordinates. This command computes the eigenvectors inthe original coordinate system.v contains the computed eigenvectors from Amodc. All other input fields are de-scribed in fsip egv. V contains the eigenvectors in the original coordinate system,both the structural part and the fluid part.The transformation depends on the formulation, p or ψ, and whether the left orright eigenvectors is desired. The following multiplications are performed, depend-ing on string1 and string2,

string1 string1

’right’ ’pr’ V rp =

[Φs 00 Φf

] [ (√

ρc2√

Ds

)−1

0

0 If

]

v

’left’ ’pr’ V lp =

[Φs 00 Φf

][ Is 0

0(√

ρc2√

Df

)−1

]

v

’right’ ’di’ V rψ =

[Φs 00 Φf

]

Is 0

0

√

c2

ρ

(√Df

)−1

v

’left’ ’di’ V lψ =

[Φs 00 Φf

]

√

c2

ρ

(√Ds

)−10

0 If

v

The superscript denotes the corresponding left or right eigenvector system. Thefirst and the last choice above shall be used in combination with the eigenvectorscomputed from Amodc from fsi mod with string=‘left’ and the second and thirdchoice in combination with the eigenvectors computed from Amodc from fsi modwith string=‘right’.

55

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substructuring using the Guyan condensation method. Computers & Structures,45(5/6):941–946, 1992.

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[9] N. Bouhaddi and R. Fillod. Model reduction by a simplified variant of dynamiccondensation. Journal of Sound and Vibration., 191(2):233–250, 1996.

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[24] Everstine G. C. A symmetric potential formulation for fluid-structure interaction.Journal of Sound and Vibration, 71(1):157–160, 1981.

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[28] C.A. Felippa, Symmetrization of the contained compressible–fluid vibration eigen-problem, Communications in applied numerical methods, Vol. 1, 241-247 (1985).

[29] F. Fahy. Sound and structural vibration. Academic Press, London, 1985.

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[37] Hong K. L. and Kim J. Analysis of free vibration of structural-acoustic coupledsystems, part ii: Two- and three-dimensional examples. Journal of Sound andVibration., 188(4):577–600, 1995.

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[41] H. Morand and R. Ohayon. Substructure variational analysis of the vibrations ofcoupled fluid-structure systems. finite elemtent results. International Journal ofNumerical Methods in Engineering, 14:741–755, 1979.

[42] D. J. Nefske, J. A. Wolf, and L. J. Howell. Structural-acoustic finite element analysisof the automobile compartment: A review of current practice. Journal of Sound andVibration, 80(2):247–266, 1982.

[43] H. Morand and R. Ohayon. Fluid structure interaction. John Wiley & sons, Chich-ester, 1995.

[44] R. Ohayon. Reduced symmetric models for modal analysis of internal structural-acoustic and hydroelastic-sloshing systems. Computer methods in applied mechanicsand engineering, 190:3009–3019, 2001.

[45] R. Ohayon and R. Valid, True symmetric formulation of free vibrations for fluid-structure interaction in bounded media, in Numerical Methods in Coupled Systems(eds R.W. Lewis, P. Bettess and E. Hinton), Wiley, Chichester, (1984).

[46] N. Ottosen and H. Peterson. Introduction to the Finite Element Method. PrenticeHall, New York, 1992.

[47] A. J. Pretlove. Free vibrations of a rectangular panel backed by a closed rectangularcavity. Journal of Sound and Vibration, 2(3):197–209, 1965.

[48] A. J. Pretlove. Forced vibrations of a rectangular panel backed by a closed rectan-gular cavity. Journal of Sound and Vibration., 3(3):252–261, 1966.

[49] J. Pan and D. A. Bies. The effect of fluid-structural coupling on sound waves inan enclosure–theoretical part. The Journal of the Acoustical Society of America,87(2):691–707, 1990.

[50] M. Petyt, J. Lea, and G. H. Koopmann. A finite element method for determiningthe acoustic modes of irregular shaped cavities. Journal of Sound and Vibration,45(4):495–502, 1976.

[51] G. Sandberg and P. Goransson, A Symmetric Finite Element Formulation for Acous-tic Fluid-Structure Interaction Analysis. Journal of Sound and Vibration 123(3),507-515 (1988).

[52] Sandberg, G.: Domain Decomposition in Structure-Acoustic Analysis, InternationalConference on Computational Engineering Science, 30 July - 3 August, pp. 857–862,1995.

58

[53] G. Sandberg. A new strategy for solving fluid-structure problems. InternationalJournal of Numerical Methods in Engineering, 38:357–370, 1995.

[54] G. Sandberg. Finite element modelling of fluid-structure interaction. TVSM 1002,Structural Mechanics, LTH, Lund University, Box 118, SE-221 00 Lund, Sweden,1986.

[55] P. Seshu. Substructuring and component mode synthesis. Shock and Vibration,4(3):199–210, 1997.

[56] V. N. Shah. Analytical selection of masters for the reduced eigenvalue problem.International Journal of Numerical Methods in Engineering, 18:89–98, 1982.

[57] W.-H. Shyu, Z.-D. Ma, and G. M. Hulbert. A new component mode synthesismethod: Quasi-static mode compensation. Finite elements in Analysis and Design,24:271–281, 1997.

[58] M. A. Tournour, N. Atalla, O. Chiello, and F. Sgard. Validation, performance,convergence and applicaion of free interface component mode synthesis. Computers& Structures, 79:1861–1876, 2001.

[59] D.-M. Tran. Component mode synthesis methods using interface modes. Computers& Structures, 79:209–222, 2001.

[60] P.-A. Wernberg. Structur-acoustic analysis; Methods, implementations and applica-tions. TVSM 1019, Structural Mechanics, LTH, Lund University, Box 118, SE-22100 Lund, Sweden, 2006.

[61] J. A. Wolf. Modal synthesis for combined structural-acoustic systems. AIAA Jour-nal, 15:743–745, 1977.

[62] J. Wandinger. A symmetric Craig-Bampton method of coupled fluid-structure sys-tems. Engineering Computation, 15(4):450–461, 1998.

[63] E. L. Wilson and E. P. Bayo. Use of special Ritz vectors in dynamic substructureanalysis. Journal of Structural Engineering, 112(8):1944–1954, 1986.

[64] O.C. Zienkiewicz and P. Bettess, Fluid–structure dynamic interaction and waveforces, International journal of numerical methods in engineering, Vol. 13, 1-16(1978).

[65] O.C. Zienkiewicz, D.K. Paul and E. Hinton, Cavitation in fluid–structure response(with particular reference to dams under earthquake loading), Journal of EarthquakeEngineering and Structural Dynamics, Vol. 11, 463-481, (1983).

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59

A Finite element routines

A.1 Routine aco2td.m

function [Ke,Me,fe]=aco2td(ex,ey,ep,eq)

% [Ke,Me]=aco2td(ex,ey,ep)

% [Ke,Me,fe]=aco2td(ex,ey,ep,eq)

%----------------------------------------------------------

% PURPOSE

% Compute element stiffness and consistent element

% mass matrices for the triangular acoustic element.

%

% INPUT: ex = [x1 x2 x3]

% ey = [y1 y2 y3] element coordinates

%

% ep = [t c raa] thickness,speed of sound and

% density

%

% eq mass inflow per unit volume and time

% (second derivative)

%

% OUTPUT: Ke : element stiffness matrix (3 x 3)

% Me : element mass matrix (3 x 3)

% fe : element load vector (3 x 1)

%----------------------------------------------------------

% LAST MODIFIED: G Sandberg 1996-03-09

% Copyright (c) Division of Structural Mechanics and

% Department of Solid Mechanics.

% Lund Institute of Technology

%----------------------------------------------------------

t=ep(1); c=ep(2); raa=ep(3);

if nargin==3; qe=0; end

%

C=[ones(3,1) ex’ ey’]; B=[0 1 0;

0 0 1 ]*inv(C); A=1/2*det(C);

Ke1=t*c*c*B’*B*A;

%

Me1=t*A/12*[2 1 1;

1 2 1;

1 1 2];

%

fe1=t*c*c*qe*A/3*[1 1 1]’;

Ke=Ke1; Me=Me1; fe=fe1;

%------------------------- end -----------------------------

60

A.2 Routine cp2s2f.m

function [he]=cp2s2f(ex,ey,ep)

% [he]=cp2s2f(ex,ey,ep)

%-----------------------------------------------------------

% PURPOSE

% Compute element coupling matrix between a 4 node

% isoparametric acoustic element and a 2 node beam element.

%

% 1-------------2 Beam

% *-------------*

% 1-------------2 Fluid

% | |

% | |

%

%

% INPUT: ex = [x1 x2] element coordinates

% ey = [y1 y2]

%

% ep = [t] thickness

%

% OUTPUT: he : element coupling matrix (6 x 2)

%-----------------------------------------------------------

% LAST MODIFIED: G Sandberg 1996-03-08

% Copyright (c) Division of Structural Mechanics and

% Department of Solid Mechanics.

% Lund Institute of Technology

%-------------------------------------------------------------

t=ep(1);

b=[ex(2)-ex(1);

ey(2)-ey(1)];

L=sqrt(b’*b);

% Form local coupling matrix according, exact integration.

% detJ = L because integration is performed for the interval [0,1].

Cle = L*[ 0 0 ;

7/20 3/20 ;

L/20 L/30 ;

0 0 ;

61

3/20 7/20 ;

-L/30 -L/20] ;

% Local to global transformation matrix G

n=b/L;

G=[n(1) n(2) 0 0 0 0;

-n(2) n(1) 0 0 0 0;

0 0 1 0 0 0;

0 0 0 n(1) n(2) 0;

0 0 0 -n(2) n(1) 0;

0 0 0 0 0 1];

Ce=G’*Cle;

he=t*Ce;

%-------------------------- end -------------------------------

62

B Calfem Examples

To illustrate the different choices of computational strategies possible in Calfem, weconsider a two-dimensional problem consisting of a rectangular fluid box, 8 m × 20 m,and, on the upper fluid boundary, a flexible structure. All other sides are rigid. Thestructure is simply supported at both ends. The problem is indicated in the figure below.

8.0

m

20.0 m

Figure 11. Two-dimensional box with a flexible top.

Assume the following data

Fluid: ρ = 1000 kg/m2

c = 1500 m/s2

Structure: E = 2.1×1011 N/m2

I = 1.59×10−4 m4

m = 50 kg/m

Below we shall go through the following steps

• define the fluid domain and solve the eigenvalue problem for the fluid domain

• define the structure domain; and solve the eigenvalue problem for the structuredomain;

• generate and solve the coupled eigenvalue problem corresponding to Eq. (3.60);



• generate and solve the reduced modal coupled system, according to Eq. (3.71) andEq. (3.73).

All necessary Matlab commands and Calfem commands are listed in the Appendix.In this section we refer to command blocks, indicated by e.g. ‘CB–1’. They can be foundin the Appendix at the corresponding marginal mark, e.g. ‘CB–1’ on page 71.

63

B.1 Fluid

To establish the fluid part of the coupled model we need to define the followingmatrices: dof fl, coord fl, and edof fl. The fluid finite elements are all 0.5 × 0.5 m.

11

2 3

22

32 4

2321 22

43 44

2020 21

141148 149

169 170

160167 168

188 189

41 42

8.0

m

20.0 m

Figure 12. The finite element mesh of the fluid domain incl. numbering.

Based on the numbering indicated in Fig. 12, the contents of these matrices are shownbelow. They are created by standard Matlab commands. See CB–1 on page 71.

dof fl

12345678...

1920212223

...188189

coord fl

0 00.5000 01.0000 01.5000 02.0000 02.5000 03.0000 03.5000 0

......

9.0000 09.5000 0

10.0000 00 0.5000

0.5000 0.5000...

...9.5000 4.0000

10.0000 4.0000

edof fl

1 1 2 23 222 2 3 24 233 3 4 25 244 4 5 26 255 5 6 27 266 6 7 28 277 7 8 29 288 8 9 30 29...

......

......

19 19 20 41 4020 20 21 42 4121 22 23 44 4322 23 24 45 44

......

......

...158 165 166 187 186159 166 167 188 187160 167 168 189 188

Table 5. Geometry and topology matrices for the fluid domain.

The finite element mesh can readily be plotted, see CB–2 on page 71 and Fig. 13.

64

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120

121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140

141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160

Figure 13. Finite element fluid mesh.

Finally, the fluid system matrices are created and the corresponding eigenvalue prob-lem is solved. The first nine eigenmodes are plotted below, Fig. 14. See further CB–3and CB–4.

75.08 150.6 188.7

203.1 227.1 241.4

295.3 305 358.6

Figure 14. The first nine fluid eigenmodes and eigenvalues (Hz). The colors indicatethe normalized pressure level.

65

B.2 Structure

To establish the structural part of the coupled model we need to define the followingmatrices: dof st, coord st, and edof st.

1

2

3 4

5

6 7

8

9 58

59

60 61

62

63

1 2 20

Figure 15. Finite element of the structural domain incl. numbering.

The numbering is shown in Fig. 15. The contents of these matrices are shown below.They are created by standard Matlab commands. See CB–5 and CB–6 on page 73.

dof st

1 2 34 5 67 8 9

10 11 1213 14 1516 17 1819 20 2122 23 2425 26 2728 29 3031 32 3334 35 3637 38 3940 41 4243 44 4546 47 4849 50 5152 53 5455 56 5758 59 6061 62 63

coord st

0 4.00000.5000 4.00001.0000 4.00001.5000 4.00002.0000 4.00002.5000 4.00003.0000 4.00003.5000 4.00004.0000 4.00004.5000 4.00005.0000 4.00005.5000 4.00006.0000 4.00006.5000 4.00007.0000 4.00007.5000 4.00008.0000 4.00008.5000 4.00009.0000 4.00009.5000 4.0000

10.0000 4.0000

edof st

1 1 2 3 4 5 62 4 5 6 7 8 93 7 8 9 10 11 124 10 11 12 13 14 155 13 14 15 16 17 186 16 17 18 19 20 217 19 20 21 22 23 248 22 23 24 25 26 279 25 26 27 28 29 30

10 28 29 30 31 32 3311 31 32 33 34 35 3612 34 35 36 37 38 3913 37 38 39 40 41 4214 40 41 42 43 44 4515 43 44 45 46 47 4816 46 47 48 49 50 5117 49 50 51 52 53 5418 52 53 54 55 56 5719 55 56 57 58 59 6020 58 59 60 61 62 63

Table 6. Geometry and topology matrices for the structural domain.

A plot of the structural finite element mesh is shown below.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Figure 16. Finite element structural mesh.

66

B.3 Coupling

In all the different coupled fluid-structure systems, according to Eqs. (3.60), (3.66),and (3.68), there is a coupling matrix present, denoted H. The topology of the couplingis shown in Fig. 17.

1

2

3 4

5

6 7

8

9 58

59

60 61

62

63

148

169

141

170

142

171 188

160

149

189

1 2 20

n

Figure 17. Structure–fluid coupling topology.

The coupling topology is defined by two matrices, edof coup st, edof coup fl,shown below. In this particular case all structural elements are in contact with thefluid; hence edof coup st=edof st. See also CB–7 on page 74.

edof coup st

1 1 2 3 4 5 62 4 5 6 7 8 93 7 8 9 10 11 124 10 11 12 13 14 155 13 14 15 16 17 186 16 17 18 19 20 217 19 20 21 22 23 248 22 23 24 25 26 279 25 26 27 28 29 30

10 28 29 30 31 32 3311 31 32 33 34 35 3612 34 35 36 37 38 3913 37 38 39 40 41 4214 40 41 42 43 44 4515 43 44 45 46 47 4816 46 47 48 49 50 5117 49 50 51 52 53 5418 52 53 54 55 56 5719 55 56 57 58 59 6020 58 59 60 61 62 63

edof coup fl

141 169 170142 170 171143 171 172144 172 173145 173 174146 174 175147 175 176148 176 177149 177 178150 178 179151 179 180152 180 181153 181 182154 182 183155 183 184156 184 185157 185 186158 186 187159 187 188160 188 189

Table 7. Topology matrices for the structure-fluid coupling.

67

Note that the element numbers and the degree-of-freedom numbers are separate forthe structural mesh and the fluid mesh, i.e. the same number occurs in both meshes.Further, the degree-of-freedom shall be given so that the outward normal in the fluiddomain is positive in a right system. The coupling matrix H is assembled by the commandassem ns from the coupling elements he, see CB–8 on page 74.

B.4 The complete system

There are three means of establishing the coupled fluid-structure system describedin this report, see Eqs. (3.60), (3.66), and (3.68). The corresponding coupled systemmatrices are readily generated in Calfem and the corresponding eigenvalue problemsolved, see CB–9, CB–10, and CB–11 starting on page 74.

The table below lists the eigenvalues from the different formulations. For comparison,the structural eigenvalues in vacuo and the fluid eigenvalues with all sides rigid are listedas well.

freq p freq psi freq sym

Eq. (3.60) Eq. (3.66) Eq. (3.68)

6.5755 6.5755 6.575520.3112 20.3112 20.311243.8516 43.8516 43.851675.9558 75.9558 75.955894.6997 94.6997 94.6997

113.0973 113.0973 113.0973130.9446 130.9446 130.9446172.6340 172.6340 172.6340187.8503 187.8503 187.8503241.9371 241.9371 241.9371254.0351 254.0351 254.0350280.0823 280.0823 280.0823

freq fl freq st

75.077 51.346150.618 115.531188.707 205.404203.093 320.993227.087 458.729241.446 462.357295.261 629.601304.956 822.890358.620 920.289384.697 1042.476384.697 1288.706391.955 1387.526

Table 8. Comparison between the first twelve eigenvalues from different fluid-structuresystems (Hz). The eigenvalues from the fluid and structure are given to the right.

Because the three systems for fluid-structure interaction are mathematically equiva-lent, the eigenvalues are the same and, from that standpoint, either can be used. Thep-system has the advantage of the fluid pressure being a more natural physical variable.For large systems and when ρ 6= 1, the ψ-system is more numerically stable, however.

The eigenmodes from the p-system are shown below, Fig. 18.

68

6.576 20.31 43.85

75.96 94.7 113.1

130.9 172.6 187.9

Figure 18. The first nine eigenmodes and eigenvalues (Hz) from the interacting fluid-structure system. The colors indicate the normalized pressure level.

The corresponding commands for generating Fig. 18 are found in CB–12, page 76.

B.5 A reduced modal form

In Sec. 3.7, the theory for the reduced modal form of the interacting fluid-structuresystem is outlined. The benefit of this procedure is the possibility to pick a limited set ofeigenmodes from the structure domain and the fluid domain, and the fact that it resultsin standard symmetric eigenvalue problems.

In this example 15 structural eigenvectors and 50 fluid eigenvectors are used. SeeCB–13, and CB–14, page 76. The eigenvalues are listed in Table 9. Note that theright and left eigenvalue system yields the same eigenvalues. The eigenmodes from thereduced system can be transformed back to the original coordinate system, see CB–15.The eigenmodes are shown in Fig. 19, see also CB–16.

69

freq pr mod freq pl mod freq p

Eq. (3.71) Eq. (3.74) Eq. (3.60)

6.7176 6.7176 6.575521.2584 21.2584 20.311246.7860 46.7860 43.851681.9979 81.9979 75.955898.6791 98.6791 94.6997

118.0772 118.0772 113.0973143.2662 143.2662 130.9446177.4479 177.4479 172.6340213.2568 213.2568 187.8503246.1991 246.1991 241.9371

Table 9. Comparison between first ten eigenvalues (Hz) from the right and left reducedfluid-structure systems. The eigenvalues from the complete system are also listed.

70

6.718 21.26 46.79

82 98.68 118.1

143.3 177.4 213.3

Figure 19. The first nine eigenmodes and eigenvalues (Hz) from the reduced interactingfluid-structure system. The colors indicate the normalized pressure level.

71

C Calfem input files

C.1 Fluid

The following set of commands generate coordinate information and the topology in-formation for the fluid. Matlab commands are used for generating the different matriciesthat are necessary to build the system matrices.

The first command block creates the following matrices: dof fl, coord fl, andedof fl.

CB–1

% ----------------------------------------------------------------------

dof_fl=[1:1:189]’;

el_no_fl=[1:1:160]’;

first_el=[1 2 23 22];

first_el_row=[];

for i=1:20

first_el_row=[first_el_row; first_el+(i-1)];

end

all_el_fl=[];

for i=1:21:160

all_el_fl=[all_el_fl; first_el_row+(i-1)];

end

edof_fl=[el_no_fl,all_el_fl];

x_coord_first_node_row=[0.0:0.5:10]’;

y_coord_first_node_row=zeros(21,1);

coord_first_node_row=[x_coord_first_node_row, y_coord_first_node_row];

coord_fl=[];

add=[zeros(21,1), zeros(21,1)+0.5];

for i=1:1:9,

coord_fl=[coord_fl; coord_first_node_row+(i-1)*add];

end

% ----------------------------------------------------------------------

To plot the finite element mesh:

CB–2

% ----------------------------------------------------------------------

plotpar=[1,1,0];

[ex_fl,ey_fl]=coordxtr(edof_fl,coord_fl,dof_fl,4);

eldraw2(ex_fl,ey_fl,plotpar,edof_fl(:,1));

72

% ----------------------------------------------------------------------

To generate the fluid mass matrix and the fluid stiffness matrix, and solve the corre-sponding eigenvalue problem:

CB–3

% ----------------------------------------------------------------------

c=1500; rho=1000;

ep_fl=[1 c rho 3];

ndof_fl=max(max(dof_fl));

K_fl=zeros(ndof_fl,ndof_fl);

M_fl=K_fl;

for i=1:length(ex_fl)

[ke_fl,me_fl]=aco2i4d(ex_fl(i,:),ey_fl(i,:),ep_fl);

K_fl=assem(edof_fl(i,:),K_fl,ke_fl);

M_fl=assem(edof_fl(i,:),M_fl,me_fl);

end

[La_fl,Egv_fl]=eigen(K_fl,M_fl);

freq_fl=sqrt(La_fl)/2/pi;

% ----------------------------------------------------------------------

To plot the first nine fluid eigenmodes, excluding the zero eigenmode:

CB–4

% ----------------------------------------------------------------------

clf; modnr=1;

for j=1:3, for i=1:3

modnr=modnr+1;

Ed=extract(edof_fl,Egv_fl(:,modnr));

Edabs=abs(Ed);

const=max(max(Edabs)); Ed=Ed/const;

h=fill(ex_fl’+10*1.1*(i-1),ey_fl’-4*1.6*(j-1),Ed’);

MOD= num2str(freq_fl(modnr));

text(10*1.1*(i-1)+4,-4*1.6*(j-1)-1, MOD)

set(h,’edgecolor’,’none’)

hold on,

end, end

axis equal, axis off

% ----------------------------------------------------------------------

73

C.2 Structure

To generate the coordinate information and topology information and to plot thefinite element mesh:

CB–5

% ----------------------------------------------------------------------

el_no_st=[1:1:20]’;

coord_st=[[0:0.5:10]’,4*ones(21,1)];

first_dof_st=[1:3:61]’;

dof_st=[first_dof_st, first_dof_st+1, first_dof_st+2];

first_edof_st=[1 2 3 4 5 6];

all_el_st=[];

for i=1:1:20, all_el_st=[all_el_st; first_edof_st+3*(i-1)]; end

edof_st=[el_no_st, all_el_st];

plotpar=[1,1,0];

[ex_st,ey_st]=coordxtr(edof_st,coord_st,dof_st,2);

eldraw2(ex_st,ey_st,plotpar,edof_st(:,1));

% ----------------------------------------------------------------------

To generate the structural mass matrix and stiffness matrix and solve the corre-sponding eigenvalue problem (Note that the structure has fixed x-displacement and y-displacement at both ends; hence the boundary condition b=[1 2 61 62]):

CB–6

% ----------------------------------------------------------------------

ndof_st=max(max(dof_st));

E=2.1e11; A=0.02; I=1.59e-4; m=A*2500;

ep_st=[E A I m];

K_st=zeros(ndof_st,ndof_st); M_st=K_st;

for i=1:length(ex_st)

[ke_st,me_st]=beam2d(ex_st(i,:),ey_st(i,:),ep_st);

K_st=assem(edof_st(i,:),K_st,ke_st);

M_st=assem(edof_st(i,:),M_st,me_st);

end

b=[1 2 61 62];

[La_st,Egv_st]=eigen(K_st,M_st,b);

freq_st=sqrt(La_st)/2/pi;

% ----------------------------------------------------------------------

74

C.3 Coupling

In all the different coupled fluid-structure systems, according to Eqs. (3.60), (3.66),and (3.68), there is a coupling matrix present, denoted H.

The coupling topology is described by two matrices, here denoted edof coup fl andedof coup st. In this particular case all structural elements are in contact with the fluid,hence edof coup st=edof st.

CB–7

% ----------------------------------------------------------------------

edof_coup_st=edof_st;

el_no_coup_fl=[141:1:160]’;

first_edof_coup_fl=[169 170];

all_el_coup_fl=[];

for i=1:1:20

all_el_coup_fl=[all_el_coup_fl; first_edof_coup_fl+(i-1)];

end

edof_coup_fl=[el_no_coup_fl, all_el_coup_fl];

% ----------------------------------------------------------------------

To generate the element coupling matricies, he, and assemble into a global couplingmatrix H:

CB–8

% ----------------------------------------------------------------------

H=zeros(ndof_st,ndof_fl);

for i=1:length(ex_st)

he=cp2s2f(ex_st(i,:),ey_st(i,:),[1]);

H=assem_ns(edof_coup_st(i,:),edof_coup_fl(i,:),H,he);

end

% ----------------------------------------------------------------------

C.4 The complete system

There are three means of establishing the coupled fluid-structure system describedin this report, see Eqs. (3.60), (3.66), and (3.68). The straightforward commands togenerate the coupled system and solve the corresponding eigenvalue problem are listedbelow.

The p-system:

CB–9

75

% ----------------------------------------------------------------------

zeroH=0*H;

M=[M_st zeroH;

rho*c^2*H’ M_fl];

K=[K_st -H;

zeroH’ K_fl];

[La_p,Egv_p]=eigen(K,M,b);

freq_p=sqrt(real(La_p))/2/pi;

% ----------------------------------------------------------------------

The ψ-system:

CB–10

% ----------------------------------------------------------------------

M=[M_st rho*H;

zeroH’ M_fl];

K=[K_st zeroH;

-c^2*H’ K_fl];

[La_psi,Egv_psi]=eigen(K,M,b);

freq_psi=sqrt(real(La_psi))/2/pi;

% ----------------------------------------------------------------------

The ψ-p-system:

CB–11

% ----------------------------------------------------------------------

zeroK=0*K_fl;

M=[M_st zeroH zeroH;

zeroH’ rho*c^(-2)*K_fl zeroK;

zeroH’ zeroK zeroK];

K=[K_st zeroH -H;

zeroH’ zeroK c^(-2)*K_fl;

-H’ c^(-2)*K_fl -(rho*c^2)^(-1)*M_fl];

[La_sym,Egv_sym]=eigen(K,M,b);

freq_sym=sqrt(La_sym)/2/pi;

% ----------------------------------------------------------------------

76

To generate a color plot of the coupled eigenmodes. (The first two lines extract theeigenvector parts belonging to the structural domain and the fluid domain, respectively):

CB–12

% ----------------------------------------------------------------------

Egv_p_st=Egv_p(1:ndof_st,:);

Egv_p_fl=Egv_p(ndof_st+1:ndof_st+ndof_fl,:);

clf; modnr=1;


hold on;

modnr=modnr+1;

Ed_st=real(extract(edof_st,Egv_p_st(:,modnr)));

Edabs_st=abs(Ed_st);

const_st=max(max(Edabs_st)); Ed_st=Ed_st/const_st;

eldisp2(ex_st+10*1.1*(i-1),ey_st-4*2.5*(j-1)+2.0,Ed_st,[1,1,0],2);

eldraw2(ex_st+10*1.1*(i-1),ey_st-4*2.5*(j-1)+2.0,[3,1,0]);

hold on;

Ed_fl=real(extract(edof_fl,Egv_p_fl(:,modnr)));

Edabs_fl=abs(Ed_fl);

const_fl=max(max(Edabs_fl)); Ed_fl=Ed_fl/const_fl;

h=fill(ex_fl’+10*1.1*(i-1),ey_fl’-4*2.5*(j-1),Ed_fl’);

MOD= num2str(freq_p(modnr));

text(10*1.1*(i-1)+4,-4*2.5*(j-1)-1, MOD)


end, end


% ----------------------------------------------------------------------

C.5 A reduced modal form

To establish the modal form of the coupled system, right eigensystem:

CB–13

% ----------------------------------------------------------------------

fp=[rho c]

list_st=[1:1:15];

list_fl=[1:1:50];

[Amodc_pr] = ...

fsi_mod(La_st,Egv_st,list_st,La_fl,Egv_fl,list_fl,H,fp,’right’);

[La_pr_mod,Egv_pr_mod] = ...

eigen(Amodc_pr,eye(length(list_fl)+length(list_st)));

freq_pr_mod=sqrt(real(La_pr_mod))/2/pi;

77

% ----------------------------------------------------------------------

To establish the modal form of the coupled system, left eigensystem:

CB–14

% ----------------------------------------------------------------------

[Amodc_pl] = ...

fsi_mod(La_st,Egv_st,list_st,La_fl,Egv_fl,list_fl,H,fp,’left’);

[La_pl_mod,Egv_pl_mod] = ...

eigen(Amodc_pl,eye(length(list_fl)+length(list_st)));

freq_pl_mod=sqrt(real(La_pl_mod))/2/pi;

% ----------------------------------------------------------------------

To transform the eigenvectors back to the original coordinate system:

CB–15

% ----------------------------------------------------------------------

[Egv_pr_red] = ...

fsi_egv(Egv_pr_mod,La_st,Egv_st,list_st,La_fl,Egv_fl,list_fl, ...

fp,’right’,’pr’);

% ----------------------------------------------------------------------

To generate a color plot of the reduced coupled eigenmodes (this is the same routineas CB–12 apart from the first two lines):

CB–16

% ----------------------------------------------------------------------

Egv_p_st=Egv_p_red(1:ndof_st,:);

Egv_p_fl=Egv_p_red(ndof_st+1:ndof_st+ndof_fl,:);

clf; modnr=1;


hold on;

modnr=modnr+1;

Ed_st=real(extract(edof_st,Egv_p_st(:,modnr)));

Edabs_st=abs(Ed_st);

const_st=max(max(Edabs_st)); Ed_st=Ed_st/const_st;

eldisp2(ex_st+10*1.1*(i-1),ey_st-4*2.5*(j-1)+2.0,Ed_st,[1,1,0],2);

eldraw2(ex_st+10*1.1*(i-1),ey_st-4*2.5*(j-1)+2.0,[3,1,0]);

hold on;

Ed_fl=real(extract(edof_fl,Egv_p_fl(:,modnr)));

Edabs_fl=abs(Ed_fl);

const_fl=max(max(Edabs_fl)); Ed_fl=Ed_fl/const_fl;

78

h=fill(ex_fl’+10*1.1*(i-1),ey_fl’-4*2.5*(j-1),Ed_fl’);

MOD= num2str(freq_p(modnr));

text(10*1.1*(i-1)+4,-4*2.5*(j-1)-1, MOD)


end, end


% ----------------------------------------------------------------------

79

fundamentals of fluid-structure interaction

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