Richard Jeans 1992 PhD Thesis

8/13/2019 Richard Jeans 1992 PhD Thesis

http://slidepdf.com/reader/full/richard-jeans-1992-phd-thesis 1/215



Abstract

This study is concerned with innovative methods for the solution of three dimensional fluid-

structure interaction problems. Solution methodologies are presented to evaluate the interaction

between submerged three dimensional thin shells, of arbitrary geometry, and acoustic radiation

in the unbounded surrounding fluid medium.

A variational boundary element formulation of the acoustic problem based on the work of Mariem

and Hamdi, [J. B. Mariem and M. A. Hamdi, Int. J. Num. Methods. Eng. 24,1251-1267

(1987)], is presented. The formulation is implemented using high order isopararnetric elements.

The advantages in using this variational formulation are, first, the manner in which a highly

singular integral operator is made amenable to numerical approximation, second, its application

to non closed thin shells, and, third, its numerical implementation leads to the formulation of a

symmetrical fluid matrix.

A collocational boundary element formulation of the acoustic problem is also presented along

with a novel solution to numerically approximate the highly singular integral operator. The

collocation method is implemented using high order isoparametric elements and a Burton and

Miller approach is used to overcome the problem of non uniqueness for closed shells at interior

resonant frequencies. This formulation allows implementation of the full set of surface Helmholtz

integral equations for the closed shell problems.

A method of formulating the acoustic problem based on the principle of wave superposition is

examined. It has been suggested that this method offers significant advantages over boundary

element methods. This study implements such a method to test this supposition, and it is

compared to the implemented boundary element methods.

Methods of accelerating the boundary element methods are tested including the use of structural

symmetry to reduce the problem size, and the use of frequency interpolation when the acoustic

solution is required over a range of frequencies.

The elastic problem is formulated using a finite element approach and is coupled to both bound-

ary element formulations of the acoustic problem. The structural equation set is reduced in

terms of eigenvectors and Lanczos vectors in order to reduce the size of the structural prob-

lem. These two methods of reduction are compared and the application of Lanczos vectors to

elasto-acoustic problems is discussed in detail.

page 2





For Nick

Page



Notation

Matrix and Vector

Q a square or rectangular matrix

matrix inverse

OT matrix transpose

I identity matrix

{} a column vector

{}T vector transpose

Structural and Acoustic Geometry

S surface of acoustic radiator

E exterior acoustic domain

D interior acoustic domain

+ denotes acoustic variable in the limit approaching S from E

denotes acoustic variable in the limit approaching S from D

A normal to the surface S

P, Q field points in E or D

p, q field points on S

r Euclidean distance between two field points

h thin shell thickness

t plate thickness

a, b spheroidal shell radii and cantilever plate dimen-c,ons

ii, v, curvilinear coordinate system

x, y, z) local coordinate system

X, Y, Z) global coordinate system

77,,v3) subelement coordinate system

e coordinate system axis vector

Iii Jacobian

ýJ31 subelement Jacobian

E small radius

Note: To avoid conflict between pressure and field point in Chapter 5, the notation ra etc is

used to denote field points.

page 5



Scalars

c acoustic speed of sound

p fluid density

p, structural density

E Young's modulus

v Poison's ratio

cp structural speed of sound

k wavenumber

w circular frequency

Functions

p pressure

v velocity

v surface normal velocity

u surface normal displacement

velocity potential function

velocity potential difference across thin shell

off far field scattered velocity potential function

Gk three dimensional Green's free space function

b delta Dirac function

Is surface integral

II variational functional

C elasticity operator

u surface displacement vector in local coordinate system

U surface displacement in global coordinate system

T vector transformation from global to local coordinate systems

R structural inertial forces

f structural surface forces

E strain vector

Q stress vector or spectral shift frequency

D local strain to stress transformation

4irc(p) external solid angle at surface point

P10,10 surface source distributions

page 6



P(cosO) Legendre polynomials

j,,, h spherical Bessel's functions

2 analytical acoustic impedance

structural acoustic impedance

f acoustic form function

A, B general linear integral operators

Numerical Methods

m number of nodes per element

n number of global nodes

ne number of elements

{Ne} vector of element shape functions

{i' } vector of the nodal velocity potentials for element j

{q5} vector of the nodal velocity potentials

{U} vector of the nodal structural displacements

1U, } vector of the nodal normal displacements

[A] area matrix usually approximated by diagonal form

[Lk] collocation matrix approximation of the Ck operator

[Mo] collocation matrix approximation of the MO operator

[Mk] collocation matrix approximation of the Mk operator

[Mk ] collocation matrix approximation of the Mk operator

[Nk] collocation matrix approximation of the Nk operator

[No] collocation matrix approximation of the JVooperator

[Nk ] variational matrix approximation of the,A/operator

[Cr] diagonal matrix of cp evaluated at the collocation points

[H], [G] general acoustic matrix

[N] shape function matrix for structural interpolation

[K] stiffness matrix

[M] massmatrix

[Z] structural impedance matrix

[Mf] fluid added mass matrix

[$(w)] dynamic stiffness matrix

[Z, ] structural impedance

[E] matrix of eigenvectors {ej}

page 7



IQ] diagonal matrix of eigenvalues

[Q] matrix of Lanczos vectors

[Tm] tri-diagonal Lanczos projection matrix

ak, 0k coeficients of Lanczos projection matrix

RM Krylov subspace

rk component of Krylov subspace

[D], [M] superposition matrices

K matrix conditioning number

a velocity reconstruction error norm

a, v imaginary and real coupling constants

Integral Operators

f-k[0l P) =is

Gk P, Q)O Q)dSqS

aGäP, )Mk[0l P)=i q Q)dsq

Sq

Mk [o] P)_I

aGý P, Q)4j Q)dsq

Sp

aGO P,`w)Mo[ýl P) =i O Q)dsq

Sq

Nk [01 P) =a2Gk P) Q)

O Q)dsqs

an önSqp

No[O] P) =a2Go P, Q)

O Q)dsq8n ön

9P

page 8





2.5 Thin shell formulation............................................................

39

2.5.1 Boundary integral formulation..............................................

40

2.5.2 Edge conditions

............................................................

43

CHAPTER 3. Collocation Method........................................................

45

3.1 Introduction......................................................................

45

3.2 Discretization....................................................................

45

3.2.1 Interpolation...............................................................

45

3.2.2 Local and Curvilinear Axes.................................................

46

3.3 Integration of Weak Singularity...................................................

47

3.4 Integral Operators................................................................

49

3.4.1 Ck Operator...............................................................

49

3.4.2 Mk Operator..............................................................

51

3.4.3 lVk Operator...............................................................

53

3.4.4 Matrix formulation.........................................................

56

3.4.5 Exterior pressure distribution..............................................

57

3.5 Thecomputer code ...............................................................

58

3.6 Numerical results .................................................................58

3.6.1 Radiation from submerged spheres .........................................58

3.6.2 Acoustic scattering from a submerged sphere ...............................59

3.6.3 Acoustic scattering from a submerged spheroid .............................60

3.6.4 Acoustic scattering from a submerged finite cylinder ........................60

CHAPTER 4. Variational Method........................................................

78

4.1 Introduction......................................................................

78

4.2 Weighted Residue Techniques and the Variational Method........................

78

4.2.1 Weighted Residual Galerkin Method........................................

78

4.2.2 Variational Method........................................................

79

4.3 Variational Boundary Integral Formulation.......................................

80

4.4 Numerical Implementation........................................................

81

4.5 Uniqueness of the Numerical Formulation ......................................... 83

4.6 Edge Boundary Conditions.......................................................

84

4.7 Computer Code..................................................................

85

page 10



4.8 Numerical Results................................................................

86

4.8.1 Spheroids..................................................................

86

4.8.2 Flat Disk

..................................................................

87

4.8.3 Flat Square Plate..........................................................

91

4.8.4 Terai s Problem............................................................

91

CHAPTER 5. The Superposition Method................................................

110

5.1 Introduction....................................................................

110

5.2 The Superposition Integral......................................................

111

5.3 Uniqueness......................................................................

113

5.4 Numerical Formulation..........................................................

115

5.4.1 Matrix Condition Number.................................................

118

5.4.2 Velocity Reconstruction Error Norm.......................................

119

5.5 Numerical Results...............................................................

120

5.6 Conclusion......................................................................

122

CHAPTER 6. Elasto-Acoustic Problem..................................................

131

6.1 Introduction .................................................................... 131

6.2 Structural Problem..............................................................

131

6.3 Fluid-Structure Interaction Force................................................

135

6.4 Coupled Equation Set...........................................................

136

6.4.1 Fluid Filled and Non Closed Shell Problems...............................

136

6.4.2 Evacuated Closed Shell Problems..........................................

137

6.5 Solution of Coupled Equation Set................................................

138

6.5.1 Structure Variable Methodology...........................................

138

6.5.2 Fluid Variable Methodology...............................................

139

6.6 Eigenvector Reduction of the Elastic Formulation................................

140

6.7 Interpolation....................................................................

142

6.8 Uniqueness and the Coupled Problem............................................

143

6.9 Elastic Thin Plate Problems.....................................................

147

6.10 Numerical Results .............................................................. 149

6.10.1 Cantilever Plate.........................................................

149

6.10.2 Spherical Shell...........................................................

150

page 11



CHAPTER 7. Lanczos vectors in elasto acoustic analysis .................................167

7.1 Introduction....................................................................

167

7.2 Lanczos Vectors

.................................................................

168

7.3 Fluid Variable Methodology.....................................................

171

7.4 Added Fluid Mass Methodology.................................................

172

7.5 Results..........................................................................

173

CHAPTER 8. Conclusions and Recomendations.........................................

191

8.1 Conclusions.....................................................................

191

8.2 Recomen dations .................................................................193

REFERENCES..........................................................................

195

APPENDIX 1 Symmetry and Half Space Problems.......................................

200

A1.1 Image Sources..................................................................

200

A1.2 Geometric Symmetries.........................................................

201

APPENDIX 2 Analytical Solutions......................................................

205

A2.1 Rigid Sphere

...................................................................

205

A2.2 Asymptotic Solutions...........................................................

210

A2.3 Elastic Sphere..................................................................

213

page 12



Figures

Page

CHAPTER2

2.1 Geometry for evaluating the discontinuity of the integral operator ..................31

2.2 Geometry for evaluating the surface Helmholtz integral equations .................33

2.3 Non-uniqueness of single layer distribution.........................................

40

2.4 Non-uniqueness of double layer distribution........................................

41

2.5 The thin shell geometry ...........................................................42

CHAPTER 3

3.1 Geometry of the 9-noded isoparametric element ....................................47

3.2 Element sub-division for singular integration.......................................

49

3.3 Geometry for evaluating the far field pressure distribution.........................

58

3.4 The different mesh geometries for the spherical problem ...........................62

3.5-6 Normalized modal impedance for the rigid sphere, 6 elements ................... 63-64

3.7-8 Normalized modal impedance for the rigid sphere, 24 elements ..................65-66

3.9-10 Normalized error of modal impedance..........................................

67-68

3.11 Plane wave backscattered form function for the rigid sphere vs frequency...........

69

3.12-13 Far field form function distribution for scattering by a rigid sphere ..............70-71

3.14 Surface pressure distribution for scattering by a rigid sphere .......................72

3.15 Far field pressure distribution for scattering by a rigid prolate spheroid .............73

3.16 Far field pressure distribution for scattering by a rigid oblate spheroid ..............74

3.17 The different mesh geometries for the cylindrical problem ..........................75

3.18 Far field pressure distribution for scattering by a rigid cylinder .....................76

3.19 Convergence of the far field scattering distribution for a rigid cylinder ..............77

CHAPTER4

4.1 Subelement division...............................................................

82

4.2 Graph of matrix assembly and factorization times.................................

88

page 13



4.3 Convergence of backscattered form function for the rigid sphere ....................92

4.4 Convergence of backscattered form function for the rigid prolate spheroid ..........93

4.5 Convergence of backscattered form function for the rigid oblate spheroid ........... 94

4.6 Open plate mesh geometries .......................................................95

4.7 8 Radial pressure amplitude on a circular disk using the variational method .......96 97

4.9 10 Radial pressure amplitude on a circular disk using the collocation method .......98 99

4.11 12 Radial pressure phase on a circular disk using the variational method .........100 101

4.13 14 Radial pressure phase on a circular disk using the collocation method .........102 103

4.15 16 Radiation impedance of a circular disk vs frequency..........................

104 105

4.17 Convergence of radial pressure amplitude on a circular disk.......................

106

4.18 19 Radiation impedance of a square plate vs frequency..........................

107 108

4.20 Near field pressure gain for a point source wave scattered by a rectangular plate. ..109

CHAPTER 5

5.1 The geometry for formulating the superposition integral..........................

112

5.2 The effect of the hybrid formulation on the single and double layer formulations...

124

5.3 Far field backscattering for a sphere using the superposition method ..............125

5.4 Far field backscattering for a spheroid using the superposition method ............126

5.5 The variation of error vs retraction of source surface for the sphere ................127

5.6 The variation of condition nuber vs retraction of source surface ...................128

5.7 The far field error vs retraction of source surface for the spheroid

.................

129

5.8 The velocity error norm vs retraction of source surface surface for the spheroid ....130

CHAPTER 6

6.1 Representation of plate problem ..................................................148

6.2 Cantilever plate geometry ........................................................153

6.3 Far field radiated pressure from point excited fluid filled elastic sphere using the varia

tional method .................................................................... 155

6.4 Far field radiated pressure from point excited fluid filled elastic sphere using the collo

cation method ...................................................................156

page 14



6.5 Far field backscattered form function for a fluid filled elastic sphere using the variational

method 1.57

6.6 Far field backscattered form function for a fluid filled elastic sphere using the collocation

method ..........................................................................158

6.7 Far field backscattered form function for a fluid filled elastic sphere using the collocation

Burton and Miller formulation...................................................

159

6.8 The surface pressure and velocity distributions at the first Dirichlet frequency for the

evacuated sphere .................................................................160

6.9 The surface pressure and velocity distributions at the first Neumann frequency for the

evacuated sphere .................................................................161

6.10 The surface pressure and velocity distributions at the first Neumann frequency for the

fluid filled sphere.................................................................162

6.11-12 Far field radiated pressure from point excited fluid filled elastic sphere using the collo-

cation method and frequency interpolation...................................

163-164

6.13-14 Far field radiated pressure from point excited fluid filled elastic sphere using the varia-

tional method and frequency interpolation .................................... 165-166

CHAPTER 7

7.1-4 Velocity response for n=0,1,2 and 3 excitation of a submerged spherical shell using

Lanczos vector reduction of the `dry structural response, and the collocation Burton

and Miller formulation.......................................................

180-183

7.5-8 Velocity response for n=0,1,2 and 3 excitation of a submerged spherical shell using

eigenvector reduction ofthe `dry

structuralresponse, and the collocation Burton

andMiller formulation

...........................................................184-187

7.9 Fluid added mass methodology with Lanczos reduction applied to the far field radiated

pressure of a point excited fluid filled spherical shell ...............................188

7.10 Far filed backscat. tered form function for an evacuated spherical shell with different

hysteretic damping...............................................................

189

APPENDIX 1

A1.1 Image source construction to model infinite rigid plane ...........................200

A1.2 Summary of image source for the acoustic problem ...............................204

page 15



APPENDIX 2

A2.1 Plot of the first four spherical Bessel s functions and their derivatives.............

208

A2.2 Geometry for the spheroidal Kirchoff problem .................................... 212

page 16



Tables

Page

CHAPTER 4

4 1 Order of Gauss integration 83

4 2 Comparison of accuracy with different orders of integration 87

4 3 Comparison of matrix assembly and factorization times 88

CHAPTER 6

6 1 Comparison

of submerged andin

vacuo quantitiesfor the

cantilever plate153

6 2 Comparison of submerged and in vacuo quantities for the evacuated spherical shell 154

CHAPTER 7

7 1 The percentage accuracy of the modal surface velocity for the fluid variable methodology

with 6 elements and Lanczos reduction of the dynamic structural matrix 176


with 24 elements and Lanczos reduction of the dynamic structural matrix 177


with 6 elements and eigenvector reduction of the dynamic structural matrix 178


with 24 elements and eigenvector reduction of the dynamic structural matrix 179

APPENDIX 2

A2 1 The resonant frequencies for the interior spherical Dirichlet and Neumann acoustic

problems 209

page 17



Introduction.

CHAPTER 1.

Introduction

1.1 Background

The acoustic analysis of submerged three dimensional structures is of great interest to en-

gineers. Such an elasto-acoustic analysis can be applied to a variety of engineering problems;

for example the determination of the sound fields generated by aircraft or underwater vehicles

and the design of transducers and acoustic shielding. The potential theory behind the acoustic

problem is also directly applicable to elasto-statics, electromagnetism, hydrodynamics, inviscid

flow and so on. In general the analysis involves solving the dynamic structural response simul-

taneously with the acoustic response, where the two are coupled by a fluid structure interaction.

Standard texts in acoustics e.g. Junger and Feit [1986], Pierce [1989]) present well estab-

lished analytical solutions for simple structures. Morse and Feshbach [1953] list eleven coordinate

systems which allow analytical treatment and Kellog [1929] presents the classical theory behind

such analysis.Early

workin

acoustics concentrated on such analytical results.As

an example

Wiener [1951] published results for thin plate problems and Spence and Granger [1951] presented

analytical results for rigid spheroids.

However analytical results are restricted to a narrow range of geometries. For the vast

majority of realistic problems, a closed form analytical solution does not exist. Before the advent

of digital computers, experimental testing and asymptotic approximations were the only analysis

tools available for arbitrary structures. The availability of high speed computers meant that

complex acoustic problems could be solved efficiently and accurately using numerical methods.

The finite element method FEM) is recognized as a extremely powerful analytical tool

that can be used to solve most well defined continuum problems. It is strongly associated with

structural analysis, but its application to other areas is wide spread. Consequently it was only

natural that it should be applied to the full elasto-acoustic problem and indeed much work has

been done in that area. A modern application of the FEM in acoustics is given by Pinsky and

Abboud [1990], who consider the transientanalysis of

theexterior

acoustics problem.

A FEM analysis of the acoustic problem is hampered by a number of difficulties. The acous-

tic problems of greatest interest are often those involving a fluid of infinite extent. Discretization

page 18



Introduction.

of such a fluid domain for the FEI is obviously difficult and almost always unsatisfactory. Even

with the use of infinite elements, the resulting equation set is large and cumbersome. Such a

method is inefficient since the variables of most interest are those on the interface between the

fluid and structural domains.

Boundary integral formulations of potential problems have been recognized for a long time.

By using the power of the new digital computers it was quickly recognized that a boundary

element method (BEM) promised an elegant and efficient solution technique for the numerical

analysis of potential problems. Regarding the acoustic analysis of rigid structures, Chen and

Scheikert [1963] published a numerical method based on the boundary layer integral formulation

and a year later Chertock[1964]

published a method based on the Helmholtz integral equation.

Over the last two decades, the BEM has emerged as the preferred solution technique for acoustic

problems in the exterior domain.

The strength of an integral formulation of the acoustic problem is the reduction of di-

mensionality; the three dimensional exterior pressure field is represented by a two dimensional

integral relationship on the surface of the structure. The elegance of the method is the math-

ematical simplicity of the resulting integral expressions. However, it was recognized by Lamb

[1945] that there is a difficulty with the classical integral formulations. At the standing wave

frequencies of the interior domain defined by the structural surface, the integral equations for

the exterior problem fail to provide a unique solution. The numerical consequences of this non-

uniqueness for rigid acoustic problems was examined by Copley [1967] and the first improved

numerical formulation to circumvent the problem was presented by Schenck [1968]. Burton and

Miller [1971] presented a composite integral formulation that was valid at all wavenumbers.

Kleinman and Roach [1974] as well as Filippi [1977] presented a mathematically rigorous analy-

sis of the uniqueness properties of the various formulations, showing the equivalence between the

boundary layer and Helmholtz integral formulations. The behaviour of BEM s at critical wave

numbers continues to be of interest and recently Dokumaci [1990] discussed the exact nature of

numerical failure of the BEM at these frequencies.

The Burton and Miller formulation and Schenck s Combined Helmholtz Integral Equation

Formulation (CHIEF) still remain today as the main numerical strategies for overcoming prob-

lems of uniqueness, each having it s own strengths and weaknesses. The Burton and Miller

formulation is robust and well tested but necessitates the evaluation of a hypersingular integral.

The CHIEF formulation on the other hand is simpler but fails at higher frequencies and involves

the solution of an over determined system of equations.

page 19



Introduction.

The first boundary element analyses were concerned with either a prescribed surface velocity

distribution or rigid body scattering. Zienkiewti cz Kelly and Bettess [1977] presented an analysis

of the general coupling between a FE-N1and BENZ derived from the FEM point of view. It was

Wilton [1978] who first showed the feasibility of coupling a structural finite element formulation

and an acoustic boundary element formulation.

Boundary integral and finite element formulations can often be applied to the same set of

problems and are often perceived as competing solution techniques. However the two method-

ologies have complementary strengths and weaknesses and their combination often results in

a very powerful solution technique. In the elasto-acoustic problem the FEM is best suited to

the accurate determination of the localized structural problem and the BEM is best suited to

the fluid problem of infinite extent. Not only do fluid-structure interaction problems suit this

category of analysis but so do a large number of other engineering problems; e.g. structure-soil

interaction structural non-linearities within homogeneous structures and so on.

A large proportion of published boundary integral formulations use constant interpolation

of the fluid variables. A popular interpolation technique uses the concept of a superparametric

element as used by Wilton [1978] and recently in the field of electro-magnetism by Ingber and

Ott [1991]. Within the superparametric element the level of geometric interpolation is higher

than the level of fluid interpolation. Mathews [1979][1986] introduced isoparametric interpola-

tion common in the FEM where both the fluid and geometric variables are interpolated by

quadratic Lagrange shape functions. Amini and Wilton [1986] implement a number of different

interpolation schemes to show the advantages in convergence efficiency and accuracy possible

with quadratic interpolation of the fluid variables.

Mariem and Hamdi [1987] presented a new variational formulation of the BEM for the thin

shell problem. In this formulation based on the work of Maue [1949] and Stallybrass [1967]

the kernel of the hypersingular integral operator is transformed using tangential derivatives

and then integrated with respect to the field point over the surface of the structure. The

resulting symmetric expression contains only a weak singularity which is amenable to numerical

integration. The disadvantages of the method is the increase in computational time needed for

thesecond

numerical integration.

Despite the advantages there seem to be very few researchers using isoparametric elements.

The variational method is idealy suited to high order fluid interpolation since the reduction

page 20



Introduction.

of the singularity is independent of numerical discretization. Of the few published papers us-

ing such interpolation techniques, Coyette and Fyfe [1989] and Jeans and Mathews [1990] used

a variational approach. Isoparametric implementations of the Collocation Burton and Miller

method have been published by Chien, Rajiyah and Atluri [1990] and W a u,Seybert and Wan

[1991]. The work by Chien et al, employed a complicated treatment of the hypersingular inte-

gral operator, dependent on the exact form of interpolation, \Vu et al employed the tangential

formulation given by Maue and then used additional regularization to integrate the remaining

Cauchy type singularity. Seybert and his co-workers have published a number isoparametric

implementations of the CHIEF methodology; e.g. Seybert, Soenarko, Rizzo and Shippy [1985].

Such

an

implementation is aided by the absence of a hypersingular integral operator, but suffers

from the inherent problems of the CHIEF methodologies.

It has already been mentioned that a large amount of research effort has been involved in

the question of the uniqueness of the integral formulation at critical frequencies. This work is

primarily on the BEM in isolation from the coupled elasto-acoustic formulation. The CHIEF and

Burton and Miller techniques seem to have been included in the coupled formulations without

qualification, although the fact that the structural formulation does not remove the ill condi-

tioning of the numerical formulation at the critical frequencies is not, in general, immediately

obvious. Huang [1984] attempted to clarify the situation by explicitly showing the theoretical

non-uniqueness at the critical frequencies of the unrelated internal acoustic problem. However

he also stated that in reality this non-uniqueness will not be present. Although this was true for

the constant fluid basis functions he was using, Mathews [1986] disproved Huang s conjecture

for isoparametric implementations of the elasto-acoustic BEM.

Like the Burton and Miller formulation, the acoustic problem of thin plates and shells has

been hampered by the need to numerically approximate the hypersingular integral operator.

A large amount of successful numerical work in this area has been published by Pierce and

colleagues; e.g. Pierce [1987], Wu, Pierce and Ginsberg [1987] and Ginsberg, Chen and Pierce

[1990]. This work used the Helmholtz intergal equation and variational procedures, but also

used axisymmetric basis functions based on an a priori understanding of the problem. Terai

[1980] applied his BEM to thin plate problems as did Coyette and Fyfe [1989], however the

approach of Coyette and Fyfe ultimately lead to the assumption of incompressible fluid to

facilitate an eigenmode analysis. Recently Kirkup [1991] has modeled the effects of acoustic

shields using a coupled FEM/BEM, but in general there seems to be little published work into

page 21



Introduction.

the application of BEM s to unbaffled thin plate problems of arbitrary geometry and even less

using isoparametric interpolation.

Without any simplifying approximations using a BEM to model the effect of the fluid on

a submerged structure leads to a non-linear eigenvalue problem. The simplest approach to a

dynamic analysis of such a structure, involves an analysis at a large number of frequency points.

The computational bottle-necks for this approach are the evaluation of the fluid impedance ma-

trix relationship at each frequency, and the large number of degrees of freedom in the structural

equation set. Wilton [1978] recognized the second of these problems and reduced the dynamic

structural matrix using the `dry eigenmodes. The problem of calculating the fluid impedance

matrix at a large number of frequency points is alleviated by frequency interpolation. Kirkup

and Henwood [1989], Schenck and Benthien [1989] and Benthien [1989] all presented and evalu-

ated interpolation techniques and Schenck and Benthien [1989] discussed in general the problems

of applying coupled BEM/FEM techniques to large scale elasto-acoustic problems. The use of

such interpolation schemes can also facilitate a modal analysis of the fluid structure interaction

problem (Kirkup and Amini [1990]). Recently Lanczos and Ritz vector techniques have been

applied to the internal fluid structure interaction problem (Moini, Nour-Omid and Carlsson

[1990], Coyette [1990]), and their use in the coupled BEM/FEM formulations promises signifi-

cant advantages over modal reduction (Jeans and Mathews [1991]).

Overcoming the problem of the hypersingular integral operator present in both Burton

and Miller and thin shell acoustic formulations, represents a large amount of research effort.

Over the years many solutions to the problem have been suggested along with an unavoidable

increase in the computational complexity of the implementation. Cunefare and Koopmann [1989]

presented a methodology they called CHI (Combined Helmoltz Integrals), where the problem

is circumvented by constraining all the field points to be interior to the surface. Koopmann,

Song and Fahline [1989] proposed a wave superposition method, where the surface pressure

distribution was reconstructed from a source distribution interior to the surface of the radiator.

Recently there has been some favorable interest in this method, with Miller, Moyer, Huang

and tTberall [1991] looking at the coupling of the FEM and Superposition method, and Song,

Koopmann and Falinline [1991] investigating the associated numerical errors.

All researchers to date recognize the problems with the retraction of the source surface in

order to circumvent high order singularities; i.e. the inherent numerical difficulties of discretizing

Fredholm integral equations of the first kind (Miller [1974]). The interior source or field points

need to be chosen carefully in order to obtain sufficiently well conditioned numerical systems.

page 22



Introduction.

Many researchers, including the author, believe that the superposition method is simply an

ill-conditioned form of the BEM, and singularities in the integral equations are an unavoidable

consequence of optimizing the numerical robustness and conditioning of the problem. This is

the motivation behind a recently submitted paper (Jeans and Mathews [1991]).

1.2 Motivation of Present Thesis

The original motivation for this thesis was to develop and refine the current state of the

art computational approaches used for the prediction of radiated sound. After three years of

research the following areas have been documented in this thesis:

(a) Development of a computationally efficient implementation of the Helmholtz in-

tegral equations for three dimensional structures of arbitrary geometry, including

thin plate problems.

i) Comparison and development of variational and collocation formulations.

ii) The use of quadratic isopararnetric elements.

iii) Frequency interpolation of fluid matrices.

(b) Coupling of a consistent FEM formulation of the thin shell and plate structural

problem to the boundary element formulations.

i) Comparison of modal and Lanczos reduction of the structural formula-

tions to improve solution efficiency.

A general review of the acoustic problem, along with the development of the various integral

formulations, is given in Chapter 2. In Chapter 3a collocation approach is presented for the

numerical solution of the Burton and Miller formulation of the exterior acoustic problem. This

work describes a novel approach to the hypersingular integral operator, based on the expression

of the integral operator given by laue [1949]. This approach is very similar to the work of Wu,

Seybert and Wan [1991], but without their additional regularization procedures. The resulting

numerical implementation is applied to spherical, spheroidal and cylindrical rigid radiation and

scattering problems.

In Chapter 4 the variational method of Mariem and Hamdi [1987] for approximation of

the hypersingular integral operator is described and numerically implemented using quadratic

isoparametric boundary elements. The resulting thin shell acoustic formulation is applied to

page 23



Introduction.

different closed and open thin shell problems and compared to the collocation formulation of

the thin shell acoustic problem.

With renewed interest in superposition formulation of the acoustic problem Chapter 5 is

devoted to an examination of these methods. It was felt that these methods were not suitable

for the general acoustic analysis of arbitrary structures due to the inherent instabilities of the

resulting integral expressions. Unfortunately to support this view point a quantitative analysis

of the method involved a substantial and disproportionate amount of research effort. However

the examination of the superposition method in this chapter necessitated introducing topics

of matrix conditioning and accuracy of the general matrix routines that might not have been

included otherwise.

The FEM is introduced in Chapter 6 and the theory behind the thin shell structural prob-

lem used in this study is outlined. The coupling procedure between the structural and acoustic

problems is described along with the various methodologies for solving the combined problem.

In this chapter eigenmode reduction of the structural problem and frequency interpolation of

the fluid matrices are described. The resulting implementations are tested with the spherical

shell problem for which there is an easily accessible analytical result and results for a submerged

cantilever plate are also presented. Significant work is also presented detailing the behaviour of

the general elasto-acoustic problem at critical frequencies.

Chapter 7 introduces the theory Lanczos vectors and describes their usefulness in coupled

elasto-acoustic formulations. The resulting implementations are again applied to the spherical

test case and compared to corresponding eigenvector reduction techniques. A fluid added mass

formulation is described using the variational formulation of the thin shell acoustic problem

and the resulting symmetric structural equation is reduced using Lanczos vectors.

The final chapter is a summary of the thesis. Conclusions are made about the consequences

and effectiveness of the presented results and finally proposals for future work are presented.

In Appendix I the application of acoustic and geometric symmetry applied to the acoustic

boundary element problem is reviewed. Such use of symmetry is of considerable interest not

only for the reduction of problem size but for the treatment of half space problems. Summarized

in Appendix II are the analytical techniques applied to the spherical elasto-acoustic problem.

page 24



. acoustic Problem.

CHAPTER 2.

Acoustic Problem

2.1 Fundamental Equations

The analysis of any acoustic problem involves the solution of a differential equation relating

fluid variables in the medium of interest, subject to certain boundary conditions. This work is

primarily concerned with the interaction of acoustic radiation with submerged structures and

the appropriate differential equation is the linear wave equation subject to a velocity boundary

condition on the submerged structure.

2.1.1 Linear Approximation

In a fundamental analysis (Pierce [1989]) of the acoustic pressure field, the linear wave

equation is developed from a consideration of mass and energy conservation. This analysis

assumes that the acoustic pressure field is a small amplitude perturbation to the ambient state,

characterized by those values that pressure, density and fluid velocity have when the perturbation

isabsent.

The fluidor acoustic medium

isassumed

to be homogeneousand quiescent;

ie the

ambient quantities are assumed to be independent of position and time with the ambient fluid

velocity equal to zero.

Consider a body of fluid in a volume V, with density p, surrounded by a surface S. The

fluid velocity at a point P is given by v(P) and the outward normal to the surface S is defined

by n. Conservation of mass requires that the net mass leaving V per unit time is equal to the

rate at which mass decreases in V. This is expressed by the relationship,

-

dt

JpdV =J pv ndS.

vs

(2.1.1)

The right hand side of this equation can be transformed to a volume integral by means of Gauss'

theorem and Euler's differential equation for conservation of mass is obtained,

a

alp 17-(Pvv)=o.

(2.1.2)

The pressure field at a point P is given by p(P). By considering the forces acting on the

fluid, and neglecting body forces and viscosity, the fluid velocity and pressure are related by,

page -95



Acoustic Problem.

p(av

+ (v V)v

This equation represents an example of Reynolds' transport theorem.

The relationship between the pressure field and density in the fluid is obtained by making the

assumption that the acoustic radiation is adiabatic in the linear approximation. Consequently

it is possible to write,

aP 2aP2_(ap),

(9t (9t ap

(2.1.4)

The real constant c is refered to as the speed of sound in the particular fluid medium and

the subscript s in Eq. (2.1.4),

indicates that the differential is evaluated at constant entropy.

Writing the pressure, density and velocity as small perturbations of the ambient state,

P=Po+p, P=Po+Pý, v=v'. (2.1.5)

the linear differential equations governing these perturbations are defined as,

op i+c2po0 - v' = 0, (2.1.6)

öv'VP'+po = 0.

2.1.2 Wave Equation

The wave equation results when v' is eliminated from Eq. (2.1.6) and Eq. (2.1.7).

From

now on the prime notation will be neglected when indicating ambient state perturbations. The

wave equation is given by,

v2pla2P

_-o.2 öt2(2.1.8)

page 26



Acoustic Problem.

Since the fluid velocity and pressure are given by a linear differential equation, it is possible

to assume an harmonic time dependency of the form e-' t, where ,;is the circular frequency of

the pressure field. The time independent fluid variables are,

p= Pe-iwt v= ive cwt. (2.1.9)

Substituting Eq. (2.1.9) into Eq. (2.1.8) results in the expression,

V2p+k2p= 0. (2.1.10)

where the wave number k is w/c. This equation is known as the reduced wave equation or

Helmholtz' equation.

2.1.3 Neumann boundary condition

The reduced wave equation is solved in terms of the Neumann boundary condition which

relates the fluid velocity to the normal velocity prescribed on S. This boundary condition is

defined by,

Op-n -

iwpvr,, (2.1.11)

where the subscript notation for the ambient density, p, is dropped and,

ap_n _n V , vn =nýv. (2.1.12)

This boundary condition is sufficient for the solution of Helmholtz' equation for a finite

body of fluid. For unbounded fluid problems another boundary condition is needed to uniquely

specify the solution to Helmholtz' equation.

2.1.4 Sommerfeld's radiation condition

For unbounded acoustic problems the pressure field must obey some boundary condition

in the far field. This boundary condition is given by Sommerfeld's radiation condition. This is

defined by,

page 27



4coustic Problem.

lien

Jkp dS=0 (2.1.13)

r--co S

The consequence of this condition is that it, allows outward travelling waves to be the only valid

physical solution for a radiated or scattered wave from a surface.

Eq. (2.1.10), Eq. (2.1.11) and Eq (2.1.13) uniquely define the acoustic problem. For

some simple geometries it is possible to solve the acoustic problem analytically, however for the

majority of realistic problems a numerical solution technique is needed. Analytical solutions for

certain problems are useful in establishing the accuracy of various numerical strategies used to

solvethe

acoustic problem.

At this stage the concept of a velocity potential function will be defined. This function

enables the fluid velocity and pressure fields to be evaluated from one potential function. The

function is defined by,

020 + 120 = 0, (2.1.14)

with,

ü= VO, p= iwpq. (2.1.15)

2.2 Integral Operators

This work is concerned with various methods of numerical solution by means of boundary

element methods. Later in this chapter the appropriate integral equations will be derived that

form the basis of any BEM. The integral operators that will combine to give an integral for-

mulation of the acoustic problem are introduced in this section and a brief discussion of their

numerical properties is presented in isolation from the physical problem.

2.2.1 Green s Function

In general when a differential equation is transformed into an integral equation, the form of

the integral equation not only depends on the governing differential equation, but the boundary

conditions specific to the problem. An integral equation not only relates an unknown function

page 28



.Acoustic Problem.

to its derivatives; ie values at neighbouring points, but also to its values at the boundary. The

boundary conditions are built into an integral equation through the form of its kernel, but for

a differential equation the boundary conditions are imposed at the final stage of solution. This

kernel is the Green s function for the problem.

For the acoustic problem, the Green s function is the fundamental solution of the inhomo-

geneous Helmholtz equation,

(02+ k2)GkP,Q)= -6(P,Q), (2.2.1)

where 6(P, Q) is the delta-dirac function. Also the Green s function must satisfy Sommerfeld*s

radiation condition, Eq. (1.1.13). The appropriate solution in three dimensions is given by,

Gk(P) Q) -

ikr

4 rr= JP- Qj. (2.2.2)

In this equation r is the Euclidean distance between the field points P and Q.

2.2.2 Discontinuities

The integral operators that are of interest are defined by,

,Ck01(P) =

isGk (1 )Q)O(Q)dSq, (2.2.3)

s

Mk[c](P) _öG9 P, Q)

q(Q)dSq, (2.2.4)S4

Mk0](P)s

GäP, )O(Q)dSq, (2.2.5)

Sp

Ar [0)(ý ) =a2Gk(P, Q)

0(Q)dSq. (2.2.6)Sananp

The function 0 is assumed to be a continuous function over S. For completeness there is also

the identity operator,

1[01(p) = «(P). (2.2.7)

An important property of these operators defined in Eq. (2.2.3-6) is that they are all solutions of

the original acoustic differential equation. Each can be thought of as the result of a continuous

distribution of sources, magnitude O(Q), over the surface of S.

page 29



Acoustic Problem.

It is useful to establish the behaviour of these integral operators as the field point P is

brought to the surface S. The limit of a field point brought to the surface in the direction

opposite to the direction of the surface normal will be denoted by p+, the limit of the field point

brought to the surface in the same direction as the surface normal will be denoted by p- .In

general points on a surface will be denoted by lower case symbols. Firstly construct a small

sphere of small radius e centered around the surface limit point. The domain of integration is

taken to be the surface S, excluding the small sphere, and that part of the small sphere, S£,

that completes the surface. The radius e is then taken to zero to evaluate the limiting value of

the integral operator. Figure (2.1) illustrates the geometry for evaluating the limit of P -- p+.

Consider the integral operator Ck.

lm,Ck[O)(P) =1ö Gk(p, 4)4(4)dSq +J

4ýesin(B)dOdyh .(2.2.8)

[IS-S, rz

P P+ S.

The polar angles 0 and 0 define the surface S. For Eq. (2.2.8) the second term on the right

hand side goes to zero in the limit and so the operator Gk is continuous across the surface S.

Consider the same limiting process for the.

Mk operator,

I aGk(n,) 2Pm Mk[01(P) = li ö anO(4)dsq+ 4ýý2 ýsin(8)d6d¢ (2.2.9)

P+ _Sc 4

Is

t

In Eq. (2.2.9) the vector r is the unit vector pointing from P to p+. This time the second term

on the right hand side tends towards a limiting value and so,

lim A4k[O](P) _ Mk[Q](P) + (l - c(P))O(P) (2.2.10)P-4p+

The quantity 4irc(p) is the external solid angle at the point p. For a smooth surface; ie one that

has a unique tangent plane, then

1c(P) =2

A definition of c(p) can be gained by considering the limit,

(2.2.11)

page 30



Acoustic Problem.

n

P048440 ...

Sc

P p+

p-

q

SE

Figure 2.1). Geometry for evaluating the discontinuity of the integral operators.

page 31



Acoustic Problem.

Jim-PPI

aGo(P q)dS(2.2.12)

P-P+ ,

This equation is continuous as P passes through the surface S. Noting the discontinuity in the

Mk operator, indicates a mathematical definition for the quantity c(p),

aco(p,)ds(P) =1+f an qSq

(2.2.13)

Using the arguments described above, it is possible to evaluate the discontinuity properties

of all the integral equations and these are summarized below,

4 [OI P+) _ ck [OI P)-

4[0](P -)i (2.2.14)

Mk [c](P+)-

(1- c(P))q(P) _ Mk [01(P) = Mk [Q](P )+ c(P)O(P), (2.2.15)

k[o](p+) + (1

- C(P))O(p) _k [o](P) = Mk [©](p-)- C(P)4(P), (2.2.16)

Nk[0l (P+)= Nk[¢](P)= Nk[q](p ), (2.2.17)

It is important to realize that Mk[q](p+) represents the limiting value of Mk[O](P) as P

tends to p+, whilst Mk [O](p) represents the principal value of the integral operation; that is the

limit value of an integration over a surface S-S,, as 6 goes to zero.

2.3 Helmholtz Integral Equations

2.3.1 Surface Helmholtz Integral Equation

The Surface Helmholtz Integral equation (SHIE) forms the basis of most boundary element

methods (BEM). This equation can be derived by considering the general acoustic geometry

shown in figure (2.2). A spherical surface, Sr, contains two closed surface Si and S. The surface

S represents the acoustic surface of interest and the surface Si represents the surface of some

acoustic source. The domain E is the volume contained by SE excluding the volume contained

by Si and S. The domain D represents the volume contained by the surface S. A small spherical

surfaceSE

surroundsthe field

pointP.

With the surface SE excluding the singular point P from the domain E, both the Green s

free space function and the velocity potential in E satisfy the reduced wave equation and so,

page 32



Acoustic Problem.

Figure (2.2). Geometry for deriving the surface Helmholtz integral equations.

page 33



,Acoustic Problem.

Q=0. (2.3.1)(4(Q)V2Gk(P, Q) - Gk(P, Q)V2o(Q)) dlE

By using Green's formula this volume integral may by converted to a combination of surface

integrals over the surfaces bounding E,

Is(P)+Is. (P)+Is, (P)+Is, (P)_0, (2.3.2)

where the integrals I(P) are of the form,

Is(P) _ O(q)oGk(P, q)

_90(q)

Gk(P, q) dSq. (2.3.3)Sqqq

The negative sign reflects the fact that the normals are defined to point into the domain E.

The SHIE for an infinite exterior domain is obtained when the radius of the surface Sr, is

taken to infinity and the radius of SE is taken to zero. By the Sommerfeld radiation condition

IS. will tend to zero. The integral Is has different values depending on the position of the field

point P. For P in D its value must be zero since IS,, is no longer a bounding surface of E. For

P on S or in E its value can be evaluated in a similar way to the limiting procedures of section

(1.2.2). The value of this integral as in the limit is given by,

O(P) PEE

Is, (P) c(P)¢(P) PES.

(2.3.4)

0 PED

The integral IS, may nowbe

seen tobe

equivalentto the

velocity potentialthat

would existin the absence of the surface S,

Is, (P) = Oi(P).

The SHIE for the infinite exterior domain can now be written as,

O(P)

o(q)0Gk(P, q)

_ao(q)

Gk(p, q) dS9+ Oi(P) _ c(P)O(P)an, Önqt0

PEE

PES

(2.3.5)

(2.3.6)

PED

Page 94





Acoustic Problem.

2.3.3 Boundary Layer Formulations

It was stated in Section 2.2 that the result of the integral operators on a continuous function

defined on the surface S, was a solution of the acoustic wave equation. Further more with the

correct definition of the Green's function the solution satisfies the radiation condition. Conse-

quently it is possible to define the velocity potential in the exterior domain, E. in terms of a

single layer distribution, p,

ö(P) = Ck {c)(P), PEE. (2.3.13)

By differentiating with respect to some normal vector, defined in the exterior domain, an ex-

pression for the dimensionless velocity field is obtained.

aý(P)-. 4[, c](P),np

PEE. (2.3.14)

When the field point, P, is taken to the surface S, the boundary layer formulation for the

acoustic problem is defined by,

[Mk-

(1- c(P))] P, pcS.

on(2.3.15)

Eq. (2.3.15) can be solved to obtain the single layer density p, and then Eq. (2.3.13) can be

used to evaluate the exterior and surface velocity potentials.

In a similar way the exterior velocity potential can be defined in terms of a double layer

distribution, a,

0 (P) _ . k[O](P)7PEE. (2.3.16)

The differentiated form is given by,

a¢(P)

- JVk[0](P),PEE. (2.3.17)

önp

Since the Vk operator is continuous across the surface S, then Eq. (2.3.17) can be solved for

PES and the surface -velocity potential is defined by,

page 36



Acoustic Problem.

0= [Mk +(1-C(P))]o.

The exterior velocity potential is given by Eq. (2.3.16).

2.4 Uniqueness of Boundary Integral Formulations

2.4.1 SHIE and DSHIE formulations

(2.3.18)

For given boundary conditions the velocity potential for the exterior problem is unique.

However it has long beenrecognized

thatwhen expressed

in termsof a

boundary integral

formulation the solution to the exterior problem may not be unique. Non-uniqueness of the

solution occurs at critical wavenumbers k, and for the acoustic problem these wavenumbers

correspond to interior resonant frequencies. It needs to be emphasized that this problem of

non-uniqueness does not imply non-uniqueness of a physical solution, but a breakdown of the

theoretical formulation at critical frequencies. A numerical implementation of an unmodified

exterior boundary integral formulation will result in ill-conditioning of the matrices at a range

of

frequencies,

centered aroundthe

critical

frequency.

The problem of non-uniqueness can be illustrated by considering the exterior Neumann

problem, Eq. (2.3.11), for a `smooth surface . There will be a unique solution as long as there

are no non-trivial solutions to the homogeneous equation,

(I+M)v=0. (2.4.1)

The non-trivial solutions to this equation occur at the eigenvalues k,. By the Fredholm Alter-

native theorems, the eigenvalue spectrum of this equation is the same as that of the transpose

Of ,

2Z-}-Mkv = 0. (2.4.2)

The eigenvalues of this equation correspond to the eigenvalues for the unrelated interior Neu-

mann problem, Eq (2.3.9). Simiarly the critical wavenumbers for the exterior Dirichlet problem

correspond to the eigenvalues of the interior Dirichlet problem.

page 37



Acoustic Problem.

For all but simple boundary geometries it is impossible to predict the values of kn. There-

fore it is highly desirable to implement a strategy that will eliminate the problem of the ill-

conditioning of the numerical formulation in the region of critical wavenumbers. Several strate-

gies have been proposed all of which unfortunately increase the computational burden of the

problem.

2.4.2 The CHIEF method

One of the first methods proposed to remove the problem of non-uniqueness was that of

Schenck [1968]. In this method the algebraic equations generated from the SHIE are combined

with additional equations generated from the interior Helmholtz relationship,

001Mk [01(P) _ ,

CkOn

(P)- OA(P), PED, (2.4.3)

evaluated at a number of interior points. The resulting overdetermined set of equations can be

solved by a least squares method.

There are several problems with this method. When some of the interior points lie on

nodal surfaces it has been shown that this method may not remove the problem of uniqueness.

Consequently, at high frequencies when the density of interior nodal surfaces is high, the choice

of interior nodal posit-ion is difficult. Several methods for choosing these nodal points have been

proposed but this adds to the complexity of the solution. For an arbitrary selection of interior

points this method cannot be relied on to remove the problem of non-uniqueness.

2.4.3 Burton and Miller s Formulation

Burton and Miller proposed [1971] that the problem of uniqueness could be overcome by

forming a linear combination of the SHIE and DSHIE. This linear combination is given by,

{ [-c(P)Z Mk] + OA k-_{ rk +a [c(P)Z +M]}ý- (Oi +a

örti .(2.4.4

On

I]

Burton and Miller demonstrated that for an imaginary coupling constant a, this formulation

should yield a unique solution for all wavenumbers.

The disadvantage of this formulation is that the kernel of the AVkoperator is highly singular

and a method needs to be used in order to integrate this operator numerically.

page 38



acoustic Problfnz.

2.4.4 Boundary layer formulations

The boundary layer formulations for the exterior acoustic problem also exhibit similar non-

uniqueness properties at the critical wavenumbers described above. A similar argument to show

this non-uniqueness to that for the SHIE and DSHIE can be used. However another argument

is illustrated in figures (2.3) and (2.4). These arguments use the jump properties of the integral

operators derived in section (1.2.2),

Single layer:

4+ _ ,Ckµ = 4-, (2.4.5)

as

+ (i - ß(p))µ TP=

aä

- c(P)u. (2.4.6)Double layer:

0+ -(1

- c(P))c _ MkO _+ c(P)O , (2.4.7)

= JVk0 =ahn (2.4.8)

The single layer formulation proves to be non-unique at eigenvalues of the interior Dirichlet

problem and the double layer formulation proves to be non-unique at eigenvalues of the interior

Neumann problem.

2.4.5 Hybrid boundary layer formulation

The established technique for overcoming the problem of uniqueness in a boundary layer

formulation is to express the surface velocity potential in terms of a hybrid combination of a

single and double layer surface distribution. The exterior velocity potential is defined by,

O(P) = [Lk + aMk] [t ](P), PEE.

Consequently the boundary integral equations defining the acoustic problem are,

4 ý_{, Ck +0 [Mk + (1

- C(p))]} v,

00=an

{ [A4k-1- c(P) )] + a.ýk}v.

(2.4.9)

(2.4.10)

(2.4.11)

If the coupling constant, a is constrained to be imaginary, then the numerical solution of

this hybrid formulation is unique at all frequencies. Again the disadvantage of this formulation

is that it requires the integration of the hypersingular kernel in the A operator.

page 39



Acoustic Problem.

Single layer distribution

Jump relationships:

0+-O-=0,

ao+ a0-an an

Boundary condition:

ao+=0.

,On

At eigenvalues of interior Dirichlet problem:

Uniqueness of exterior problem

From (1) and (4)

0- =as

ýo.

= µ54o,

Not unique solution

Not at eigenvalues of interior Dirichlet problem:

a0=0.ý ý

On

Unique solution

From (2), (3) and (6)

From (2), (3) and (8)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

Figure (2.3). Mathematical illustration of the non-uniqueness of the single layer

distribution at eigenvalues of the interior Dirichlet problem

page 40



Acoustic Problem.

Double layer distribution

Jump relationships:

0+-0-=0-,

ao+ 00-

an an

(1)

(2)

Boundary condition:

ao+an =0

Z* 0+ = 0,

Andao-

= 0,

(3)

Uniqueness of exterior problem. (4)

From (2) and (3). (5)

At eigenvalues of interior Neumann problem:

as= o, 0-00.

0 o,

= Not unique solution

Not at eigenvalues of interior Neumann problem:

00

,_=0.ý

On

v=0,

Unique solution

(6)

From (1), (4) and (6). (7)

(8)

From (1), (4) and (8). (9)

Figure (2.4). Mathematical illustration of the non-uniqueness of the double layer

distribution at eigenvalues of the interior Neumann problem

Page 41



Acoustic Problem.

As the boundary layer formulation are related to the SHIE and the DSHIE, this hybrid

formulation is analogous to the Burton and Miller formulation described above.

2.5 Thin shell formulation

A prime motivation for this project was the analysis of thin shell acoustic problems. In this

work a thin shell is defined as a shell for which the through shell displacement field is assumed

to be constant. Acoustically this means that the normal derivative of pressure is the same on

both sides of the shell.

2.5.1 Boundary integral formulation

The geometry of the thin shell is illustrated in figure (2.5). On such a shell three closely

associated points may be defined. The points are p, p-, and p+, where p represents a point

midway through the thickness of the shell, p+ is a point on one surface, and p- is the corre-

sponding point on the other surface. The normal np is defined to be in the direction from p-

to p+. The Green's function and the normal derivative of Green's function at these points will

have the following simple relationships:

agS(g+anq

Gk(P, q4 ,

aGk(P,l)anq±

önq

= Gk (P, 4),

=OGk(P,

anq

E

np+

n

P-

n'

D

Figure (2.5). The thin shell geometry.

S+

S-

(2.5.1)

Page 42



4couslic Problfm.

Using these relationships, the SHIE formulation for the thin shell problem may be written as

O(P)= Oi(P)+ o(q+)aGk(P,g+)

SI+ aTZq+

+ (q)aGk(P,)an9-

_

o(q)Gk(P,

q+) dSq+9+

C)6(q)Gk(P,q+) dSq

.9-

(2.5.2)

By using the relationships in Eq. (2.5.1) and by setting gy(p) -- q(p+)-

0(p-), the SHIE

and DSHIE formulations for the thin shell become,

0(P) = Mk{41ý](P) + Q1(P),

a

n)= Nk[ý)(P) +

a¢a(P)

PEE, (2.5.3)

PEE. (2.5.4)

The surface domain S is now the surface of one side of the thin shell. Since S has been

redefined the domains D and E need also to be redefined. The domain D becomes the domain

whose interface with S contains the points p-, and E becomes the domain whose interface with

S contains the points p+. Both D and E are sub-domains of the exterior domain surrounding

S. Taking the limit of P to p in Eq. (2.5.3) and Eq. (2.5.4);

C(P)O+(P) + (1- c(P))O-(P) _ Mk[ ](P) + ci(P), PES, (2.5.5)

ao(P)=Nk[,l(P) +

ao (P)PES. (2.5.6)

an anp

2.5.2 Edge conditions

When considering thin shells which do not enclose an interior volume it is important to

consider the behaviour of the integral equations at the edge of this shell. It has been proposed

in previous work (WVarham [1988) that by taking the limit of P to p in from D and E, results

in two equations that give a condition on I. However if the limits are taken correctly then,

lim Mk[ ýD](P) _ -(P) =Mý[ý](P) + (1- c+(P))(O+(P) - -(P))p-.

P+

HM. vfk[I](P) _ 0-(P) =Mk[ i](P) -

(1- c_(P))(O+(P) -

p-

(2.5.7)

(2.5.8)

page 43



Acoustic Problem.

which leads to the identity,

ýý+ P) + c- p) - 1)ß P) = 0. 2.5.9)

This equation simply states the fact that c_ p) =1- c+ p) and gives no condition on D. A

more sophisticated argument is needed to define the edge conditions for the open plate problem.

When the thickness of the plate is taken to zero the edge around the plate becomes an

additional boundary. Consequently there needs to be a supplementary boundary condition

specified on this boundary. In his recent paper Martin [1991] discusses the behaviour of one-

dimensional hypersingular integral equations over finite intervals and this idea is discussed in

more detail. This edge boundary condition is arbitrary in the mathematical sense, but for this

case is governed by the original physical problem. Continuity of the pressure difference across

the plate means that Dmust be zero at the edge of the shell;

Dp) = 0, p on the edge. 2.5.10)

In past work eg. Pierce [1987]), this edge boundary condition has been satisfied by the

choice of appropriate fluid basis functions. In most numerical work for arbitrary thin plates e.g.

Terai [1981] and Warham [1988]) the edge boundary condition is essentially ignored since there

are no nodes on the edge of the plate. In the numerical work described in later chapters, the

presence of nodes on the edge of the plate means that this boundary condition must be imposed

upon the numerical formulation.

Page44



Collocation Mc fhod.

CHAPTER 3.

Collocation Method

3.1 Introduction

In this chapter a collocation method is described for the solution of radiation and scattering

problems for a submerged body of arbitrary geometry. A new way of numerically integrating

the hyper-singular kernels that occur in the boundary integral formulation is presented, and

the method is shown to be independent of the interpolation used for the fluid and geometrical

variables.Results for

a selection of scattering and radiation problems showthe

validity and

accuracy of the method.

The problem of uniqueness at critical wavenumbers is circumvented by implementing the

method of Burton and Miller, described in the previous chapter. High order quadrilateral

elements are used to interpolate for the acoustic and geometric variables.

3.2 Discretization

3.2.1 Interpolation

A number of connecting elements are defined on the surface domain. The nodal points

defined by these elements form the set of collocation points at which the boundary integral

equations are satisfied. Within the elements, the approximate fluid and geometric variables are

related to the element nodal values in terms of element shape functions. The nodal positions

of the element are defined in terms of the global Cartesian axes X, Y, Z), the local Cartesian

axes, x, y, z), define the normal and tangential planes at the Gauss integration points and the

curvilinear axes, ý, rl, ) define the element.

The approximations of global position and fluid variables are given in terms of the elemental

shape functions and element nodal values by,

mmm

X N1 ß,ý1)Xi, Y ý, ýI)= N1 ß,ýl)Yi,2 e,

ý1)_ Ný ý, ý1)z,. X3.2.1)

1=1 , =11=1

Nj ý, 77)01,90 ß, 77)

_ 11)and

3.2.2)

=1

an

1=1

In the definitions above, m. represents the number of nodes per element.

Page 45



Collocation.Method.

For this study 9-noded quadrilateral elements were used and the appropriate shape functions

are given by,

Ni = 4ýý - 1)ý1ýý]- 1),

N4 = 2ýý- 1)(1 - 7?),

N7 = 4ýýý- 1)i](7l + 1),

Na = zý1 -ý2)77(ýl - 1),

1'ß'5 (1 -0)(1

- q'),

N8=2(1_ý2)7J(1l+1),

X3 = 4ý(e + 1)i1(i7- 1),

N6= 2ý(ý+1)(1-112

N9 =4b+ 1)71(71+ 1),

Figure (3.1) shows the geometry for the 9-noded element. The approximation of a variable

within an element can be written in terms of the vector product,

0(ý,77)=

{Ne IT

loj

},(3.2.3)

where {Ne} is the vector of the element nodal shape functions defined at. (ý, i) and {QJ} is the

vector of nodal values of the variable 0 for element j.

3.2.2 Local and Curvilinear Axes

The curvilinear axes within the element are defined by,

where taking W=X,Y, and

2,

(gÄ( aY Z-e =ý -ex +a eY + a- eZ,

oÄ( aY aZ (3.2.4)e=

a, 7eX+- ä-eY, - (9ýeZ,

e( = ee x e,

aw _ y, 71) V1. (3.2.5)

The local axes vectors are derived from these curvilinear ones with the relationships,

ex = eý,

eZ = eý,

ey = e, x e,

(3.2.6)

where the hat notation for the vectors indicates normalization with respect to the global axes

system. By noting the orthogonality of the local axes, it is possible to write the Jacobian matrix

for the transformation between the local and curvilinear coordinate systems as,

Page 46



Collocation Method.

7

A5

12

8

9

ýy

x6

3

Z

Y

X

7

4

1

Figure 3.1). Geometry of the 9-noded isoparametric element.

Page 47

23



Collocation Method.

8dý

__iii

äý J21

0 aor

J-- dy (3.2.7)

With,

ax ax Oyii= =ex ee, Jai=ý=et e,,. J22==_ey

From these relationships the following Jacobian may be formed:

IJI= J11J22.

A vector that appears later in this chapter is defined by,

nq x Vqq.

this can be expanded in the local coordinate system as,

aoq x 7y 0=ax

ex -y ey.

eý. (3.2.8)

(3.2.9)

(3.2.10)

(3.2.11)

The relationship given in Eq. (3.2.7) allows this vector to be evaluated in terms of the curvilinear

axes thus enabling a numerical approximation.

3.3 Integration of Weak Singularity

There exist several numerical methods that enable a weak singularity to be integrated

within an element. For this study the inverse distance singularity scheme of Lachat and Watson

[1976] was used. In the collocation method a singularity will occur at the nodal points within an

element. In the Lachat and Watson scheme the singular element in the curvilinear axes system

is replaced by two, three, or four triangular elements with the singularity always at the same

corresponding vertex. These triangular elements are collapsed rectangular elements and the

number of subelements depends on whether the singularity is at a corner, mid-side, or center

node. When the Jacobian relating the integration over element and sub-element is calculated it

is found that the O(r-1) singularity is removed. Figure (3.2) shows the division of the 9-noded

element depending on the singular node.

page 48



Collocation Method.

1

2

Corner node

0

1

42

3

Center node

1

2

3

Mid-side node

Singularity

2

4c

3,4

Sub-element

Figure (3.2). Element sub-division for singular integration.

Page 49B133L

LONDIN.UNý/



Collocation Method.

The shape functions for the four-noded subelement, defined by the subelement axes ý,, and

r), are,

N13= 4(1 +ßs)(1 + als),

N3 =4(1

-ßs)(1

- ýIs)>

\ .4 =1 1-ßs)(1+7)s),

=4(1 +ý3)(1 -CJs)

The Jacobian is defined by,

IJ(ýs,ýs)I=äý äý

ýs qs

where,

o3ai18

as

ii=11

aSs

(3.3.1)

(3.3.2)

The superscript indicates the subelement nodal coordinate with respect to the 4,77axes. The

other terms in Eq. (3.3.1) are defined in a similar way.

Noting that ý3 = ý4 and r73 = q4, an expression for the subelement Jacobian can be

evaluated,

ýJ3(i3,0

I=-1

,q, + 1)-4,

(3.3.3)

where i is the number of the subelement and Ai is the area in the curvilinear space of that subele-

ment. The integration of a singular element can be transformed into an expression, removing

the singularity,

1subel

r(ý

1,

i1Jsdýsdiis (3.3.4)

Here r represents the distance between the singular node and the integration point in the

curvilinear space. In the modified integration scheme, the curvilinear points are interpolated

using the subelement shape functions defined above This transformation allows numerical

implementation using a simple Gauss integration.

3..4 Integral Operators

The object of any numerical method is to transform the governing equations into a system

of equations that is amenable to numerical solution. For a boundary element method this

page 50



Collocation Method.

involves approximating the integral equations, presented in Chapter 2, by discretized matrix

equations. This has to be done so that high order singularities and problems of non-uniqueness

of the numerical formulation are removed or circumvented. The integral operators that form

the components of the integral equations will be discussed individually.

3.4.1 £k Operator

This operator is defined by.

Gk [01(P)_

fGk(PQ)(Q)dSq.

(2.2.3)

The kernel of this integral operator contains singularities O(r-1) and there is no need to trans-

form this operator for numerical discretization. Using the element shape functions the,Ck

operator may be approximated by the following numerical expression,

ne

Gk[OI(P) _f Gk (P, q){Ne}T dS; {ý }. (3.4.1)

j =lS

Inthis equation, ne

isthe number of elements.

The integration isperformed numerically

inthe

curvilinear system using the method of Lachat and Watson described above. The surface field

point is interpolated using the element shape functions and the area element dSj is given by,

dS, -- IJ I<dq. (3.4.2)

When Eq. (3.4.1) is evaluated at each of the set of collocation points {Pt}, a system of

equations can be assembled. These equations can be written in matrix form in the following

way,

{, Ck[OI (Pi)l=

[Lk]{ }.

3.4.2 Mk Operator

This operator is defined by.

Mk[q5](P) =JaGäPýQ)q(Q)dsq.

Sq

(3.4.3)

(2.2.4)

page 51



Collocation alethod.

For this integral operator, the kernel contains singularities O(r-` ) and so a way is needed to

avoid the integration of high order singularities. The operator can be rewritten in the following

way,

[Mk -c(P)1]0= [Mk-Mol0+ [Mo-c(P)ll0.

Eq. (3.4.4) can be approximated in terms of the element shape functions,

n° aGk(P, q)_

3G0(ß.4)Mk ýýý(P) E Is, (

ön an{Ne}T dS; 10i }+

j=1 9 an,

iaý ÖG0(P, q)eT

anIN e dS; {0i }- c(p)O(P).

is

i

(3.4.4)

(3.4.5)

The second term on the right hand side of Eq. (3.4.5) contains singularities of O(r-2). This

singularity is ignored and Eq. (3.4.5) is integrated numerically using the modified numerical

integration technique. By taking the field point P to each of the collocation points a system of

equations can be assembled. These can be expressed in matrix form by,

{[Mk- C(P)1]I =

[Mk-

Cp]l01-

[Alk-_

lo]Lq} + [M0-

Cp]101. (3.4.6)

The diagonal matrix [Cp] is the value of c(P) evaluated at the collocation node points. Consider

the following definition of the last term in Eq. (3.4.6),

mil - cl

m21

[Mo-Cp]{O}=Mil

Mni

M12...

mli

m22 - c2 ... m2i

mat 772t; c;

mn2 mni

min Oi

M2n

02

ln2n Oi

mnn - Cn 0n

(3.4.7)

The O(r-2) singularities are distributed by the element shape functions to the diagonal elements

of this matrix. These matrix elements can be accurately defined by using a row sum procedure.

The definition of c(P) was given in Chapter 2 by,

j 0GoP, )ds.is

(3.2.13)

page 52



Collocation Method.

By using this relationship the diagonal elements of the matrix defined in Eq (3.1.7) are given

by,

mii - ci - -1 - mij.

j i

(3.4.8)

Therefore not only are the singular elements evaluated but c(p) is also implicitly evaluated.

The A4 operator may be numerically evaluated in a similar way. By redefining it as,

[Mk + c(P)Z] [Mk + Mo] - [Mo - c(P)Zj ß,

the operators in the numerical approximation are given by,

Tn, aGk (P, q) aGoýP, qý

e 7'

k[0](P)

-E On+

anIN } dSj { }-

? =1 ifp 9

n` r aGGO(P,Q)

eT

JS an{N } dSj {0j }- c(p)O(P).

j=1 q

This leads to the following matrix expression;

{ [MT + C(P)Z] 01 =[1ý1k + Cp]{0}

= [Mk + M0]{0} -[silo

-Cp]{q}

(3.4.9)

(3.4.10)

(3.4.11)

The first matrix on the right hand side of Eq. (3.4.11) contains singularities O(r-1) and the

O(r-2) singularities in the second matrix are evaluated using the row sum procedure described

above.

3.4.3 )/k Operator

This operator is defined by,

, i-ý[¢](P) =

a2Gk(P, Q)O(Q)dSq.

ön öns9P

(2.2.6)

It is this operator that, need the greatest amount of consideration when implementing a

numerical method since there are high order singularities that need integrating. Based on a

page 53



Collocation 11(thod.

proof shown by Maue [1949], it is possible to convert the expression in Eq. (2.2.6) into tangential

derivatives using the basic vector identity,

(np- Vq)(n9 - Vq) = (nn

-nq)(Vq 0)-

(nq x Vq) (np x Vq). (3.4.12)

By integrating by parts, the transformed expression for the Nk operator is given by,

Nk01(P)-J{(nP

. nq)k2Gk(P, q)O(q) -(nq xV

q0(q)) (nP x C9)Gk(P, q)] dSq+

S (3.4.13)is

n9 O9 x (q)(np x VgGk(P, q))] dSq.

The second term on the right hand side of Eq. (3.4.16) can be transformed by Stoke's theorem

into an integral around the edge of S. Therefore for closed surfaces it is equal to zero and can

be neglected. The transformed expression for the 111,Eoperator is finally given by,

Nk[01(P) =f [(np ' n9)k2Gk(P, 4)o(q) - (n9 X Vq (9)) . (np x Vq)Gk(P, q)] dSq. (3.4.14)s

The validity of this expression has been shown in greater detail by Stallybrass [1967] and Mitzner

[1966]. Eq. (2.2.6) can be rewritten in the following way,

Nk0=

[.Nk-

Vol 0+ %Voc (3.4.15)

The first operator on the right hand side contains terms of order O(r-1) and the second terms

of order O(r-2). Using the element shape functions Eq. (3.4.15) can be approximated by the

following expression,

Nk [01 (P)-J l(nP

j=1

nq)k2Gk(P, q){Ne}T _

(nq X Vq{Ne}T) (npx V9)(Gk(P, q) -Go(P, q))dS9 10i }-

ný/J

(nq x Vq{Ne}T) (np x V)Go(P. q)dSj { }.

1=1 J)

0i

(3.4.16)

page 54



Collocation.11ethod.

The O(-2) singularity in the second term of this expression is ignored for the time being and

Eq. (3.4.16) is evaluated at each collocation point to assemble the following matrix equation;

{-Nk[OJ(PP)} _ [, k]M _ [:Vk - .Vo){¢} + [_1 o]{yh}. (3.4.1 7)

This represents the matrix equation for the Nz operator, and no further modification is

made to integrate the singularity occurring in the [N0] term. It will be shown that there is

cancellation of the inaccuracies due to the numerical integration of these terms.

Consider thesingular

integrationover one element

for the A öoperator.

Thesingularity

is separated from the rest of the element by a disk segment of small radius e. The singular

integration over the element can be separated into a non singular integration over the element

minus the disk segment, Iý, and and a singular integration over the disk segment, Iý ;

Ij (nq XVq (4))-(np x V, GO(P,q))dS9 = I,

j,Ij,

.(3.4.18)

s,

The integration Iý is assumed to be accurate within the limits of the numerical integration

scheme. The inaccuracies for the element integration are assumed to be contained within the

integration Ij2. The small disk segment is assumed to be flat and the vector T is assumed to be

in the plane of the disk, perpendicular to the normal. The gradients within Eq. (3.4.18) can be

expressed as,

(nqx

Oq4(r, e))=

ö«r, B)(n9

X ), (3.4.19)3G0(r, B)

(np xV gGo(r,B)) = Or

(np x T),

The integration in Eq (3.4.18) can be rewritten as,

e Eä0(r, 6)2oGo(r, 0)Ij

-

In

8r ar rI JI drd9. (3.4.20)n

where the Jacobian IJI can be taken as constant. Within the disk segment it is assumed that 0

has a linear dependency on the global position vector and so in Eq (3.4.20) the variation of 0 is

independent from r and given by,

page 55



Collocation.1lfthod.

00)= pose

0Or

r-0

(OX

r=0sine y

r_0

(3.4.21)

Using this relationship Ij can be represented by,

Ii =Ii, +IB1jr.

(3.4.22)

For all elements j that contain the singularity there will be similar errors in Ir due to the

inadequate singular integration. However since the summation of the Ie terms is equal to zero

theseerrors cancel each other out.

Therefore it issafe

toevaluate the singular

terms inthe

[N. ]

matrix using the Lachat and Watson inverse distance singularity scheme.

In a recent paper by Wu, Seybert and Wan [1991] a very similar collocation method has

been described. This shows that Maue s equation needs C continuity at the collocation points

and to achieve this on CO continuous elements the collocation points are put inside the elements

to form an over determined set of equations. To achieve the integration of the Cauchy principal

value integral shown in Eq. (3.4.18) they use additional regularization.

This study uses a much more simple method. By integrating Eq. (3.4.18) for singular

elements directly it is recognized that there will be errors in the matrix approximation of the

X, operator. However this work has shown that there will be cancellation of these errors when

the acoustic problem is solved and so the sophisticated treatment of Eq (3.4.18) in the work by

Wu et al is unnecessary.

3.4.4 Matrix formulation

It is now possible to write the matrix approximation for the Burton and Miller boundary

integral equation, Eq. (2.4.4);

([Mk-

Cp] + a[Nk]) f01=

QLk} + a[Mk + cr]) {o}-foil -a

(90i

an ön

Eq. (3.4.23)represents a matrix equation of

the form,

[H]x =

(3.4.23)

(3.4.24)

page 56



Collocation Method.

where the matrix [H], is full, complex and non-symnmetric. In the past considerable effort

has been made to symmetrize this matrix equation (Mathews [1986]). For this study this non

symmetric equation set was solved directly using a standard LU-factorization technique followed

by forward and backward substitution. The imaginary coupling constant a takes the value i/k,

since this has been shown by previous researchers (Burton and Miller [1971]) to be the optimum

value. For the numerical integration 3x3 Gauss integration was used for both the non singular

elements and for the Lachat and Watson subelements.

3.4.5 Exterior pressure distribution

The exterior pressure distribution is given by,

Os P) _ Mk[0](P) - £k[01(P). PEE. (3.4.25)

This expression can be approximated numerically by using the element shape functions,

(P)_

öGk(P, q) {Ne}dS {( )- Gk (P, q){Ne}dSai

(3.4.26)

=1

fSa9(is

a=1 i=1 I

Since the field point P is exterior to the boundary surface there is no singular kernel in this

expression that needs special consideration.

For the far field pressure distribution this numerical expression can be further simplified.

By considering the geometry shown in figure (3.3) it is possible to write,

Off(P) _

eikRný /ý a

kiý)f

4ýRiknq - ReikR. 9{Ne}dS {¢ý }-J ei

q{: ý e}dS anJs S

(3.4.27)

where R represents the far-field field point vector and q represents the boundary surface point.

In vector form this expression can be written as

o3- lMkf J

{0}-

{L k }

an(3.4.28)

page 57



Collocation Method.

Figure (3.3) Geometry for evaluating the far field pressure distribution.

3.5 The computer code

The numerical implementation of the collocation method was coded in FORTRAN predom-

inantly on an IBM RS6000 machine in a UNIX environment. The code was written to primarily

evaluate the numerical method presented above and it was found that in core memory space

was large enough to run problems of a large enough size to demonstrate accuracy for a number

of simple geometries.

Use was made of structural geometry in order to reduce the problem size considerably. This

was done by specifying a degree of rotational symmetry about the global Z axis and in most

cases only one axisyinmetric quarter of the acoustic structure is actually discretized.

3.6 Numerical results

3.6.1 Radiation fron submerged spheres

One axisymmetric quarter of a sphere with radius a is discretized into a number of elements

of equal area. Figure (3.4) shows the different mesh geometries for the spherical problem, A

surface velocity with a Legendre distribution is specified on the sphere,

ao aooan an P(cos9).

The resulting normalized acoustic impedance, Z, is calculated,

(3.6.1)

page 58



Collocation fcihod.

Z-ko/0'0. (3.6.2)

and further normalized with respect to P, to form the normalized modal acoustic impedance.

The results for various discretizations and modes are plotted with respect to wavenuinber

in figures (3.5-8). These results are compared to the analytical solution (Junger and Feit [1986])

and show high accuracy and good convergence.

The effectiveness of the of the Burton and 1-filler formulation in removing the problem of

non-uniqueness at critical frequencies is shown in figures (3.9-10). The error in the normalized

modal impedance is plotted at the internal acoustic eigenvalues for the interior spherical problem,

for ka less than 10. These eigen-frequencies can be calculated analytically. The error plots show

quite clearly the high errors for the SHIE and DSHIE formulations around the appropriate

critical frequencies of the interior problem. These plots also show that with the Burton and

Miller formulation, the error in modal impedance for these modes is less than 0.1 for n=0,

less than 1 for n=1 and n=2 and less than 5 for n=3.

3.6.2 Acoustic scattering from a submerged sphere

Figure (3.11) shows the convergence of the backscattered far-field form function for scat-

tering of a plane wave by a rigid sphere compared to an analytical solution. The incident wave

is defined by,

Oi = eikz (3.6.3)

and the backscattered far-field form function is given by,

2R 10,=

aa(3-6.4)

This plot shows good convergence to the analytical solution and the low and high frequency

characteristics of plane wave scattering. At low frequency the form-function depends on k2 as

Raleigh scattering is predominant. In this frequency regime the acoustic wavelength is large

compared to the scatterer and the form-function depends only on the volume of the sphere. At

higher frequencies the form-function converges to the Kirchoff limit. In this frequency regime

it is the specular reflections from the illuminated surfaces that dominate and for the sphere the

page 59



Collocation Ifcthod.

Kirchoff approximation predicts a frequency independent form-function, dependent on the area

of the illuminated surface. This frequency plot also shows the characteristic oscillations due to

the interference of Franz waves.

Figures (3.12-13) show the scattering distribution in the Y-Z plane for various frequencies.

The axis magnitudes for these plots is the Y-Z components of the far-field pressure amplitude

with unit magnitude incident plane wave. Again the numerical results are compared to an

analytical solution.

Figure (3.14) shows the amplitude of the surface pressure in the Y-Z plane for the

same unit amplitude incident plane wave, at two frequencies for different discretizations. The

plotted points in this figure correspond to the nodal points on the surface mesh. These points are

compared to the analytical solution. It is worth noting that for 24 elements the far-field accuracy

seems to be greater than the surface accuracy. This indicates a degree of error `smoothing when

calculating the far-field quantities.

3.6.3 Acoustic scattering from a submerged spheroid

The figures (3.15-16) show the far-field scattering distributions for oblate and prolate

spheroids. One axisymmetric quarter of the spheroid is discretized and the plane-wave is taken

to be incident end on to preserve the axisyinmetry of the problem.

In the absence of any sophisticated meshing algorithm, the mesh for the spheroids was

calculated by simply scaling the spherical mesh in the X, Y, Z directions. The comparison to

the numerical results was provided by the program written by S.W. Wu, for his PhD project

[1990]. This program was written to solve for purely axisymmetric acoustic problems using high

order line-elements. The program was run using 20 such elements and the assumption made that

this represented a converged solution. There seems to be a high degree of accordance between

the results. These test cases show the efficiency of the model for geometries other than the

simple spherical problem described above.

3.6.4 Acoustic scattering from a submerged finite cylinder

The last set of numerical results concerns the far-field scattering distribution from a finite

cylinder of length L and radius a. The mesh geometries for this problem are shown in Figure

(3.17). Again only one axisynunetric quarter of the cylinder is discretized and the plane wave is

incident. along the Z axis. The numerical results are compared to the results of S.W. Wu using

44 axi-syrrunetricline elements.

page 60



Collocation.fethod.

Figure 3.18) shows the far field scattering results for a cylinder with length 4a and 8a.

Both plots in the figure show good agreement with the results of S.W. \Vu. Figure 3.19) shows

the convergence of the scattering distribution from the L=8.0 cylinder. This figure shows the

importance of a high density of elements around the surface discontinuities, as is the case for

the 104 element mesh.

page 61



Collocation Method.

a) 6 Elements

c) 96 Elements

b) 24 Elements

Figure 3.4). The different mesh geometries for the spherical problem

page 62



Collocation Method.

1.5

Z

1.0

n=O

0.5

0.0

-0.5

-1.0`0

1.5

Z

1.0

0.5

0.0

-0.5

1n

246g ka 10

1 .V

02468 ka 10

Analytical Real . Imaginary A.

Figure 3.5). Normalized modal impedance for the rigid sphere, modes 0 and 1.

One axisymmetric quarter of the sphere, radius a, is discretized into 6 elements.

page 63



Collocation Method.

1.5

Z

1.0

n=2

0.5

0.0

-0.5

-1.002

1.5

Z n=3

1.0

0.5

0.0

-0.5

-1.0

-1.502468 ka 10

Analytical Real . Imaginary



4 6 g ka 10

page 64



Collocation Method.

1.5

Z

1.0

n-0

0.5

0.0

-0.5

-1.00

1.5

Z

1.0

0.5

0.0

-0.5

1

2 4 6 g ka 10

n=1

02468 ka 10

Analytical Real . Imaginary &.

Figure (3.7). Normalized modal impedance for the rigid sphere, modes 0 and 1.


page 65



Collocation Method.

1.5

Z

1.0

0.5

0.0

-0.5

1A

n=2

-I. V

0246

1.5

Z n=31.0

0.5

0.0

-0.5

-1.0

-1.502468 ka 10

Analytical Real . Imaginary A.



g ka 10

page 66



Collocation Method.

1040 0

0103 n=0

LOL

102ee

101eo

e0100

000ee

dý 0 A

10-1 epe aoA

sse0

o0a®e e ®e G

0

10 -2 00

10-3

10-402468 ka 10

10400

103 n=1LO

L 102eeA

101 0

o AAA

100 eeA000A, & A 8

es010 -1 00 oQs:

Oam O ta NW MI

102 -

'Do

103

10-402468 ka 10

Coupled . SHIE o. DHSIE '& .

Figure (3.9). Normalized error of modal impedance for the rigid sphere, modes

0 and 1. One axisymmetric quarter of the sphere, radius a, is discretized into

24 elements.

page 67



Collocation Method.

104

103

OV

L 102

101

10°

10-1

10-2

10-3

, n-4

n=2

A

A

0

A

0e0°°00

Ao

g °o °o

00 0° ® ®

oo000

0

sIV

02468 ka 10

104

103 n=3LA00`

10 2es

101Ls e

000 AS 00ae10

0CPA

°0 o49A

10 -10o0 Le

00

00 MEN

o

10 -2 00

103

10-402468 ka 10

Coupled . SHIE o. DHSIE.

Figure (8.10). Normalized error of modal impedance for the rigid sphere, modes

2 and 3. One axisymmetric quarter of the sphere, radius a, is discretized into

24 elements.

page 68



Collocation Method.

o

0

0

0

0

0

0

0

0

0

0

kn

Itt

M

N

. --4

O 00 ýO ýt NO 00 N

ri - r-+ .-OOOOO

uo Jz)un, uuao3

6 elements 0.24 elements .

Figure (4.11). Plane wave backscattered form function for therigid sphere.

One axisymmetric quarter of the sphere, radius a, is discretized into 6 and 24

elements.

page 69



Collocation Method.

ka=0.1

0.015

0.010

0.005

0.000

-0.005

-0.010

-0.0154--

-0.020 -0.015 -0.010

ka=1

0.80.6

0.4

0.2

0.0

-0.2

-0.4-0.6

-0.8

-1.0

-0.005 0.000 0.005

Analytical Numerical

Figure S. 12). Far field form function distribution for scattering of an incident

plane wave by a rigid sphere. The plane wave is incidentfrom left

to right.One

axisyinmetric quarter of the sphere, radius a, is discretized into 24 elements and

the distribution is calculated at ka = 0.1 and 1.0

page 70

-0.8 -0.6 -0.4 -0.20.0 0.2 0.4



Collocation Method.

ka=3

1.5

1.0

0.5

0.0

-0.5

-1.0

-1.5

-1.5

ka=5

1.5

1.0

0.5

0.0 X

-0.5

-1.0

-1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Analytical Numerical O.

Figure 3.13). Far field form function distribution for scattering of an incident

plane wave by a rigid sphere. The plane wave is incident from left to right. One

axisymmetric quarter of the sphere, radius a, is discretized into 24 elements and

the distribution is calculated at ka = 3.0 and 5.0

page 71

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0



Collocation Method.

ka=3

1.5

1

0.5

0

-0.5

-1

-1.5

-2

2

1.5

1

0.5

0

-0.5

-1

-1.5

-2

-2.5

-1.5 -1 -0.5 0 0.5 1 1.5

ka=5

-2 -1.5 -1 -0.50 0.5 1 1.5

Analytical 24 Elements 13.96 Elements

Figure 3.14). Amplitude of surface pressure for plane wave scattering by a rigid

sphere. The plane wave is incident from left to right. One axisymmetric quarter

of the sphere, radius a, is discretized into 6 and 24 elements and the distribution

is calculated at ka = 3.0 and 5.0

page 72



Collocation Method.

a=1.0 b=0.2k=1.0

0.20

0.10

0.00

-0.10

-0.20 .ý-

-0.40

.1

-0.20 0.00

a=1.0 b=0.2 k=3.0

0.8

0.6

0.4

0.2

0.0

-0.2

-0.4

-0.6

-0.8_1.8

0.20 0.40

-1.4 -1.0 -0.6 -0.20.2 0.6 1.0 1.4 1.8

S. W. Wu Numerical O.

Figure 8.15. Farfield pressure distribution for scattering of an incident plane

wave by a rigid oblate spheroid. The plane wave is incident from lefto right.

One axisymmetric quarter of the spheroid is discretized into 96 elements and

the distribution is calculated at k=1.0 and k=3.0. The results of S. W. Wu were

calculated using 20 axisymmetric line elements.

page 73



Collocation Method.

a=0.2 b=1.0 k=1.0

0.02

0.01

0.00

-0.01

-0.02

-0.02 -0.01 0.00 0.01

a=0.2b=1.0k=3.0

0.05

0.03

0.01

-0.01

-0.03

-0.05 -

-0.05

S. W. Wu Numerical

Figure 3.16). Farfield pressure distribution for scattering of an incident plane

wave by a rigid prolate spheroid. The plane wave is incident from left to right.

One axisymmetric quarter of the spheroid is discretized into 96 elements and

the distribution is calculated at k=1.0 and k=3.0. The results of S. W. Wu were

calculated using 20 axisymmetric line elements.

Page 74

-0.03 -0.010.01 0.03 0.05



Collocation Method.

a) 72 Elements

b) 88 Elements

c) 104 Elements

Figure 3.17. The differentmesh geometries for the cylinder problem

page 75



Collocation Method.

a=1.0 L=4.0 k=1.0

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.7 -0.5 -0.3 -0.1 0.1 0.3

a=1.0L=8.0 k=1.0

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.6 -0.4 -0.20 0.2 0.4

0.5

0.6

S. W. Wu Numerical O.

Figure 3.18). Farfield pressure distribution for scattering of an incident plane

wave by a rigid cylinder. The plane wave is incident from left to right. The

distribution is calculated at k=1.0 for L=4.0 and L=8.0. One axisymmetric

quarter of the cylinder is discretized into 72 elements for L=4.0 and 104 elements

for L=8.0. The results of S. W. Wu were calculated using 44 axisymmetric line

elements.

page 76



Collocation Method.

\oö

Ö

0

0

II

0

ea

i .`

ýi

1

NO

0

N9

9

Ö

S. W. Wu.72 Elements f

.88 Elements 13.104 Elements .

Figure 8.19). Convergence

ofthe farfield

scatteringdistribution from

the cylin-der, L=8.0, a=1.0, at a wave number of k=1.0.

yý4

1N

.w.

page 77

ýt NO C14 lqt ýcÖCCÖÖÖ



I ariaftonal.110hod.

CHAPTER

Variational Method

4.1 Introduction

In this chapter a variational method is described that enables the accurate numerical ap-

proximation of the hyper-singular Yk operator. This method is based on the work of Mariem

and Hamdi [1987]. Previous chapters have indicated that the.A operator is important for two

reasons. First it is a main constituent of the DSHIE for closed body acoustic problems, which

when coupled with the SHIE, forms the Burton and Miller formulation circumventing the prob-

lem of uniqueness. Second it forms the basis of the thin shell acoustic formulation described in

Chapter 2.

The variational method is considered since it provides an elegant solution to the problem

of integrating the hypersingular kernel as well as resulting in a symmetric matrix equation. The

resulting numerical formulation is also independent of the type of numerical interpolation used.

Subsequent chapters will show that the variational formulation couples to the elastic formulation

of the thin shell in an efficient and symmetric way.

The disadvantage with the method is that it involves an extra integration which increases

the computational size of the problem. Results are presented for thin shell scattering and

radiation problems that show the validity and accuracy of the method. Both closed and non

closed thin shells are considered and comparisons are made between the variational method and

the collocation method described in Chapter 3.

4.2 Weighted Residue Techniques and the Variational Method

The purpose of this section is to outline the principles behind the collocation and variational

methods. The arguments that follow are taken from Zeinkiewicz [1989] and are presented to

highlight the relationship between the two methods.

4.2.1 Weighted Residual-Galerkin Method

Thegeneral acoustic problem may

be

expressed

by the linear equation,

A[O] p) =8ý P) 4.2.1)

page 78



Variational .11cthod.

Eq. (4.2.1) n-iust be satisfied at every point, p, on some surface domain, S. It follows therefore

that;

isu.,p)A[¢](P)dSp =J u'(P)B ön

(P)dSp,is

(4.2.2)

where w is a weighting function and Eq. (4.2.2) must be satisfied for all w. The fluid variables

may be approximated in terms of the global shape functions,

{N9}T {0},a0 : 1, -g IT fan

(4.2.3)

If the approximated values of the fluid variables are inserted into Eq. (4.2.2) then it is clearly

impossible for this expression to be satisfied for all uw. nstead an approximation is made and u'

is replaced by a finite set of weighting functions so that,

w= wj, j=1 to n. (4.2.4)

Eq. (4.2.2) will now yield a set of n simultaneous equations given by,

fw(P)A[{'ßr9}T

}](p)dS= J w(P)13 {N}T'91]

(P)dS. (4.2.5)s

This equation represents an integral of error residuals and clearly the equation set generated

depends on the set of weighting functions that are chosen.

If the weighting functions are chosen so that,

wi = b(P- Pj), (4.2.6)

where pj is the set of nodal points then Eq. (4.2.5) will correspond to the collocation method

described in Chapter 3. If however the weighting functions are chosen so that,

Wj _ :ý'J

,

then Eq. (4.2.5) is the Galerkin weighted residual method.

(4.2.7)

page 79



Variational.f0hod.

4.2.2 Variational Method

For the variational method some scalar functional is generated using some a priori guess

as to the physical nature of the problem. For some physical problems this scalar functional

may correspond to the potential energy or the energy dissipation of the system. The solution

of the problem is achieved by minimizing this functional with respect to the unknown variable.

Consider the BEM formulation for the thin shell acoustic problem defined in Chapter 2,

a4 P)

_ Alk [ýDl P)- 4.2.8)an

The appropriate functional is defined by,

n=I ý3O- 2ýA1

[<D]p) dSp. 4.2.9)9n

The minimization of this functional gives the variational formulation of the problem. Due to

the symmetry of the J1/k operator, the stationary value problem can be evaluated in a straight

forward manner,

bII =I b a P)-

b DA-k[ DIP) dSp = 0. 4.2.10)

It can be seen that this expression is identical to the Galerkin formulation of the thin shell

problem. In general if the governing integral operator is self adjoint then the variational and

Galerkin formulations will be identical. The whole subject of variational methods is discussed

in great detail in the book by Morse and Feshbach [1953].

4.3 Variational Boundary Integral Formulation

The full boundary integral equation for the thin shell problem is defined by,

ao p)

_ A,k[ýJ P)

as P)P

2.5.6)

where 1 is the pressure difference across the shell. Following the method shown in Chapter 3

the Nk operator can be transformed into tangential derivatives;

page 80



Variational If0hod.

[(tp *f9)k2Gk(P, q)4(q) -

(n9 X Vq (q))-

(nP x Vq)C'k(P,q)] dSq. (4.3.1)

For the variational formulation, Eq. (2.5.6) is multiplied by 6D(p) and integrated with

respect to p over the shell surface S,

Ibý(P)

aanP)dsp =

Ibý(p)i1

k[<D](p)dSp+ &D(p)aý (P)

dSp. (4-3.2)s

is

p

The expression for the Alk operator given in Eq. (4.2.1) can be further transformed by

integrating by parts to obtain the expression,

All [6,ý] =ff

[(np' nv)k26D(P)(D(4') -

(np x GPb(D(P)) ' (nq x Vq(D(q))] Gk(p, q)dSgdSp.ss

(4.3.3)

Eq. (4.3.3) contains singularities of O(r-1) and consequently it is possible to integrate this

equation accurately using the modified integration scheme of Lachat and Watson [1976].

In arriving at Eq. (4.3.3) integrations around the edge of the thin shell are discarded. The

justification for this in Chapter 3 was that for closed thin shells there is no edge and consequently

these integrations must be equal to zero. For non-closed thin shells this is not the case. However

in Chapter 2 it was shown that the pressure difference around the edge of the shell must be

equal to zero. Therefore for these shells the edge integration must also be equal to zero.

4.4 Numerical Implementation

The discretization of the variational formulation is implemented using the same order of

interpolation as that for the collocation method. The Vk. operator can be numerically approxi-

mated by,

ýe ne

E {bit T l [(npnq)k`2{N

}{Ng }T

_i=1 j-1 I>

(nP X OPlApj)-(n9 X Vq{NgIT)]Gk(P,9)dSqdSp W}.

(4.4.1)

page 81



Variational Alethod.

In Eq. 4.4.1) the gradient of the shape function is expressed in terms of the local coordinate

axis system. In order to evaluate these gradients it is necessary to use the relationship shown

in Eq 3.2.11). The Jacobian matrix for the system, defined in Eq. 3.2.7), then enables the

gradient to be expressed in derivatives of the curvilinear system.

For the integration in Eq. 4.4.1), the point p ranges over Si and the point q ranges over Sj.

The integration with respect to p is performed using simple Gauss integration. The singularities

in the above numerical approximation will occur at these Gauss integration points when i=j.

It is these auto-influence elements that need to be integrated using the Lachat and Watson

singular integration scheme. For the integration with respect to q, the element, will be divided

as shownin figure 4.1)

Singularity at Gausspoint

Figure 4.1). Subelement division.

The expression in Eq. 4.4.1) can be assembled into a matrix formulation of the variational

problem,

[N{}

-A

a0

-190`

= 0. 4.4.2).älß an

where the matrix [A] represents the numerical approximation,

ne

r

{Ne}{Ne}TdS 4.4.3)o}T [A]{O} =E {b4ti}T

Isi

j=1

In practice this banded symmetric matrix may be further approximated by the diagonal matrix,

page 82



Variational Method.

aii =i' ,(4) dSq.

S

(4.4.4)

The matrix equation in Eq. (4.4.2) is symmetric and is factorized into a LLT form using

the standard Choleski factorization technique (Jennings [1977)). The pressure differences are

then evaluated by means of forward and backward substitution. The order of Gauss integration

used to calculate the numerical results in this chapter are summarized in Table (4.1). Chapter

2 showed that the pressure distribution exterior to the thin shell is given by,

0 (P) _ (4.4.5)

and following similar arguments presented in Chapter 3 the far field pressure distribution is

approximated by the vector expression,

Osf(R) _{111kf}

ý, PEE.

Element p Integration q Integration

Singular 3x3

Non - singular 2x2

j Integration within subelement

4x4x4t

2x2

Table (4.1) Order of Gauss integration

4.5 Uniqueness of the Numerical Formulation

(4.4.6)

The variational method was introduced as a way of formulating the acoustic problem for

thin shells. In general there is no problem of uniqueness of the numerical solution for thin

shells since the interior domain for the shell tends to zero. Since the density of interior resonant

frequencies is proportional to this volume, the numerical formulation for non closed thin shells

is unique for all frequencies.

For rigid thin shells that enclose a finite interior domain there will be problems of uniqueness

at the resonant frequencies of this interior domain. Since the variational formulation of the

page 83



Variational Method.

problem is equivalent to the double boundary layer formulation detailed in Chapter 2, the

numerical solution will become ill-conditioned at eigenvalues of the interior Neumann problem.

Althougha

hybrid boundary layer formulationovercomes these problems of uniqueness, the

resulting boundary layer potential is no longer equivalent to the pressure difference across the

shell. The reason for this is that a natural variational formulation of the hybrid problem does

not exist, and the corresponding Galerkin formulation is non-symmetric.

In this chapter no effort is made to circumvent the problems of uniqueness that arise for

rigid closed thin shells. However Chapter 6 will demonstrate that for the elastic, closed thin

shells, the formulation is unique at all frequencies.

4.6 Edge Boundary Conditions

In Chapter 2 it was shown that for non closed thin shells it is necessary to enforce an

edge boundary condition. The physical reality of this is that the pressure difference across the

shell must be zero at the edges. Since this boundary condition is not implicit in the collocation

or variational numerical approximation of the .k operator, then it must be imposed on the

resulting matrix equation set.This

matrix equation set canbe

representedby,

[H]{ýD} = {y}, (4.6.1)

where [H] is the matrix approximation of the collocation or variational formulation and {y} is

the appropriate right hand side of this matrix equation;

[A] {an -

0 } Variational,(4.6.2)y} =

a0_

Collocation.an an

}

Defining { EJ and {(DI} as those elements of the vector that correspond to edge or interior

nodes, Eq. (4.6.1) can be rewritten as,

HII HIE (b I yI

YEEI HEE

41>E

(4.6.3)

The interior pressure differences can now be written in terms of the known edge pressure

differences and the known vector {yl},

page 84



Variational.fethod.

[HIf]Oýj}-

{y }-

[HIE]{(ýE}. (4.6.4)

If (D is equal to zero on the edge of the plate, this equation reduces to,

[HII]{(DI}=

{yI}. (4.6.5)

Imposing the edge boundary condition in this way is equivalent to making the shape function

for an edge node equal to zero so that there is no contribution to the interpolated pressure

difference within the element from this node. Consequently the way the pressure difference goes

to zero, depends on the other shape functions within the edge element.

This problem has not been delt extensively before in previous work. (Mariem and Hamdi

[1987], Terai [1980], Warham [1988]). These studies have used constant value elements and

for these cases there are no edge nodes and the edge boundary condition is satisfied when

distributing out the pressure differences from the element constant value to the nodal values.

The numerical results in this chapter show that the method described above satisfies the

edge boundary condition adequately. However it is found that there are larger numerical errors

at these edge points due to the imposition of the edge boundary condition. One possible im-

provement not implemented in this study would be the modification of the shape functions in

the edge elements, so that the edge boundary condition is included in the numerical formulation

implicitly and thus more efficiently.

4.7 Computer Code

The variational method was coded like the collocation method using FORTRAN in a UNIX

environment and many of the same routines used in the collocation method program were used

with little modification. Again all matrix routines are performed in core, however the symmetry

of the matrices in the variational method means that they can be stored more efficiently in a

triangular form, reducing the strain on memory requirements.

The most striking feature of the variational method is the increase in computational time

needed to assemble the matrix equation set. This increase in computational time is clearly

due to the extra integration. For both methods, but especially the variational method the

page 85



Variational Method.

computational effort needed to generate the numerical matrices will depend heavily on the

order of Gauss integration used.

Table 4.2) shows a comparison of accuracy for different orders of integration for the 6 ele-

ment test case using the variational and collocational method with a constant normal velocity

boundary condition at ka = 1.0. The singular integrations represent the integrations within the

Lachat and Watson subelements and the non singular integrations represent all other numeri-

cal integrations. This data allows the selection of the optimum integration scheme, balancing

computational speed with accuracy. Such an integration scheme seems to be 2x2 non singular

integration and 3x3 singular integration. However for this relatively small problem the timing

contribution due to the singular integration is high. For large problems the ne contribution from

the non-singular integrations will dominate the timing however accuracy will depend strongly

on the order of singular integration.

Some timing data is shown in table 4.3), for both the collocational and variational methods,

applied to the thin shell formulation. For this direct comparison of timings, 3x3 integration

is used within the subelements of the singular integration and 2x2 integration is used oth-

erwise. This data is shown graphically in figure 4.2). The assembly time for the variational

matrix is clearly greater than that of the collocation matrix and both assembly times show the

same dependency of approximately ne.There is a significant difference in the times of matrix

factorization. Whilst for both methods the factorization times are significantly less than the re-

spective assembly time, the factorization timings show a n3 dependency. For large problems for

which n.e» 100, this time factor will become the dominant factor, thus favouring the variational

method with the quicker symmetric Choleski factorization.

4.8 Numerical Results

4.8.1 Spheroids

The thin shell formulation is used to calculate the backscattered form function for different

spheroidal geometries with an end on incident plane wave. The form function in given by,

_

2Rb Off

a2 0t

4.8.1)

where for the spherical case of a=b this equation reduces to Eq 3.6.4). The spherical and

spheroidal mesh geometries are the same as those in Chapter 3.

page 86



Variational 3ffthod.

Method Non Singular Singular Timing sx 10-2 Accuracy ( )

Collocation 2x2 2x2 219 1.55

3x3 376 0.74

4x4 579 1.05

3x3 2x2 365 0.68

3x3 509 0.08

4x4 723 0.23

4x4 2x2 589 1.04

3x3 730 0.30

4x4 937 0.60

Variational 2x2 2x2 333 1.38

3x3 509 0.65

4x4 732 0.23

3x3 2x2 1157 0.65

3x3 1550 0.81

4x4 2204 0.43

4x4 2x2 3111 0.32

3x3 3847 0.43

4x4 4942 0.56

Table (4.2) Comparison of accuracy with different orders of integration

page 87



Variational Method.

Method Elements Nodes

CPU

Assembly

Timing in seconds

Factorisation

x 1O-

Total

Collocation 3 19 172 1 173

6 33 367 3 370

12 61 1039 7 1046

24 113 3071 49 3120

48 217 9680 457 10137

96 417 36708 5598 42306

Variational 3 19 205 0 205

6 33 518 1 519

12 61 1437 10 1447

24 113 4348 24 437248 217 15258 179 15437

96 417 58364 1205 59569

Table (4.3) Comparison of matrix assembly and matrix factorization times

1UUU(o

10000

CPUSeconds

x10 2

1000

100

1n

Assembly

vc

c

V

Factorisation

IV10

Number of Elements100

Figure (4.2). Matrix assembly and matrix factorization times for the variational,

(v), and collocation, (c), methods

page 88



l ariulion(il. 1lcthod.

For those closed geometries there will be problems of uniqueness at interior resonant fre-

quencies and 110method is ii pleiuented to reinme these problems. This is likely to account. for

the breakdown of the numerical results with respect to frequency

sincethe

range of wavenuin-bers over which the numerical problem is ill-conditioned at a critical wavenumber increases wit Ii

frequency until there is an overlap. Loss of accuracy at low frequencies is unlikely to be evident

unless the vavenuiuber is very close to the critical wavenuniber.

Figure (4.3) shows the results calculated using both the variational and collocation method

compared to the analytical solution for a sphere. The 6 element results show high accuracy at

low frequencies with a breakdown of both numerical solutions starting at ka = 2.5. There is

no significant difference in accuracy between the two methods. The 24 element results show

no breakdown of the solutions below ka =5 but the there is slightly better accuracy for the

variational results at the higher frequencies.

Figures (4.4-5) show the two methods compared to the results generated by S.W. Wu s

axisymmet ric method using 20 high order elements applied to prolate and oblate spheroids

respectively. In the case of the prolate spheroid, the breakdown of both numerical results occurs

at the same order of frequencies, and there is no significant difference in accuracy between them.

Since the mesh geometries for the spheroids are simply a scaled spherical mesh, higher accuracy

would be obtained if a more intelligent mesh were used that reflected the geometry and boundary

conditions of the specific spheroid.

For oblate spheroid there is no distinct breakdown of the numerical methods for both mesh

densities. The 24 element case shows high accuracy for all frequencies. For the 6 element case the

accuracy of both methods degenerates at about ka = 3.0 with the variational method showing

slightly higher accuracy.

4.8.2 Flat Disk

Figures (4.7-14) show the numerical calculation of the dimensionless radial surface pres-

sure on a circular disk radiating with constant normal velocity without a baffle. The results

are calculated using the mesh geometries shown in figure (4.6) using both the variational and

collocation methods. The equation for the surface pressure field for a thin shell was presented in

Chapter 2. For a flat continuous surface with no incident wave Eq. (2.5.5) gives a relationship

for the surface pressure,

2ýdý+(P)+0-(P)) _ .Mk[ý](P) = 0. (4.8.2)

page 89



Variational Mcihod.

Therefore the dimensionless surface pressure is given by,

2k (D (4.8.3)

an/o

where(11..

represents the constant normal surface velocity. Figure (4.7-10) show the amplitude0

of this quantity and figure (4.11-14) show the phase.

The numerical results are compared to results extracted from work by Weiner [1951], which

were calculated from diffraction data published by Leitner [1949]. Similar results have also been

published by \ Vu, Pierce and Ginsberg [1987] who use an axisymmetric variational procedure.

All results show convergence between 20 and 80 elements and there is little that separates

the accuracy between the collocation and variational methods. Overall there is a high degree

of accuracy. It is clear from these graphs however, that accuracy decreases significantly at the

edges of the disk. The accuracy at the edge does not seem to improve as the density of elements

is increased.

Figures (4.15-16) show the radiation reactance and resistance of the disk calculated using

both the variational and collocation methods. The radiated power and rate of radiation of

kinetic energy for the disk are given by,

11_1

2iwP

Is

JS ondS

(4.8.4)a

ý=1(9

a2k;,ßpsn

dS

The radiation impedance is defined using Eq. (4.8.4) by,

or= or, Ui =11

*(4.8.5)

where 0r is the positive radiation resistance, representing radiation damping, and o-i is the

radiation reactance that is usually positive; representing added fluid mass rather than fluid

damping.

In both figures the numerical results compare well to the results of Weiner [1951] and

Bouwkanip [1941], with a slightly higher degree of accuracy with the variational method. The

page 90



Variational.fethod.

resistance results from the work by Bouwkarnp were calculated for the complementary aperture.

The results are plotted on logarithmic scales in figure (4.15) to show the linear low order fre-

quency dependency of the radiation reactance and the higher order frequency dependency of the

radiation resistance.

The final result for the flat disk, figure (4.17), shows the convergence of the radial pressure

amplitude for the three mesh discretizations. The 28 element mesh is distinguished by the high

density of elements near the edge, however all the mesh geometries show similar inaccuracies

at the edge of the plate. To improve the accuracy at the edge of the disk it seams that a more

sophisticated way of imposing the edge boundary condition needs to be developed.

4.8.3 Flat Square Plate

Figure (4.18-19) show the radiation resistance and reactance calculated for the square plate

of side length L. The numerical results are calculated using both the variational and collocation

methods and are compared to the results extracted from work by « arham [1988].

There is good agreement using all methods but the non logarithmic plot shows that there

is higher accuracy with the variational method. The results from \Varham correspond to his

results for an asymptotic fine mesh. Comparison with the results for his `practical mesh show

that there is a significantly higher degree of accuracy and rate of convergence for both the

collocation and variational methods. This would be expected since Warham s method uses a

piecewise constant element discretization.

4.8.4 Terai s Problem

Terai s paper [1980] provides a comparison between his numerical results and an experi-

mental result. Figure (4.20) shows the comparison between the collocation and variation results

and the measured result extracted from Terai s work. The results are the pressure gain in dB

around a rectangular plate due to a point source. The results are calculated at a radius of 0.31

from the origin, in the X-Z plane, and the source is at a distance of 0.5 from the origin along

the Z axis. The rectangular plate in the X-Y plane has dimensions 0.3 x 0.2.

Both the collocation results and the variational results agree well with the measured values,

and show a significantly higher degree of accuracy than Terai s results.

page 91



Variational Method.

1.2

1.0

0.8

f

0.6

0.4

0.2

nn

a/b=16 Elements

v. v

0

1.2

1.0

1 2 3 ka 4

0.8f

0.6

0.4a/b=124 Elements

0.001234 ka 5

Analytical Variational ---- -- - .Collocation ----

Figure (4.3). The far field backscattered form function for a sphere discretized

into 6 elements and 24 elements. The numerical results are calculated using the

variational and collocation thin shell formulations.

page 92

0.2



1.2

1.0

0.8f

0.6

0.4

0.2

nn

Variational Method.

ii

t

5

alb=0.5 6 Elements

v. v

U

1.2

1.0

0.8

f

0.6

0.4

.,,,

a/b=0.5 24 Elements0.2

0.0 ,0123 ka 4

S. W. Wu variational --------- .Collocation ----

Figure (4.4). The far field backscattered form function for a prolate spheroid

discretized into 6 elements and 24 elements. The numerical results are calculated

using the variational and collocation thin shell formulations with an end on

incident plane wave.

1 2 3 ka 4

page 93



Variational bfethod.

1.5

w

. `

ý.0

f

0.5

0.0L0

1.5 r

. -,

1.0

f

0.5alb=2 24 Elements

0.001234 ka 5

S. W. Wu Variational ......... .Collocation ----

Figure (4.5). The far field backscattered form function for an oblate spheroid

discretized into 6 elements and 24 elements. The numerical results are calculated

using the variational and collocation thin shell formulations with an end on

incident plane wave.

a/b=2 6 Elements

1 2 3 4 ka 5

Page 94



Variational Method.

a) 20 element disk

c) 80 element disk

b) 28 element disk

d) 36 element plate

e) 24 element Terai problem plate

Figure 4.6). The thin plate mesh geometries

page 95



Variational.%ffihod.

2.0

2

IPI 43

.... i

1.0

ka=1

0.00.0 0.2

Wiener.

0.4 0.6

Variational

0.8 r/a 1.0

Figure The dimensionless radial pressure amplitude on a circular disk

radiating with constant normal velocity without a baffle. The variational results

were calculated with one quarter of the disk discretized into 20 elements.

page 96



Variational Method.

2.0

IPI

1.0

0.0L-0.0 0.2

Wiener.

0.4 0.6

Variational

0.8 r/a 1.0

Figure 4.8). The dimensionless radial pressure amplitude on a circular disk

radiating with constant normal velocity without a baffle. The variational results


page 97



Variational Method.

2.0

:::::....:...

2

3IPI

1.0

5

ka=1

4

0.01-0.0 0.2

Wiener.

0.4 0.6 0.8 r/a

Collocation ---------

1.0

Figure (4.9). The dimensionless radial pressure amplitude on a circular disk

radiating with constant normal velocity without a baffle. The collocation results


page 98



Variational.fffhod.

2.0

IPI

1.0

0.01 --0.0 0.2

Wiener

0.4 0.6

Collocation

0.8 r/a 1.0

Figure (4.10). The dimensionless radial pressure amplitude on a circular disk

radiating with constant normal velocity without a baffle. The collocation results


page 99



Variational.fethod.

90

S

60

30

0

4

3

-30

-60

ka=1

-90 -I0.0 0.2

Wiener.

0.4 0.6 0.8r/a 1.0

Variational --------- .

Figure 4.11). The dimensionless radial pressure phase on a circular disk radiat-

ing with constant normal velocity without a baffle. The variational results were

calculated with one quarter of the disk discretized into 20 elements.

2

page 100



Variational Mdhod.

90

5

60

4

0

30

---- ------------............................................................................

...............9

-30

a

-60

ka=l

-900.0 0.2

Wiener

0.4 0.6 0.8 r/a1.0

Variational --------- .

Figure (4.12). The dimensionless radial pressure phase on a circular disk radiat-

ing with constant normal velocity without a baffle. The variational results were

calculated with one quarter of the disk diýcretized into 80 elements.

page 101



Gariational Method.

90

5

60

4)

30

0

4

.....

3

1-----1111*1................:::.-30

-60

ka=1

-900.0 0.2

Wiener

0.4 0.6 0.8 r/a 1.0

Collocation --------- .


ing with constant normal velocity without a baffle. The collocation results were


..

........................................................................................-- ........... ..........

page 102



Variational Method.

S

40

a

ý ....

10

4

3

'4.

\ -1

ka=1

-90 `-0.0 0.2

Wiener

0.4 0.6 0.8r/a 1.0

Collocation --------.


ing with constant normal velocity without a baffle. The collocation results were


page 103



l ariational Method.

10

1

.1

01

001

0001

0

0

Variational Collocation --------- .

Bouwkamp ----Wiener O.

Figure 4.15). The radiation impedence of a circular disk radiating with constant

normalvelocity without a baffle. The variational and collocation results were

calculated with one quarter of the disk discretized into 20 elements. The results

are plotted on a logarithmic scale to highlight the low frequency dependence.

page 104

11 ka 10



Variational affthod.

1.5

resistance

ý,,,.~ýý

v,.ir..

1.0

i

0.5

ýd.

reactance

I

0.0 `0 246 ka 8

Variational Collocation--------- .

Bouwkamp Wiener O.

10

Figure (4.16). The radiation impedance of a circular disk radiating with constant

normal velocity without a baffle. These are the same results as the previous figure

but plotted on a linear scale.

page 105



Variational Method.

1.0

ka= 1

0.8

IPI

0.6

0.4

0.2

0.0-

0.0

3k

Di

ka=4

2.0

IPI

1.0

a

0.00.0 0.2 0.4 0.6 0.8 1.0 ka 1.2

Wiener.

20 Elements o. 28 Elements o. 80 Elements .

Figure /x.17). The dimensionless radial pressure amplitude for different mesh

densities at ka =1 and ka = 4. The results are calculated using the variational

Method.

0.2 0.4 0.6 0.8 r/a 1.0

page 106



Variational Method.

10 1

100

10-1 ;ý

resistance

10 -2

10

.1

Variational

1L /ý 10

Collocation --------- .Warham ----

Figure 4.18). The radiation impedence of a square plate radiating with constant

normal velocity without a baffle. The variational and collocation results were

calculated with one quarter of the plate discretized into 36 elements. The results

are plotted on a logarithmic scale to highlight the low frequency dependence.

reactance

page 107



Variational Method.

1.5

1.0

0.5

An

r

. reactance

i

resistance

f.

ýrý

U.V0 L/%, 3

Variational Collocation --------- .Warham ----

Figure (1.19). The radiation impedance of a square plate radiating with constant

normal velocity without a baffle. These are the same results as the previous figure

but plotted on a linear scale.

page 108



Variational Method.

4

dB

I.;

0 I

-4 `0

Source

90 180 270 0 360

Microphone

Measured Variational --------- .Collocation

----

Figure (4.20). Nearfield pressure gain for a point source wave scattered by a

rectangular plate at k= 18.44. The numerical results are calculated using 24

elements and the measured result is extracted from the work by Terai.

page 109

iv



The Superposition Method.

CHAPTER 5.

The Superposition Method

5.1 Introduction

Superposition methods have long been used as a bench test for other numerical analyses of

acoustic fields. Recently Koopmann Song and Fahnline [1989] suggested that a superposition

method could be extended into a general solution technique for calculating acoustic fields. In

the work of Koopmann et a1[1989] Song et al [1991] and Miller et aI [1991] a complex radiator

is replaced by an array of simple monopole sources of unknown magnitude constrained to lie on

a surface interior to the body of the radiator. The magnitude of the simple sources is calculated

by equating the normal velocity prescribed on the surface of the radiator to that generated by

the array of simple sources. This is performed at the same number of points as there are simple

sources and consequently a system of equations is generated. The solution of this equation set

gives the magnitude of the simple sources and from these values an exterior pressure field can

be calculated.

Since the superposition method removes the need for singular integration techniques is

has been suggested that the method improves on the traditional BEM. The author however

supports the view of Katz [1987] who suggests that some [superposition] methods spoil the

whole approach ...they will work with some cases but are not general and they will produce

ill conditioned systems in some cases. It was decided that the superposition method was worth

investigating in order to establish that in a hierarchy of integral methods the BEM represents

the limit in terms of accuracy and conditioning of the superposition method.

Although the superposition method described above circumvents the problem of uniqueness

of solution found in BEMs a formulation which constrains the interior point sources to be on

an interior surface does become ill-conditioned at another set of critical wavenutnbers. These

wavenumbers correspond to the eigenvalues of the unrelated Dirichlet problem interior to the

source surface. The superposition integral is now equivalent to a single layer source distribution

and as such can be shown to have a non-unique solution at these eigenvalues. Although it is

possible to reduce the interior source surface so that any critical wavenumbers lie outside the

frequency range of interest there is also a potential loss in numerical stability. In this chapter

a numerical strategy will be presented that circumvents this uniqueness problem allowing the

page 110




optimum choice of interior surface to be made, in order to obtain the maximum improvement

in numerical stability of the solution.

The problem of uniqueness has been overcome in this study by considering the superposition

integral in terms of a hybrid combination of single and double layer potentials. A similar solution

to the problem of uniqueness in B.E..N1 s is discussed by Filippi [1977], and Colton and Kress

[1983].

As noted by Koopmann et al [1989] it is possible to use derive a superposition integral either

by assuming that the source distribution and Green s function are constant over each interior

element or by allowing the source distribution and the Green s function to vary in some manner

and the kernel function to be evaluated using Gaussian quadrature. In this paper the first

method is denoted as the Point source Superposition Method (PSNI). Results are also presented

for the second method which is denoted as the Integrated source Superposition Method (ISM);

the source distribution and geometry are approximated by the use of quadratic interpolation

functions defined using 9 noded surface elements.

5.2 The Superposition Integral

For a body radiating with a prescribed surface velocity it is desirable to calculate the exterior

pressure field. The principle of wave superposition shows that the acoustic solution for some

radiating body is equivalent to the acoustic solution of some source distribution interior to the

body. If both systems satisfy the same Neumann boundary condition on the surface of the body

then the pressure distributions generated by both are equivalent since an exterior pressure field

is unique for a prescribed boundary condition. Koopmann et al proposed that the solution of

the equivalent superposition formulation of a problem offers many significant advantages over a

B. E. M solution of the same problem. The validity of the superposition method can be shown in

the following way.

Consider a continuous distribution of sources contained in a volume Q. This source distri-

bution is interior to a closed fictitious surface S on which there is a prescribed normal velocity

distribution u,,. The volume enclosed by S is denoted D and the volume exterior to S is denoted

E. This geometry is shown in figure (5.1).

page 111




E

S

Figure (5.1). . The geometry for formulating the superposition integral.

Applying the principle of mass conservation (Pierce [1989] Chapter 4 p162) to the volume

enclosed by S leads to a modified reduced wave equation defined by,

v2 p(r) + k2P(r) = iwpgo(r), rED. (5.2.1)

A time dependence of e-'wt is assumed and qo(r) is the interior source strength defined by,

q(r) rES2,

qo(r) = (5.2.2)

0 rEDexcluding ft

Using Eq. (5.2.1) the modified interior Helmholtz integral equation can be developed in a similar

way to the standardinterior Helmholtz

equation andis

givenby,

ýiwpu(ro)Gk(r, ro) - p(ro)

öGk

ö

(nr,ro) d'Sr, + iwp4(ro)Gk(r, ro)dVraJs ra

fn(1

- c(r))(r)p(r),

Gk (r, ro) =1eikr

47rrT=

IT- vol

(5.2.3)

The unrelated exterior Helmholtz equation is independent of the interior source distribution and

identical to the exterior Helmholtz equation for a real surface S with prescribed normal velocity

distribution u,,. This equation is defined by,

page 112



The,cuperpositaon Method.

ÖGk(r, ro)

SPro)

a 1.ý pun(ro)Gk dS ro = C(r )P(r ). (5.2.4)

n*o

In the source distribution model of the vibrating body, the surface S is a construction

indicating where the normal velocity distribution is prescribed. Consequently the pressure and

velocity distributions across the surface must, be continuous. When r is taken to be on S, Eq.

(5.2.3) and Eq. (5.2.4) combine to give,

P(r) =fipq(ro)Gk(r,

ro)dVro. (5.2.5)

Eq. (5.2.5) is the superposition integral. Associated with this equation is its differentiated form,

defined by,

2dn(r) =

In

q(ro)ÖG3(ro d\0.

0

5.3 Uniqueness

(5.2.6)

In the derivation of Eq. (5.2.5) and Eq. (5.2.6) the Helmholtz integral equations are cir-

cumvented and for a source domain that contains no interior volume, numerical implementations

of these equations are unique at all frequencies. However the choice of volume S2 s arbitrary

and for ease of numerical implementation it is best to locate the interior source distribution

over an interior surface S of small thickness 6r. However the superposition integral will now

exhibit non-uniqueness at critical wave numbers. This non-uniqueness can be demonstrated in

the following manner.

For the case where Sl is chosen to be a thin shell, Eq. (5.2.5) and Eq. (5.2.6) become,

P(r) =I iwpq(ro)Gk(r, ro)brrdSfo. (5.3.1)IS/

un(r) =J 4(ro)8GO(r ro)

brdS;.o.

(5.3.2)is,

o

page 113




In the limit as the thickness of the surface S' becomes zero, these equations show the pressure

distribution being defined in terms of a single layer potential. Using the integral operator

notation,it is

possibleto

writeEq. (5.3.1)

andEq. (5.3.2) in

terms of a singlelayer

potential

along with an associated expression for the pressure distribution in terms of a double layer

potential. These expressions are defined by,

P(r) - iwpCk[ 's](r), (5.3.3)

u(r) = Mk [t3](r), (5.3.4)

P(r) = 2Wp ik [11d](r), (5.3.5)

um(r) _Yk[0d](r), (5.3.6)

Consider the case of the single layer potential formulation for u =0 on S. Since the

pressure field exterior to S' is unique for all wavenumbers then u, =0 and p+ =0 on S. The

superscript indicates the pressure or velocity evaluated on the exterior (+) or interior (-) surface

of S'. By consideration of the surface discontinuities in the integral operators Gk and Mk, it is

possible to write Eq. (5.3.3) and Eq. (5.3.4) in the limit as r tends to r0 on S' from the exterior

and interior domains. These equations are,

p+(ro) = P-(r0) = iwprk[0, ](ro), (5.3.7)

un (ro) = um(ro)- 0s(ro) =

(Mr

-2{ s](ro).(5.3.8)

Eq. (5.3.7) shows that the pressure distribution is continuous across the single layer surface

and consequently if p+ =0 then p- = 0. From Eq. (5.3.8) with u,+, = 0, v, is non-zero when

un # 0. Since p- =0 this only occurs at eigenvalues of the Dirichlet problem interior to S. At

these frequencies, 0, will have a non-trivial solution for a zero velocity distribution on S. This

means that t,,, is not unique at these critical wavenumbers and the numerical implementation

of Eq. (5.3.4) will become ill-conditioned. A similar treatment for the double layer potential

shows that in Eq. (5.3.6), 1d is non-unique at the interior eigenvalues of the Neumann problem

interior to S.

To eliminate the problem of uniqueness at critical wavenumbers, a hybrid combination

of single and double layer potentials is used. The formulation for the superposition problem

becomes,

page 114



The Superposition.1Method.

P(?) = iwp(. Ck + iTlMk) [v,7(r), (5.3.9)

utz(r) = (Mk + ii \ (5.3.10)

The value of the real constant 77n Eq. (5.3.9) and Eq. (5.3.10) is chosen to be 1/k to compensate

for the order of frequency terms in the integral operators. The advantage of the superposition

method using this hybrid formulation is that there is no high order singularity in the .A operator

since the source surface S is not coincident with the boundary surface S. It should be noted

however that in regions where there is a high density of critical wavenumbers, a high degree of

accuracy is needed to reduce the range of wav-enumbers over which the non-hybrid operators are

ill-conditioned, so that the hybrid formulation is efficient.

5.4 Numerical Formulation

A numerical implementation of the superposition integral has several advantages over a

BEM. A major factor in any BEM is the numerical treatment of the singularities that inevitably

occur in the formulation. In the superposition method the source distribution is interior to the

body and therefore not coincident. Consequently there is no singularity in the Green s function

of Eq. (5.3.9) and Eq. (5.3.10).

In Eq. (5.3.9) and Eq. (5.3.10) the surface S can be discretized into a number of elements,

n. The nodes of the interior source elements must each have a corresponding surface node. Using

the definitions of previous researchers the corresponding node on the interior source surface is

known as the self node. This definition can be extended to refer to the self element which

is thesource

surface element that contains the self node. Eq. (5.3.10) can be approximated

numerically by,

n

1ý,Gk(r, ro) 02Gk(r. ro)

un(r) =E -ý 7(ro)dSro7=1m Ö12ranr.

For the PSM it is assumed that the kernel of Eq. (5.4.1) and (r0) is constant within the

element and it can be written in matrix form by,

{un} = DA{ ;}, (5.4.2)

page 115



The Supfrposition.Method.

With,

r"n; r-ni r. nj Gk(r) Ge(r)Dij = Gk(r)

r+i77

rr

(G(r)

-T-r 7zß.nj C . 4.3)

r=ri - rj. r=Irl.

and A is the diagonal area matrix.

In Eq. (5.4.2) {u} is the vector containing the n values of u evaluated at the points r;

on the boundary surface that correspond to the n nodal points defined on the interior surface,

rj. The source surface nodal points are defined at the centroid of the source surface element.

The vector {0} contains the n values of ý,j evaluated on the interior source surface and .

4j ,

corresponds to the amplitude of point source radiators situated at the interior nodal positions.

The discretized form of Eq. (5.3.9) is defined in a similar way to Eq. (5.4.2),

{p}-

(5.4.4)

with,

r"njM; j = Gk(r) + iriG' (r) (5.4.5)

r

Eq. (5.4.2) and Eq. (5.4.4) can be combined to give an expression for p in terms of ums,. his is

written as,

{p} = MD-1{u} (5.4.6)

In the PSM the principal assumption is that the interpolation of the source distribution is

given by,

V= b(ro, rj)Zj, (5.4. i)

where 6(ro, rj) is one at the nodal point defining the element and zero otherwise, and V)j is the

nodal value of the source distribution for the element. For the ISM a different assumption is

made. Within the element the source distribution is given by.

page 116



Thf Superposition Method.

VI)= {1Ve}T{ a}, (5.4.8)

where {Ne} is the vector of element shape functions and {ýJ} is the vector of element nodal

values of the source distribution. The numerical approximation given by Eq. (5.3.10) now

becomes,

un(T) -

(ýGk(rro)

+ .ýa2GkI. Te}TdSr°{2J}. (5.4.9)S

Onr (9nrOnrJ=1 ý°

The numerical integration in Eq. (5.4.9) can be performed using simple Gaussian quadrature

since there is no singularity for retracted source surfaces. A matrix equation set can be assembled

for the acoustic formulation eliminating 0, the global vector of nodal source strengths,

{p} = MIDI'{un} (5.4.10)

In this study the interior nodal points for both the PSM and ISM were generated in the

same way. Since only the nodal point for the element is needed in the PSM, these nodal points

can correspond to the nodal points for the ISM. To optimize the conditioning of the D and DI

matrices, the definition of the boundary and source surface nodal points is critical. Following

previous work the interior source points are defined by,

Tj = Ti -d1

2, (5.4.11)

where d is defined as the retraction distance. The relationship in Eq. (5.4.11) is not sufficient

to ensure optimum conditioning. The boundary nodal points have to be defined so that,

.ITi

-TjIj,

4i > ITi-'I=i (5.4.12)

These relationships ensure that there is optimum symmetry and diagonal dominance of the

equation set. This requirement is important in reducing the degree of ill conditioning of the

formulation, however such a restriction on the positioning of the boundary surface nodal points

will be independent of the applied boundary conditions.

page 117



The Super position Method.

For radiation problems u is simply the prescribed surface normal velocity. For plane wave

scattering u1. can be shown (Junger and Feit [1986]) to be equal to,

PI-

&I r

wp(5.4.13)

where uj is the fluid particle velocity on the boundary surface in the absence of any interior

source distribution, generated by the incident pressure wave. The incident pressure wave has

amplitude pl. and wave vector kI.

5.4.1 Matrix Condition Number

A well recognized failing of the superposition method is the ill conditioning of the generated

matrices. This lack of conditioning is due to the numerical instability of Fredholm equations of

the first kind (Arfken [1985], Miller [1974]). One loss of conditioning is due to the loss of diagonal

dominance as the source surface is retracted from the boundary surface. Computationally the

loss of conditioning means that small changes in the surface velocity distribution can have a

large effect on the source distribution. Following Golub and Van-Loan [1983] a conditioning

number can be calculated that gives a measure of the sensitivity of a linear system. The matrix

problem for the PSM or ISM can be stated as,

{un}= D{O}. (5.4.14)

If the boundary velocity distribution is perturbed by an amount {6u } then for this linear

system,

{b} = D-1{bu, }, (5.4.15)

and from the properties of vector and matrix norms,

11 11< JID-111

16un11(5.4.16)1011 11011

tun II IIDIIIIVII. (5.4.1 7)

page 118



The Superposition.11Eihod.

Therefore the perturbation in un is related to the perturbation in L- by means of the

condition number n;,

IIa'+'II<K

IIbun II(5.4.18)

IIv'II IIti II

= IIDIIIID-111. (5.4.19)

In this study the conditioning number is calculated in terms of Euclidean matrix norms.

The equality in Eq. (5.4.18) states that the conditioning of the matrix equation set is related

to t;. However, the accuracy of the matrix equation solution need not be directly related to this

quantity. This condition number can not reflect the improvement in solution accuracy that is

possible if the symmetry relationship for nodal point placement is observed. The superposition

problem defined in Eq. (5.4.14) can be written in terms of a singular value decomposition,

{un} = XS2YT {b} (5.4.20)

The square matrices X and Y represent orthogonal matrices and the diagonal matrix Q2, rep-

resents the eigenvalues of the symmetric matrix, DT D. The solution to the problem is given

bY,

lo}Y =Q-lf, un}X, (5.4.21)

where the subscript X and Y represents the projection of the vector onto the respective orthog-

onal matrix. If the matrix D is nearly symmetric then the source distribution and the boundary

velocity distribution will be expressed in terms of the same orthogonal matrix. An eigenvalue

analysis (Strang [1988]) of Eq. (5.4.21) shows that when D is symmetric the eigenvalues of

this equation are perfectly conditioned. Therefore increasing the symmetry of the superposition

matrix, significantly improves the overall conditioning of the numerical formulation.

5.4.2 Velocity Reconstruction Error Norm

In order to select the optimum position of interior nodal points a measure of the solution

accuracy is needed. This may be done by an a priori knowledge of the boundary pressure

distribution however this is clearly not always possible. Another measure of the accuracy of

the superposition method is the extent to which the prescribed normal boundary velocity is

page 119



Thf Superpo. iiion .110hod.

reconstructed by the source distribution. By using the numerical approximations of either the

PS-N1or ISM the reconstructed surface normal velocity is given by,

it,, (r) = {D(r)}T D-1{ u, }. (5.4.22)

When r corresponds to a boundary surface nodal point, Ti, then clearly,

(5.4.23)

For other points there will however be some difference in the prescribed and calculated surface

normal velocities and a measure of this difference is the velocity reconstruction error norm,

Ilun- unllo

Ilun llo

where,

(5.4.24)

un -it ll, =

j(u-

-(5.4.25)

This error norm will be closely linked to the error in the boundary surface pressure and can

be used to select the optimum interior nodal positions without an a priori knowledge of the

boundary surface pressure distribution.


The superposition formulations were applied to spherical and spheroidal scattering and

radiation problems. These boundary surfaces were chosen due to the cylindrical symmetry that

exists for appropriate boundary conditions. In all cases one quarter of the boundary surface

is discretized, making use of the problem symmetry to reduce the problem size. In order to

optimize the symmetry of the resulting superposition matrices the surface nodal points are

distributed as evenly as possible over the boundary surface.

Figure (5.2) contrasts the non uniqueness characteristics at critical wavenumbers of the

single and double layer formulations with that of the hybrid formulation. A constant normal

velocity distribution is prescribed for a sphere with radius a and the source surface is retracted

page 120



The Superposition J10hod.

by 0.5a. These graphs show the variation of the condition number, n, and velocity reconstruction

error norm, a, compared to ka for the spherical problem. For the spherical geometry the critical

wavenumbers will occur at,

k a - d) = kn. 5.5.1)

For the breathing mode problem, ko is equal to it and 4.493 for the single layer and double layer

problems respectively. The loss of conditioning is clearly shown at these points along with the

resulting loss in numerical accuracy for the single and double layer formulations. The hybrid

formulation removes the problem of non uniqueness at these frequencies.

The potential accuracy of both the PSM and ISM for calculating the back scattered form

function for both spheres and spheroids is shown in figures 5.3-4). In figure 5.3) the PSNN1and

ISM applied to the spherical problem is compared to the analytical result and the collocation

BEM result. The retraction distance is taken to be 0.5a. The superposition method results show

very high accuracy due to the high degree of symmetry between the spherical boundary and

source surfaces. The symmetry of the resulting matrices significantly improves the conditioning

of the problem so that high accuracy is possible. The reduced size of the source surface elements

compared to the corresponding boundary surface elements also allows the superposition methods

to have a more accurate surface representation for this particular example.

Figure 5.4) shows the same comparison for the spheroidal back scattering problem. The

spheroid has an aspect ratio of alb = 0.5, where a and b represent the minor and major radii

respectively and a retraction distance of 0.4b is used. A converged solution is taken to be the

result of an axisymmetric BEM solution with a large number of elements and these results are

calculated by a program written by Wu [1990] using 40 quadratic line elements. The PSM

results show satisfactory accuracy whilst the BEM results show high accuracy compared to the

less accurate results of the ISM. The results of the ISM suggests that the solution accuracy is

dependent on the conditioning of the superposition matrices.

In order to illustrate that the accuracy of the superposition method is dependent on the

retraction distance, the far field scattered pressure was calculated for the problem of a plane wave

incident upon a sphere. An error quantity, E, is defined as the error of the far field back scattered

form function with respect to the result for a 96 element BEM calculation. The variation of

the log of this error together with the log of the velocity reconstruction error norm, a has been

page 121



The SupErposition Method.

plotted in figure 5.5) as a function of d/a, for a ka = 1. Figure 5.5a) shows the results for the

point superposition method and figure 5.5b) for the integrated superposition method.

The condition number of the solution matrix for both superposition methods is given in

figures 5.6a) and 5.6b). The results for the spherical and spheroidal geometries are presented.

Comparsion of the error measures in figure 5.5a) with the condition number of the matrix

generated with the spherical P5M1 in figure 5.6a) indicates little correspondence between these

quantities. As the velocity reconstruction error norm, a is less than the far field error, the ma-

trices are conditioned such that the accuracy is dependent only on the order of the interpolation

of the source distribution. As the source surface is moved closer to the boundary surface, both

error quantities increase until the velocity reconstruction error norm, a is greater than the error

in the far field back scattered form function, F.

The results for the ISM, shown in figure 5.5b), indicate that between d/a = 0.9 and 0.5 the

trend in the conditioning of the problem, shown in figure 5.6b) and the error indicators a and e is

similiar. For this range of retraction distances the degree of symmetry in the problem increases,

until at about d=0.5 the problem became sufficiently well-conditioned to allow very accurate

solutions to be attained. The far field error, a, increases as d tends to zero, while the error in

the internodal velocity reconstruction remains small. The accuracy in the formulation could be

increased for these low retraction ratios by simply using the singular integration techniques used

in the BEM.

The velocity reconstruction error, a, and the far field back scattered form function error,

e, were also calculated for the spheroidal problem. The values of a and e were calculated as a

function of d/b at kb =1 for both the point superposition and integrated superposition methods.

The results are given in figures 5.7-8). For both methods the variation of a is similar, revealing

a minimum for a particular retraction ratio, before the numerical error due to small d became

significant. The value of d/b where this occurs decreases with increasing n. The errors for the

PSM show there is a greater range of retraction distances over which there is low error reflecting

the greater symmetry of the corresponding superposition matrices.

5.6 Conclusion

A numerical solution to acoustic problems has been described along with a strategy to

overcome problems of uniqueness using a hybrid formulation. The superposition formulation

was shown to be valid and the breakdown of the numerical formulation at the eigenvalues of

page 122



Thf Superposition Method.

the interior source surface was demonstrated. The accuracy of the formulation was measured

through use of a velocity reconstruction error norm.

The numerical results of the superposition method applied to radiation and scattering

problems are presented for spherical and spheroidal surfaces. The numerical results indicate

that with careful choice of boundary and source surface nodal points that it is possible to attain

solutions with high accuracy to both classes of problems. The results demonstrate that the

accuracy of the superposition method unlike the surface integral approaches depends to a

significant degree on the conditioning of the problem rather than the accuracy of the source

representation.

It should be noted that both formulations tested are numerical approximations to Fredholm

equations of the first kind and therefore subject to a inherent instability in the solution process.

As demonstrated in the results the conditioning of the formulation will deteriorate as as the

source surface surface is retracted from the boundary surface. Despite the deterioration of the

condition number accurate solutions are possible for a certain range of retraction distance

however once out of this range large errors occur in the calculated nodal surface velocities.

The numerical conditioning of the problem can be optimized by reducing the source sur-

face retraction distance and therefore increasing the diagonal dominance of the formulation.

However as the surface retraction distance is reduced close to the boundary surface then the

solution accuracy will deteriorate once again due to the inadequate integration of the source sin-

gularity. Another factor that will increase the conditioning of the formulation thus for specific

problems giving a high solution accuracy is the degree of symmetry in the resulting equation

set. However as noted by Song et aI [1991] the symmetry of the formulation will depend on the

surface nodal points the retraction distance and the interpolation of the source distribution.

The numerical experiments indicate that for optimal solution accuracy and problem con-

ditioning then the source nodal points need to be placed at a fixed distance along the normal

from the boundary nodal points. If the source nodal points are placed at any other locations

then a dramatic loss of solution accuracy will occur.

The results also indicate that in order to make the solution accuracy solely dependent on

the discretization of the problem while optimizing the numerical conditioning of the formulation

then the boundary and source surfaces need to be coincident.

page 123



The Sunervosition.Method.

8

6

4

2

0

-2

-46. ý

6

4

2

0

-2

A

Single layer

K0

OOOOOOO00000000000

a

78 6.280 6.282 6.284 6.286 ka 6.288

Double layer

OK

O0OOOO

a

''''''

-It8.982 8.984 8.986 8.988 ka 8.990

Figure (5.2). The effect of the hybrid formulation on the elimination of criti-

cal wavenumbersfor

the single layer and double layer PSMformulations.

The

results are for a sphere with constant normal velocity distribution and d=0.5.

The solid line represents the hybrid formulation.

page 124




aa

aaM

il>, C

C wCn

¢ as a º..,

a

13

N

SC

0N

LO

1:

O

r

LC)6

PNO 00 CO It NOO

ÖÖÖOO

uorppun WJod

Figure (5.3). The far field form function for plane wave backscattering for a

sphere. The superposition results are compared against a BEM and analytical

result.

page 125




aad

134

13

m13

0a

4 1340

a

U,N

Gý

.1

0

N

LO

0

C3

L6

NOcO

uoip: un,4 W. o,4

c0Ö

IT6

MN

NC

3 M a .

0 4 a

N6

J°o°0

Figure 5.4). The far field form function forplane wave

backscattering for

aspheroid with a/b = 0.5. The superposition results are compared against a BEM

result and results calculated by S. W. Wu.

page 126



1

0

-1

-2

-3

-4

-50

1

0

-1

-2

The Superposition Afethod.

r`

St

PSM

1

Sý

C

a

.00.20.4 0.6 0.8 d/a to

-3

-4

-50.0

Figure 5.5). Thevariation of e and a against d/a

fora sphere calculated using

the PSM and ISM. The log of the error values is plotted. a) PSM with n= 33.

b) ISM with n=6.

page 127

0.2 0.4 0.6 0.8 d/a 1.o




20

ýd

20

ýd

10

T- 1

3 n=33 pSM- n=113

- n=417

- n=33 Sphere

10

0.2 0.4 0.6 0.8 dlb 1.o

0-0.0

ý- n=6n=24

ý- n=96-°- n=6 Sphere

ISM

0.2 0.4 0.6 0.8 d/b 1.0-0.0

Figure (5.6). The log of the superposition matrix condition number as a function

of d/b for the prolate spheroid with alb = 0.5. (a) PSM. (b) ISM.

page 128




PSM

`

-2

-3

-4`-0.0

1

0

-1

-2

-3

-4`-0.0 0.2 0.4 0.6 0.8 d/b 1.0

Figure (5.7). The far field error as a function of d/b for the prolate spheroid

with alb = 0.5.. (a) PSM. (b) ISM.

----- n=33

n=113

n=417

111 i

Y

i

i

1ir

0.2 0.4 0.6 0.8 dlbto

ISM

`/ ..

4

4`d

.

ii

n=6n=24n=96

page 129




8

PSM

6

öbD

04

2

0

6

b1DO

4

---- n=33n=113n=417

/

J`

el

0.2 0.4 0.6 0.8 d/b 1.o

ISM

-2 --0.0

8r

n=6n=24n=96

R

r

2

0

0.2 0.4 0.6 0.8 dlb 1.0-2`

0.0

Figure (5.8). The velocity reconstruction error norm as a function of d/b for

the prolate spheroid with a/b = 0.5. (a) PSM. (b) ISM.

page 130



Elasto-Acoustic Problem.

CHAP TER 6.

Elasto-Acoustic Problem

6.1 Introduction

Previous chapters have been concerned with the acoustic analysis of rigid submerged struc-

tures. When a submerged structure is elasticly deformed by the acoustic pressure acting on its

surface, the acoustic model needs to be expanded to account for this fluid-structure interaction.

The established technique for modeling the elastic deformation is the Finite Element Method

(FEM). An elasto-acoustic analysis is performed by coupling the BEM to a FEM structural for-

mulation. The resulting equation set can be solved for the structural displacement and surface

pressure in terms of the structural and acoustic excitation. In this chapter the elasto-acoustic

analysis of thin shells will be presented.

6.2 Structural Problem

For an extensive treatment of the FEM the reader is referred to specialized texts, e.g.

Zienkiewicz and Taylor [1989], Hughes [1987], Hinton and Owen [1979]. The following section

is an outline of the thin shell FE11 used in this study. For the thin shell domain, Sl, of density

p, there exists a equilibrium equation relating the applied surface forces, f, the inertial forces

R and the local displacement vector, u,

Cu+R=f, (6.2.1)

where C is the elasticity operator. The FEM is formulated from the minimization of the following

energy functional,

II= 2J

.CUT

(Cu+R-f)) (6.2.2)

The local displacement vector in Eq. (6.2.1) will have six degrees of freedom; three dis-

placement and three rotational components,

page 131



Elasto- Acoustic Problem.

uT

uy

u,zu=0.T

eyeZ

It is related to the global displacement vector by means of a transformation matrix,

u=TU,

6.2.3)

6.2.4)

where T is the matrix of directional cosines and U is the displacement vector expressed in the

global coordinate system.

The structural domain is modeled by Mindlin type thin shell elements, defined in three

dimensional space. These elements are derived from a three dimensional continuum element

using a degeneration technique. The degeneration relies on two assumptions; first the normals to

the mid surface of the element remain straight after deformation and second the stress component

normal to the shell is constrained to zero.

The thin shell elements are defined geometrically by the same set of interpolation functions

used in the acoustic boundary element problem. This conformity between the acoustic and

structural meshes simplifies the coupling between the two formulations, however it is believed

that in most cases, refinement of the resulting formulation would be most efficiently achieved

by different acoustic and structural surface meshes.

Using the shapefunctions defined in Chapter 3,

thelocal displacement

vector canbe

in-

terpolated from the element nodal displacements. For the element j, this interpolation can be

written as a matrix relationship,

-71{vj}.6.2.5)

For nine noded elements the shape function matrix N will have dimensions 6x 54 since { UP}

has six components per node. The local displacement vector given by Eq. 6.2.5) represents the

mid plane displacement vector defined at = 0. The through thickness displacement field is

related to the mid surface displacements by,

page 132




ýr = ur+zOy,

icy=tcy -SOX

üz = Uz,

(6.2.6)

where the local z variable ranges from-h/2 to h/2, where h is the thickness of the shell.

These equations reflect the assumption of constant through thickness displacement. The strain

displacement differential operator matrix is defined by,

Exa

ax 0 0

Eya

ay ux

Ery - ay ax0 uy (6.2.7)

E12 aZ

0x

fl,

Eyz 0 a aaz ay

Eq. (6.2.7) can be written in terms of the mid plane displacements using Eq. (6.2.6),

ax0 0 0 zäx

Exy

0 ay 0 z ay 0 0 ux

\ä ä o -z

äzä 0 yfxy

E= =

yO

xo ä x

0

y

1 0 (6.2.8)

L

0 0 ä-1 0 0

-txz 0 0 0 0z7yz 0 0 0 0 0

y

The pseudo-strains yaZ and -tiyz in Eq (6.2.8), are defined at this stage to eliminate the nu-

merical difficulties associated with the drilling degree of freedom, 0, that can arise in the final

structural formulation. The structural strains can now be written in terms of the element nodal

displacements,

E=ýBý J L

)

where,

[Bý= AIM.

The operator9 represents the differential matrix operator in Eq. (6.2.8).

The local stress-strain relationship is defined by,

(6.2.9)

(6.2.10)

page 133




a= Dc, (6.2.11)

where D, the reduced constitutive equation, enforces the assumption of zero normal stress. This

matrix can be shown to be given by,

1 v 0 0 0 0 0

1 0 0 0 0 0

D1

E

-v2

µ O

µ

0

O

µ

0

0

0

0

0,

0

where,

v= Poisson's ratio,

E= Young's modulus,

1-v

2

(6.2.12)

(6.2.13)

The constant K defines the pseudo-strain / pseudo-stress relationship and takes a small value of

10-4.

The inertial forces in the structural domain are approximated by,

Ps 0 0

0 ps 0

_0-0

p,0 0 0

0 0 0

0 0 0

= -W2[ps]u,

o 0 0 üx0 0 0 uy0 0 0 üZ

0 0 0 9x '

o 0 0 ey

o 0 0 9y

(6.2.14)

and the structural equilibrium relationship can now be defined using the variational principle

shown in Eq. (6.2.2). This results in a structural equation set defined by,

([K]- w2[M]) {U} = {1 }

,

where the stiffness and mass matrices are assembled from the elemental components;

(6.2.15)

page 134




[k ]=in

[m ý= in [MTp,[Jdv,

and,

{ F }= j[rs7]T7fdS

(6.2.16)

(6.2.17)

The expression for the applied body forces in Eq. (6.2.17) is integrated over the surface

of the thin shell, S. This assumes that there are no internal body forces due for example

to thermal expansion. The numerical integrations in Eq. (6.2.16-17) are performed by using

Gaussian quadrature and the local-curvilinear relationships derived in Chapter 2.

6.3 Fluid-Structure Interaction Force

When a submerged elastic thin shell is vibrating, the fluid exerts a force on the surface of

the shell. This force is the fluid-structure interaction force, and it serves to couple the elastic

and acoustic formulations. The applied surface force vector in Eq. (6.2.17) consists of two

components; the applied external forces and the fluid structure interaction force,

{F} = {FA}-

{FI}. (6.3.1)

The force F1 acts in the opposite direction to the normal, and the negative sign in Eq. (6.3.1),

shows that work is done on the structure by the fluid. The force Fj is defined in terms of the

pressure difference across the shell, bp,

{Fj} =f[N]n6pds.

(6.3.2)

,

The modified structural equationis

now givenby,

(K- w2Al) {U} + CA {bp} = {FA}

,(6.3.3)

page 135




where the area matrix A can be approximated by the same diagonal area matrix defined in Chap-

ter 4. The matrix C represents the transformation of the normal values to the corresponding

normal vectorsin the

global coordinate system,

{FI} = C{FI }, (6.3.4)

where FI is the normal components of the interaction force. For the conforming acoustic and

structural meshes used in this study the coupling matrix C has dimensions nx 6n and is simply

composed of the normal direction cosines. If different structural and acoustic meshes are used

this matrix must accommodate the two sets of interpolation functions.

6.4 Coupled Equation Set

6.4.1 Fluid Filled and Non Closed Shell Problems

The acoustic formulation for the thin shell problem was derived in Chapter 2. This formu-

lation depends on the assumption that the fluid density is equal on both sides of the shell. This

acoustic formulation is valid for non closed thin shells; i.e. shells that do not enclose an interior

volume, and closed thin shells that enclose a volume of fluid with the same density as the fluid

surrounding the shell. This class of problem will be called fluid filled shell problems (FFSP).

The collocation and variational acoustic equation sets are given respectively by,

(6.4.1k{6p} PW2CT{u}än

1

x {bp} = pw2ACT {U} +Al

(6.4.2)

ý

(9n

I

Combining the acoustic equation sets in Eq. (6.4.1) and (6.4.2), with the structural equation

set in Eq. (6.3.3) gives the combined elasto-acoustic equation sets for the collocation and

variational acoustic formulations,

(K- w2M) CA U FA

(6.4.3)CT

-Nk/w2p

bp of

(K- w2M) CA U FA

ACT-Nk/w2p

(ýp)=

(u)(6.4.4)

page 136




with,

ap,2UI =an

w p. (6.4.5)

The variational BEM/FEM formulation in Eq. (6.4.4) is symmetric. This is due to the

consistent formulation of the structural and acoustic equation sets. The coupled equation set is

the result of minimizing an energy functional that is symmetric with respect to bp and U. This

functional is given by,

II =2 [{U}T (K - w2M){U} - {bp}T Nk {bp}] + {U}T AC{bp}. (6.4.6)

Eq. (6.4.4) can be derived from the minimization problem,

an=u

Off=O(bp)

(6.4.7)

Thecollocation

BEM/FEM formulation isnon symmetric and cannot

be formulated interms

of a similar variational procedure. Insted it can be thought of as the result of minimizing the

functional in Eq. (6.2.2) subject to a constraint given in Eq. (6.4.1).

6.4.2 Evacuated Closed Shell Problems

The other class of elasto-acoustic problem considered in this study is the evacuated closed

shell problem (ECSP). For this problem the thin shell encloses a domain in which the acoustic

pressure is identically zero. Consequently the pressure difference across the surface of the shell

is simply the exterior surface pressure. The acoustic equation set for the exterior problem is

given in Chapter 3. Neglecting acoustic excitation, the Burton and Miller formulation of the

exterior acoustic problem is given by,

([Mk-

Cpl + aNk) {p+}= W2p

(Lk + a[11ý1k + Cp]) CT {U}. (6.4.8)

The coupled ECSP elasto-acoustic problem is now given by,

(K- w2M) CA U FA

GGG-H/w2p

(np) (u7)(6.4.9)

page 137




where the matrices G and H represent the left and right hand side matrices in Eq. 6.4.8) and

ul represents the acoustic excitation for the Burton and Miller problem.

The coupled equation set in Eq 6.4.9) is non symmetric. In previous work this equation set

has been symmetrized by using the energy functional for the acoustic problem Zienkiewicz et al

[1977], Mathews [1979][1986]). This procedure differs from the true variational elasto-acoustic

formulation in that the functional is reconstructed using the non symmetric fluid impedance ma-

trices. The variational elasto-acoustic formulation uses the variational principle as the starting

point for the method.

The formulations in Eqs. 6.4.3) 6.4.4) and 6.4.9) will be referred to as the collocation

coupled method, CCM), the variational coupled method VCM) and the Buton and Miller

coupled method, BMCM). With the correct definitions of G and H, all three formulations may

be generalized by Eq. 6.4.9).

6.5 Solution of Coupled Equation Set

The coupled BEM/FEM equation set may be solved directly. However this is computa-

tionally expensive since it requires the factorization of a 7n x 7n equation set. The normal

procedure is to substitute one equation set into the other. The structure variable methodology,

SVM), involves substituting the fluid equation set into the structural equation set. This method

used by Zienkiewicz et al [1977], results in a fluid modified elastic formulation. The fluid vari-

able methodology involves substituting the structural equation set into the fluid equation set.

Derivation of this method was first given by Wilton [1978] with subsequent work by Mathews

[1979][1986].

6.5.1 Structure Variable Methodology

Neglecting acoustic excitation, the elimination of the fluid variables from Eq. 6.4.9) gives,

K-w2M+w2 pCAH-1GCT) {U} = FA. 6.5.1)

Eq. 6.5.1) is

a matrixequation

of size6n

x6n

and canbe

rewritten

in

a more concise way as,

K- w2[.1I - Mf]) {U} = FA, 6.5.2)

page 138



Elasio-Acoustic Problem.

where Alf is the added fluid mass matrix. This is a misleading name since the fluid interaction

actually represents a frequency dependent mass and damping term. The form of this fluid mass

matrix depends on the acoustic formulation;

Alf = pCANk 1CT (CCM), (6.5.3)

_1Il = pCANk -TACT (N,CM), (6.5.4)

Aff = pCAH- GCT (BMCM). (6.5.5)

Only equation (6.5.4) for the fluid stiffness term derived using the variational method is

symmetric, the others are only symmetric in the continuous limit. These terms defined by Eq.

(6.5.3) and (6.5.5) are not exactly symmetric for the following reason presented by Hartmann

[1989].

The generalized collocation acoustic formulation,

He{p} = w2pG{u },

(6.5.6)

expresses the coupling between the surface displacement and pressure distributions, given by a

combination of the SHIE and DSHIE. However the coupling is only satisfied at a finite number

of collocation points. Consequently the interpolated displacements and pressure distributions

are not strictly compatible; i.e. the identity,

lun}T A{PI = iP}T Alu, },

is not satisfied since the matrix identity,

{. V 9}T {1 \*9}H-1G=

[{N9}T {: v9}H-1G]T,

holds only at the collocation points.

(6.5.7)

(6.5.8)

The possibility of symmetrizing the fluid stiffness matrix has been mentioned before. How-

ever Mathews [1979] and Tullberg and Bolteus [1982] find that the highest accuracy is possible

with the unmodified forms of A f given by Eq. (6.5.3) and Eq. (6.5.5).

page 139




6.5.2 Fluid Variable Methodology

Eliminating the structural variables from Eq. (6.4.9) gives the modified fluid equation set.

H+GC [K_w2M]-1 CA {bp} =GCT

[K_W2M]-1FA+uj. (6.5.9)

L0 p

This equation set is considerably smaller than the modified structural equation set and it rep-

resents a matrix equation of size nxn. Writing the normal applied force as f and neglecting

acoustic excitation, this equation set can be written in terms of fluid and structural impedance

matrices,

(I + ZfZ3 1) {6p} = ZfZ» 1 n, (6.5.10)

where,

-iwZs 1= CT [K - w2M] -1 CA, (6.5.11)

and,

Zf = -iwNki (CCM), (6.5.12)

Zf= Vk -'A (VCM), (6.5.13)

Zf = -iwH-1G (B-N1CM), (6.5.14)

Only the VCM results in both symmetric fluid and structural impedance matrices.

The fluid variable method is the preferred formulation of the elasto-acoustic problem. The

surface pressure distribution is evaluated using Eq. (6.5.9) and the corresponding displacement

field is found using Eq. (6.3.3). There are two main advantages with this methodology; first the

size of the coupled formulation is significantly smaller than the structural variable formulation

and second for fluid filled thin shell problems the exterior pressure distribution given in Chapter

4, is independent of the displacement. field. The most computationally expensive part in solving

Eq. (6.5.9) is the backward and forward substitution of the structural formulation into the

page 140




acoustic formulation. This computational burden can be significantly reduced by reducing the

size of the structural equation set.

6.6 Eigenvector Reduction of the Elastic Formulation

The reduction of the structural equation set can be performed by a variety of transfor-

mations. These include static condensation [Guyan 1965], Lanczos reduction and Eigenvector

reduction. In connection with the elasto-acoustic problem modal reduction is the most estab-

lished technique for reducing the size of the structural equation set. The structural eigenproblem

is defined by,

ýIi- w M]{ek} _0

The eigenvectors, ek, correspond to the n3 non trivial solutions to this equation at the eigenval-

ues, wk. The set of orthogonal eigenvectors, E, are normalized so that the following identities

hold,

ETME = I,(6.6.1)

ET KE= S2,

where the diagonal matrix Q is the set of structural eigenvalues.

Using these identities the inverse of the dynamic stiffness matrix can be written in terms

of a modal summation,

[K- w2M]-i -

By assuming that,

n, fek}fek }T

k=1 k

W «Wm,

it is possible to write,

m[K

- w21\f]-1

T_ n+ {ekl f fk}T2-w2

64)-1 kk=mß-1

k

m<n,

(6.6.2)

(6.6.3)

(6.6.4)

page 141




The second summation in Eq. (6.6.4) is a frequency independent static correction term. If m is

large enough then this term can be neglected.

Eq. (6.6.4) can be used in the FVM in order to significantly reduce the computational

burden of the backwards and forwards substitution of the structural equation set into the fluid

formulation. To accurately reconstruct the fluid modified structural response, the number of

`dry' eigenvectors used must be carefully selected. This is not always straight forward since the

structural eigenmodes of a complex structure can be unpredictably modified by the fluid. The

structural inverse impedance in Eq. (6.5.11) can be approximated by,

m{en }{e

IT-2WZs1-ý

ký k (6.6.5)uJ -W2

k=1 k

where

{en}=

CT {ek }. (6.6.6)

Using Eq. (6.6.5) has the added advantage that the individual eigenvectors may be `col-

lapsed' to their normal degrees of freedom only once, insted of at each frequency point.

6.7 Interpolation

Even with careful use of structural symmetry after the calculation of the `dry' eigenmode

solution, a large part of the total computational time needed to calculate the components of

the FVM is taken by the assembly of the acoustic matrices. Benthien [1989] proposed that

the acoustic matrices could be calculated at a reduced number of frequency points and then

interpolated for intermediate points. This interpolation scheme is implemented for the FFSP in

this study.

At wavenumbers greater than one the dominant frequency term of the H matrix is the

exponential term of the Green's free space function. By using the transformation,

hi' (k)_ e-ikhij (k), r=1r1-vu, (6.7.1)

the interpolation scheme is defined by,

page 142



Elasto-. 1coustic Problem.

h'ij(k)-

h' (k1) +k-ki

(hij (k2) - h'ij(ki))k2

-kl

(6.7.2)

At low frequencies the fluid matrices can be assumed to be constant. Such interpolation

schemes allow very significant improvements in computation times and recently researchers (eg

Kirkup [1991]) have used interpolation techniques in order to calculate elasto acoustic eigen-

modes.

6.8 Uniqueness and the Coupled Problem

Theproblem of a rigid submerged

thinshell

thatencloses an

interior domaincan

be

con-

sidered as the special case of the general elasto- acoustic problem. The DSHIE acoustic integral

equation for the closed thin shell consists of the equation for the exterior problem and the in-

terior problem. Denoting p- and p+ as the interior and exterior nodal pressure distributions

these equations are,

H{p+} - w2p+G+{'u}= {2ll},

H{p-} - w2p-G-{u} = 0, (6.8.2)

where,

H= Nk and G± =[±i

+ Mk.

(6.8.3)

Thevector u

is thenormal

components of the nodal surface velocity distribution and p± is the

density of the fluid exterior and interior to the thin shell respectively. For the purposes of this

study it is assumed that the interior and exterior waveneumbers are the same; ie the acoustic

speed of sound in both the exterior and interior domains is equal.

The structural equation set for the elasto-acoustic problem can be written as,

S(w){U} + CT A ({p+}-

{p- }) = CA{ f }, (6.8.4)

where the dynamic stiffness matrix is denoted by S(w). The excitation forces and the fluid-

structure interaction forces act on the normal direction degrees of freedom of the structure. It

page 143




is advantageous to factorize out the other degrees of freedom. Denoting the normal degrees of

freedom with the subscript n and the others with m, Eq. (6.8.4) can be rewritten as,

Snn Snm un+A0

p+ p-_

(A 0f(6.8.5)

5'mn smm um 0000000

It follows that the structural impedance equation is given by,

Zs{il} + {p+}-

{p-} = ff }, (6.8.6)

where,

-2WZ3 -A-1 [Snn

-SnmSrnmSmn]

.(6.8.7)

The generalized fluid filled thin shell problem is better illustrated by making the following

transformation,

bp=p+-p-,

P=p +P

and,

(6.8.8)

6p=p -p(6.8.9)

p++p .

Using these transformations it is possible to write the coupled formulation as,

Z I0 ü f

-iw[bpMk +2 pI] Nk 0 bp j= eil (6.8.10)

-iw[pMk + 2bpI] 0 Nk ui

Consider the coupled formulation given in Eq. (6.8.10), neglecting for the moment the fluid

variable p,

Z{ } + f6p}= ff}, (6.8.11)

Page 144




-ýwSpMk +2 pI {u} + :ýýk{6p}-{ý; }. (6.8.12

At eigenvalues of the interior Neumann problem, both the following homogeneous equations

have non-trivial solutions,

Nk{v} = 0, (6.8.13)

Mk + 2I {v} = 0. (6.8.14)

Assuming that bp # p, then at the interior Neumann eigenvalues Eq. (6.8.12) defines ü uniquely

with respect to the fluid variable bp, but bp is not uniquely defined with respect to the surface

normal velocities, ii, However the structural equation set ,Eq. (6.8.11), uniquely defines both

it,, and 6p, and consequently in the coupled formulation both are defined uniquely at frequencies

corresponding to the Neumann eigenvalues.

When bp = p; ie for the "evacuated" thin shell problem, Eq. (6.8.12) defines neither ufz or

bp uniquely and the coupled formulation no longer has a unique solution. The coupling of the

exterior and interior pressure fields by the structural equation set, ensures that for p- 0 0, the

coupled formulation is unique for all frequencies.

By considering the uncoupled expression for the second fluid variable, p, given in Eq.

(6.8.10), is is possible to see that at the interior Neumann eigenvalues it is not uniquely de-

fined since,

Nk{p} = iwIPMT

+26PI

{Ti} + {ti7}. (6.8.15)

Thus Eq. (6.8.10) is uniquely defined fort and bp but not for p. It is, however, possible to

use a definition of p, derived from the SHIE expressions for the interior and exterior acoustic

problems, that is uniquely defined in terms of p for all frequencies,

21{p} = Mk{bp} - iwSpLk{ü} +pi. (6.8.16)

page 1/5



Elasto-. acoustic Problem.

The advantage of this formulation is that it gives p in terms of the other fluid and structural

variables, without the need for matrix factorization.

Consider the specific example of a closed shell containing the same fluid in the exterior and

interior domains. Neglecting acoustic excitation the FVM and SVM elasto-acoustic equation

sets are respectively,

(Z71+Zd 1) {bp} = Zs 1{f}, (6.8.17)

(Zs+Zf){t} ={f}" (6.8.18)

At eigenvalues of the interior Neumann problem the inverse of the fluid impedance matrix is

ill conditioned and the fluid impedance matrix is ill defined. In Eq. (6.8.17) the structure

impedance matrix removes the ill conditioning from the left hand side. Formulated as in Eq.

(6.8.18), the SVM will be ill defined at the critical frequencies. However it can be rewritten as,

(z7'z3+ i) {ü} = Z1{f}. (6.8.19)

It is perhaps computationally unfeasible to formulate the SVM in this way, but it illustrates the

uniqueness of the SVM at the critical frequencies. A similar argument for the evacuated thin

shell fails since both ZI and Zf1 are ill defined at the eigenvalues interior Neumann problem.

In order to obtain a unique solution for the evacuated closed thin shell problem, it is necessary

to use the Burton and Miller formulation of the exterior acoustic problem.

Huang [1984]

proposedthat whilst the evacuated fluid interaction problem was in theory ill

conditioned at internal critical wavenumbers, in practice this is ill conditioning is not seen. His

reasons were that discretization errors remove the degeneracy of the eigenvalues of the H and

G+ matrix in Eq. (6.8.1-2). However the numerical results in this Chapter along with previous

results ('1Mlathews 1986]), show that this proposition is not valid for the high order fluid and

surface interpolation.

Since the elasto-acoustic problem does not break down at the interior Neumann eigenvalues,

as long as p- is non-zero, it is interesting to consider the physical properties of the shell at these

critical wavenumbers. Following the notation of Junger and Feit [1986], modal pressure and

surface velocity are related by,

page 146




Wn= fn,

+-nPn-+

2-t-

7-

fn,Zn

n n

_

Zn

Pn = fnZn +4- Zn ,

where,

Z- ip- c_jn(k-a)

(6.8.20)

(6.8.21)

(6.8.22)

(6.8.23)

and the other impedance relationships are defined on page 161 and 233 of Junger and Feit.

For the fluid filled thin shell at the eigenvalues of the interior Neumann problem,

in'(k a) = 0

Assuming p- 00 and c- 00 then the following relationships will hold;

wn=o,

Pn =1,

Pn -- -fn

(6.8.24)

(6.8.25)

(6.8.26)

(6.8.27)

At the interior Neumann eigenvalues the spherical shell will show a rigid response for that

mode, whilst at interior Dirichlet eigenvalues it will display a response independent of the interior

fluid.

6.9 Elastic Thin Plate Problems

The treatment of rigid plate problems has already been introduced in Chapter 4. For the

collocation method,the

pressure

difference distribution is

approximated

in Eq. (4.6.5) by,

HII{5pi}=

{uj}. (6.9.1)

page 147




The pressure difference on IF is constrained to be zero. The consequence of this constraint is

that in general for the rigid plate problem the reconstructed velocity on the edge of the plate is

not equal to the specified interface velocity,

{UE} 0 HEI{SPI}. 6.9.2)

For the elasto acoustic plate analysis the edge pressure difference can be constrained to be

zero in a similar way. As in the rigid plate problem, constraining the pressure difference to be

zero on the edge in this way will result in a discontinuity in the displacement field at the edge

ofthe

plate.

The results for the rigid plate acoustic problem in chapter 4, indicate that the acoustic

problem is not well defined at the edges. This is especially the case for nodes that are situated

on the edge of the plate. The plate problem can be further investigated by extending the analysis

of closed thin shells. Figure 6.1) represents a closed thin shell divided into two regions, Sl and

S2. The interface between the two regions is the line F. Assume that there are two different

boundary conditions specified for Sl and S2;

U1 = U0,

bp2= bpo.

r

Figure 6.1. Representation of plate problem.

6.9.3)

If Sl is assumed to be -igid, and öpo is zero, then the acoustic equation set can be written,

page 1, 8

F S7 n+




Hll H1` bpi

_u° (6.9.4)

H21 H22 0 U2

For the case where r does not contain any surface nodes then this equation may be rear-

ranged to give a solution in terms of the known quantities,

Hip 0 bPi_

uoH21

-1 u2 0(6.9.5)

Continuity of pressure and displacement over IF is satisfied by the interpolation functions.

If howeverthe edge nodes are taken

intoconsideration then

Eq. (6.9.5)results

inan over

determined set of equations for bpl ;

Hll{bpl} = {uo}, (6.9.6)

Hrl{bpl} = {uo}. (6.9.7)

Ignoring Eq. (6.9.7) gives the solution technique given in Eq. (6.9.1). An improvement on this

might be the solution of Eq. (6.9.6) and Eq. (6.9.7) by a least squares method.


The elasto-acoustic methodologies were fully implemented in FORTRAN. This was done by

coding the FEM and coupling the resulting matrices into the existing acoustic code. The large

size of the resulting structural matrices and the lack of an out of core solution procedure meant

that only problems of a moderate complexity could be modeled. Ther majority of numerical

results in this chapter correspond to the analysis of submerged elastic spherical shells since

analytical solutions are well established and straight forward. The other problem modeled was

the cantilever plate.

6.10.1 Cantilever Plate

The cantilever plate shown in figure (6.2) was studied with dimensions; a=0.4064, b=

0.2032 and t=0.00267. The plate has constitutive material properties; E=0.2068 x 1012,

v=0.3 and p, = 7830. These parameters were chosen so the results could be compared to

those of Coyette and Fyfe [1989]. The rigid wall anchoring the plate at X=0 was incorporated

page 149




into the model by assuming acoustic reflective symmetry in the Y-Z plane. The plate was

discretized into 24 elements. The plate is submerged in water (c = 1500 and p= 998) and the

first four eigenvalues are calculated. These results are shown in Table (6.1) and compared to

the results of Coyette and Fyfe and the experimental results of Lindholm et al [1965].

The numerical results are calculated in two ways. First the pressure distribution due a

point load is calculated over a range of frequencies and the eigenvalues are extracted graphically

(VCM). Second the fluid added mass is calculated in the incompressible limit (c = oo, k= 0)

and an eigenvalue analysis is done on the resulting structural equation set (FAM). The acoustic

equation set was formulated using the variational method.

The agreement between all results in Table (6.1) is satisfactory. However the success of

the incompressible results shows that the analysis is at low frequencies and consequently not a

rigorous test of the formulation.

6.10.2 Spherical Shell

The 24 element spherical shell illustrated in Chapter 2 is used to model a number of elasto-

acoustic radiation and scattering problems. Again only one quarter of the shell is discretized

and the structural symmetry is incorporated into the FEM by fixing the appropriate degrees of

freedom on the symmetry planes.

Structural damping is included through a hysteretic approximation. This involves intro-

ducing a complex component to the stiffness matrix,

KD = K(1 - i-y).

If modal reduction is used for the structural equation set, then this damping can be introduced

by making the `dry eigenvalues complex,

[KD-W

2M]-1 N

r` {ek } {ek }TE

k-wk(1 - 17) - w2

Unless specified thedimensionless

structural constantsfor

the spherical shell are,

h/a = 0.01, p3/p=7.67, v=0.3,

(6.10.1)

page 150




cp c = 3.53(1- iy/2), 7=0.01.

The thickness of the shell is denoted h and the structural speed of sound, cp is defined by,

Cp =

Ps(1

E-

v2)

1/2(6.10.2)

Table (6.2) shows a numerical modal analysis of the in vacuo and submerged evacuated

spherical shell compared to the results extracted from Junger and Feit [1986] p282. The eigen-

value results indicate which structural eigenvalue corresponds to the axisymmetric analytical

mode, and the undamped dimensionless frequency is defined by,

SZ = wa/cp. (6.10.3)

An eigenvalue analysis is performed on the `dry' structure to extract the numerical eigen-

values for the 6 and 24 element discretizations. The results show that the numerical `dry'

eigenvalues converge and the agreement with the analytical results is satisfactory. The sub-

merged eigenvalues were evaluated from a graphical frequency plot, to three significant places,

and it was found that there is a slight discrepancy between the coded analytical and Junger and

Feit submerged eigenvalues. Again these results show good accuracy and convergence.

The second set of results showing the ratio of tangential to radial displacement are more

informative. The results for the lower branch modes again show good accuracy for the in vacuo

and submerged ratios, however the numerical calculation of the submerged ratios rapidly looses

accuracy for the higher modes of the upper branch. There are two contributory factors to this

inaccuracy. First there is the inaccuracy of the dry eigenvector and second the inaccuracy of the

fluid modification of the eigenvector. Whilst a refined mesh would reduce the inaccuracy, it may

be more efficient to refine the structural and acoustic problems independently. It is the author's

belief that such a methodology would balance the increased complexity with much improved

efficiency.

Figures (6.3-4) show the pressure gain in the far field from a fluid filled elastic spherical

shell excited by a unit point force. The pressure gain is calculated as the ratio between the far

field pressure and 1pPa For the variational formulation there is high accuracy up to ka = 3.

The contribution to the scattering from the lower branch modes is over approximated, however

the structural damping for the test case is very small. For the collocation method however,

page151




there is a similar accuracy in the underlying response, but the excitement of the lower branch

modes is less accurate. After ka ;zzý.5 there should be no contribution to the far field radiation

from the lower branch modes. Although excitation by a point source is a severe test of the

formulation, these results indicate that the consistent formulation of the variational BEM and

FEM may have the edge in accuracy over the coupled collocation BEM/FEM formulation.

The results in figures (6.5-6) show the backscattering form function for the fluid filled

elastic spherical shell. A comparison of the collocation and variational results shows very little

difference in accuracy and close agreement with the analytical result up to ka = 8. The difference

between these results and the radiation results for the point excited spherical shell indicate that

the accuracy of the structural response may be a limiting factor for consistent structural and

fluid meshes. Figure (6.7) shows the back scattered form function for the evacuated thin shell,

calculated using the collocation Burton and Miller formulation.

The behaviour of the coupled SHIE and Burton and Miller formulations applied to the

evacuated spherical shell, at the first Dirichlet critical frequency is shown in figure (6.8).

Figure (6.9) compares the coupled DSHIE and Burton and Miller formulations for the same

problem at the first Neumann critical frequency. Both set of results disprove Huang's conjecture

and show the need for a modified formulation for the evacuated thin shell problem. The

results in figure (6.10) however support the proposal made in this study that for a fluid filled

elastic thin shell, the coupling of the exterior and interior problems by the elastic formulation

removes problems of uniqueness at the critical frequencies. The results of figure (6.10) show

that at the first Neumann frequency the surface pressure difference will be equal to the applied

excitation pressure.

Figures (6.11-14) indicate the advantages of frequency interpolation. The results for the

collocation and variational methods both indicate that a frequency interpolation step of Aka =

1.0 is possible before significant loss in error. Such a frequency interval in these plots represents

the calculation of fluid impedance matrices at approximately 1% of frequency points.

page 152



Elasto-acoustic Problem.

.A-*o

NL...

_ý

Z

x

Figure 6.2. Cantilever plate geometry.

Mode Theory

In Vacuo

C F Numerical Experimental

Submerged

C F FAM VCM

1 13.8 13.8 13.9 5.1 6.1 6.0 6.0

2 59.3 59..E 60.0 29.8 34.8 34.0 33.1

3 83.9 899.1 89.2 34.4 40.8 40.0 40.1

4 194 198.5 198.4 99.1 120.2 116.0 112.6

Table (6.1). Comparison of submerged and In vacuo quantities for the cantilever

plate.

page 153




Undamped dimensionless

natural

frequency

Eig e avalue In Vacuo Submerged

Mode n=6 n=24 n=6 n=2.4 J F n=6 n=24 Anal. J F

Lower 1 1 1 0.000 0.000 0.000 0.000 0.000 0.000 0.000

2 2 2 0.713 0.702 0.701 0.322 0.323 0.324 0.318

3 5 5 0.909 0.835 0.830 0.424 0.407 0.407 0.392

4 7 7 1.237 0.896 0.881 0.700 0.467 0.465 0.461

Upper 0 11 25 1.613 1.613 1.610 1.270 1.270 1.270 1.220

1 15 27 1.979 1.975 1.980 1.800 1.800 1.800 1.820

2 19 39 2.733 2.724 2.720 2.290 2.300 2.300 2.550

3 29 51 3.744 3.642 3.640 3.370 3.360 3.360 3.420

4 38 68 4.707 4.614 4.600 4.250 4.390 4.390 4.420

Ratio of tangential to radial displacement

In Vacuo Submerged

Mode n=6 n=24 J F n=6 n=24 J F

Lower 1 1.000 1.000 1.000 1.000 1.000 1.000

2 0.275 0.271 0.270 0.252 0.250 0.250

3 0.117 0.122 0.123 0.107 0.116 0.117

4 0.118 0.069 0.070 0.113 0.067 0.068

Upper 0 0.000 0.000 0.000 0.000 0.000 0.000

1 -3.500 -0.500 -0.500 -0.669 -0.668 -0.646

2 -0.593 -0.616 -0.616 -53.700 -153.0 -1.080

3 -0.608 -0.673 -0.680 -22.400 -77.8.0 -3.280

4 -0.688 -0.679 -0.713 -34.200 -110.0 -5.500

Table (6.2). Comparison of submerged and In vacuo quantities for the evacuated

thin shell.

page 154




130

120

dB

110

100

90

80

70

ci

Analytical

' Variational

.11 ka 10

Figure (6.3). The ; ar field back radiated pressure from a fluid filled sphere ex-

cited by a point unit force. The sphere is discretized into 24 elements and the

variational thin shell formulation was used to model the acoustic response. The

structural equation set is reduced using 100 eigenvectors.

page 155



Elasio-acoustic Problem.

130

120

dB

110

100

90

80

70

r,

F

1

Analytical

° Collocation

1 1 ka 10

Figure (6.4). The far field back radiated pressure from a fluid filled sphere ex-

cited by a point unit force. The sphere is discretized into 24 elements and the

collocation thin shell formulation was used to model the acoustic response. The

structural equation set is reduced using 100 eigenvectors.

page 156




10

f

1

.1

01

i

/i

/

,ý'

'/

i

i

Analytical

'. '. Variational

Rigid

.0011 1 ka 10

Figure (6.5). The far field baclcscattered form function for a fluid filled elastic

sphere. The sphere is discretized into 24 elements and the variational thin shell

formulation was used to model the acoustic response. The structural equation

set is reduced using 100 eigenvectors.

page 157




10

f

1

I

.1

.01

ii

.41

i

.i

Analytical

Collocation

Rigid

.0011 1 ka 10

Figure (6.6). The far field backscattered form function for a fluid filled elastic

sphere. The sphere is discretized into 2.4 elements and the collocation thin shell

formulation was used to model the acoustic response. The structural equation

set is reduced using 100 eigenvectors.

page 158




10

f

1

1

.01

1' ,,

i

i

i

i

i

ii

1i

i

i

ii

ii

i,

i

.; NX,

Analytical

Numerical

Rigid

1 1 ka 10

Figure (6.7). The far field backscattered form function for an evacuated elastic

sphere. The sphere is discretized into 24 elements and the Burton and Miller

collocation formulation was used to model the acoustic response. The structural

equation set is reduced using 50 Lanczos vectors.

page 159




1.2

1.0

0.8

0.6

0.4

0.2

nn

Real Pressur Difference

"

0.62

I\'

0.55

nAQ. v v. -to

3.0416 3.1416 ka 3.2416 3.0416

0.7

0.3

-0.1L-

3.0416

-0.25

"

"

Imaginary Press e Difference

3.1416 ka 3.2416

Imaginary D splacement

40

-0.30

_nZS

"" ". "tý.. l..

""

"

-V. J

3.1416 ka 3.2416 3.0416

Analytical.

BYM ---------

3.1416 ka 3.2416

SHIE f.

Figure (6.8). The surface pressure and velocity distributions in the region of the

first Dirichlet critical wavenumber for an evacuated elastic spherical shell. The

surface is discretized into 24 elements and is excited by unit constant pressure.

Real Displa cement

"

0

page 160




1.1

1.0

09

Real Pressur Difference

40--- -------------------

03

0.2

Al

w

-----------------

Imaginary Press e Difference

4.4934 ka 45934.3934 4.4934 ka 45934 4.3934

0.05

0.00

_nnS

Real Disp

4.3934

40

-0.1

-0.2

-0.34.4934 ka 4.5934 43934 4.4934

ka 4.5934

Analytical_.

B&M --------- .DSHIE f.

Figure (6.9). The surface pressure and velocity distributions in the region of the

first Neumann critical wavenumber for an evacuated elastic spherical shell. The


Imaginary Dis acement

page 161




1.03

1.00

Real PrLure

Difference0.001

0.000

0.97 -4.3934

0.03

nnni

Imaginary Press re Difference

0.003

Imaginary Displacement

0.00

Displacement

-v. w i

4.4934 4.5934 4.3934 4.4934 4.5934

-0.03-

4.3934

0.000

-0.0034.4934 4.5934 4.3930 4.4930 4.5930

Analytical Collocation f.

Figure (6.1 D). The surface pressure and velocity distributions in the region of the

first Neumann critical wavenumber for a fluid filled elastic spherical shell. The


page 162




130

dB

110

90

Analytical

. ka=0.1

130

dB

110

70 -

0.1 1.1 ka 2.1

Analytical

Oka=0.2

700. 1.1 ka 2.1

Figure 6.11). The far field back radiated pressure from a fluid filled sphere

excited by a point unit force. The results are calculated using the collocation

method with frequency interpolation at a) Oka = 0.1 and b) Oka = 0.2.

90

page 163




130

dB

110

90

Analytical

Oka=1.0

130

dB

110

700.1 1.1 ka 2.1

90

..... --°...... Analytical

Aka=-2.0

700.1 1.1 ka 2.1

Figure (6.12). The far field back radiated pressure from a fluid filled sphere

excited by a point unit force. The results are calculated using the collocation

method with frequency interpolation at (a) Oka = 1.0 and (b) Oka = 2.0.

page 164




130

dB

110

90

............... Analytical

Oka=O.

130

dB

110

700.1 1.1 ka 2.1

90

............... Analytical

Oka-0.2

700.1 1.1 ka 2.1

Figure 6.13). The far field back radiated pressure from a fluid filled sphere

excited by a point unit force. The results are calculated using the variational

method with frequency interpolation at a) Oka = 0.1 and b) Oka = 0.2.

page 165



Elasto-acoustic Prvblem.

130

dB

110

90

..... -. .. - Analytical

Oka=1.0

130

dB

110

70

0.1 1.1 ka 2.1

90

. .............. Analytical

Aka=2.0

70 '-

0.1 1.1 ka 2.1

Figure (6.14). The far field back radiated pressure from a fluid filled sphere

excited by a point unit force. The results are calculated using the variational

method with frequency interpolation at (a) Oka = 1.0 and (b) Oka = 2.0.

page 166



Lanc: os vectors in elasto-acoustic analysis.

CHAP TER 7.

Lanczos vectors in elasto-acoustic analysis

7.1 Introduction

This chapter evaluates the use of Lanczos vectors in the coupled Boundary Element Method

and Finite Element Method with a view to improving the currently available solution techniques,

for exterior field problems, in terms of computational speed and accuracy. It has been shown

that the fluid variable method is the most economical solution technique for these exterior

field problems. In the fluid variable method, the structural equation set is substituted into the

acoustic equation set, the structural displacement vector is subsequently eliminated and solution

of the resultant matrix equation set yields the normal surface pressure. The disadvantage of

solving the coupled system in this manner is that considerable computational effort is expended

in a forward and backward substitution of the dynamic structural matrix into the acoustic

equation set. Therefore it is advantageous to use a reduced basis technique to represent the

dynamic structural matrix, in order to reduce the computational burden of this forward and

backsubstitution process, required at each

frequency.

In Chapter 6 an eigenvector reduction of the dynamic structural matrix was used in order

to ease this computational burden. The difficulty in using the modal approach, is the large num-

ber of eigenvectors required to accurately represent the dynamic response of the `dry structure.

Recent work (Nour-Omid and Clough [1984], Chen and Taylor [1989] and Nour-Omid and Regel-

brugge [1989]) on the application of Lanczos vectors to the solution of interior fluid-structure

interaction problems has shown far fewer Lanczos vectors are required to represent the dynamic

structural matrix. The increased accuracy of these vectors is due in part to the manner in which

the loading vector is incorporated into their generation; the starting vector for the Lanczos al-

gorithm, as will be demonstrated later in the chapter, is the dynamic deflection shape. Not only

are far fewer Lanczos vectors required to accurately represent the submerged elastic structure,

these vectors are also, computationally less expensive to generate than eigenvectors.

In this chapter two different methods for which the application of Lanczos vectors are used

to solve the coupled fluid-structure interaction problem will be presented. First the solution of

the exterior acoustic problem for evacuated closed thin elastic shells is studied. The collocation

method is used and to ensure that the acoustic formulation is valid throughout the entire fre-

quency range, the methodology of Burton and Miller is implemented. The finite element model

page 167





Lanc: os rectors in elasto-acoustic analysis.

KQ=K-a2M. (7.2.3)

This spectral shift improves the Lanczos representation of the structural equation set by forcing

the starting vector to be a more appropriate dynamic displacement shape.

It can be shown (Nour-Omid and Clough [1984], Chen and Taylor [1989] and Nour-Omid

and Regelbrugge [1989]) that the algorithm for generating successive Lanczos vectors is given

by,

ro = Kam FA, (7.2.4)

qo = 0, (7.2.5)

,Qj _

ýrý11L1rý-i)1ý2.

(7.2.6)

rj-1qj = a

(7.2.7)

rj = Kv Mqj, (7.2.8)

aj = qý MTA, (7.2.9)

ri = ri - aj qj -3jgj-i, (7.2.10)

where the set of m Lanczos vectors are defined by,

m

Qm = qi. (7.2.11)

Using this set of Lanczos vectors it is now possible to reduce the dynamic structural equa-

tion. By writing,

1u} = Q,n x}. (7.2.12)

and premultiplying Eq. (7.2.1) by QmMKý 1, the transformed dynamic structural equation is

given by,

page 169



Lancxos vectors in elasto-acoustic analysis.

[(Qm1ý1(wm ý2QmMKo MQm)-w2Q MAC-1,11Q,

n]{x} = Qm-ý1KQ1 F }. (7.2.13)

From the orthogonal properties of the Lanczos vectors, Eq. (7.2.13) can be greatly simpli-

fled. The left hand side can be transformed by noting that,

QmMK; 1MQ,n = Tm, (7.2.14)

QT= Im, (7.2.15)

where the tri-diagonal matrix Tm is defined by,

a1 /32

32 az 03

TI. (7.2.16)

Qm-1 am-1 Qm

Qm am

The right hand side of Eq. (7.2.13) is given simply by,

QT MKS 1{FA}= {9,,, , (7.2.17)

where, if el is defined as the first column of the identity matrix Im, g,,, is given by,

{g,,, } = ßi{el}. (7.2.18)

Finally the reduced dynamic structural equation can be written as,

[im + (0,2- w2) Tm] {x} _ {g,,,}. (7.2.19)

There are several numerical points that need to be considered before numerically imple-

menting a Lanczos algorithm. Although in theory the orthogonalization of each Lanczos vector

page 170



Lanczos rectors in elasto-acoustic analysis.

with the previous two should ensure the orthogonality of the complete set, in practice this is not

the case. Because of rounding errors orthogonality is lost as the iteration proceeds. One way

to overcome this problem would be to performa

full

reorthogonalization at each step.

However

this is computationally expensive and unnecessary. A more satisfactory method would be to

examine the degree of orthogonality at each step and when necessary, a Gram-Schmidt reorthog-

onalization could be performed. In this study a method proposed by Simon [1982] was used,

that iteratively generates a test vector that represents the degree of orthogonality of the current

Lanczos vector against all other Lanczos vectors. This vector is tested to indicate the degree of

orthogonality at each step, and if necessary full reorthogonalization is performed. It was found

in thisstudy

that therounding errors

in the Lanczosalgorithm

becameserious only when

the

value of o corresponded too closely to a `dry eigenvalue.

The second point is the determination of the point at which enough Lanczos vectors have

been generated to ensure convergence of the reduced system to the full dynamic response. Previ-

ous researchers have also evaluated error estimates of the accuracy of the reduced system which

can be used to determine a point at which the algorithm should stop. However the accuracy

of such indicators in purely structural analysis has been called into doubt (Chen and Taylor

[1989]) and for the coupled fluid-structure problem there is another factor to be considered.

The interaction of a surrounding fluid with a structure can often modify the `dry eigensolution

of the structure in an unpredictable way. Therefore it is only with a prior understanding of

the problem that an adequate number of Lanczos vectors can be chosen. For this reason an

automatic termination of the Lanczos algorithm is very difficult. For the purposes of this study

the number of Lanczos vectors used in the analysis of the structural response is specified for

each problem.

7.3 Fluid Variable Methodology

The full coupled elasto-acoustic formulation for an evacuated closed shell was given in

Chapter 6,

7K- w2M CA

(U)

_ FaG

-H/w2pbp -0

(7.3.1)

If the dynamic structural matrix is reduced using Lanczos vectors Eq. (7.3.1) becomes,

page 171



Lanczos vectors in elasto-acoustic analysis.

(In+(o2_w2)Tm Qm111 'ý 1CA x gmGCTQm

-H/w2p

bp -0(7.3.2)

Following the fluid variable methodology the `participation' factors x are eliminated and

the fluid-structure interaction equation is given by,

wH p+

GCQm [I, + 0-2 - w2) Tm] -1QT %I K47-1CT A {bp}=

GCQr [I,, + (cr2

_W2)

T"]- I{9m}.(7.3.3)

Defining,

CQm = Om, (7.3.4)

Qm-11Ko1CT=

[CKo 1MQm]T- O'm (7.3.5)

[Im + (0.2 - w2) Tmý (7.3.6)

Eq. (7.3.3) can be written more concisely as,

H+GemS-1® A {bp} = GO,, S-'{9m}. (7.3.7)

7.4 Added Fluid Mass Methodology

In Chapter 6 the elasto-acoustic problem for an fluid filled or open thin shell was derived

in terms of a fluid added mass term,

(K- w2 [M

- Mf(w)]) {U} = {Fa}, (7.4.1)

where by formulating the acoustic problem in terms of the variational method, ßllis the com-

plex, symmetric and frequency dependent `fluid mass' matrix,

page 172




Mf(w) = pCANk-1 ACT (7.4.2)

The symmetry of the added fluid mass term makes it more amenable to Lanczos decom-

position. Algorithms are available for Lanczos decomposition of non symmetric equation sets

(Rajakumar and Rogers [1991], Nour-Omid et al [1991] and Golub and Van Loan [1983]), but

are not considered in this study. Eq. (7.4.1) can be rewritten in terms of Mf evaluated at some

shift frequency o. This gives,

[K- w2 (Ms

- Mf (o)) + w2 (Mf (w) - Mf (o))] {U} = {Fa }. (7.4.3)

The third term in the left hand side of Eq. (7.4.3) will be a small correction term, assuming

the added fluid mass and fluid damping has a small variation with respect to frequency in the

region of a.

A Lanczos decomposition is performed using Eq. (7.4.3) with,

Ka =K-cT2(M8-M1(u)),

M=ms -Mf(Q).

(7.4.4)

(7.4.5)

In the first instance, if the small correction term in Eq. (7.4.3) is neglected the Lanczos reduction

will be given by,

[(0,2_w2)Tm + Im] {Xm1

= t9m}(7.4.6)

where T,,,, x,,, and g,,, will be complex.

The numerical solution of Eq. (7.4.6) will be extremely efficient for a range of frequencies.

However its usefulness as an accurate model will depend on the assumption that the third term

in the left hand side of Eq. (7.4.3) is negligible.

The work performed in this study indicates that for ka « 1. where a is the same order as

the size of the structure, the assumption that the added fluid mass is frequency independent is

page 173




valid and leads to little loss in accuracy. However, for ka -- 1 and ka >1 it will be shown that

this assumption is inadequate.

7.5 Results

To test the different methodologies the spherical shell described in Chapter 6 was used. Hys-

teretic damping was implemented numerically by modifying the stiffness matrix before Lanczos

decomposition.

Figures (7.1-4) show the results of the fluid variable methodology applied to different dis-

cretizations withdifferent

numbers of Lanczos vectors to characterize the `dry' structural re-

sponse. The nodal normal velocities were normalized firstly by P(cosO) and then further nor-

malized with the formula,

1Olog12

Ivn(n + 1)I2exact

(7.5.1)

The modal analytical solution was taken from the work by Geers and Felippa [1983] and

corresponds to the solutions obtained for a submerged, lightly damped, vibrating elastic sphere

excited by loads that are functions of the Legendre polynomial.

The results in figures (7.1-4) show that only 5 Lanczos vectors are needed to accurately

represent the dynamic deflection of the structure. For more complicated shell geometries care

needs to be shown in choosing the number of Lanczos vectors in order not to exclude higher

structural modes that may be significantly perturbed by the influence of the fluid interaction.

Tables (7.1-2) show in more detail the degree of accuracy of the results shown in figures (7.1-4).

The solutions for the higher modes would converge with the use of a finer mesh.

Figures (7.5-8) show the results of an eigenvector decomposition of the dynamic structural

matrices. These results also show good accuracy and convergence, however a higher number of

eigenvectors are needed than Lanczos vectors for convergence. Figure 7 shows the absence of a

structural resonance with 50 eigenvectors, a resonance that is accurately reproduced with only

5 Lanczos vectors.

The added mass formulation was tested in the far field region only, with the fluid mass

correction being evaluated at the lowest frequency. The same structural constants were used

page 17!



Lanczos rectors in elasio-acoustic analysis.

for the fluid filled shell as previously. The acoustic speed of sound in water was assumed to

be 1500m/s. Figure 7.9) shows the results for the far-field pressure radiated at 9=0 by a

fluid filled sphere excited by a point force at 0=0. The pressure is measured in dB using one

micropascal at one metre as the reference level. The exact analytical series solution is given on

the same figure.

It was shown in Chapter 6 that the elasto-acoustic problem for fluid filled or open shells is in

general well defined at all frequencies. However, for fluid filled shells, the added fluid mass will

will not be well defined at the eigenvalues of the interior Neumann problem. This problem can

be circumvented by calculating the added fluid mass term away from these critical frequencies.

It may be observed from Figure 9 that for low frequencies, that is ka «1 the agreement

with the analytical solution is excellent. The difference between using 10 or 40 Lanczos vectors

is very small, indicating that the solution converges for a small number of Lanczos vectors.

However as the frequency increases the converged numerical solution displays a loss in accuracy

in comparison to the analytical solution. This loss in accuracy was expected in this frequency

regime as the third term on the right hand side in Eq. 7.4.3) is no longer negligible at these

higher frequencies. For this methodology to be effective at these higher frequencies, high order

frequency terms would need tobe

added to the formulation.

It was thought that use of the Lanczos methodology would be particularly beneficial for

scattering problems. The reduction of the dynamic structural matrix is performed in terms of

the incident acoustic wave and FA in equation 7.2.4) is defined by,

{FA}=ACT{p}

where p; is the incident pressure.

7.5.2)

Figure 7.10) shows the results for the far field back scattering of a plane wave by an

evacuated spherical shell with the same parameter values as used for the radiation problems.

One quarter of a sphere was discretized into 24,9-noded elements. The results, calculated

using the fluid-variable methodology, are presented as a plot of form function, a commonly

used scattering measure, against wavenumber. Twenty Lanczos vectors were used to model the

elastic structural response. The results obtained are compared to an analytical series solution

given by Junger and Feit [1986], for two different values of structural loss factor. Higher solution

accuracy was attained with increased structural damping. It appears therefore, that considerable

accuracy, is needed in the determination of the structural elastic response.

page175




w/ nß-1) n=0 n=1 n=2 n=3

0.1 0.12 2.13 5.06 21.30

7r 1.33 2.49 10.10 48.70

2ir 0.42 4.96 11.00 9.08

10 1.64 18.30 9.12 1.88

a) 6 elements and 5 Lanczos vectors

w/ n+1) n=0 n=1 n=2 n=3

0.1 0.12 2.13 5.06 21.30

7r 1.33 2.95 9.05 46.40

2ir 0.39 4.94 13.10 7.65

10 0.79 10.50 9.09 2.04

b) 6 elements and 20 Lanczos vectors

Table 7.1). The percentage accuracy of the modal surface velocity for the fluid

variable methodology with 6 elements and Lanczos reduction of the dynamic

structuralmatrix.

page 176



Lanczos vectors in elasto-acoustic analysts.

w/ n+1) n=0 n=1 n=2 n=3

0.1 0.01 0.20 0.30 1.46

ir 1.56 2.31 6.38 21.90

2ir 0.72 0.21 0.91 1.96

10 0.32 0.97 10.00 0.26

a) 24 elements and 5 Lanczos vectors

w/ n+1) n=0 n=1 n=2 n=3

0.1 0.01 0.20 0.30 1.46

7r 1.56 2.34 6.46 24.60

2ir 0.73 0.20 0.73 1.15

10 0.31 1.03 9.50 0.98

b) 24 elements and 20 Lanczos vectors

Table 7.2). The percentage accuracy of the modal surface velocity for the fluid

variable methodology with 24 elements and Lanczos reduction of the dynamic

structuralmatrix.

page 177




w/ n+1) n =0 n=1 n=2 n=3

0.1 0.12 2.46 5.06 21.30

T 0.55 2.60 6.37 42.00

27r 0.02 5.19 18.30 10.90

10 0.88 12.20 9.19 1.77

a) 6 Elements and 50 eigenvectors

w/ n+1) n=0 n=1 n=2 n=3

0.1 0.12 2.46 5.05 21.30

7r 0.56 2.38 6.06 42.20

2ir 0.01 5.15 12.30 10.40

10 0.63 17.00 9.87 1.85

b) 6 Elements and 100 eigenvectors

Table 7.3). The percentage accuracy of modal surface velocity for the fluid

variable methodology with 6 elements and eigenvector reduction of the dynamic

structural matrix.

page 178



Lanc: os vectors in elasto-acoustic analysis.

w/ n+1) n=0 n=1 n=2 n=3

0.1 0.01 0.52 0.29 2.43

7r 0.79 1.20 3.49 37.40

21r 0.35 0.06 0.87 17.20

10 0.14 0.93 10.30 16.10

a) 24 Elements and 50 eigenvectors

w/ n+1) n=0 n=1 n=2 n=3

0.1 0.01 0.52 0.29 1.45

T 0.79 1.20 3.44 15.40

2ir 0.35 0.05 0.86 1.92

10 0.14 0.93 10.10 0.76

b) 24 Elements and 100 eigenvectors

Table 7.4). The percentage accuracy of modal surface velocity for the fluid

variable methodology with 24 elements and eigenvector reduction of the dynamic

structuralmatrix.

page 179




10

dB

-10

-30

.1

30

20

10

dB

0

-10

In

n=

1

20

0

dB

-20

i stU_40

L

10 10.1

n=

40

1

n=

20

dB

0

-'. v

.11 (0/3 10

.20L

.11 U4 10

Figure (7.1). Velocity response for n=0,1,2 and 3 excitation of a submerged

spherical shell discretised into 6 elements using the fluid variable methodology

and 5 Lanczos vectors. w is the dimensionless frequency w/c. The velocities are

normalised with respect to v(n + 1)exact.

62 10

page 180




10

dB

-10

-30 L

.1

30

20

10

dB

0

-10

In

n.

1

20

0

dB

-20

fiL2i1-40

10 `

w 10.1

1

n=

40

20

dB

0

-w

.11

n=

Yii[C

H2O

`

wý3 10(r

.11 w/4 10




normalised with respect to v(n + 1)exact.

62 10

page 181




10

dB

-10

-30L

.1

30

20

10

dB

0

-10

-20

.l1 W3 10

-20 L

.11 oY4 10




normalised with respect to v(n + 1)exnct

n=

1

20

0

dB

-20

ee ssss , in

w 10

-VtJ.

1

n=

40

1 o/2 10

n=20

dB

0

page 182




10

dB

-10

-30

.l

30

20

10

dB

0

-10

In

n=

20

0

dB

-20

1 10 10.11

/2 10

n=

-w

.l

40

20

dB

0

i 4. [ L

-If(ý

lii[] [) CL44[]SLYi[1

1 °3 10 oY4 10




normalised with respect to v(n + 1)esact"

n.

page 183




10

dB

-10

-30

.1

40

20

dB

0

-20L

.1

n=

40

20

dB

0

scisa cttý

1

-20Fscsss 14 gcsss aii

co/3 10.11

oY4 10



and 50 eigenvectors. w is the dimensionless frequency w/c. The velocities are

normalised with respect to v(n + 1)ezact"

n=

20

0

dB

-20

'41ýA0l"l[[

1 co 10.11

c&2 10

n=

page 184




10

dB

-10

-30L

.1

40

20

dB

0

-20L

.1

n=

40

20

dB

0

1 °3 10

20

.11U4 10




normalised with respect to v(n + 1)eract.

n=

20

0

dB

-20nil

[[]fL i i

-40

l ]lLC Ot

1w 10.11

(o/2 10

n=

page 185




10

0

dB

-10

-20

_'n

n=

.1110 10

.11OY2 10

40

20

dB

0

n-

zu

0

dB

-20

-40

f.n

40

1

-20L

.1

20

dB

0

IA

n-

13 10.11

aY4 10




normalised with respect to v (n + 1)exact

page 186




10

dB

-10

-30

.1

40

n: 0

zu

0

dB

-20

-40

Kn

nil

n: 2

20

dB

0

-20L

.1

40

20

dB

0

LL t-7Ffi2I2Flit[72fLGi2L7SSi3t1

1

------ -20

w/3 10.11

oY4 10




normalised with respect to v(n + 1)exact"

1 10 10Ti

1 o'2 10

page 187




ööÜÜ

bOO

o

oa

CIO

Q0o

o°°

00a

8

V

L

8

8

8

8OOOO0 O, 00ý4

v--4 -d'Lý

Figure (7.9). Far field radiated pressure at 0=0 for a fluid filled steel spherical

shell, excited by a point force at 0=0. One quarter of the sphere is discre-

tised into 24 elements and the results were calculated using the fluid added mass

methodology.

page 188




8

3

Analytical

yß. 01

f

6

4

2

oý0 1 ka 2

Analytical

yam.f

2

1

oý0 1 ka 2

Figure (7.10). Far field backscattered form function for an evacuated spherical

shell with different hysteretic damping, (a) y=0.01,(b) y=0.1. The structural

response is modeled using 20 Lanczos vectors.

page 189



Conclusions and Recommendations.

CHAP TER 8.

Conclusions and Recommendations

8.1 Conclusions

For the numerical acoustic analysis of arbitrary three dimensional structures the boundary

element method is becoming established as the preferred solution technique. In this study a

collocation and variational BEM have been presented. Both methods make use of isoparametric

elements but unlike many previous BEM e.g. Terai [1980]), they are independent of the order of

interpolation used. The difficulty in implementing the Burton and Miller composite formulation

or the thin shell acoustic formulation has been the accurate evaluation of the hyper singular

integral operator. Both the collocation and variational formulations presented make use of the

conversion to tangential derivatives first presented by Maue [1949].

After conversion to tangential derivatives there still remains a Cauchy type O r-2) singu-

larity. In order to implement a collocation form of the hypersingular integral operator, Chapter

3 gave an argument for essentially ignoring the degree of this singularity. It was shown that

for a collocation point on the surface with strict Cl continuity, there will be cancellation of

errors in the assembled matrix equation. It is assumed that with just Co continuity, there is

still sufficient continuity of first order derivatives to ensure numerical accuracy. The resulting

collocation approximation of the Ak operator is used both in thin shell and Burton and Miller

formulations.

The thin shell formulation, presented in Chapter 4, usesthe variational procedure of Mariem

and Hamdi [1987] to evaluate the JVk operator. A comparison of computational effort needed

to form this operator by using the variational and collocational methods shows the increased

burden imposed by the extra integration for the variational method. However in terms of matrix

factorization the advantages of a symmetric formulation are shown. The timing data indicates

that with a n3 computational time dependency, for large problems the matrix factorization will

become the computational bottle neck.

The Burton and Miller formulation is implemented using the collocational method and

tested with rigid sphere, spheroids and cylinders. The effectiveness of the formulation at re-

moving problems of uniqueness is shown, and the numerical results show good accuracy and

page 190




convergence. The accuracy of the results for the cylinder are encouraging since they show the

validity of the formulation applied to structures with surface discontinuities.

In Chapter 4 the variational and collocation thin shell formulations are tested for a range

of closed and open thin shells. Both formulations again show good accuracy and convergence

and there seems to be little to separate the accuracy of either method for the rigid problems. It

must be noted however that the formulations were compared using relatively simple geometries

and gentle boundary conditions.

An extensive analysis of the superposition method is given in Chapter 5. It is shown

that although accuracy is possible using retracted surface boundary element methods, there are

severe problems of conditioning and stability unless the source surface and boundary surfaces

are coincident. The main objective of this work was to disprove the belief that superposition

methods offer advantages over boundary element methods for arbitrary acoustic problems. The

problem of numerical singularities have to be dealt with rather than avoided.

The finite element method for thin shells is introduced in Chapter 6 and the various coupling

methodologiesfor

elasto-acoustic analysis are presented.The

collocation and variational acousticformulations are coupled to the FEM using the structural variable method. The modal reduction

of the structural equation set is described along with frequency interpolation of the fluid matrices.

The problem of uniqueness at critical frequencies for the closed elastic acoustic sell is clarified,

and it is shown that the thin shell elastic formulation couples the exterior and interior acoustic

problems except when the shell is rigid or the shell is evacuated. In the case of the evacuated

thin shell uniqueness problems are shown to occur in theory and results are presented that

disprove Huang sconjecture,

that theseproblems occur

fornumerical

testcases.

Resultsare also

presented to show that the elasto-acoustic problem for fluid filled thin shell is not ill conditioned

at critical frequencies. The advantages of fluid interpolation are also clearly shown.

The comparison of the collocation method and variational method coupled with the struc-

tural equation set, is more revealing than the similar comparison for the rigid body case in

Chapter 4. For the far field results for scattering from an elastic sphere show there is little

difference in accuracy. The results for the point loaded sphere however show that fluid modifica-

tion of higher eigenmodes is significantly more accurate with the variational formulation of the

acoustic problem. For the collocation coupled BEM/FEM formulation, the point loading results

show the inaccurate excitation of higher eigenmodes, which in the analytical and variational

page 191




cases have very high modal impedance. This indicates the advantages of having a consistent

FEM and BEM formulation.

Chapter 7 examines the use of Lanczos vectors in the coupled boundary element and finite

element structural acoustics problems in an infinite domain. Two methodologies were described

along with a review of the principles of Lanczos vectors. The results of the numerical imple-

mentation of these two methods together with a comparison of the more traditional eigenvector

approach were presented. The results from the fluid variable methodology show that even with

a small number of Lanczos vectors and therefore, a large reduction in the size of the dynamic

structural matrix, high accuracy is still possible. The results obtained using the added mass

methodology reveal a loss in accuracy at the higher frequencies. however, the method s speed

and simplicity allows the character of the dynamic frequency response to a particular excitation

force to be very quickly ascertained.

The numerical results presented Chapter 7 show that using Lanczos vectors instead of eigen-

vectors to form a reduced basis for the dynamic structural matrix offers significant improvements

in accuracy and computational efficiency. The examples presented also show that far fewer Lanc-

zos vectors are needed than eigenvectors to accurately calculate the elastic structural response.

The use of Lanczos vectors for the solution of structural acoustics problems in infinite domains,

offer the advantage of a considerable increase in computational efficiency with increased solution

accuracy in comparison to the eigenvector modal methods.

The numerical methods presented in this thesis have been validated and their accuracy

shown. It is felt that they represent the state of the art in acoustic BEM formulations. The

variational method is felt to be the most robust and accurate method but suffers from the

increased computational burden of the extra integration. The collocation method shows com-

parable accuracy in the presented results and is more efficient than the variational method.

However the results for the point loaded elastic sphere show that it may not be as accurate as

the variational method for coupled problems.

8.2 Recommendations

Comparison of the variational and collocation methods shows that they both have compli-

mentary strengths and weaknesses. A Galerkin method could be implemented by collocating at

element Gauss points and then distributing the resulting non square matrix to the nodal points

by using the Gauss weighting factors. Such a procedure for the thin shell acoustic formulation

page 192




would be equivalent to the variational formulation but would be more efficient since the second

integration would be more efficient as a matrix multiplication. Further more, by definition, there

would be Cl continuity at the collocation points since they are interior to the element.

The work presented in this thesis represents an investigation and implementation of the

basic building blocks of a coupled FEM/BEM elasto-acoustic analysis. Within the next three

years the research group at Imperial College hopes to refine and expand the techniques in order

to produce a totally integrated structural acoustics software package. This package will be

written from scratch in a highly modular and object orientated way. The numerical heart of the

program will be based around the methods presented in this thesis, along with other state of

the art procedures necessary for a robust and practical software package. The proposed project

can be divided into several distinct subtasks:

a) Development and refinement of analytical solutions

b) Wetted surface mesh generation

c) Generation of the fluid matrices using BIE

Ensure that the program can handle mixed boundary conditions

Development of out-of-core matrix utilities

d) implementation of the fluid matrix interpolation technique

e) generation of the structural matrices using FEM

f) Implementation of the Lanczos vector generation algorithm

g) Implementation of the coupling algorithm

h) Implement the biconjugate gradient semi-iterative solution method and compare

with the normal LU type factorization solution method

i) Post processing graphics incorporating all the tools necessary for presentation of

results including X-Y plotting and hard copy postscript output.

r Full interactive 3-D colour contoured display of the scattered pressure field

At this point a complete working three-dimensional code will exist and research can begin

on:

page 193




j) Dynamic substructuring techniques

in order to model interior ballast tanks and structure

k) Adaptivity using error indicators and mesh refinement to optimize solution perfor-

mance

1) Enhance the computational performance through optimization of key portions of the

code

m) Enhancement of the biconjugate gradient semi-iterative solution method through

using as trial solution vectors the solution obtainedfrom

a system where thefluid is

treated as incompressible.

The code will be developed using the C and C++ languages within a UNIX environment,

and the graphics facilities will be implemented using X-Windows.

page 194





[18] P. J. T. Filippi. Layer potentials and acoustic diffraction. Journal of Sound and Vibration,54:473-500,1977.

[19] T. L. Geers and C. A. Felippa. Doubly asymptoticapproximations

for

vibrationanalysis

of submerged structures. Journal of the Acoustical Society of America, 73: 1152-1159,

1983.

[20] J. H. Ginsberg, P. Chen, and A. D. Pierce. Analysis using variational principles of the

surface pressure and displacement along an axisymmetrically excited disk in a baffle.

Journal of the Acoustical Society of America, 88(1): 548-559, July 1990.

[21] G. H. Golub and C. F. Van Loan. Matrix Computations. North Oxford Academic Pub-

lishing Co., 1983.

[22] R. J. Guyan. Reduction of stiffness and mass matrices. AIAA Journal, 3(380), 1965.

[23] F. Hartmann. Introduction to Boundary Elements. Springer-Verlag, 1989.

[24] H. Huang. Helmholtz integral equations for fluid structure interaction. In Advances in

Fluid Structure interaction AMD Vol 64, pages 19-38. ASME, New York, 1984.

[25] M. S. Ingber and R. H. Ott. An application of the boundary element method to the mag-

netic field integral equation. IEEE Transactions on antennas and propagation, 39(5): 606-

611,1991.

[26] J. H. James. Fortran program for vibration and sound radiation of spherical shell. Tech-

nical Report ARE TM N1 86501, Admiralty Research Establishment, 1986.

[27] J. H. James. Calculation of transient response from time harmonic spectrum. Technical

Report ARE TR 89308, Admiralty Research Establishment, 1989.

[28] R. A. Jeans and I. C. Mathews. Solution of fluid-structure interaction problems using

a coupled finite element and variational boundary element technique. Journal of the

Acoustical Society of America, 88(5): 2459-2466,1990.

[29] R. A. Jeans and I. C. Mathews. Comparison of the collocation and variational proceduresfor providing numerical approximations to the hypersingular acoustic integral operator.In proceedings of the Second Workshop on Reliability and Adaptive Methods in Compu-

tational Mechanics, Cracow, Poland, 1991.

[30] R. A. Jeans and I. C. Mathews. Use of lanczos vectors in structural acoustics problems.In Structural Acoustics ASME/WAM NCA Vol. 12 /AMD-Vol. 128. ASME, New York,

1991.

[31] R. A. Jeans and I. C. Mathews. The wave superposition method as a robust technique for

computing acoustic fields. Submitted to: Journal of the Acoustical Society of America,

1992.

[32] A. Jennings. Matrix computation for engineers and scientists. Wiley (London), 1977.

[33] M. C. Junger and D. Feit. Sound, structures and their interaction. MIT Press (Cam-

bridge), second edition, 1986.

[34] C. Katz. Murphy s law in boundary element implementations. In C. Brebbia, editor,

Proceedings of the 9th international conference on boundary element methods. Springer-

Verlag, 1987.

[35] O. D. Kellogg. Foundations of potential theory. Springer-Verlag, 1929.

[36] S. M. Kirkup. The computational modeling of acoustic shields by the boundary and shell

element method. Computers and Structures, 40(5): 1177-1183,1991.

page 196



[37] S. M. Kirkup and S. Amini. Modal analysis of acoustically-loaded structures via integral

equation methods. Computers and Structures, 40 5): 1279-1285,1991.

[38] S. M. Kirkup and D. J. Henwood. Techniques for

speeding upthe boundary element

solution of acoustic radiation problems. In Proceedings of the ASME winter meeting, SanFransisco, 1989.

[39] R. E. Kleinman and G. F. Roach. Boundary integral equations for the three-dimensionalhelmholtz equations. SIAM Review, 16 2): 214-235, April 1974.

[40] G. H. Koopmann, Limin Song, and J. B. Fahnline. A method for computing acousticfields based on the principle of wave superposition. Journal of the Acoustical Society ofAmerica, 86 6): 2433-2438, December 1989.

[41] J. C. Lachat and J. O. Watson. Effective numerical treatment of boundary integral

equations: A formulation for three-dimensional elastostatics. International Journal for

Numerical Methods in Engineering, 10:991-1005,1976.[42] H. Lamb. Hydrodynamics. Dover, New York, 1945.

[43] A. Leitner. Diffraction of sound by a circular disk. Journal of the Acoustical Society ofAmerica, 21: 331-334,1949.

[44] J. B. Mariem and M. A. Hamdi. A new boundary finite element method for fluid-

structure interaction problems. International Journal for Numerical Methods in Engi-

neering, 24: 1251-1267,1987.

[45] P. A. Martin. End-point behaviour of solutions to hypersingular integral equations. Pro-

ceedings of the Royal Society of London Series A, 432:301-320,1991.

[46] I. C. Mathews. Sound radiation from vibrating elastic structures of arbtrary shape. PhDthesis, Department of Aeronautics, Imperial College, London, 1979.

[47] I. C. Mathews. Numerical techniques for three-dimensional steady-state fluid-structure

interaction. Journal of the Acoustical Society of America, 79 5): 1317-1325,1986.

[48] A. W. Maue. Zur formulierung eines allgemeinen beugungsproblems durch eine Integral-

gleichung. Z. Phys., 126:601-618,1949.

[49] G. F. Miller. Fredholm equations of the first kind. In L. M. Delves and J. Walsh, editors,Numerical solution of integral equations, chapter 18, pages 175-188. Clarendon press,Oxford, 1974.

[50] R. D. Miller, H. Huang, E. T. Moyer Jr., and H. Uberall. The analysis of the radi-ated and scattered acoustic fields from submerged shell structures using a modal finite

element /boundary element formulation. In Proceedings of the ASME 6Vinter Anual Meet-

ing. ASME, New York, 1989.

[51] R. D. Miller, E. T. Moyer Jr., H. Huang, and H. Überall. A comparison between theboundary element method and the wave superposition approach for the analysis of the

scattered fields from rigid bodies and elastic shells. Journal of the Acoustical Society ofAmerica, 89 5): 2185-2196,1991.

[52] K. M. Mitzner. Acoustic scattering from an interface between media of greatly different

density. Journal of Maths and Physics, 7 11): 2053-2060,1966.

[53] S. Moini, B. Nour-Omid, and H. Carlsson. Harmonic and transient analysis of fluid

structure interaction problems using lanczos coordinates. Technical report, IBM Scientific

Center, 1990.

page 197



[54] P. M. Morse and H. Feshbach. Methods of theoretical physics. McGraw-Hill (New York),

international student edition, 1953.

[55] B. Nour-Omid and R. W. Clough. Dynamic analysis of structures using lanczos coordi-

nates. Earthquake Engineering and Structural Dynamics, 12:565-577,1984.

[56] B. Nour-Omid, W. S. Dunbar, and A. D. Woodbury. Lanczos and arnoldi methods for

the solution of convection-diffusion equations. Computer Methods in Applied Mechanics

and Engineering, 88:75-95,1991.

[57] B. Nour-Omid and M. E. Regelbrugge. Lanczos methods for dynamic analysis of damped

structural systems. Earthquake Engineering and Structural Dynamics, 18:1091-1104,

1989.

[58] D. M. Photiadis. The scattering of sound from fluid loaded plates. Journal of the Acous-

tical Society of America, 85(6): 2440-2451, June 1989.

[59] A. D. Pierce. Stationary variational expressions for radiated and scattered acoustic power

and related quantities. IEEE Journal of Oceanic Enginering, OE-12(2): 404-411,1987.

[60] A. D. Pierce. Acoustics. Acoustical Society of America, 1989.

[61] P. M. Pinsky and N. N. Abboud. Transient finite element analysis of the exterior struc-

tural acoustics proplem. Journal of Sound and Vibration, 112:245-256,1990.

[62] C. Rajakumar and C. R. Rogers. The lanczos algorithm applied to unsymmetric gener-

alized eigenvalue problem. International Journal for Numerical Methods in Engineering,

32: 1009-1026,1991.

[63] H. A. Schenck. Improved integral formulation for acoustic radiation problems. Journal

of the Acoustical Society of America, 44:41-58,1968.

[64] H. A. Schenck and G. W. Benthien. The application of a coupled finite-element boundary-

element technique to large scale structural acoustic problems. In C. A. Brebbia and J. J.

Connor, editors, Advances in boundary elements. Springer, New York, 1989.

[65] H. A. Schenck and G. W. Benthien. Numerical solution of acoustic-structure interaction

problems. Technical Report 1263, Naval Ocean Systems Center, 1989.

[66] A. F. Seybert, B. Soenarko, F. J. Rizzo, and D. J. Shippy. An advanced computational

method for radiation and scattering of acoustic waves in three dimensions. Journal of the

Acoustical Society of America, 77:362-368,1985.

[67]H. D.

Simon. The lanczosalgorithm

forsolving symmetric

linearsystems.

Technical

Report PAM-74, Center for Pure and Applied Mathematics, University of California,

1982.

[68] Limin Song, G. H. Koopmann, and J. B. Fahnline. Numerical errors associated with the

method of superposition for computing acoustic fields. Journal of the Acoustical Society

of America, 89(6): 2625-2633,1991.

[69] R. D. Spence and S. Granger. The scattering of sound from a prolate spheroid. Journal

of the Acoustical Society of America, 23(6): 701-706, November 1951.

[70] M. P. Stallybrass. On a pointwise variational principle for the approximate solution oflinear boundary value problems. Journal of Maths and Mechanics, 16(11): 1247-1286,

1967.

[71] M. P. Stallybrass. On a pointwise variational principle for the approximate solution oflinear boundary value problems. Journal of Mathematics and Mechanics, 16(11): 1247-

1287,1967.

page 198



[72] G. Strang. Linear Algebra and its Applications. Harcourt Brace Jovanoývich, third edition,1988.

[73] T. Terai. On calculationof sound

fields

aroundthree dimensional

objectsby integral

equation methods. Journal of Sound and Vibration, 69(1): 71-100,1980.

[74] O. Tullberg and S. Bolteus. A critical study of different boundary element matrices. In

C. A. Brebbia, editor, Boundary Element Methods in Engineering, Proc. Ph Int. SeminaSouthampton, pages 621-635. Berlin Heidelberg New York: Springer-Verlag, 1982.

[75] V. K. Varadan and V. V. Varadan V. V. Computation of rigid body scattering by prolate

spheroids using the t-matrix approach. Journal of the Acoustical Society of America,

71(1): 22-25, January 1982.

[76] A. G. P. Warham. The helmholtz integral equation for a thin shell. Technical Report

DITC 129/88, National Physical Laboratories, 1988.

[77] M. F. Werby and R. B. Evans. Scattering from objects submerged in unbounded andbounded oceans. IEEE Journal of Oceanic Engineering, OE-12(2): 380-394,1987.

[78] F. M. Wiener. On the relationship between the sound fields radiated and diffracted by

plane obstacles. Journal of the Acoustical Society of America, 23(6): 697-700,1951.

[79] D. T. Wilton. Acoustic radiation and scattering from elastic structures. International

Journal for Numerical Methods in Engineering, 13: 123-138,1978.

[80] S. W. Wu. A fast robust and accurate procedure for radiation and scattering analyses of

submerged elastic axisymmetric bodies. PhD thesis, Department of Aeronautics, Imperial

College, London, 1990.

[81] T. W. Wu, A. F. Seybert, and G. C. Wan. On the numerical implementation of a cauchy

principal value integral to insure a unique solution for acoustic radiation and scattering.Journal of the Acoustical Society of America, 90(1): 554-560,1991.

[82] X. F. Wu. Faster calculations of sound radiation from vibrating cylinders using variationalformulations. Journal of Vibration, Acoustics, Stress and Reliability in Design, 111:101-

107, January 1989.

[83] X. F. Wu, A. D. Pierce, and J. H. Ginsberg. Variational method for computing surface

acoustic pressure on vibrating bodies, applied to transversely oscillating disks. IEEE

Journal of Oceanic Engineering, OE-12(2), 1987.

[84] O. C. Zienkiewicz, D. Kelly,and

P. Bettess. The coupling of finite

element andbound-

ary solution procedures. International Journal for A umerical Methods in Engineering,

11:355-375,1977.

[85] O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method, volume 1. McGraw-Hill

(London), fourth edition, 1989.

page 199



Symmetry and Half Space Problems.

APPENDIX 1.

Symmetry and Half Space Problems

A 1.1 Image Sources

A important result in potential theory is the uniqueness of the exterior pressure field,

surrounding a radiator. The implication of this result is that the exterior pressure field can be

uniquely specified by the boundary conditions on the surface of the radiator. One important

consequence of this theory is that scattering and radiation problems are inter-changeable (Wiener

[1951]). This theory has another important consequence for the modeling of acoustic problems in

semi-infinite media, bounded by a zero pressure or zero velocity plane. Such half space problems

are of great interest in submarine and environmental acoustics, where the effect of rigid or free

surfaces correspond to the effect of the surface of the ground or water respectively.

The conceptual device of image sources is used to simplify the half space problem (Pierce

Ch. 5 [1989]). The plane surface is replaced by an image source so that the boundary condition

on the plane surface is preserved. This concept for the rigid surface is shown in figure (A1.1).

un=0

roý

S

Rigid Boundary T

un =0

_/1

Image Source

r

Imaginary Boundary T

Figure (A1.1). Image source construction to model infinite rigid plane.

Consider the exterior acoustic pressure for the source and image source. The pressure and

velocity fields can be written as single and double layer source distributions and since S is the

image of S the combined effects of the source distributions can be gathered under one integral,

page 200




1j eikr eskp_ 4r

Jrr, µ(ro)dS,

s

=1((ikr

- 1) ör=k,, ikr' -1 (9r' ik,.u

47r r2 öne+

I2 an es

(A1.1.1)

01.1.2)

where µ and a are the single and double layer source distributions defined at the surface source

points r0. Consider next the limit of these source distributions on the rigid plane. If the normal

n is defined as the normal to the plane, then from the symmetry of S and S',

r=T

or ÖTV on r.

an an

(A1.1.3)

Consequently the normal velocity on the surface of the imaginary plane is zero and the pressure

is twice the pressure than if the image surface were absent.

A semi-infinite acoustic domain bounded by a rigid plane, can be modeled by modifying

the Greens free space function according to the reflective symmetry in the plane. A pressure

release plane can be modeled by an antisymmetric image surface, so that the exterior pressure

and velocity fields are given by,

eikr eikr'

p- p(ro)ds, (A1.1.4)4r r r'

=1((ikr

- 1) ör ikr' -1 ör'

2 aneikr -

r/2 aneikr ý(ro)dS. (A1.1.5)u 4r

Is

r

In general the appropriate boundary condition can be imposed upon a ficticious surface by

modifying the Green's free space function according to Eq. (A1.1.1-2) or Eq. (A1.1.4-5). This

modification of the Green's free space function is appropriate for both the indirect and direct

formulations of the boundary element method.

A1.2 Geometric Symmetries

Consider a symmetric acoustic radiator defined by a repeating surface So, with surface

points ro. A geometric transformation describes the whole surface from So,

r; =Ti ro, i=1... n, -1, (A1.2.1)

page 201




where n, is the order of symmetry and T is the geometric transformation operator. The area

Jacobian in the integral formulation of the acoustic problem is invariant under the symmetry

transformationand so

the boundary layer formulationof the acoustic problem

isgiven

by,

n -1eikr,

p=- p(r,, )dS, (A1.2.2)47r

S. ri-0

n, -1

ýn =47r

(ikrr2 1) ýneikr, ý(ro)dS. (A1.2.3)

JS, -0 s

Again the geometric formulation is accounted for by modifying the Green's free space func-

tion and the same procedure can be applied to the SHIE and DSHIE formulations of the acous-

tic problem. The appropriate modifications of the Green's functions are summarized in figure

(A1.2).

In this study two symmetry transformations were implemented. The majority of prob-

lems analyzed were asymmetric and so rotational symmetry was implemented to account for

axisymmetric geometric symmetries. The appropriate transformation matrix is given by,

f cosO; sing; 0X

7rotro - _sin9; cos0i 0Y, (A1.2.4)

001Z

where,

2iriei= -, (A1.2.5)

n,

Reflectional geometry was also implemented and used in the cantilever plate problem of

Chapter 6, to model the half-space problem defined by the rigid wall holding the cantilever

plate. The transformation for a reflection in an arbitrary plane, r, with normal vector (1, in, n)

is given by,

1- 212 -21m -21n dz X

ef_ -2m1Tr T°--2n1

1- 2m2

-2nm

-2mn1- 2n2

dy

dZ

Y

Z(A1.2.6)

0 0 0 1 1

where (dt, dy, dZ) is the vector from the plane to the origin in the direction of the normal to the

plane. A fourth coordinate is added so that the reflection transformation can be written in terms

page 202




of a matrix multiplication. Up to three reflection planes with orthogonal normals can be used

to generate image sources. For reflection planes that are not orthogonal, an infinite number of

imagesources are generated.

The

set of symmetrictransformations for

three reflection planes,

a, b and c, are given by,

ra, b,e =7ä ,f )a yreJ)b

TTref )Croe a, b, c = 0,1. A1.2.7)

The order of the symmetry is given by 23 = 8.

page 203




U =o

Rigid Boundary

PressureRelease Boundary

Gk*_GkI+ Gk2

Geometric Symmetry

Figure (A1.2). Summary of image sources for the acoustic problem. The modi-

fied Green s function is denoted by G*.

page 204

un o



Analytical Series Solution for the Sphere.

APPENDIX 2.

Analytical Series Solution for the Sphere

A2.1 Rigid sphere

Throughout this study the spherical radiator is used to validate the acoustic numerical

methods since an analytical solution for the spherical problem is easily available. These results

are covered in great depth by Junger and Feit [1986], and they are summarized in this appendix.

The SHIE for the exterior problem is given by,

p(r) =

(p(ro)Or0) T_w2pwGk(r,ra) dSro, rEE, (A2.1.1)jo

where w is the radial displacement. If the coordinate system of the problem allows variable

separation of the Greens free space function, then an analytical solution is available. In spherical

polar coordinates the Greens free space function can be written in terms of spherical Besse]

functions and Legendre polynomials,

00 n

(2n + 1)cos m(0 - 00)k(r, O,01r, Bo Oo) =_ik Z

(n- m)1

47rn=o m=-n

(n + m) (A2.1.2)

Pn (cose)PP (coseo)jn(kro)hn(kr), r> ro

The coordinate system (r, 6, ¢) describes the exterior domain and (ro, Bo,0o) describes the

surface of the sphere. For coordinate systems where an analytical solution exists, the Green's

function in Eq. (A2.1.1) can be written as,

9(rß ro) = Gk (r, ro) + r(r, ro), (A2.1.3)

where F is a solution to the homogeneous Helmholtz' equation and the normal derivative of r

cancels the normal derivative of Gk on S. Consequently Eq. (A2.1.1) can be rewritten as,

P(r) = -,. )2p

iswg(r, ro)dSra,

,Js

r> r0. (A2.1.4)

page 205




Taking the general case of ro =a the correct expression for g(r, r0), satisfying the homoge-

neous Neumann boundary condition and the radiation condition, is given by,

00 n

9(r, e, 0ja, 0,0o) =147rka2ZZ (n

- m)(2n + 1)cos m(¢ - 40)

n-o m. -n

(n + m).(. X2.1.5)

Pn (cosO)PP (cosOo)h'ý(ka)

r>a.n(

The exterior pressure field given in Eq. (A2.1.4) can now be written in terms of this modified

Green's function,

hn(kr)

ý)=2

oo n (n-rn)

( 2n + 1)(r B,47rk

n=0m=-n(n + m) hn (ka)

j2x

rJcos m(0 - 0)P(cosB)P(cosG0)sinBodBod0o, r a..

0(A2.1.6)

The radial displacement in Eq. (A2.1.6) can also be written as a summation of modal

displacements,

00 ný

W=1:1: WmmnuPP '(CO89o)COS

m'0. (A2.1.7)

nl=0 m'=-nl

When the modal expansion of w is substituted into Eq (A2.1.7), the orthogonality of the func-

tions simplifies the expression for the exterior pressure field,

2 °O n ftn kr

p(r, B, q) =-wkP EE WmnPn (cos9)cos mO h, (ka) ,r>a.,

(A2.1.8)

n-0 m--nn

where,

wmn =1(n - m)i (2n + 1) 2r Z Pn (cos90)cos m w(a, 90,ýo) sin90dO0dýo. (A2.1.9)

4ir n+m . ,0

j0()o

The pressure on the surface of the sphere can be expressed in terms of the modal impedance,

page 206




00 n

P+(a, B,¢) = Z+,W,,,,,Pn (cosB)cos MO, (.12.1.10)Z

n-0 m--n

with,

zn = iPCh,(kQ)

(A2.1.11)

n()

The acoustic problem interior to thespherical shell can

beanalyzed

ina similar way.

The modified Green's function in Eq. (A2.1.5) must be constructed so that it satisfies the

homogeneous Neumann boundary condition and be finite as r-0. The corresponding surface

pressure on the interior surface is,

00 n

-(a, 8,0) = Zn Wmn P,, (cos9)cos MO, (A2.1.12)

n-0 m=-n

with,

?pCý;+(ka).

(A2.1.13)n

A comparison of Eq. (A2.1.11) and Eq. (A2.1.13) shows that whilst the exterior pressure

distribution is composed of a superposition of traveling waves, the interior pressure distribution is

a superposition of standing waves. The eigenvalues of the interior spherical problem correspond

to the solutions to,

zn (kD) = 0,

1

Ln=0(kN) .

(A2.1.14)

The interior eigenvalues for the spherical problem are tabulated in Table (A1.1) and the

frequency dependence of j and j;, is shown in figure (2.1).

page 207




1.0

0.5

0.0

-0.50.000

0.4

0.2

0.0

-0.2

-0.4

-0.6`-0.0000 3.1416 6.2832 ka 9.4248

Figure (2.1). Plot of the first four spherical Bessel's functions and the deriva-

tives.

------------- ---------------1

3.142 6.284 ka 9.426

:. te ýý1

, ý' ý.

-------------- ------------- ---- - ----

Jý

page 208




Dirichlet eigenvalues kD where j((kD), ) = 0.

1 n=0 n=1 n=2 n=3 n=4 n=5 n=6

1 3.141593 4.493409 5.763459 6.987932 8.182561 9.355812 10.512835

2 6.283185 7.725252 9.095011 10.417119 11.704907 12.966530 14.207392

3 9.424778 10.904122 12.322941 13.698023 15.039665 16.354710 17.647975

4 12.566371 14.066194 15.514603 16.923621 18.301256 19.653152 20.983463

Neumann eigenvalues kN where jn((kN)I) = 0.

1 n=0 n=1 n=2 n=3 n=4 n=5 n=6

1 4.493409 2.081576 3.342094 4.514100 5.646704 6.756456 7.851078

2 7.725252 5.940370 7.289932 8.583755 9.840446 11.070207 12.279334

3 10.904122 9.205840 10.613855 11.972730 13.295564 14.590552 15.863222

4 14.066194 12.404445 13.846112 15.244514 16.609346 17.947179 19.262710

Table (A2.1). The resonant frequencies for the interior spherical Dirichlet and

Neumann acoustic problems.

For any prescribed velocity distribution on the surface of the sphere the surface pressure

distribution may be calculated using Eq. (A2.1.10). For scattering from rigid spheres the surface

velocity must be equal to zero. Therefore the fluid particle velocity of the incident wave must

cancel that of the scattered wave. From the uniqueness of the external acoustic problem, the

scattered pressure field is obtained by solving the Helmholtz' reduced wave equation with the

virtual acceleration boundary condition,

iv, = -fi i=I

apion S. (A2.1.15)

An incident plane wave can be expanded in terms of spherical harmonics by using an addition

theorem,

00

p(r, O)

=P; E(2n + 1)i P(cosO)jn(kr). (A2.1.16)

n-0

Substituting Eq (A2.1.16) into Eq. (A2.1.15), Eq. (A2.1.8) gives an expression for the axisym-

metric scattered pressure field,

page 209




00 hp5(r, O) = -Pi

E(2n+ 1)i Pn(cos9)

hn(ka)jn(kr) (. 2.1.17)

n. 0

The external pressure distribution is given by,

00 hn(kr).P(r, O)= p, + pi = PsE(2n + 1)inPn(cos8)

[n(kr)_

h, ka 7(kr) (A2.1.18)

n-0

li

n()

A2.2 Asymptotic solutions

The frequency dependence of the far field scattered pressure for an arbitrary geometry can

be approximated by asymptotic solutions. Consider Eq. (A2.1.17) in the far field,

00 1

psf (R, B) -ZPkR

Rj: (2n + 1)Pn(cos8)

hn(kr ,kR» n2 + 1. (A2.2.1)

n-0 n()

At low frequencies the first two terms of this series expansion are O((ka)2) and the n=2 term

is only O((ka)5). Taking the first two terms gives the far field pressure at low frequencies,

ikRep;

f (R, 8) =P3R

k2a33cosO

-1 k3a3 9 1, kR » 1. (A2.2.2)

The commensurate influence of monopole and dipole radiation in the far field, and the

consequent k2 dependence of the radiation amplitude is typical of Rayleigh Scatterers. A more

general consideration of such scatterers shows that at low frequencies the far field pressure

amplitude has the form,

ffk2V

k3L3«1, kR » 1.aR

where V is the volume and L is the characteristic dimension of the scatterer.

(. 2.2.3)

In the high frequency regime an approximation due to Kirchoff is appropriate. In this plane

wave approximation it is assumed that each surface element radiates as if it were in an plane

baffle,

page 210




dpP (R) -pw(ro)exp(ikIR - roI )dS,

kra » 1. (A2.2.4)

The far field back scattered pressure is given by integrating Eq. (A2.2.4) over the illuminated

surface of the scatterer,

_iP, kjf (R)2irR

n elexp(2ikIR- r0I)dS, kro » 1. (A2.2.5)

Is.

In Eq. (A2.2.5) the plane wave is incident along the direction of Z-,. Consider now the case of

a spheroid shown in figure (A2.1). For end on incidence the problem is axysymmetric and so

the cylindrical coordinate system, (q, 0, Z) is appropriate to form the Kirchoff integral. In this

coordinate system, Eq. (A2.2.5) becomes,

ptf (R)Rh-i k nze6T1(Z)dF,

kL» 1, (A2.2.6)Pi

is

where ,

2

dI' =1+ azdZ, (A2.2.7)

2

nz öZ+

az(A2.2.8)

b(Z) = 2ikZ. (A2.2.9)

Eq. (A2.2.6) now becomes,

pof (R)Rk

ýZeaikZdZ kL » 1, (A2.2.10)

J,

where for the spheroid,

p'f (R) R_ ka2 0ZezikzdZ

,kb » 1,

pi b2

fb(.2.2.11)

page 211




Inctuerli rruve

Figure A0. L). Geometry for the spheroidal Kirchof problem.

The integral in Eq. A2.2.11) can be done in parts, and neglecting the terms of O kb)-1

or higher the high frequency far field back scattering amplitude is given by,

pof R)R a2 kb » 1.Pi

I= I2l,A2.2.12)

This is the result used in Chapter 4, and for the special case of the sphere, with b=a, it

reduces to the spherical result used in Chapter 3. If higher frequency terms are included an

interference pattern is introduced into the farfield result. This interference corresponds to a

path difference of 2b; ie the path difference between a specular wave at Z=-b and one at

Z=0. The numerical results contained in this thesis show such an interfence pattern, however

the period of the fluctuations are much less than those predicted by the Kirchoff approximation.

The accepted explanation for the interference pattern eg Varadan et al [1982]) is that they are

a result of the interference between the specular reflection at Z= -b and a Franz or creeping

wave originating at the circular cross section at b=0. The predicted period of the interference

is given by,

A kb)27

,D -+2 A2.2.13)

where D is one half the perimeter of the spheroid in the X=0 plane. For the various aspect

ratios used in this study this period is,

page 212




alb = 0.5, z(kb) -- 1.42,

alb = 1,0(kb) = 1.22,

alb = 2,0(kb) = 2.84,

A2.3 Elastic sphere

J(ka) = 0.71,

J(ka) = 1.22, (A2.2.14)

0(ka) = 1.42.

For axisymmetric boundary conditions, there is a readily accessible analytical solution for

the dynamic elastic response of the spherical elastic shell. This solution is described in greater

depth in many standard texts, and is summarized in this appendix. Membrane and flexural

effects are included in the analytical model which can be derived from Hamilton s variational

principle. Many of the thin shell approximations used for the finite element derivation are

used for the analytical result; ie the deformation can be described in terms of the mid-shell

displacements and the mid-shell normal is remains straight after deformation.

Axisymmetric, nontortional motions are considered and the shell displacements can be

described in terms of the tangential and radial components; ur and ue. A free dynamic analysis

of the shell results in an expression for the dimensionless natural frequencies, (5l )2 and (52,, )2,

which are roots of a quadratic in Q2,

524- [1+3v+A-/32(1-v-An-vAn)]

52+(AT-2)(1-v2)+

(A2.3.1)

ß2 [An-4,n+An(5-v2)-2(1-v2)] =0,

where,

=ca, Q =

h2

1222An = n(n + 1). (A2.3.2)

P

The radial or normal and the tangential displacements can be expanded in terms of Legendre

polynomials,

00

ur =1: WWPn(µ),

n=O

°°P2)1/2

dPn(p) (A2.3.3)

n=0dp

p=COSO.

page 213




For forced vibration of the spherical shell, the modal radial amplitude is related to the

modal force amplitude by,

ýn =_

fn

__

IP, cp h [c2-

(c(1))2] [92-

(9(2»2,(A2.3.4)

2wwn 9 a[92- (1- QZ)(v + An - 1)]

where the modal excitation force is given by,

(2n + l) 1fn =2f f(µ)P (µ)dp. (A2.3.5)

i

Finally the modal radial and tangential amplitudes are related by,

V_

[32(v+A- 1)+(i+ v)] (A2.3.6)

For a spherical shell vibrating in an acoustic fluid, the excitation force in Eq. (A2.3.5)

consists of the apllied force, f', modified by the fluid-structure interaction force, f'. The modal

normal velocity of the submerged spherical shell is given by,

Z,n=fn -. fn, (A2.3.7)

where,

fn= Pn - Pn = ýz - zn ] Wn. (A2.3.8)

Therefore by combining Eq. (A2.3.7) and Eq. (A2.3.8) the surface pressure and velocity

fields for the excited spherical shell are given by,

°O fa

un (a, e)=Zn_ PP (cose),

[Zn + Zn - Zni=0

0o a

P(a, 0) =Z

Zn+ fn

_P1 (COSB).

i-0 `Zn+ Zn - zn

For the point excited spherical shell the modal excitation force is defined by,

(A2.3.9)

page 214




a_-(2n + 1)F

(.-2.3.10)"47ra2

and the resulting far field pressure field is,

pff(R, B)

-

Fpce'kR (-i)n(2n+ 1)Pn(cos8)kR» 1. (A2.3.11)

=o(2n + Z+ - -n)h' (ka)'7razkR J n

The case of scattering from an elastic spherical shell is more complicated. It is necessary

to consider both the rigid body scattering component and the elastic radiated component. The

surface pressure on the rigid body due to an incident plane wave is given in Eq. (A2.1.18) with

Documents

Richard Jeans 1992 PhD Thesis