The Boundary Element Method inAcoustics - An internship report

Fabio Kaiser

Studiengang: Elektrotechnik-Toningenieur der Universitat fur Musik unddarstellende Kunst und der Technischen Universitat, Graz, Osterreich

Supervision: M. A. Frank Schultz, TU Berlin

Technische Universitat Berlin, Fakultat IInstitut fur Sprache und Kommunikation, Fachgebiet Audiokommunikation

October 7, 2011

Abstract

This report documents the work done during a 3-month internship at theAudio communication group of the TU Berlin. Basically the theory of theboundary element method, applied to acoustics is introduced and simple testsimulations are described.

For any correspondence about this text please refer to the author (fabio kaiser@gmx.at)or the supervisor (frank.schultz@tu-berlin.de).

1

Contents

1 Introduction 3

2 Basic theory of acoustics 52.1 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 72.4 The interior and exterior problem . . . . . . . . . . . . . . . . 8

3 The boundary element method. Part I. Helmholtz integralequation 93.1 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1.1 HIE for interior problems . . . . . . . . . . . . . . . . . 103.1.2 HIE for exterior problems . . . . . . . . . . . . . . . . 14

3.2 Indirect Method . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Existence and Uniquness . . . . . . . . . . . . . . . . . . . . . 16

3.3.1 CHIEF . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.2 Burton-Miller formulation . . . . . . . . . . . . . . . . 18

4 The boundary element method. Part II. Numerical imple-mentation 194.1 Discretization and collocation . . . . . . . . . . . . . . . . . . 194.2 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . 214.3 Solving a system of equations . . . . . . . . . . . . . . . . . . 224.4 Post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . 224.5 Computational issues . . . . . . . . . . . . . . . . . . . . . . . 22

5 Implementation and examples 255.1 OpenBEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2.1 Error measures . . . . . . . . . . . . . . . . . . . . . . 275.2.2 Pulsating sphere . . . . . . . . . . . . . . . . . . . . . 285.2.3 Piston on a sphere with disc . . . . . . . . . . . . . . . 28

6 Conclusions and Outlook 37

A Divergence theorem and Green’s identities 38

2

1 Introduction

In acoustics, the linearized wave equation describes the propagation of wavesin fluids. The Helmholtz equation is the time-harmonic equivalent, also calledreduced wave equation. Both are partial differential equations (PDE), thewave equation is a so called hyperbolic PDE and the Helmholtz equationis an elliptical PDE. By solving these PDE’s the propagation of sound canbe predicted. Analytical methods to solve these problems are available andhave been studied for long time. Still, these methods suffer from the factthat solutions can be found for a limited class of problems only.

Therefore and with the rise of computers in the 1970’s, numerical methodsfor solving PDE’s have been developed. The most popular methods are thefinite element method (FEM), the finite difference method (FDM) and theboundary element method (BEM). These techniques allow to calculate thesound field of any acoustic scenario and the inaccuracy of the result is onlydetermined by restrictions of computational resources. All are based on thediscretization of space into little pieces like in a puzzle. The FEM and FDMare so called domain based methods because the whole space of interest needsto be discretized. The BEM on the other hand discretizes only the boundariesof the domain and therefore reduces the dimensionality of the computationsby one. This is also why the FEM is usually used in the structural domainand the BEM is used for the treatment of infinite domains.

Outline of the BEMThe basic idea of the BEM is the reformulation of the Helmholtz equation

into a (boundary-) integral equation that is mathematically equivalent. Twointegrals arise, one which is defined on the boundary of the domain and onewhich relates the boundary solution to points in the field. Basically thereformulation represents the acoustic field as a superposition of the fieldsof elementary sources (monopoles and dipoles) located at the boundary ofthe domain. The reformulation can only be applied to classes of PDE’swhere a fundamental solution can be found. It is therefore not applicable toevery physical problem compared to the FEM and FDM, which are nearlyuniversally applicable. However, if a fundamental solution can be found theBEM can be applied and it states an easier to use and computationally moreefficient method.

The integral equation then is discretized into small elements over the sur-face of the domain and a numerical integration is conducted on each element.As mentioned this is the main advantage of the BEM over domain methods.If the radiation of an vibrating object should be calculated the domain is of

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infinite extend. In FEM this infinite space has to be discretized. In BEMonly the surface of the objects have to be discretized in order to computethe sound field on every point in the domain. The Sommerfeld radiationcondition is inherently fulfilled. However, this reduction of dimensionality ofthe approach also results in difficulties of non-uniqueness which do not existin the original problem.

The discretization process results in a linear system of equations or amatrix equation which has to be solved for. The resulting matrices are fullyoccupied, complex and unsymmetrical [Mec08]. This means that the storagenecessary is high and the computation time is longer compared to symmetric,sparse matrices which arise in the FEM.

The BEM is often referred to as the boundary integral equation method,especially at the beginning of the development. This is still the case to-day when the derivation and analysis of the method is addressed instead ofdiscussing the implementation and/or application.

Applications The BEM can generally be applied to every kind of radiationand scattering problems and it can also be applied to interior eigenvalueproblems. The main applications are found in structural analysis and noiseradiation but also in aero-acoustics, bioacoustics, the automotive industry,enviromental acoustics and architectural acoustics [Cis91]. The BEM hasalso been applied to the simulation of head-related transfer functions [Kat01][Kre09]. It is nowadays an essential part of product design where the BEMis used to simulate the acoustical behaviour of a product and based on theresults the product might be redesigned, e.g. loudspeaker design.

In this work the BEM is applied in the framework of headphone design...In chapter 2 the basic theory of acoustics is shortly reviewed. This is

necessary to form the basis for chapter three, the derivation of the boundaryintegral equations. Additionally the problems arising with the formulationsare shortly addressed and the existing solutions presented. Chapter 4 in-troduces then the discretization by elements process and the formulation ofthe matrix equations. After these theoretical chapters in chapter 5 the im-plementation used in this work is presented and several examples are shown.This report ends with a summery of the work that has been done and an out-look about what is still open. The main references for this work are the twobooks on BEM in acoustics [Cis91] and [Wu00] and a chapter O.5 in [Mec08]which provides a good overview. Books covering a wider range of theoreti-cal issues are [Bee92] [Che92] and [Sau10]. A book treating implementationissues more directly is [Bee08].

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2 Basic theory of acoustics

In this chapter the basic theory of linear acoustics is shortly introduced.The wave equation has been developed in order to model the propagation ofsound. This is the starting point of all further derivations in the context ofthis work.

2.1 Wave equation

The homogeneous wave equation is a linear partial differential equation ofsecond order

1

c2

∂2p

∂t2−∇2p = 0 (2.1)

where c is the speed of sound, p is the sound pressure and ∇2 is theLaplace operator, in cartesian coordinates defined by

∇2 ≡ ∂2

∂x2+

∂2

∂y2+

∂2

∂z2. (2.2)

This equation is linear because the result is independent of where andwhen the wave equation is excited. It is partial because there are morethen one variables (dimensions)and it is of second order because the secondderivation is applied. It describes a real or complex function p(t, x1, . . . , xn)in space and time and relates the second derivate of space with the secondderivate in time. The acceleration is proportional the curvature.

The solutions are waves which can interfere and which are independentof each other, i.e. they do not influence one and another. If some excitationis considered the inhomogeneous wave equation can be written as

1

c2

∂2p

∂t2−∇2p = f(t, x1, . . . , xn) (2.3)

where f is considered as the excitation.

Helmholtz equation Sometimes it is easier to consider only time-harmonicprocesses, i.e., p(t, x1, . . . , xn) = Rep(x, t)eiωt. This can be injected intoEq. 2.1 which yields the so called Helmholtz equation

∇2p+ k2p = 0 , (2.4)

where k = ωc

= 2πλ

is the wavenumber. The Helmholtz equation is alsocalled reduced wave equation because only time-harmonic processes are con-sidered.

5

(a) (b)

Figure 1: Green’s function, a) real part, b) imaginary part.

For the free-field case or simple geometries solutions can be found an-alytically. If the geometry becomes more complex it can get very difficultor impossible to find analytical solutions. Numerical approaches have beendeveloped in order to solve for this problem.

2.2 Green’s function

The Green’s function is the particular solution for the Delta-distribution δas an excitation

(∇2 + k2)G(r|r′) = −δ(r − r′) . (2.5)

The solution in three-dimensional space is [?])

G(r|r′) =e−ik|r−r

′|

4π|r − r′|. (2.6)

It is the solution to a point source in free-space at r = r′. It can beseen as a spatio-temporal transfer function (or impulse response) equivalentto a impulse response in the time-domain. This is why a solution to anyinhomogenity can be found through the convolution with the Green’s function[Wik11a]. Therefore, it is called the fundamental solution of the Helmholtzequation. In Fig. 1 the real and imaginary part of Eq. 2.6 are plotted. Wewill see that in order to apply the BEM formulations a fundamental solutionof the governing equation is necessary.

6

2.3 Boundary conditions

As it is the case with every kind of PDE, also the wave equation can only besolved subject to certain boundary conditions. These can either be active orpassive. Active means that there is an object that vibrates and therefor isthe source of sound energy. In acoustics there are two possibilities, either thesound pressure is prescribed at some point or surface in space or the particlenormal velocity. If the sound pressure is prescribed it is called a Dirichletboundary condition

p = p (2.7)

where p indicates known values, e.g. p = 0 which corresponds to a pres-sure release case. If the velocity is prescribed it is called a Neumann boundarycondition

v = v , (2.8)

and if v = 0 it corresponds to a rigid body. It has to be pointed out thatin the active case the normal velocity of the vibrating structure is equal tothe normal velocity of the fluid in contact with the structure [Wu00, p. 3].

Passive boundary conditions apply when an object reflects from a passivesurface (e.g. an absorbing surface). In general form it can be written

α p+ β vn = γ (2.9)

where α, β, γ are arbitrary complex functions. They can be called generalboundary conditions because the Dirichlet and Neumann boundary condi-tions can be generated as a special case by setting e.g. α = 0, β = 1 andγ = vn in Neumann case. Other names are impedance boundary conditionsor Robin boundary conditions [Wik11c]. If we reorder for the velocity Eq.2.9 is written as

vn = −αβp+

γ

β(2.10)

where the left term on the right side αβ

is an acoustic admittance Y andthe right term is a forced or prescribed velocity vs

vn = −Y p+ vs . (2.11)

In this case vs can be seen as the velocity of the vibration of a structureand Y p = va as the velocity of the fluid [Kol11] [Mar99]. One can imagine avibrating structure with absorbing material stitched to it.

7

Figure 2: Exterior, interior and a combined problem.

Sommerfeld radiation condition For exterior problems, where the do-main extends to infinity, the Sommerfeld radiation condition has to be ful-filled in addition to the boundary conditions

limR→∞

[R∣∣ ∂p∂R− ikp

∣∣] = 0 (2.12)

where R is the radius of a big sphere in spherical coordinates which includesthe radiating or scattering object. It basically means that any radiated orscattered acoustic wave has to converge towards zero when the radius extendsto infinity.

2.4 The interior and exterior problem

There are two basic classes of problems in acoustics,the interior and the ex-terior problem. For the exterior problem the sound sources are located insidea region and the region of interest is defined to be outside the surface includ-ing the sound sources. The exterior problem includes infinity. Radiation andscattering problems fall into this category. The interior problem is definedexactly the other way around. Fig. 2 plots the defined regions.

In the scope of the derivation of the HIE we have to distinguishe theproblems.

8

3 The boundary element method. Part I.

Helmholtz integral equation

The problem defined in the previous chapter is a boundary value problem, i.e.,a PDE is solved by make use of certain boundary conditions. The BEM is anumerical method for the solution of this problem. The method is not directlyapplied on the Helmholtz equation but on a reformulation as a boundaryintegral equation [Wei11b]. This is the first step in the derivation of theBEM equations.

In this chapter it is shown how we can get from a PDE to an integralequation. Two possibilites have devoleped in BEM’s, the direct and theindirect method.

3.1 Direct Method

The direct method deals with functions in the integral equation that are phys-ically meaningful (sound pressure and velocity) instead of fictitious densityfunctions. The solutions directly yields the unknown values on the boundary,therefor the name.

The integral equation is derived from the divergence theorem in whichcertain substitutions lead to Green’s identities (see Appendix A). If one ofthe functions in Green’s second identity is taken to be the unknown andthe other the fundamental solution of the Helmholtz equation, the boundaryintegral equation is obtained. The derivaiton follows [Bee92, Appendix D]and [Wil99, Ch. 8].

Green’s second identity yields an integral equation for two scalar fields φand ψ ∫∫∫

Ω

(φ∇2ψ − ψ∇2φ) dV =

∫∫S

(φ∂ψ

∂n− ψ ∂φ

∂n) dS . (3.1)

Now, if it is assumed that the functions φ and ψ are solutions to theHelmholtz equation, i.e. they have to satisfy the homogeneous Helmholtzequation on the surface and in the volume

∇2φ+ k2φ = 0

∇2ψ + k2ψ = 0 (3.2)

the left hand side of Eq. 3.1 becomes

9

Figure 3: The region V with its surface S and the outward normal n. Theonly restriction on the shape is that the derivate has to be continuous.

φ∇2ψ − ψ∇2φ = φ(−k2φ)− ψ(−k2ψ) = 0. (3.3)

and the volume integral vanishes:∫∫S

φ∂ψ

∂n− ψ ∂φ

∂ndS = 0. (3.4)

Eq. 3.4 is satisfied if φ and ψ have no singularities on the surface S. TheHIE’s for interior and exterior problems can be derived from here.

3.1.1 HIE for interior problems

The basic definitions of the geometry for derivation of the interior problemare shown in Fig. 3.

Let’s consider the case where ψ has a singularity somewhere in the domainat r = r′.

∇2ψ + k2ψ = −δ(r − r′) (3.5)

The function ψ(r) is a solution of the inhomogenuous Helmholtz equationand satisfies the Sommerfeld radiation condition. The field point r covers allspace. Solving Eq. 3.5 yields the free-space Green’s function, also called thefundamental solution :

G(r|r′) =e−ik|r−r

′|

4π|r − r′|=ik

4πh0(k|r − r′|), (3.6)

10

Figure 4: The region V with its surfaces So and Si and the outward normaln. The evaluation point r = r′ inside the volume is excluded. From [Wil99].

where k the wavenumber and h0 the spherical Hankel function of first kindand zeroth-order. Now, three cases can be distinguished, depending on wherethe evaluation point r′ is exactly located.

Case 1: r′ inside S As the point r′ lies inside the volume V (Fig. 4),the Green’s identity can no longer be applied because it has a singularityat r = r′. Therefore the geometry has to be modified by excluding thesingularity. Green’s second identity is therefore redefined for the volume Vwith a boundary S = So + Si where Si defines a small sphere inside V andSo is the outer surface as before. The outward normal of Si points towardr′. Now we let the radius ε of the small sphere Si go to zero. With ψ = Gthis yields

∫∫S

(φ(r)

∂G(r|r′)∂n

−G(r|r′)∂φ(r)∂n

)dSo+ lim

ε→0

∫∫S

(φ(r)

∂G(r|r′)∂n

−G(r|r′)∂φ(r)∂n

)dSi = 0.

(3.7)

We now have to evaluate the limes of the two integrals [Wil99] [CW07].The Green’s function on Si becomes

G(r|r′) =e−ikR

4πR=e−ikε

4πε, (3.8)

and the normal derivative of G is given by

∂G(r|r′)

∂n= − 1

4π

∂e−ikε/ε

∂ε= − 1

4π

e−ikε(ikε− 1)

ε2. (3.9)

11

The first integral to evaluate in Eq. 3.7 can than be written as

limε→0

∫∫S

(φ(r)

∂G(r|r′)

∂n

)dSi = lim

ε→0(− 1

4π

e−ikε(ikε− 1)

ε2)

∫∫S

φ(r)dSi. (3.10)

Noting that the function φ(r) is continuous within the sphere and thesurface of the sphere is 4πε2 it can further be written

limε→0

(− 1

4π

e−ikε(ikε− 1)

ε2)4πε2φ(r) = φ(r)|r=r′ = φ(r′). (3.11)

The second integral to evaluate in Eq. 3.7 becomes

limε→0

∫∫S

(−G(r|r′)

∂φ(r)

∂n

)dSi = lim

ε→0(eikε

4πε)

∫∫S

∂φ(r)

∂ndSi. (3.12)

As ∂φ∂n

is continuous about r = r′ it can be taken out of the integral. Asbefore this yields

limε→0

(e−ikε

4πε)4πε2

φ(r)

∂n= 0. (3.13)

The results of Eqs. 3.11 and 3.13 inserted in Eq. 3.7 yield the interiorHelmholtz equation for interior problems [Mec08, Ch. O.5]

φ(r′) =

∫∫S

G(r|r′)∂φ(r)

∂n− φ(r)

∂G(r|r′)

∂ndSo (3.14)

Case 2: r′ on S If the evaluation point r′ is located at the surface of S0

the surface Si doesn’t span a whole sphere anymore but spans that part ofthe sphere with a radius of ε centered at r′ that lies inside the volume V(Fig. 5). This means that with∫∫

S

dSi ∼ Ω(r′)ε2 (3.15)

where Ω(r′) is the solid angle subtended at r′ by the volume V [CW07][Wik11b], Eq. 3.11 becomes

limε→0

(− 1

4π

e−ikε(ikε− 1)

ε2)φ(r)

∫∫dSi =

Ω(r′)

4πφ(r′). (3.16)

where

12

Figure 5: The region V with its surfaces So and Si and the outward normaln. The evaluation point r = r′ on the surface is excluded. From [Wil99].

Ω(r′) =1

ε2

∫∫S

dSi (3.17)

and hence

Ω(r′)

4πφ(r′) =

∫∫S

G(r|r′)∂φ(r)

∂n− φ(r)

∂G(r|r′)

∂ndSo (3.18)

is the surface Helmholtz integral equation for interior problems. If thesurface r′ is located on is smooth then Ω(r′)

4π= 1

2.

Case 3: r′ outside S The evaluation point lies anywhere outside thevolume and hence no singularity exists within or on the surface. This meansEq. 3.4 applies, which yields

0 =

∫∫S

G(r|r′)∂φ(r)

∂n− φ(r)

∂G(r|r′)

∂ndSo (3.19)

the exterior Helmholtz intergral equation for interior problems.

Combining Eqs. 3.14, 3.18 and 3.19 the full expression of the Helmholtzintegral equation for interior problems is written as

∫∫S

(G(r|r′)

∂φ(r)

∂n− φ(r)

∂G(r|r′)

∂n

)dS =

Ω(r′)

4πφ(r′) (3.20)

13

with

Ω(r′)

4π=

0, r′ ∈ Ve12, r′ ∈ S .

1, r′ ∈ Vi

(3.21)

for smooth surfaces. In acoustics we deal with the sound pressure andvelocity. Therefor we substitute φ = p and use Euler’s equation

∂p(r)

∂n= −iρ0ckvn(r) (3.22)

where ρ0, c, k, vn(r) are the air density, speed of sound, wavenumber k = ωc

and the particle velocity in the outward normal direction, respectively. Eq.3.20 then becomes

∫∫S

(iρockvn(r)G(r|r′)− p(r)

∂G(r|r′)

∂n

)dS =

Ω(r′)

4πφ(r′) (3.23)

It has to be noted that the integration is with respect to the field point rand the variable r′ is the point where the sound pressure is evaluated. TheHIE states that the pressure at any point inside the volume can be computedfrom the known pressure and normal velocity at it’s surface [Wil99, p. 257].

3.1.2 HIE for exterior problems

Basically the approach to obtain the formulation of the HIE for exterior prob-lems is equivalent to the interior case. We just have to redefine the regionswhere Green’s second identity should be applied. The surface S is again anarbitrarily shaped body with the normal pointing outwards the surface (Fig.3). As before φ and ψ both satisfy the homogenous Helmholtz equation inthe region V, which is defined to be outside the surface S extending to infin-ity. At infinity the Sommerfeld radiation condition has to be satisfied by φand ψ. This means that Eq. 3.4 also applies, but with sign changed as thenormal is defined in the same way as for the interior problem.∫∫

S

ψ∂φ

∂n− φ∂ψ

∂ndS = 0. (3.24)

Similar to the interior case we define a small sphere with surface Si aroundthe point r′ with radius ε either outside, on or inside the surface So so thatthe total surface S becomes

S = So + Si. (3.25)

14

The function φ is again chosen to be the free space Green’s function sothat Eq. 3.24 becomes equivalently to Eq. 3.7

∫∫S

(G(r|r′)∂φ(r)

∂n−φ(r)∂G(r|r

′)

∂n

)dSo+ lim

ε→0

∫∫S

(G(r|r′)∂φ(r)

∂n−φ(r)∂G(r|r

′)

∂n

)dSi = 0.

(3.26)

The result of the limes is the same as before:

limε→0

∫∫S

(G(r|r′)

∂φ(r)

∂n− φ(r)

∂G(r|r′)

∂n

)dSo =

Ω(r′)

4πφ(r′) (3.27)

where

Ω(r′)

4π=

0, r′ ∈ Ve12φ(r′), r′ ∈ S

φ(r′), r′ ∈ Vi

(3.28)

with Ω(r) the solid angles as defined in Eq. 3.17.The Helmholtz integral equation for exterior problems is written as

∫∫S

(φ(r)

∂G(r|r′)

∂n−G(r|r′)

∂φ(r)

∂n

)dS =

Ω(r′)

4πφ(r′). (3.29)

This is identical to the interior problem except for the sign changed be-cause the surface normal is also pointed outwards. We can write Eq. 3.29using sound pressure and velocity as

∫∫S

(p(r)

∂G(r|r′)

∂n− iρockvn(r)G(r|r′))

)dS =

Ω(r′)

4πp(r′). (3.30)

Using the HIE for exterior problems it is possible to calculate the soundpressure radiated by sources inside S, everywhere outside the surface S byknowing the pressure and the velocity on the surface. If the evaluation pointis located on the same side of the surface as the sources do, then the HIEyields a null field. This is valid for the interior and the exterior problem. Todetermine the sound pressure anywhere on the side of the sources would givean inverse problem. The HIE does not solve the inverse problem [Wil99, Ch.8].

15

3.2 Indirect Method

Single and double layer potential see [CW07] [Mec08, Ch. O.5.1] or simplesource formulation see [Wil99, Ch. 8.7]

An alternative formulation of a boundary integral equation is found bythe so called indirect method. This method deals rather with density distri-butions than with the functions of interest directly, therefore the name. It isalso called the potential-layer approach [Mec08] because the sound pressurecan be represented as a single-layer potential

p(r) =

∫∫S

σ(r′)G(r, r′) (3.31)

or a double layer potential

p(r) =

∫∫S

ψ(r′)∂G(r, r′)

∂n) (3.32)

where σ and ψ are density distributions. It can be thought as a layer ofmonopoles or dipoles in contrast to the HIE which contains both monopolesand dipoles. In the case of single layer potential this representation leads toa boundary integral equation:

σ(r)

2−∫∫

S

σ(r′)∂G(r, r′)

∂ndSr′ = jωρ0vn (3.33)

where r ∈ S. This is the density if the exterior Neumann problem isconsidered. For the formulations of other problems and the double-layerapproach see [Mec08, p. 1061] and [CW07, Ch. 2.4].

The single and double-layer approaches yield boundary integral equationswhich are ill-posed and therefore need special treatment in order to obtain asolution. In acoustics the direct method is generally preferred.

3.3 Existence and Uniquness

There is one major shortcoming of the BEM which is due to mathemati-cal properties of the derivation. It can be shown that the HIE for exteriorproblems doesn’t have a unique solution at certain frequencies. These char-acteristic eigenfrequencies can be associated with the corresponding interiorDirichlet problem. If we consider an exterior problem with Dirichlet bound-ary condition the issue can be explained. The surface HIE becomes∫

S

iρωvndS = −1

2p−

∫S

p∂G

∂ndS (3.34)

16

with the surface normal pointed outwards. Further we have a look at theinterior Dirichlet problem. With the surface normal unchanged the interiorHIE becomes ∫

S

iρωvndS =1

2p−

∫S

p∂G

∂ndS . (3.35)

It can be seen that both equations share the same left-hand side. Thelatter equation is for an interior problem and therefore has eigenfrequen-cies. The exterior problem does not have any eigenfrequencies but it sharethe same left-hand side which determines the one of the coefficient or sys-tem matrices. If the interior problem is evaluated at resonance frequenciesthe coefficient matrix will become singular so will the one for the exteriorproblem. It should be pointed out that this issue for the exterior problemarises purely from the mathematical approach and doesn’t have any physicalmeaning. Basically two methods were suggested to overcome this problem[Wu02].

3.3.1 CHIEF

One simple method to overcome the non-uniqueness problem was suggestedby Schenk in 1968 [Sch68] and he called that method the combined Helmholtzintegral equation formulation (CHIEF). The idea is to create an overdeter-mined system of equations by which the interior problem can be distinguishedfrom the exterior problem. This is done by adding or combining the HIE forseveral interior points with the set of surface HIE’s. The additional or con-straint equations can be referred to as CHIEF equations and CHIEF point.The interior HIE for exterior problems is∫∫

S

(p(r)

∂G

∂n− iρockvn(r)G

)dS = 0 . (3.36)

This equation enforces the zero pressure condition inside the surface andthere can be seen as a constraint to the surface HIE. This is usually enoughfor the exterior problem to have a unique solution. A problem arises whenthe CHIEF point is located at an interior nodal surface of an eigenfrequency.The constraint effect is gone because the interior pressure is zero for the in-terior problem, as well. Further with increasing frequency the nodal surfacesbecome more densely and so the location of CHIEF points becomes hard tochoose [Wu00, p. 27].

17

3.3.2 Burton-Miller formulation

Another popular technique to overcome the non-uniqueness problem is theso called Burton-Miller formulation [Bur71]. It has been shown that a linearcombination of the HIE and its normal derivative yield a unique solution overthe whole frequency range.

CBIE + β HBIE = 0 (3.37)

where CBIE is the conventional boundary integral and HBIE its derivative.The constant β has to be complex, e.g. i

k[Li11, p. 154]. The drawback is

that a the normal derivative of the HIE is a hyper singular integral. Regu-larizations have to be used [Cis91].

18

4 The boundary element method. Part II.

Numerical implementation

In chapter 3 the Helmholtz integral equation has been derived. In order tosolve this integral the idea is to discretize the surface into small elements,represent the geometry and the variables by shape function, numerically inte-grate over the elements and add together the results. This chapter describeshow to get from the HIE to a matrix equation.

4.1 Discretization and collocation

In fact there are two stages of discretization. First the surface of the objectin consideration has to be discretized. And second the boundary variableshave to be discretized (in acoustics the sound pressure p and the particlenormal velocity vn). In principal this can be done independently but inpractice the geometry and the variables are mostly discretized in the sameway which yields so called isoparametric elements. The variables can thenbe represented as

φ =n∑i=1

φiNi(ξ1, ξ2), (4.1)

where φ can either be the geometric variables (x, y, z-coordinates), thesound pressure p or the particle normal velocity vn. The index i indicates thenodal points, n is the number of nodes in that element and Ni are the shapefunctions defined on a master element with local coordinates ξ1, ξ2 [Wu00, p.55].

The shape of the elements in three-dimensions can either be of triangularor quadrilateral shape (Fig. 6 ). The geometry is represented by the nodesand the shape functions which interpolate between the nodes. The number ofnodes on an element determines the order of the shape functions. A constantelement places one node at the centroid of the element. A linear elementplaces nodes at the corners of the element. A quadratic element places nodesat the corners and in the middle of the edges. As each element has a similarshape the integration is generalized and made on a parent or master element.This means that every real element is transformed into local coordinates ofthe master element for integration. It has to be pointed out that part of thenumerical error in the result is due to the approximation of the boundary.Therefore higher-order elements might be preferable. For a comprehensivetreatment of elements and shape functions the reader is referred to standardBEM textbooks, e.g., [Bee92, Ch. 2] or the standard FEM book [Zie89].

19

(a)

(b)

Figure 6: Real element (left) and parent element (right) a) Triangular b)Quadrilateral. From [Bee08].

In a third step the coordinate r′ in Eq. 3.29 is placed on the nodes ofthe geometry. This is called collocation or nodal collocation. The HIE thenbecomes

Cp =N∑j=1

∫Sj

p∂G

∂ndS − iρ0ck

N∑j=1

∫Sj

vnGdS . (4.2)

The coordinate variables are left out for readability. Finally inserting Eq.4.1 yields

Cp =N∑j=1

n∑i=1

pijdij −N∑j=1

n∑i=1

vn,ijmij . (4.3)

where

dij =

∫Sj

∂G

∂nNi dS (4.4)

are the dipole terms and

mij = iρ0ck

∫Sj

GNi dS (4.5)

20

are the monopole terms. If we now assume constant element, i.e. N = 1,a matrix equation can be set up

Cp = Dp−Mvn , (4.6)

where C is a N × N diagonal matrix containing the solid angles, p is avector of length N containing the sound pressure at the collocation nodes, Dis the N ×N dipole matrix, M is the N ×N monopole matrix and vn is avector containing the particle normal velocities at the collocation nodes. Nstates the number of collocation points. Combining C and D into D yields

Dp = Mvn . (4.7)

If now either p or vn is known and the matrix equation is ordered in thestandard way, the matrix equation reduces to

Ax = b , (4.8)

where matrix A and vector b contains the knowns and vector x containsthe unknowns.

4.2 Numerical integration

The monopole and dipole integrals of Eqs. 4.4 4.5 have to be evaluatednumerically. This can be done by standard Gaussian quadrature. In the 2Dcase the Gaussian quadrature is∫

Sj

∂G

∂nNidS =

∫ 1

−1

∂G

∂nNiJdξ =

l∑k=1

wkf(ξk) (4.9)

where l is the number of Gaussian points on the element, f = ∂G∂nNiJ , ξk

is the k-th Gaussian point and wk is the corresponding weight. For moredetails see [Wu00, p. 36] [Wei11c].

The integrals contain a singularity at r = r0 and therefore have to betreated more carefully. Basically they can be treated by Gaussian quadraturebut the convergence rate is very slow. The Green’s function and it’s derivativeare of order O(1/r) as r approaches zero. The singularity is therefore weak.One way of removing the singularity is to introduce polar coordinates. Foran overview how to apply the numerical integration schemes see [Wu00, p.56-60]. A more detailed treatment can be found in [Bee92, Ch. 7.5]. Forthe special treatment of near-singular kernels, which is also implemented inOpenBEM, see [CH01].

21

4.3 Solving a system of equations

As mentioned in the introduction the discretization process results in a sys-tem of linear equations or matrix equation in which the system matrix isnon-symmetric, complex and fully populated. The non-symmetry is a prod-uct of the approximation of the solution of the integral equation by numericalmethods. When the discretization of the boundary becomes finer, the sym-metry increases. Further it has been noticed that linear elements yield moresymmetric matrices than quadratic ones [Bee08, Ch. 7.3].

Different geometries with a high amount of elements can lead to huge ma-trices which need much computation time and they even can exceed the mem-ory space available. Further when a solution at several frequencies should becalculated, the computation time can explode.

The most straight-forward technique and most applied for solving thematrix system is the Gauss elimination. It is a direct solver like the LUdecomposition with back-substitution. The numerical effort for direct solveris of order N3 which is why systems containing a high amount of equationscan lead to extreme long computation times [Mec08].

For bigger systems iterative solvers should be used, because the order ofeffort is approximately N2.

A detailed description of the Gauss elimination and iterative solvers canbe found in [Bee08, Ch. 7 and 8]. An overview can be found in [Mec08, Ch.O.5.3]

4.4 Post-processing

Post-processing basically deals with the presentation of the results obtainedfrom the procedure of the previous chapters. This includes the graphical dis-play of results as well as the calculation of other properties than the soundpressure and velocity (e.g. impedance, sound power, intensity). It has tobe pointed out that some literature also classifies the computation of pointsinside the domain as a step of post-processing. In [Bee08, Ch. 9] a compre-hensive chapter on post-processing is provided.

4.5 Computational issues

There are two issues determining the computational efficiency, computationtime and memory requirements. The computation time of the system ma-trices is of order M2 and the time the Gauss elimination requires for solvingthe matrix equation is of order N3, where M is the number of nodes and Nis the number of elements.

22

Mostly the memory requirements of the system matrices is the biggerproblem, because dependent on the memory available the mesh of a geometrycould be too fine and therefore cannot be saved at all. As the matrices arefully-populated and complex the memory requirement for a 64-bit computeris

B = 16M2 , (4.10)

where B is the memory space in bytes. This means that a personal computerwith 2GB memory space cannot store more than 11180 nodes, with 4GB itis 15811 nodes and with 12GB it is 27386 nodes.

The rule of thumb of six elements per wavelength for the FEM applies alsofor the BEM. It originates from Shannon’s theorem that says two points perwavelength have to be used to avoid spatial aliasing, but practice, especiallyfrom FEM applications, has shown that in between 6 and 10 should be used(for a detailed treatise on this issue see [Mar02]). This means that we canapproximately determine a frequency limit up to which the results are correct.A simple procedure is to take the maximum edge length, λmax, of all elementsto compute the frequency limit [Kat01]

fmax =c

6λmax. (4.11)

Fig. 7 shows a comparison of a sphere with radius a = 1m and one witha = 0.07m. One way to reduce computational time is to use different meshesfor different frequency ranges. At lower frequencies coarser meshes can beused and at higher frequencies finer meshes have to be used. Another possi-bility is to use symmetric properties of the geometry to reduce the numberof unknowns. Half-space formulations or axi-symmetric formulations of theHIE can be applied to reduce computation time [Cis91] [Juh93].

23

0 0.5 1 1.5 2

x 104

0

0.5

1

1.5

2

2.5

3x 10

4

Frequency, Hz

Nu

mbe

r of ele

ments

, N

Nmax,2GB

Nmax,4GB

Nmax,12GB

sphere, r=0.07m

sphere, r=1m

Figure 7: Frequency limit dependent on geometry size and number of ele-ments. Example with two sphere of different radius. The horizontal linesindicate the limit of elements due to memory for a 2GB, 4GB and 12GBmemory space.

24

5 Implementation and examples

This chapter gives an overview of the implementation of the BEM used inthis work and further discusses standard radiation examples.

5.1 OpenBEM

The software used in this work is OpenBEM which is an open source tool-box for Matlab based on the PhD theses by Peter Juhl and Vicente Cu-tanda Henriquez [Juh93] [CH01]. It is available for download at http:

//www.openbem.dk/. The version used here is 07/11. OpenBEM imple-ments the conventional BEM for 2D, 3D and axisymmetric external or in-ternal problems. Several test-cases are provided in order to get to know thecode. It is also possible to use the CHIEF method to overcome the non-uniqueness problem. In the frame of this work a few functions were added inorder to summarize OpenBEM functions or to provide further possibilities.The OpenBEM and additional functions are introduced in the following. Anoverview of the software by the authors is given in [CH10]. Examples ofcommercial software are Ansys, Sysnoise, FMBEM.

The main steps of BEM are listed here and further explained in the fol-lowing:

• Mesh generation

• Import and check mesh

• Define boundary conditions

• Solve for surface variables

• Solve for field variables

• Display and evaluate results

Mesh generation This step is one of the most crucial parts of the wholeprocess. In practice it can be one of the most time consuming parts. Nor-mally CAD programs are used to create computer models of objects whichmostly can be can be exported into a mesh format (e.g. STL). The tun-ing of parameters for the mesh creation is mostly limited in pure CADprograms which is why commercial software mostly provide an own meshgeneration tool. Open source software for the generation of meshes can befound, but each program has it’s own deficiencies. In this work Meshlab(http://meshlab.sourceforge.net/) was used to create simple objects.

25

Figure 8: Mesh model of a sphere with 160 elements.

Import and check mesh The code is so far limited to import STL formatwhich should be sufficient for most applications. The function import meshimports the mesh to Matlab, checks the geometry (meshcheck) and calculatesthe estimation of frequency limit.

Define boundary conditions In a next step the boundary conditions(Ch. 2) have to be defined for every node. Dirichlet, Neumann or impedanceboundary conditions can be defined. In case of impedance bc’s the forced orprescribed velocity and the admittance have to be defined. Also mixed bc’s,pressure and velocity, can be implemented. In this case the matrices have tobe reordered [CH10].

Solve for surface variables After defining the boundary conditions thematrix equation 4.8 can be solved. This step is implemented in the pro-cess surface values function. First the system matrices have to be computed,this is implemented in the TriQuadEquat function of OpenBEM. Memoryspace can be reduced when the boundary conditions are applied directlywith creating the monopole and dipole matrices. This means that only oneof the two has to be saved and therefore reduces memory requirements. InOpenBEM this is not the case. The surface variables are then calculated bymatrix inversion (the backslash function is used in Matlab).

The CHIEF method described in Ch. 3 can be used for exterior problemsto calculate the surface variables. Because the resulting matrix is not squareanymore the pseudo-inverse has to be applied. This is also implemented in

26

the backslash function of Matlab.

Solve for field variables After the surface variables are calculated, pointsin the domain (field points) can be calculated by applying the exterior orinterior HIE for exterior or interior problems (C = 1). This is implementedin the process field points function which is based on the points function ofOpenBEM.

Display and evaluate results The surface variables can be plotted overthe radiating objects, encoded in color by the plotresult function from OpenBEM.

5.2 Examples

In order to evaluate the code first a standard test problem, the pulsatingsphere, is observed and then a piston on a sphere with a disc in front of it issimulated. This problem states a prestep to the problem of a dummy headwith a headphone on.

5.2.1 Error measures

In order to evaluate the performance of the BEM we introduce a commonerror measure [Mar08]. An error function is defined as

e(x) = p(x)− p(x) (5.1)

where p(x) is the approximate surface pressure computed by the BEMand p(x) is the exact solution obtained by the analytical solution. In order togain one value for the performance the error norm is used. The discrete errorfunction is evaluated at all collocation points on the surface and at possiblefield points. This yields

||e||m =

(1

N

N∑i=1

||e(xi)||m) 1

m

(5.2)

where N is the number of points and m is the specific norm. Mostly theeuclidean norm is used where m = 2. Further the relative error will be used

em =||e||m||p||m

(5.3)

where ||p||m stands for the discrete norm of the analytical solution. The errornorm can be defined for any set of points. It can be the surface nodes of theradiating object or only the field points or both together.

27

N time [s] memory [MB] fmax [Hz]160 2.4 0.2 89640 25.2 3.3 193

1280 97.6 13.2 347

Table 1: Pulsating sphere. Number of elements N vs. calculation time,memory requirements (for one frequency) and frequency limit.

5.2.2 Pulsating sphere

This is a typical test problem to evaluate numerical methods in acoustics.We consider the radiation from a sphere with radius a and surface S intoinfinite space E. The surface of the sphere is vibrating equally with velocityv0. This radiation problem is posed as a boundary value problem:

∇2p+ k2p = 0 ∀x ∈ E , (5.4)

∂p

∂n= −iωρ0v0 ∀x ∈ S . (5.5)

The solution to this problem is given by:

p(r, ω) = ρ0cv0ka2 ka− i((ka)2 + 1)r

eik(r−a) , (5.6)

where r ≥ a. The solution satisfies the Sommerfeld radiation condition[Wil99, p. 213].

Figure 9 shows a comparison between the analytical solution, the numer-ical solution and the numerical solution using the CHIEF method. Figure 10compares different mesh sizes. The radius is a = 1m, the velocity amplitudev0 = 1m

sand the sound pressure and the air density correspond to a temper-

ature of 20. Table 1 shows the calculation time and memory requirement ofdifferent mesh sizes.

5.2.3 Piston on a sphere with disc

This scenario states a preliminary example to the simulation of a head witha headphone on. The head is simply modelled by a sphere with a psiton capon it and the headphone is simply a disc which should model the diaphragmof the headphone. Figure shows the mesh models.

The problem set is as follows. On the sphere with radius a is a pistonof angle α located around the positive x-axis and this piston vibrates with anormal velocity v0. In front of the piston there is a thin disc of radius b with

28

102

103

104

125

130

135

140

145

150

155

160

165

170

175

frequency (Hz)

SP

L (

dB

)

fmax

=347.2071

analytical

BEM

CHIEF

(a)

102

103

104

0

1

2

3

4

5

6

7

frequency (Hz)

rel. e

rror

(%)

fmax

=347.2071

analytical

BEM

CHIEF

(b)

Figure 9: BEM simulation of a pulsating sphere using N=1280 elements. a)SPL, b) relative error over octave bands at a point in 1m distance. Theapproximate frequency limit of the mesh is indicating by the vertical dashedline.

29

102

103

104

120

130

140

150

160

170

180

190

frequency (Hz)

SP

L (

dB

)

analytical

N=160

N=640

N=1280

Figure 10: BEM simulation of a pulsating sphere with different meshes. SPLover octave bands at a point in 1m distance.

the x-axis as it’s origin line and with a distance d to the sphere. The disccannot be arbitrarily small because it is a known problem that this leads toinstabilities in the system matrices [CH01].

In a first step everything is assumed to be rigid and in a second step thedisc is set to have a certain impedance.

Rigid sphere and disc The rigid scenario is basically a Neumann prob-lem. This means that the normal velocity of all nodes is prescribed. Basedon Eq. 4.7 the Neumann problem yields a matrix equation

p = D−1Mvn (5.7)

where D is the dipole matrix, M the monopole matrix with the expres-sion −iωρ0 included, vn is the prescribed velocity and p is the sound pressureat the collocation points we are looking for. The velocity is zero for everynode on the surface except at the nodes that are covered by the piston onthe sphere where it is v0.

In this simulation one node located at the positive x-axis is chosen tovibrate with v0 = 1m

s. The size of the meshes are N1 = 1280 for the sphere

30

102

103

104

0

500

1000

1500

frequency (Hz)

cond

N=160

N=640

N=1280

Figure 11: Condition numbers of the dipole matrix of different mesh sizes Nover frequency.

and N2 = 540 for the disc (Fig: 5.2.3). This yields frequency limits of 4960Hzand 4562Hz. The distances from the sphere to the disc are chosen to be in5cm steps and the frequency is chosen to be 3461Hz which is exactly thefrequency with half wavelength of 5cm.

Figs. 13, 14 and 15 show the resulting sound field in between the pistonand the disc from a top view. The sound pressure level (SPL) is plotted onetime for the case where no disc is prevalent and the other times where thedisc is located at different distances. One can observe the changes in thesound field due to the disc.

Rigid sphere and disc with surface impedance In this scenario wehave to apply impedance boundary conditions to the surface nodes. Theimpedance boundary condition solved for the normal velocity is

vn = −αβp+

γ

β(5.8)

In matrix notation it becomes

vn = −Ep + d (5.9)

31

−0.1−0.05

00.05

0.10.15 −0.1

−0.05

0

0.05

0.1

−0.1

−0.05

0

0.05

0.1

y

x

z

Figure 12: Mesh model of the sphere (N1 = 1280) and the disc (N2 = 540).

where d is a vector containing the known values γβ

and E is a diagonalmatrix containing the known values α

β. Substituting Eq. 5.9 into Eq. 4.7

yields

Dp = M (d−Ep) . (5.10)

If we set γβ

to be the surface admittance Y = 1Z

and αβ

to be the forced

vibration of the structure vs Eq. 5.11 becomes [Cis91]

p = (D + MY )−1(Mvs) . (5.11)

In this scenario xs is zero everywhere except at the piston nodes and Yis approximately zero everywhere except at the nodes of the disc.

The simulation is basically equivalent to before but this time the disc hasa certain surface impedance. Figure 16 shows again the sound field in SPLin between the sphere and the disc. As expected, it can be seen that the twofields are equivalent.

32

−0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

x [m]

y [

m]

−0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

x [m]

y [m

]

SP

L

85

90

95

100

105

110

115

120

125

130

135S

PL

85

90

95

100

105

110

115

120

125

130

135

Figure 13: Sound pressure field in SPL. Rigid disc at a distance d = 5cm.

33

−0.05 0 0.05 0.1 0.15

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

x [m]

y [m

]

−0.05 0 0.05 0.1 0.15

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

x [m]

y [m

]

SP

L

70

80

90

100

110

120

130

SP

L

70

80

90

100

110

120

130

Figure 14: Sound pressure field in SPL. Rigid disc at a distance d = 10cm.

34

−0.05 0 0.05 0.1 0.15 0.2

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

x [m]

y [m

]

SP

L

70

80

90

100

110

120

130

−0.05 0 0.05 0.1 0.15 0.2

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

x [m]

y [m

]

SP

L

70

80

90

100

110

120

130

Figure 15: Sound pressure field in SPL. Rigid disc at a distance d = 15cm.

35

−0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

x [m]

y [m

]

SP

L

85

90

95

100

105

110

115

120

125

130

135

−0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

x [m]

y [m

]

SP

L

85

90

95

100

105

110

115

120

125

130

135

Figure 16: Sound pressure field in SPL. Disc with a surface impedance ofZ = Z0 = ρ0c at a distance d = 5cm.

36

6 Conclusions and Outlook

This report has summarized the theory of the Boundary Element Methodand shown simple examples to which the implementation could be appliedsuccessfully. Especially the application of impedance boundary conditionswas made possible which opens up possibilities for further investigations.

One main conclusion was found that after understanding the theory andthe implementations the main challenge in the daily practice of using theBEM is the creation of suitable mesh models. In the frame of this work it wasnot possible to find an environment where mesh models can easily be createdand edited and all that for free. One software found which at least allowedto create some models is Meshlab (http://meshlab.sourceforge.net/).

A second point which could not be addressed is the issue of CHIEF points.How many and where is dependent on the geometry. For a simple sphere thecharacteristic frequency is know to be kr = π. But for more complex shapesit is not trivial to find out the characteristic frequencies. Especially at higherfrequencies more CHIEF points have to be used but also the density of thenodal surfaces increases.

Further, if precise computations should be done up to high frequencies, agood estimate of the time needed should be found. The general complexityis found with O(f 6) but the real time needed is dependent on the CPU.

Finally, it should be said that the engagement with the BEM was hardwork worthwhile because a lot about acoustics in general had to be consideredand therefore the knowledge and experience with acoustics could be extended.

37

A Divergence theorem and Green’s identities

The divergence theorem (also called Gauss theorem or Gauss-Ostrogradskytheorem) states that the volume integral of the divergence of a vector field isequal to the surface integral of the outward normal component of the vectorfield. This relation is expressed by∫∫∫

V

∇·A dV =

∫∫S

A·n dS. (A.1)

where A is any continuously differentiable vector field, n is the unit outwardnormal to the surface and dV and dS are a differential volume element anda differential surface element, respectively. The physical interpretation is,the flow of energy (e.g. of heat) through the surface is equal to the creationor destruction of energy inside the volume. If no source or sink of energyis prevalent inside the volume the overall flow through the surface is zero[Wei11a]. The divergence of A and the scalar product of A with the normalvector in cartesian coordinates are defined by

∇·A =∂Ax∂x

+∂Ay∂y

+∂Az∂z

(A.2)

A·n = ∂Axnx + ∂Ayny + ∂Aznz (A.3)

Green’s first identity The vector field A can now be replaced by theproduct of two scalar fields φ∇ψ where ∇ is the gradient given by

∇ =∂

∂x+

∂

∂y+

∂

∂z(A.4)

Eq. A.1 then becomes with Eqs. A.2 and A.3

∫∫∫V

[∂

∂x(φ∂ψ

∂x) +

∂

∂y(φ∂ψ

∂y) +

∂

∂z(φ∂ψ

∂z)

]dV =

=

∫∫S

[φ∂ψ

∂xnx + φ

∂ψ

∂xny + φ

∂ψ

∂xnz

]dS. (A.5)

Applying the product rule and rearranging yields

∫∫∫V

φ

[∂2ψ

∂x2+∂2ψ

∂y2+∂ψ2

∂z2

]+

[∂φ

∂x

∂ψ

∂x+∂φ

∂y

∂ψ

∂y+∂φ

∂z

∂ψ

∂z

]dV =

=

∫∫S

[φ∂ψ

∂xnx + φ

∂ψ

∂xny + φ

∂ψ

∂xnz

]dS. (A.6)

38

With∂ψ

∂xnx +

∂ψ

∂yny +

∂ψ

∂znz =

∂ψ

∂n(A.7)

the derivative with respect to the outward normal and using vector no-tation again Eq. A.6 becomes∫∫∫

V

(φ∇2ψ +∇φ· ∇ψ) dV =

∫∫S

φ∂ψ

∂ndS (A.8)

This is generally known as Green’s first identity.

Green’s second theorem The term ∇φ· ∇ψ in Eq. A.8 is undesirableand the operator ∇2 alone under the integral could make the volume integralvanish. Hence, we can try to get rid of the second term on the left-hand side.Eq. A.8 rewritten with the functions φ and ψ interchanged∫∫∫

V

(ψ∇2φ+∇ψ· ∇φ) dV =

∫∫S

ψ∂φ

∂ndS (A.9)

and then subtracted from each other yields

∫∫∫V

(φ∇2ψ − ψ∇2φ

)dV =

∫∫S

(φ∂ψ

∂n− ψ ∂φ

∂n

)dS , (A.10)

Green’s second identity or Green’s symmetric identity. Here φ and ψ haveto be twice continuously differentiable [Bee92]. From Eq. A.10 the HIE canbe derived.

39

References

[Bee92] Beer, G.; Watson, J.O. (1992): Introduction to finite and boundaryelement methods for engineers. Wiley.

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