The Boundary Element Method in Acoustics - An internship report

Transcription

1 The Boundary Element Method in Acoustics - An internship report Fabio Kaiser tudiengang: Elektrotechnik-Toningenieur der Universität für Musik und darstellende Kunst und der Technischen Universität, Graz, Österreich upervision: M. A. Frank chultz, TU Berlin Technische Universität Berlin, Fakultät I Institut für prache und Kommunikation, Fachgebiet Audiokommunikation October 7, 2011

2 Abstract This report documents the work done during a 3-month internship at the Audio communication group of the TU Berlin. Basically the theory of the boundary element method, applied to acoustics is introduced and simple test simulations 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

3 Contents 1 Introduction 3 2 Basic theory of acoustics Wave equation Green s function Boundary conditions The interior and exterior problem The boundary element method. Part I. Helmholtz integral equation Direct Method HIE for interior problems HIE for exterior problems Indirect Method Existence and Uniquness CHIEF Burton-Miller formulation The boundary element method. Part II. Numerical implementation Discretization and collocation Numerical integration olving a system of equations Post-processing Computational issues Implementation and examples OpenBEM Examples Error measures Pulsating sphere Piston on a sphere with disc Conclusions and Outlook 37 A Divergence theorem and Green s identities 38 2

4 1 Introduction In acoustics, the linearized wave equation describes the propagation of waves in fluids. The Helmholtz equation is the time-harmonic equivalent, also called reduced wave equation. Both are partial differential equations (PDE), the wave equation is a so called hyperbolic PDE and the Helmholtz equation is an elliptical PDE. By solving these PDE s the propagation of sound can be predicted. Analytical methods to solve these problems are available and have been studied for long time. till, these methods suffer from the fact that solutions can be found for a limited class of problems only. Therefore and with the rise of computers in the 1970 s, numerical methods for solving PDE s have been developed. The most popular methods are the finite element method (FEM), the finite difference method (FDM) and the boundary element method (BEM). These techniques allow to calculate the sound field of any acoustic scenario and the inaccuracy of the result is only determined by restrictions of computational resources. All are based on the discretization of space into little pieces like in a puzzle. The FEM and FDM are so called domain based methods because the whole space of interest needs to be discretized. The BEM on the other hand discretizes only the boundaries of the domain and therefore reduces the dimensionality of the computations by one. This is also why the FEM is usually used in the structural domain and the BEM is used for the treatment of infinite domains. Outline of the BEM The basic idea of the BEM is the reformulation of the Helmholtz equation into a (boundary-) integral equation that is mathematically equivalent. Two integrals arise, one which is defined on the boundary of the domain and one which relates the boundary solution to points in the field. Basically the reformulation represents the acoustic field as a superposition of the fields of elementary sources (monopoles and dipoles) located at the boundary of the domain. The reformulation can only be applied to classes of PDE s where a fundamental solution can be found. It is therefore not applicable to every physical problem compared to the FEM and FDM, which are nearly universally applicable. However, if a fundamental solution can be found the BEM can be applied and it states an easier to use and computationally more efficient method. The integral equation then is discretized into small elements over the surface 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 3

5 infinite extend. In FEM this infinite space has to be discretized. In BEM only the surface of the objects have to be discretized in order to compute the sound field on every point in the domain. The ommerfeld radiation condition is inherently fulfilled. However, this reduction of dimensionality of the approach also results in difficulties of non-uniqueness which do not exist in the original problem. The discretization process results in a linear system of equations or a matrix equation which has to be solved for. The resulting matrices are fully occupied, complex and unsymmetrical [Mec08]. This means that the storage necessary 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 today when the derivation and analysis of the method is addressed instead of discussing the implementation and/or application. Applications The BEM can generally be applied to every kind of radiation and scattering problems and it can also be applied to interior eigenvalue problems. The main applications are found in structural analysis and noise radiation but also in aero-acoustics, bioacoustics, the automotive industry, enviromental acoustics and architectural acoustics [Cis91]. The BEM has also been applied to the simulation of head-related transfer functions [Kat01] [Kre09]. It is nowadays an essential part of product design where the BEM is used to simulate the acoustical behaviour of a product and based on the results 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 boundary integral equations. Additionally the problems arising with the formulations are shortly addressed and the existing solutions presented. Chapter 4 introduces then the discretization by elements process and the formulation of the matrix equations. After these theoretical chapters in chapter 5 the implementation 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 outlook about what is still open. The main references for this work are the two books 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 theoretical issues are [Bee92] [Che92] and [au10]. A book treating implementation issues more directly is [Bee08]. 4

6 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 of sound. This is the starting point of all further derivations in the context of this work. 2.1 Wave equation The homogeneous wave equation is a linear partial differential equation of second order 1 2 p c 2 t 2 2 p = 0 (2.1) where c is the speed of sound, p is the sound pressure and 2 is the Laplace operator, in cartesian coordinates defined by 2 2 x y z. (2.2) 2 This equation is linear because the result is independent of where and when the wave equation is excited. It is partial because there are more then one variables (dimensions)and it is of second order because the second derivation is applied. It describes a real or complex function p(t, x 1,..., x n ) in space and time and relates the second derivate of space with the second derivate in time. The acceleration is proportional the curvature. The solutions are waves which can interfere and which are independent of each other, i.e. they do not influence one and another. If some excitation is considered the inhomogeneous wave equation can be written as 1 2 p c 2 t 2 2 p = f(t, x 1,..., x n ) (2.3) where f is considered as the excitation. Helmholtz equation ometimes it is easier to consider only time-harmonic processes, i.e., p(t, x 1,..., x n ) = Re{p(x, t)e iωt }. This can be injected into Eq. 2.1 which yields the so called Helmholtz equation 2 p + k 2 p = 0, (2.4) where k = ω = 2π is the wavenumber. The Helmholtz equation is also c λ called reduced wave equation because only time-harmonic processes are considered. 5

7 (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 analytically. If the geometry becomes more complex it can get very difficult or impossible to find analytical solutions. Numerical approaches have been developed in order to solve for this problem. 2.2 Green s function The Green s function is the particular solution for the Delta-distribution δ as an excitation ( 2 + k 2 )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 be seen as a spatio-temporal transfer function (or impulse response) equivalent to a impulse response in the time-domain. This is why a solution to any inhomogenity can be found through the convolution with the Green s function [Wik11a]. Therefore, it is called the fundamental solution of the Helmholtz equation. In Fig. 1 the real and imaginary part of Eq. 2.6 are plotted. We will see that in order to apply the BEM formulations a fundamental solution of the governing equation is necessary. 6

8 2.3 Boundary conditions As it is the case with every kind of PDE, also the wave equation can only be solved subject to certain boundary conditions. These can either be active or passive. Active means that there is an object that vibrates and therefor is the source of sound energy. In acoustics there are two possibilities, either the sound pressure is prescribed at some point or surface in space or the particle normal velocity. If the sound pressure is prescribed it is called a Dirichlet boundary condition p = p (2.7) where p indicates known values, e.g. p = 0 which corresponds to a pressure release case. If the velocity is prescribed it is called a Neumann boundary condition v = v, (2.8) and if v = 0 it corresponds to a rigid body. It has to be pointed out that in the active case the normal velocity of the vibrating structure is equal to the normal velocity of the fluid in contact with the structure [Wu00, p. 3]. Passive boundary conditions apply when an object reflects from a passive surface (e.g. an absorbing surface). In general form it can be written α p + β v n = γ (2.9) where α, β, γ are arbitrary complex functions. They can be called general boundary conditions because the Dirichlet and Neumann boundary conditions can be generated as a special case by setting e.g. α = 0, β = 1 and γ = v n in Neumann case. Other names are impedance boundary conditions or Robin boundary conditions [Wik11c]. If we reorder for the velocity Eq. 2.9 is written as v n = α β p + γ β (2.10) where the left term on the right side α is an acoustic admittance Y and β the right term is a forced or prescribed velocity v s v n = Y p + v s. (2.11) In this case v s can be seen as the velocity of the vibration of a structure and Y p = v a as the velocity of the fluid [Kol11] [Mar99]. One can imagine a vibrating structure with absorbing material stitched to it. 7

9 Figure 2: Exterior, interior and a combined problem. ommerfeld radiation condition For exterior problems, where the domain extends to infinity, the ommerfeld radiation condition has to be fulfilled in addition to the boundary conditions [ lim R p R R ikp ] = 0 (2.12) where R is the radius of a big sphere in spherical coordinates which includes the radiating or scattering object. It basically means that any radiated or scattered acoustic wave has to converge towards zero when the radius extends to infinity. 2.4 The interior and exterior problem There are two basic classes of problems in acoustics,the interior and the exterior problem. For the exterior problem the sound sources are located inside a region and the region of interest is defined to be outside the surface including the sound sources. The exterior problem includes infinity. Radiation and scattering problems fall into this category. The interior problem is defined exactly the other way around. Fig. 2 plots the defined regions. In the scope of the derivation of the HIE we have to distinguishe the problems. 8

10 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 a numerical method for the solution of this problem. The method is not directly applied on the Helmholtz equation but on a reformulation as a boundary integral equation [Wei11b]. This is the first step in the derivation of the BEM equations. In this chapter it is shown how we can get from a PDE to an integral equation. Two possibilites have devoleped in BEM s, the direct and the indirect method. 3.1 Direct Method The direct method deals with functions in the integral equation that are physically meaningful (sound pressure and velocity) instead of fictitious density functions. The solutions directly yields the unknown values on the boundary, therefor the name. The integral equation is derived from the divergence theorem in which certain substitutions lead to Green s identities (see Appendix A). If one of the functions in Green s second identity is taken to be the unknown and the other the fundamental solution of the Helmholtz equation, the boundary integral 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 φ and ψ (φ 2 ψ ψ 2 φ) dv = (φ ψ ψ φ ) d. (3.1) Ω Now, if it is assumed that the functions φ and ψ are solutions to the Helmholtz equation, i.e. they have to satisfy the homogeneous Helmholtz equation on the surface and in the volume 2 φ + k 2 φ = 0 the left hand side of Eq. 3.1 becomes 2 ψ + k 2 ψ = 0 (3.2) 9

11 Figure 3: The region V with its surface and the outward normal n. The only restriction on the shape is that the derivate has to be continuous. φ 2 ψ ψ 2 φ = φ( k 2 φ) ψ( k 2 ψ) = 0. (3.3) and the volume integral vanishes: φ ψ ψ φ d = 0. (3.4) Eq. 3.4 is satisfied if φ and ψ have no singularities on the surface. The HIE s for interior and exterior problems can be derived from here HIE for interior problems The basic definitions of the geometry for derivation of the interior problem are shown in Fig. 3. Let s consider the case where ψ has a singularity somewhere in the domain at r = r. 2 ψ + k 2 ψ = δ(r r ) (3.5) The function ψ(r) is a solution of the inhomogenuous Helmholtz equation and satisfies the ommerfeld radiation condition. The field point r covers all space. olving Eq. 3.5 yields the free-space Green s function, also called the fundamental solution : G(r r ) = e ik r r 4π r r = ik 4π h 0(k r r ), (3.6) 10

12 Figure 4: The region V with its surfaces o and i and the outward normal n. The evaluation point r = r inside the volume is excluded. From [Wil99]. where k the wavenumber and h 0 the spherical Hankel function of first kind and zeroth-order. Now, three cases can be distinguished, depending on where the evaluation point r is exactly located. Case 1: r inside As the point r lies inside the volume V (Fig. 4), the Green s identity can no longer be applied because it has a singularity at r = r. Therefore the geometry has to be modified by excluding the singularity. Green s second identity is therefore redefined for the volume V with a boundary = o + i where i defines a small sphere inside V and o is the outer surface as before. The outward normal of i points toward r. Now we let the radius ɛ of the small sphere i go to zero. With ψ = G this yields ( φ(r) G(r r ) G(r r ) φ(r) ) do + lim ɛ 0 ( φ(r) G(r r ) G(r r ) φ(r) ) di = 0. (3.7) We now have to evaluate the limes of the two integrals [Wil99] [CW07]. The Green s function on i becomes and the normal derivative of G is given by G(r r ) = e ikr 4πR = e ikɛ 4πɛ, (3.8) G(r r ) = 1 e ikɛ /ɛ 4π ɛ = 1 e ikɛ (ikɛ 1). (3.9) 4π ɛ 2 11

13 The first integral to evaluate in Eq. 3.7 can than be written as lim ɛ 0 ( G(r r )) φ(r) di = lim( 1 e ikɛ (ikɛ 1) ) φ(r)d ɛ 0 4π ɛ 2 i. (3.10) Noting that the function φ(r) is continuous within the sphere and the surface of the sphere is 4πɛ 2 it can further be written lim( 1 ɛ 0 4π e ikɛ (ikɛ 1) ɛ 2 )4πɛ 2 φ(r) = φ(r) r=r = φ(r ). (3.11) The second integral to evaluate in Eq. 3.7 becomes lim ɛ 0 ( G(r r ) φ(r) ) di = lim( eikɛ ɛ 0 4πɛ ) φ(r) d i. (3.12) As φ is continuous about r = r it can be taken out of the integral. As before this yields lim( e ikɛ φ(r) )4πɛ2 = 0. (3.13) ɛ 0 4πɛ The results of Eqs and 3.13 inserted in Eq. 3.7 yield the interior Helmholtz equation for interior problems [Mec08, Ch. O.5] φ(r ) = G(r r ) φ(r) ) φ(r) G(r r d o (3.14) Case 2: r on If the evaluation point r is located at the surface of 0 the surface i doesn t span a whole sphere anymore but spans that part of the sphere with a radius of ɛ centered at r that lies inside the volume V (Fig. 5). This means that with d i Ω(r )ɛ 2 (3.15) where Ω(r ) is the solid angle subtended at r by the volume V [CW07] [Wik11b], Eq becomes where lim( 1 ɛ 0 4π e ikɛ (ikɛ 1) )φ(r) ɛ 2 d i = Ω(r ) 4π φ(r ). (3.16) 12

14 Figure 5: The region V with its surfaces o and i and the outward normal n. The evaluation point r = r on the surface is excluded. From [Wil99]. and hence Ω(r ) 4π φ(r ) = Ω(r ) = 1 ɛ 2 G(r r ) φ(r) ) φ(r) G(r r d i (3.17) d o (3.18) is the surface Helmholtz integral equation for interior problems. If the surface r is located on is smooth then Ω(r ) 4π = 1 2. Case 3: r outside The evaluation point lies anywhere outside the volume and hence no singularity exists within or on the surface. This means Eq. 3.4 applies, which yields 0 = G(r r ) φ(r) ) φ(r) G(r r d o (3.19) the exterior Helmholtz intergral equation for interior problems. Combining Eqs. 3.14, 3.18 and 3.19 the full expression of the Helmholtz integral equation for interior problems is written as ( G(r r ) φ(r) ) ) φ(r) G(r r d = Ω(r ) 4π φ(r ) (3.20) 13

15 with Ω(r ) 4π = 0, r V e 1 2, r. 1, r V i (3.21) for smooth surfaces. In acoustics we deal with the sound pressure and velocity. Therefor we substitute φ = p and use Euler s equation p(r) = iρ 0ckv n (r) (3.22) where ρ 0, c, k, v n (r) are the air density, speed of sound, wavenumber k = ω c and the particle velocity in the outward normal direction, respectively. Eq then becomes ( iρ o ckv n (r)g(r r ) p(r) G(r r ) ) d = Ω(r ) 4π φ(r ) (3.23) It has to be noted that the integration is with respect to the field point r and the variable r is the point where the sound pressure is evaluated. The HIE states that the pressure at any point inside the volume can be computed from the known pressure and normal velocity at it s surface [Wil99, p. 257] HIE for exterior problems Basically the approach to obtain the formulation of the HIE for exterior problems is equivalent to the interior case. We just have to redefine the regions where Green s second identity should be applied. The surface is again an arbitrarily shaped body with the normal pointing outwards the surface (Fig. 3). As before φ and ψ both satisfy the homogenous Helmholtz equation in the region V, which is defined to be outside the surface extending to infinity. At infinity the ommerfeld radiation condition has to be satisfied by φ and ψ. This means that Eq. 3.4 also applies, but with sign changed as the normal is defined in the same way as for the interior problem. ψ φ φ ψ d = 0. (3.24) imilar to the interior case we define a small sphere with surface i around the point r with radius ɛ either outside, on or inside the surface o so that the total surface becomes = o + i. (3.25) 14

16 The function φ is again chosen to be the free space Green s function so that Eq becomes equivalently to Eq. 3.7 ( G(r r ) φ(r) )) ( φ(r) G(r r do + lim G(r r ɛ 0 ) φ(r) )) φ(r) G(r r di = 0. (3.26) The result of the limes is the same as before: ( lim G(r r ɛ 0 ) φ(r) ) ) φ(r) G(r r d o = Ω(r ) 4π φ(r ) (3.27) where Ω(r ) 4π = 0, r V e 1 2 φ(r ), r φ(r ), r V i (3.28) with Ω(r) the solid angles as defined in Eq The Helmholtz integral equation for exterior problems is written as ( φ(r) G(r r ) G(r r ) φ(r) ) d = Ω(r ) 4π φ(r ). (3.29) This is identical to the interior problem except for the sign changed because the surface normal is also pointed outwards. We can write Eq using sound pressure and velocity as ( p(r) G(r r ) ) iρ o ckv n (r)g(r r )) d = Ω(r ) 4π p(r ). (3.30) Using the HIE for exterior problems it is possible to calculate the sound pressure radiated by sources inside, everywhere outside the surface by knowing the pressure and the velocity on the surface. If the evaluation point is located on the same side of the surface as the sources do, then the HIE yields a null field. This is valid for the interior and the exterior problem. To determine the sound pressure anywhere on the side of the sources would give an inverse problem. The HIE does not solve the inverse problem [Wil99, Ch. 8]. 15

17 3.2 Indirect Method ingle and double layer potential see [CW07] [Mec08, Ch. O.5.1] or simple source formulation see [Wil99, Ch. 8.7] An alternative formulation of a boundary integral equation is found by the so called indirect method. This method deals rather with density distributions than with the functions of interest directly, therefore the name. It is also called the potential-layer approach [Mec08] because the sound pressure can be represented as a single-layer potential p(r) = σ(r )G(r, r ) (3.31) or a double layer potential p(r) = ψ(r ) G(r, r ) ) (3.32) where σ and ψ are density distributions. It can be thought as a layer of monopoles or dipoles in contrast to the HIE which contains both monopoles and dipoles. In the case of single layer potential this representation leads to a boundary integral equation: σ(r) 2 σ(r ) G(r, r ) d r = jωρ 0 v n (3.33) where r. This is the density if the exterior Neumann problem is considered. For the formulations of other problems and the double-layer approach see [Mec08, p. 1061] and [CW07, Ch. 2.4]. The single and double-layer approaches yield boundary integral equations which are ill-posed and therefore need special treatment in order to obtain a solution. 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 mathematical properties of the derivation. It can be shown that the HIE for exterior problems doesn t have a unique solution at certain frequencies. These characteristic eigenfrequencies can be associated with the corresponding interior Dirichlet problem. If we consider an exterior problem with Dirichlet boundary condition the issue can be explained. The surface HIE becomes iρωv n d = 1 2 p p G d (3.34) 16

18 with the surface normal pointed outwards. Further we have a look at the interior Dirichlet problem. With the surface normal unchanged the interior HIE becomes iρωv n d = 1 2 p p G d. (3.35) It can be seen that both equations share the same left-hand side. The latter equation is for an interior problem and therefore has eigenfrequencies. The exterior problem does not have any eigenfrequencies but it share the same left-hand side which determines the one of the coefficient or system matrices. If the interior problem is evaluated at resonance frequencies the coefficient matrix will become singular so will the one for the exterior problem. It should be pointed out that this issue for the exterior problem arises purely from the mathematical approach and doesn t have any physical meaning. Basically two methods were suggested to overcome this problem [Wu02] CHIEF One simple method to overcome the non-uniqueness problem was suggested by chenk in 1968 [ch68] and he called that method the combined Helmholtz integral equation formulation (CHIEF). The idea is to create an overdetermined system of equations by which the interior problem can be distinguished from the exterior problem. This is done by adding or combining the HIE for several interior points with the set of surface HIE s. The additional or constraint equations can be referred to as CHIEF equations and CHIEF point. The interior HIE for exterior problems is ( p(r) G ) iρ ockv n (r)g d = 0. (3.36) This equation enforces the zero pressure condition inside the surface and there can be seen as a constraint to the surface HIE. This is usually enough for the exterior problem to have a unique solution. A problem arises when the 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 interior problem, as well. Further with increasing frequency the nodal surfaces become more densely and so the location of CHIEF points becomes hard to choose [Wu00, p. 27]. 17

19 3.3.2 Burton-Miller formulation Another popular technique to overcome the non-uniqueness problem is the so called Burton-Miller formulation [Bur71]. It has been shown that a linear combination of the HIE and its normal derivative yield a unique solution over the whole frequency range. CBIE + β HBIE = 0 (3.37) where CBIE is the conventional boundary integral and HBIE its derivative. i The constant β has to be complex, e.g. [Li11, p. 154]. The drawback is k that a the normal derivative of the HIE is a hyper singular integral. Regularizations have to be used [Cis91]. 18

20 4 The boundary element method. Part II. Numerical implementation In chapter 3 the Helmholtz integral equation has been derived. In order to solve this integral the idea is to discretize the surface into small elements, represent the geometry and the variables by shape function, numerically integrate over the elements and add together the results. This chapter describes how 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 object in consideration has to be discretized. And second the boundary variables have to be discretized (in acoustics the sound pressure p and the particle normal velocity v n ). In principal this can be done independently but in practice the geometry and the variables are mostly discretized in the same way which yields so called isoparametric elements. The variables can then be represented as φ = n φ i N i (ξ 1, ξ 2 ), (4.1) i=1 where φ can either be the geometric variables (x, y, z-coordinates), the sound pressure p or the particle normal velocity v n. The index i indicates the nodal points, n is the number of nodes in that element and N i are the shape functions defined on a master element with local coordinates ξ 1, ξ 2 [Wu00, p. 55]. The shape of the elements in three-dimensions can either be of triangular or quadrilateral shape (Fig. 6 ). The geometry is represented by the nodes and the shape functions which interpolate between the nodes. The number of nodes on an element determines the order of the shape functions. A constant element places one node at the centroid of the element. A linear element places nodes at the corners of the element. A quadratic element places nodes at the corners and in the middle of the edges. As each element has a similar shape the integration is generalized and made on a parent or master element. This means that every real element is transformed into local coordinates of the master element for integration. It has to be pointed out that part of the numerical error in the result is due to the approximation of the boundary. Therefore higher-order elements might be preferable. For a comprehensive treatment of elements and shape functions the reader is referred to standard BEM textbooks, e.g., [Bee92, Ch. 2] or the standard FEM book [Zie89]. 19

21 (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 is placed on the nodes of the geometry. This is called collocation or nodal collocation. The HIE then becomes N Cp = p G N j d iρ 0ck v n Gd. (4.2) j j=1 The coordinate variables are left out for readability. Finally inserting Eq. 4.1 yields where Cp = N j=1 n p ij d ij i=1 d ij = j N j=1 j=1 n v n,ij m ij. (4.3) i=1 G N i d (4.4) are the dipole terms and m ij = iρ 0 ck GN i d j (4.5) 20

22 are the monopole terms. If we now assume constant element, i.e. N = 1, a matrix equation can be set up Cp = Dp Mv n, (4.6) where C is a N N diagonal matrix containing the solid angles, p is a vector of length N containing the sound pressure at the collocation nodes, D is the N N dipole matrix, M is the N N monopole matrix and v n is a vector containing the particle normal velocities at the collocation nodes. N states the number of collocation points. Combining C and D into D yields Dp = Mv n. (4.7) If now either p or v n is known and the matrix equation is ordered in the standard way, the matrix equation reduces to Ax = b, (4.8) where matrix A and vector b contains the knowns and vector x contains the unknowns. 4.2 Numerical integration The monopole and dipole integrals of Eqs have to be evaluated numerically. This can be done by standard Gaussian quadrature. In the 2D case the Gaussian quadrature is j G 1 N G id = 1 N ijdξ = l w k f(ξ k ) (4.9) where l is the number of Gaussian points on the element, f = G N ij, ξ k is the k-th Gaussian point and w k is the corresponding weight. For more details see [Wu00, p. 36] [Wei11c]. The integrals contain a singularity at r = r 0 and therefore have to be treated more carefully. Basically they can be treated by Gaussian quadrature but the convergence rate is very slow. The Green s function and it s derivative are 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. For an overview how to apply the numerical integration schemes see [Wu00, p ]. A more detailed treatment can be found in [Bee92, Ch. 7.5]. For the special treatment of near-singular kernels, which is also implemented in OpenBEM, see [CH01]. k=1 21

23 4.3 olving a system of equations As mentioned in the introduction the discretization process results in a system of linear equations or matrix equation in which the system matrix is non-symmetric, complex and fully populated. The non-symmetry is a product of the approximation of the solution of the integral equation by numerical methods. When the discretization of the boundary becomes finer, the symmetry increases. Further it has been noticed that linear elements yield more symmetric matrices than quadratic ones [Bee08, Ch. 7.3]. Different geometries with a high amount of elements can lead to huge matrices which need much computation time and they even can exceed the memory space available. Further when a solution at several frequencies should be calculated, the computation time can explode. The most straight-forward technique and most applied for solving the matrix system is the Gauss elimination. It is a direct solver like the LU decomposition with back-substitution. The numerical effort for direct solver is of order N 3 which is why systems containing a high amount of equations can lead to extreme long computation times [Mec08]. For bigger systems iterative solvers should be used, because the order of effort is approximately N 2. A detailed description of the Gauss elimination and iterative solvers can be 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 obtained from the procedure of the previous chapters. This includes the graphical display of results as well as the calculation of other properties than the sound pressure and velocity (e.g. impedance, sound power, intensity). It has to be pointed out that some literature also classifies the computation of points inside the domain as a step of post-processing. In [Bee08, Ch. 9] a comprehensive chapter on post-processing is provided. 4.5 Computational issues There are two issues determining the computational efficiency, computation time and memory requirements. The computation time of the system matrices is of order M 2 and the time the Gauss elimination requires for solving the matrix equation is of order N 3, where M is the number of nodes and N is the number of elements. 22

24 Mostly the memory requirements of the system matrices is the bigger problem, because dependent on the memory available the mesh of a geometry could be too fine and therefore cannot be saved at all. As the matrices are fully-populated and complex the memory requirement for a 64-bit computer is B = 16M 2, (4.10) where B is the memory space in bytes. This means that a personal computer with 2GB memory space cannot store more than nodes, with 4GB it is nodes and with 12GB it is nodes. The rule of thumb of six elements per wavelength for the FEM applies also for the BEM. It originates from hannon s theorem that says two points per wavelength have to be used to avoid spatial aliasing, but practice, especially from 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 can approximately determine a frequency limit up to which the results are correct. A simple procedure is to take the maximum edge length, λ max, of all elements to compute the frequency limit [Kat01] f max = c 6λ max. (4.11) Fig. 7 shows a comparison of a sphere with radius a = 1m and one with a = 0.07m. One way to reduce computational time is to use different meshes for different frequency ranges. At lower frequencies coarser meshes can be used and at higher frequencies finer meshes have to be used. Another possibility is to use symmetric properties of the geometry to reduce the number of unknowns. Half-space formulations or axi-symmetric formulations of the HIE can be applied to reduce computation time [Cis91] [Juh93]. 23

25 2.5 3 x 104 N max,12gb sphere, r=0.07m sphere, r=1m Number of elements, N N max,4gb N max,2gb Frequency, Hz x 10 4 Figure 7: Frequency limit dependent on geometry size and number of elements. Example with two sphere of different radius. The horizontal lines indicate the limit of elements due to memory for a 2GB, 4GB and 12GB memory space. 24

26 5 Implementation and examples This chapter gives an overview of the implementation of the BEM used in this work and further discusses standard radiation examples. 5.1 OpenBEM The software used in this work is OpenBEM which is an open source toolbox for Matlab based on the PhD theses by Peter Juhl and Vicente Cutanda Henriquez [Juh93] [CH01]. It is available for download at http: // The version used here is 07/11. OpenBEM implements the conventional BEM for 2D, 3D and axisymmetric external or internal problems. everal test-cases are provided in order to get to know the code. It is also possible to use the CHIEF method to overcome the nonuniqueness problem. In the frame of this work a few functions were added in order to summarize OpenBEM functions or to provide further possibilities. The OpenBEM and additional functions are introduced in the following. An overview of the software by the authors is given in [CH10]. Examples of commercial software are Ansys, ysnoise, FMBEM. The main steps of BEM are listed here and further explained in the following: Mesh generation Import and check mesh Define boundary conditions olve for surface variables olve for field variables Display and evaluate results Mesh generation This step is one of the most crucial parts of the whole process. In practice it can be one of the most time consuming parts. Normally CAD programs are used to create computer models of objects which mostly can be can be exported into a mesh format (e.g. TL). The tuning of parameters for the mesh creation is mostly limited in pure CAD programs which is why commercial software mostly provide an own mesh generation tool. Open source software for the generation of meshes can be found, but each program has it s own deficiencies. In this work Meshlab ( was used to create simple objects. 25

27 Figure 8: Mesh model of a sphere with 160 elements. Import and check mesh The code is so far limited to import TL format which should be sufficient for most applications. The function import mesh imports the mesh to Matlab, checks the geometry (meshcheck) and calculates the 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 impedance boundary conditions can be defined. In case of impedance bc s the forced or prescribed velocity and the admittance have to be defined. Also mixed bc s, pressure and velocity, can be implemented. In this case the matrices have to be reordered [CH10]. olve for surface variables After defining the boundary conditions the matrix equation 4.8 can be solved. This step is implemented in the process surface values function. First the system matrices have to be computed, this is implemented in the TriQuadEquat function of OpenBEM. Memory space can be reduced when the boundary conditions are applied directly with creating the monopole and dipole matrices. This means that only one of the two has to be saved and therefore reduces memory requirements. In OpenBEM this is not the case. The surface variables are then calculated by matrix inversion (the backslash function is used in Matlab). The CHIEF method described in Ch. 3 can be used for exterior problems to calculate the surface variables. Because the resulting matrix is not square anymore the pseudo-inverse has to be applied. This is also implemented in 26

28 the backslash function of Matlab. olve for field variables After the surface variables are calculated, points in the domain (field points) can be calculated by applying the exterior or interior HIE for exterior or interior problems (C = 1). This is implemented in the process field points function which is based on the points function of OpenBEM. Display and evaluate results The surface variables can be plotted over the 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 pulsating sphere, is observed and then a piston on a sphere with a disc in front of it is simulated. This problem states a prestep to the problem of a dummy head with a headphone on Error measures In order to evaluate the performance of the BEM we introduce a common error 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 BEM and p(x) is the exact solution obtained by the analytical solution. In order to gain one value for the performance the error norm is used. The discrete error function is evaluated at all collocation points on the surface and at possible field points. This yields e m = ( 1 N N i=1 e(x i ) m ) 1 m (5.2) where N is the number of points and m is the specific norm. Mostly the euclidean norm is used where m = 2. Further the relative error will be used e m = e m p m (5.3) where p m stands for the discrete norm of the analytical solution. The error norm can be defined for any set of points. It can be the surface nodes of the radiating object or only the field points or both together. 27

29 N time [s] memory [MB] f max [Hz] Table 1: Pulsating sphere. Number of elements N vs. calculation time, memory requirements (for one frequency) and frequency limit 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 into infinite space E. The surface of the sphere is vibrating equally with velocity v 0. This radiation problem is posed as a boundary value problem: 2 p + k 2 p = 0 x E, (5.4) p = iωρ 0v 0 x. (5.5) The solution to this problem is given by: p(r, ω) = ρ 0 cv 0 ka 2 ka i ((ka) 2 + 1)r eik(r a), (5.6) where r a. The solution satisfies the ommerfeld radiation condition [Wil99, p. 213]. Figure 9 shows a comparison between the analytical solution, the numerical solution and the numerical solution using the CHIEF method. Figure 10 compares different mesh sizes. The radius is a = 1m, the velocity amplitude v 0 = 1 m and the sound pressure and the air density correspond to a temperature of 20. Table 1 shows the calculation time and memory requirement of s different mesh sizes Piston on a sphere with disc This scenario states a preliminary example to the simulation of a head with a headphone on. The head is simply modelled by a sphere with a psiton cap on it and the headphone is simply a disc which should model the diaphragm of the headphone. Figure shows the mesh models. The problem set is as follows. On the sphere with radius a is a piston of angle α located around the positive x-axis and this piston vibrates with a normal velocity v 0. In front of the piston there is a thin disc of radius b with 28

30 analytical BEM CHIEF PL (db) f max = frequency (Hz) (a) 7 6 analytical BEM CHIEF 5 rel. error (%) f max = frequency (Hz) (b) Figure 9: BEM simulation of a pulsating sphere using N=1280 elements. a) PL, b) relative error over octave bands at a point in 1m distance. The approximate frequency limit of the mesh is indicating by the vertical dashed line. 29

31 analytical N=160 N=640 N= PL (db) frequency (Hz) Figure 10: BEM simulation of a pulsating sphere with different meshes. PL over 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 disc cannot be arbitrarily small because it is a known problem that this leads to instabilities in the system matrices [CH01]. In a first step everything is assumed to be rigid and in a second step the disc is set to have a certain impedance. Rigid sphere and disc The rigid scenario is basically a Neumann problem. This means that the normal velocity of all nodes is prescribed. Based on Eq. 4.7 the Neumann problem yields a matrix equation p = D 1 Mv n (5.7) where D is the dipole matrix, M the monopole matrix with the expression iωρ 0 included, v n is the prescribed velocity and p is the sound pressure at the collocation points we are looking for. The velocity is zero for every node on the surface except at the nodes that are covered by the piston on the sphere where it is v 0. In this simulation one node located at the positive x-axis is chosen to vibrate with v 0 = 1 m s. The size of the meshes are N 1 = 1280 for the sphere 30

32 1500 N=160 N=640 N= cond frequency (Hz) Figure 11: Condition numbers of the dipole matrix of different mesh sizes N over frequency. and N 2 = 540 for the disc (Fig: 5.2.3). This yields frequency limits of 4960Hz and 4562Hz. The distances from the sphere to the disc are chosen to be in 5cm steps and the frequency is chosen to be 3461Hz which is exactly the frequency with half wavelength of 5cm. Figs. 13, 14 and 15 show the resulting sound field in between the piston and the disc from a top view. The sound pressure level (PL) is plotted one time for the case where no disc is prevalent and the other times where the disc is located at different distances. One can observe the changes in the sound field due to the disc. Rigid sphere and disc with surface impedance In this scenario we have to apply impedance boundary conditions to the surface nodes. The impedance boundary condition solved for the normal velocity is v n = α β p + γ β (5.8) In matrix notation it becomes v n = Ep + d (5.9) 31

33 z x y 0.05 Figure 12: Mesh model of the sphere (N 1 = 1280) and the disc (N 2 = 540). where d is a vector containing the known values γ and E is a diagonal β matrix containing the known values α. ubstituting Eq. 5.9 into Eq. 4.7 β yields Dp = M(d Ep). (5.10) If we set γ to be the surface admittance Y = 1 and α to be the forced β Z β vibration of the structure v s Eq becomes [Cis91] p = (D + MY ) 1 (Mv s ). (5.11) In this scenario x s is zero everywhere except at the piston nodes and Y is approximately zero everywhere except at the nodes of the disc. The simulation is basically equivalent to before but this time the disc has a certain surface impedance. Figure 16 shows again the sound field in PL in between the sphere and the disc. As expected, it can be seen that the two fields are equivalent. 32

34 y [m] PL x [m] y [m] PL x [m] Figure 13: ound pressure field in PL. Rigid disc at a distance d = 5cm. 33

35 y [m] PL x [m] y [m] PL x [m] Figure 14: ound pressure field in PL. Rigid disc at a distance d = 10cm. 34

36 y [m] PL x [m] y [m] PL x [m] 70 Figure 15: ound pressure field in PL. Rigid disc at a distance d = 15cm. 35

37 y [m] PL x [m] y [m] PL x [m] Figure 16: ound pressure field in PL. Disc with a surface impedance of Z = Z 0 = ρ 0 c at a distance d = 5cm. 36

38 6 Conclusions and Outlook This report has summarized the theory of the Boundary Element Method and shown simple examples to which the implementation could be applied successfully. Especially the application of impedance boundary conditions was made possible which opens up possibilities for further investigations. One main conclusion was found that after understanding the theory and the implementations the main challenge in the daily practice of using the BEM is the creation of suitable mesh models. In the frame of this work it was not possible to find an environment where mesh models can easily be created and edited and all that for free. One software found which at least allowed to create some models is Meshlab ( 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 the characteristic frequency is know to be kr = π. But for more complex shapes it is not trivial to find out the characteristic frequencies. Especially at higher frequencies more CHIEF points have to be used but also the density of the nodal surfaces increases. Further, if precise computations should be done up to high frequencies, a good estimate of the time needed should be found. The general complexity is 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 hard work worthwhile because a lot about acoustics in general had to be considered and therefore the knowledge and experience with acoustics could be extended. 37

39 A Divergence theorem and Green s identities The divergence theorem (also called Gauss theorem or Gauss-Ostrogradsky theorem) states that the volume integral of the divergence of a vector field is equal to the surface integral of the outward normal component of the vector field. This relation is expressed by A dv = A n d. (A.1) V where A is any continuously differentiable vector field, n is the unit outward normal to the surface and dv and d are a differential volume element and a differential surface element, respectively. The physical interpretation is, the flow of energy (e.g. of heat) through the surface is equal to the creation or destruction of energy inside the volume. If no source or sink of energy is 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 normal vector in cartesian coordinates are defined by A = A x x + A y y + A z z (A.2) A n = A x n x + A y n y + A z n z (A.3) Green s first identity The vector field A can now be replaced by the product of two scalar fields φ ψ where is the gradient given by = x + y + z 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 = [ = φ ψ x n x + φ ψ x n y + φ ψ ] x n z d. Applying the product rule and rearranging yields (A.4) (A.5) V [ ] 2 ψ φ x + 2 ψ 2 y + ψ2 + 2 z 2 = [ φ ψ x x + φ ψ y y + φ ] ψ z z [ φ ψ x n x + φ ψ x n y + φ ψ x n z dv = ] d. (A.6) 38