Fast Multipole BEM Modeling of Head Related Transfer Functions of a Dummy Head and Torso

Transcription

1 Fast Multipole BEM Modeling of Head Related Transfer Functions of a Dummy Head and Torso P. Fiala 1,2, J. Huijssen 2, B. Pluymers 2, R. Hallez 3, W. Desmet 2 1 Budapest University of Technology and Economics, Department of Telcommunications, Magyar tudósok körútja 2., H-1117, Budapest, Hungary fiala@hit.bme.hu 2 K.U.Leuven, Department of Mechanical Engineering, Celestijnenlaan 300 B, B-3001, Heverlee, Belgium 3 LMS International N.V., Interleuvenlaan 68, B-3000, Leuven, Belgium Abstract Head related transfer functions (HRTF) are used in 3D auralization to synthesize binaural sound from a monaural source. In our paper, we present a Fast Multipole Boundary Element (FMBEM) method used to simulate the pressure field scattered from a dummy head and torso mesh illuminated by incident wave fields from different directions. The FMBEM implementation is based on the diagonal form of the acoustic Green s function s multipole expansion. This method is optimal for high frequencies, and reduces to a conventional BEM in the lower frequency range. For the case of scattering problems with complex sources, often several excitations need to be modeled which is computational demanding. In the paper we present an SVD based reduction method to reduce the number of independent RHS vectors and to adjust the tolerances for optimal performance. The modeled HRTFs are validated by comparison to conventional BEM results in the lower frequency range and to measurement data in the range up to 10 khz. 1 Introduction Head related transfer functions (HRTF) are used in 3D auralization to synthesize binaural sound from a monaural source. Due to the complex geometry of the human head and torso and the wide audible frequency range from 20 Hz to 20 khz, the modeling of HRTF is a challenging issue in acoustics. The conventional method of HRTF simulation is signal based modeling starting from measurements. These simplified models decompose the HRTF into a sequence of simplified transfer functions describing the (1) shoulder echo, (2) head shadow and delay and (3) pinnae echoes [1]. All of the components are expressed using simple closed form formulas with the variables f frequency, φ azimuth and θ elevation. In a computational acoustics framework, boundary element methods applied to exterior scattering problems are the most feasible to model the scattering of incident wave fields from the head geometry. Recent applications in this topic involve single pinnae models without the head [2] and complete head model without torso up to 20 khz [3]. The typical mesh complexity is several ten-thousands of DOF and the typical computation times are about hours on a multiprocessor environment. In the present paper, a fast multipole boundary element technique is presented that is capable to model the acoustic behavior of the human head in the whole frequency range between 20 Hz and 20 khz. The aim of the paper is to demonstrate how the fast multipole boundary element method can be used to compute the HRTF of the dummy head. The paper briefly introduces the numerical model and studies its performance compared to conventional boundary element computations. The main emphasis is on the efficient FMBEM computations 2301

2 2302 PROCEEDINGS OF ISMA2010 INCLUDING USD2010 source receiver Γ Ω p tot receiver R φ p inc source R φ (a) (b) Figure 1: Modeling of HRTF with the (a) direct and (b) reverse approach with several independent excitations. The modeling results are compared to measurements carried out in 1994 in the MIT Media Lab Perceptual Computing [4] on a KEMAR dummy head microphone. 2 Modeling the transfer functions using Boundary Elements There are two main approaches to model the direction characteristics of the human head. The direct approach computes the sound pressure at the two ear positions when the head is illuminated by an acoustic source located at a given source position, typically characterized by its azimuth and elevation. In a boundary element context, the direct approach involves the solution of an acoustic scattering problem with several right hand sides (one for each source direction) and two receiver positions, as shown in Fig. 1(a). Its advantage is that it inherently produces the sound pressure field around the whole human head and torso, which has valuable physical meaning, and it is capable to model the head s response to complex acoustic excitations, e.g. response to pass-by noise generated by a real vehicle source [5]. The reverse approach makes use of acoustic reciprocity and computes the directional characteristics by expressing the acoustic pressure field on a sphere surface around the head due to two point source excitations located at the two ear positions. The indirect approach involves the solution of a scattering problem with two right hand sides (two ear locations) and extensive post processing (several response locations). Its disadvantage is that the resulting pressure field around the head and torso due to the point sources at the ears is not physical, and the resulting transfer functions can only be related to point source excitations on the surrounding sphere. In the present paper, the direct approach is applied, due to its capability to model arbitrary excitations. In a scattering problem, we are searching for the total pressure wave field p tot around an acoustic scatterer object Γ, excited by an a-priori known external incident wave field source p inc. The total sound field is decomposed into the incident and scattered sound fields as p tot = p inc + p sca where the scattered wave field satisfies the Helmholtz equation and the Sommmerfeld radiation condition in the external domain Ω. The boundary conditions are prescribed on the scatterer s boundary surface Γ for the total wave field p tot. For the case of an acoustically hard scatterer, the boundary condition can be written as q tot (x) = p tot(x) = 0 which reduces the scattering problem to a radiation problem written for the scattered field in the form 2 p sca (x) + k 2 p sca (x) = 0, x Ω, q sca (x) = q inc (x), x Γ (1) The boundary element method-based solution of the Helmholtz equation is originated from the acoustic representation formula (Helmholtz integral): c(ξ)p sca (ξ) = q G (ξ, x)p sca (x)dx p G (ξ, x)q sca (x)dx (2) Γ where ξ denotes an arbitrary point of the exterior domain Ω, p G and q G denote the pressure Green s function of the full space and its normal derivative, respectively, and c(ξ) = 1 if ξ Ω, c(ξ) = 0 if ξ / Ω and c(ξ) = 1/2 if ξ Γ. Γ

3 MEDIUM AND HIGH FREQUENCY TECHNIQUES 2303 The numerical solution of Eq.(2) is performed by discretizing the boundary surface and the field variables using polynomial interpolation (shape) functions and transforming the representation formula into a system of linear equations in the form 1 2 p sca = Hp sca Gq sca (3) where the vectors p sca and q sca contain the nodal values of the surface field variables and H and G are the frequency dependent, fully populated acoustic surface system matrices. After the linear system has been solved, the external sound pressure p sca can be computed by evaluating the integral (2) for x Ω: p sca (ξ) = q G (ξ, x)p sca (x)dx p G (ξ, x)q sca (x)dx (4) Γ Γ As the system matrices H and G are in general non-symmetric and fully populated, the direct solution of the system (3) is not feasible for large wave numbers k that require a fine discretization of the boundary surface Γ. The application of iterative solvers with the full BEM matrices [6] may partly overcome this problem, but one soon faces the high O(N 2 ) computational cost and memory requirements of the successive evaluation of the matrix-vector products. The fast multipole boundary element method (FMBEM) is based on an iterative solution of the system of equations, where the matrix-vector products are accelerated by means of a multilevel fast multipole algorithm (MLFMA). Originally, the MLFMA was developed for the simulation of particle interactions [7] in the late eighties, and has been later further developed to handle the case of electrostatic potential problems [8], as well as electromagnetic [9], acoustic [10, 11, 12] and elastodynamic wave propagation phenomena [13]. The multilevel algorithm reduces the computational cost of an iteration to O(N log b N) or even further O(N), making the FMBEM feasible to solve high wave number problems. For the theoretical background of the FMBEM the reader is referred to the mentioned papers. In the subsequent text discussions regarding the application of the FMBEM presuppose basic knowledge of the method. 3 Model description The geometry of the dummy head is shown in Fig. 2. The dimensions of the dummy head are m in the x, y and z directions, respectively. The total surface area is m 2. In order to cover the frequency range between 20 Hz and 12 khz, different meshes of the dummy head are used. A coarse mesh (mesh #1) consisting of elements is used in the low frequency domain f < 3 khz. A refined version of the coarse mesh (mesh #2) consisting of elements is used in the range 3 khz < f < 6 khz, and finally, a fine mesh (mesh #3) containing elements is used in the high frequency range 6 khz < f < 12 khz. The coarse mesh does not contain detailed information of the pinnae, but gives a good geometrical description of the head and the torso. The fine mesh, shown in Fig. 2 gives an excellent representation of the complicated ear geometry. The maximal frequency f max applicable to the different meshes was determined by the rule of thumb f max = c 7d max, where c = 343 m/s is the speed of sound and d max is the maximal element side length. Due to the complex shape of the human head around the ears, the boundary mesh of the dummy head is highly diverse with respect to the element length. Fig. 3 displays the histograms of the maximal frequency related to the separate elements of the meshes. Note the factor of 4 5 between the smallest and highest applicable frequencies for the case of the fine mesh. A BEM formulation with constant elements is applied, with one DOF per element, located at the element center. For the case of the coarse mesh, a conventional boundary element method can be applied to compute the transfer functions below 3 khz. However, in order to compare the performance and accuracy of the conventional and fast multipole BEM applications, the low frequency transfer functions are computed with the

4 2304 PROCEEDINGS OF ISMA2010 INCLUDING USD2010 (a) (b) Figure 2: (a) Side view of the dummy head model and (b) closeup view of the fine mesh #3. Number of elements mesh #1 mesh #2 mesh # Frequency [Hz] Figure 3: Histogram of the maximal frequency applicable to separate boundary elements of the three meshes. The dashed lines show the actual frequency limits used.

5 MEDIUM AND HIGH FREQUENCY TECHNIQUES 2305 (a) (b) Figure 4: (a) Level-4 and (b) level-7 clustering of the dummy head mesh. The green box denotes a freely chosen source cell, the red boxes are its near field, and the blue boxes are the interaction list. FMBEM as well. For the case of the finer meshes, the application of the conventional BEM is not feasible any more, only the FMBEM is applied. 3.1 The cluster tree The fast multipole method divides the problem mesh into a set of clusters hierarchically distributed in a cluster tree. Based on the clustering of the mesh, the matrix-vector product Gq is evaluated as Gq = G nf q + (Gq) mp (5) where the near field contributions are described by the sparse submatrix G nf and the far field contributions are effectively computed by the multipole algorithm. The effectiveness of the multipole method is governed by the radius of the near field limit that is usually selected proportional to the wavelength. In the high frequency domain, where the near field limit is small, the multilevel algorithm can effectively reduce the computational times and memory requirements of the problem. In the low frequency domain, where the near field limit is large, the solution reduces to the conventional BEM algorithm with full system matrices. The clustering of the head mesh is performed so that kd (lmax) 0.5, where d (lmax) denotes the cluster size at the leaf level. This criterion results in cluster trees with frequency dependent depths. For the case of the coarse mesh #1 at low frequencies (f 300 Hz), a 3-level tree is used. For the case of the finest mesh #3 at high frequencies (f 10 khz), the depth of the cluster tree is 7. The clustering at level 4 and level 7 is plotted in Fig. 4. The clustering of the mesh determines the sparsity structure of the near field sparse BEM matrices G nf and H nf. The sparsity structure is demonstrated in Fig. 5 for two cases. The first case is the clustering of the coarse mesh #1 at low frequencies, where only a 4-level deep cluster tree is allowed. In this case, the sparsity index (ratio of nonzero and total matrix elements) is 15%. The second demonstrated case is the 6-level clustering of the fine mesh #3 at a relative low frequency, where the 7-level clustering is still not allowed. The number of nonzero elements in the sparse matrix is , resulting in a sparsity index of

6 2306 PROCEEDINGS OF ISMA2010 INCLUDING USD2010 (a) (b) Figure 5: The sparsity structure of the near field bem matrices G nf and H nf for the case of the (a) coarse mesh #1 with 4-level clustering and the fine mesh #3 with 6-level clustering. φ = 0 o φ = +45 o φ = +90 o Figure 6: The incident pressure derivative wave field q inc,i for point sources at different directions

7 MEDIUM AND HIGH FREQUENCY TECHNIQUES The excitation The excitation used is in agreement with the measurement procedure reported in [4]. During the measurements, the dummy head was set on a turn table, and the excitation was provided by a loudspeaker placed at a distance of 1.5 m from the central point of the interaural line. The direction characteristics were recorded by rotating the head around the vertical z axis. In the numerical simulation, the loudspeaker was replaced by an acoustic monopole source located at 1.5 m from the head center, as show in Fig. 1(a). The head s response has been computed with an angular resolution of 1 o over the azimuth range 0 φ < 360 o, resulting in 360 excitations of the head model. As it can be seen in Fig. 1(a), φ = 0 corresponds to a frontal incidence, and φ = 90 o corresponds to an excitation incoming from the left. The incident pressure derivative wave field due to the point source excitation is displayed in Fig. 6 for three directions and for the maximal frequency applicable to the fine mesh. 4.1 Solving with multiple right hand sides For the case of a particular boundary condition set, the system of equation can be written in the form Aλ = b, where λ contains the unknown field variables, the matrix A is assembled from the columns of H and G, while the excitation b is the linear combination of the columns of H and G and the known field variables. In our application, it is desired to solve the system of equations Aλ = b with several right hand side vectors: Aλ i = b i, i = 1... m, or in matrix form AΛ = B, where Λ and B are n m matrices. In order to avoid m independent calls to the iterative solver, it is advantageous to express the m excitation vectors as a linear combination of a few basis vectors. For this purpose, as proposed in [14], the excitation matrix B is factorized by means of its singular value decomposition: B = UΣV H, where the columns of the n p matrix U contains the orthonormal left singular vectors of B, the p p diagonal matrix Σ contains the real positive singular values σ i in descending order, and the columns of the m p matrix V are the orthonormal right singular vectors of B. An approximation of the excitation matrix can be written as B = Ũ ΣṼH + E (6) where the matrices denoted by a tilde are the submatrices of U, Σ and V corresponding to the q largest singular values σ 1, σ 2,... σ q and E contains the contribution of the p q smallest singular values σ q+1, σ q+2... σ p. The error norm of the approximation is E 2 = σ q+1. The linear systems of equations with multiple right hand sides can now be solved with the approximate excitation vector B = Ũ ΣṼH in two steps. First, the systems AY = Ũ are solved, then, in the second step, the n q solution matrix Y is used to recover the final approximate n m solution Λ as Λ = Y S where S = ΣṼH. The important question is how the truncation parameter q of the singular value decomposition and the tolerances δ j of the reduced systems Ay j = ũ j have to be set if we want that the final solutions λ i have the backward relative errors ɛ i defined as b i A λ i ɛ i (7) b i The tolerances δ j of the reduced system are defined as ũ j Ay j ũ j = ũ j Ay j = r j δ j (8) The final residual can be expressed as B A Λ = Ũ S + E AY S = R S + E (9)

8 2308 PROCEEDINGS OF ISMA2010 INCLUDING USD2010 where the columns of the residual matrix R are the residual vectors r j of the reduced system. The residual norm corresponding to the i-th right hand side vector b i can be estimated as b i A λ i b i = R s i + ε i b i q r j s ji b i j=1 = q j=1 r j s ji + ε i b i + ε i b i q j=1 δ j s ji b i + σ q+1 b i ɛ i (10) This means that the tolerances and the truncation parameter have to satisfy the inequality q δ j s ji + σ q+1 ɛ i b i (11) j=1 One appropriate solution of the inequality can be found in the following way [14]: The truncation parameter is the smallest q such that σ q+1 β min ( b i ɛ i ) (12) i where 0 < β < 1, and the tolerances of the reduced system are found as ( ) bi ɛ i δ j = α j min (13) i where 0 < α j < 1. The parameters α j and β balance the contributions of the terms in the left hand side of Eq. (11) in the total error ɛ i b i. The obvious choice would be α j = β = 1/(q + 1), but as shown in [14], this choice is too conservative and leads to unnecessary low tolerances δ j. Instead, the heuristic choice α j = α = 0.7 and β = 0.5 is advised. s ji 4.2 Decomposition of the excitations The singular value decomposition technique was used to decompose the 360 right hand side excitation vectors b i = q inc,i into orthonormal excitation vectors ũ j. The backward tolerance of the iterative solver was defined as a constant ɛ i = ɛ = 10 2 for all of the excitation directions. During the singular value decomposition, the accuracy parameters (defined in equations (12) and (13)) were set to α = 0.4 and β = 0.4. These values are a bit lower more conservative than those suggested by [14]. The number of selected singular values (reduced right hand side vectors) is shown in Fig. 7. Obviously, the number of reduced right hand side excitation vectors increases with frequency. It is worth mentioning that the mesh changes around Hz and Hz show no effect on the singular value decomposition of the excitations. This shows that the meshes represent the excitation well even at the highest applicable frequencies. At the highest frequency, the number of selected singular values is still 5-6 times smaller than the number of original right hand side vectors. Three elements of the reduced basis at the maximal frequency are plotted in Fig. 8. The most significant base vector ũ 1 contains mainly larger wavelength excitation patterns. The small wavelength patterns are found in the least significant base vector ũ 83 whose singular value σ 83 is two orders of magnitude smaller than the largest σ 1. This can be explained with the Neumann scattering problem type. For the case of scattering problems, the shortest apparent excitation wavelengths are found where the surface normal is orthogonal to the direction of incidence. For the case of the Neumann problem, the differentiation with respect to the normal vector suppresses these patterns in the excitation (as can be seen in Fig. 6), so they are not so important in the singular value decomposition. The tolerances δ j of the reduced systems have been determined by equation (13). The singular values and the tolerances are plotted in Fig. 9 for the case of the highest frequency and the finest mesh #3. Note that not only a significant number of right hand side vectors have been removed (from 360 to 83), but the tolerances of the reduced system are also significantly relaxed for the excitations ũ j with smaller singular values.

9 MEDIUM AND HIGH FREQUENCY TECHNIQUES 2309 #singular values (q) mesh #1 mesh #2 mesh # Frequency [Hz] Figure 7: Number of selected reduced right hand side vectors as a function of excitation frequency j = 1 j = 20 j = 43 σ 1 = σ 20 = σ 83 = Figure 8: Real part of the SVD-reduced excitation fields ũ j for the largest applicable frequency of the fine mesh

10 2310 PROCEEDINGS OF ISMA2010 INCLUDING USD σ j /σ 1 δ j j Figure 9: Largest relative singular values σ j /σ 1 of the excitation matrix and derived tolerances δ j of the reduced systems of equations for the highest applicable frequency of the fine mesh #3. The overall tolerance is defined as ɛ i = ɛ = Iteration count log 10 residual Iteration count log 10 residual (a) j (b) j Figure 10: Logarithm of the relative backward error of the reduced systems Aỹ j = ũ j in each iteration step for (a) the coarse mesh #1 at f Hz and (b) the fine mesh #3 at f Hz. 5 The iterative solution The iterative solver used in the computations is nonrestarted GMRES [15] preconditioned with a sparse approximate inverse preconditioner [16]. The sparsity structure of the preconditioner is based on the near field sparsity structure of the system matrices. 5.1 Convergence of the multiple right hand side problem The overall convergence of the total reduced set of linear systems is demonstrated in Fig. 10. The figure shows the logarithm of the relative backward error in each iteration step of all the systems of the reduced set at two selected frequencies. The most impressive result is the significant reduction on the required number of iterations for the systems with higher indices j (smaller singular values). Comparing Fig. 9 with Fig. 10, it can be seen that even a very slight relaxation of the tolerance δ j can lead to a large reduction of the iteration count. After the reduced systems have been solved, the solutions of the original systems with 360 right hand side vectors can be recovered using the singular value decomposition, and the final relative backward errors can be compared to the initially prescribed value ɛ = This comparison is presented in Fig. 11 for the coarse mesh and the fine mesh, for several selected frequencies. As it can be seen from the results, the

11 MEDIUM AND HIGH FREQUENCY TECHNIQUES 2311 (a) Backward relative error Hz 1378 Hz 2067 Hz 2756 Hz φ [ o ] (b) Backward relative error Hz 4134 Hz 4823 Hz 5857 Hz φ [ o ] Figure 11: Recomputed relative backward error of the original system Aλ i = b i as a function of angle of incidence for the case of the (a) coarse mesh #1 and (b) refined mesh #2 at several frequencies choice of accuracy parameters α = β = 0.4 resulted in acceptable resultant relative backward errors. For lower frequencies (689 Hz), the choice seems conservative, but near to the upper frequency limit of the meshes, the final residual norm is in very good agreement with the prescribed tolerance. 5.2 Computation times Tab. 1 compares the computation times and accuracy of the conventional BEM and the FMBEM algorithms for the low frequency case, where the conventional BEM is still feasible to solve the scattering problem. For the case of the conventional BEM, t mat denotes the total time to generate the matrices G and H. As can be seen these matrix generation times are approximately constant, as they only depend on the number of elements and Gaussian points. The column denoted by t inv contains the time spent on solving the system of equations p i = H 1 Gq i for all the 360 right hand sides. In the FMBEM part, t sparse denotes the time spent to generate the sparse BEM matrices H nf and G nf. The data in this column show a decreasing tendency with frequency, what can be explained by the frequency dependent clustering. At the lower frequencies, the cluster tree has only 3 4 levels, resulting in denser near field matrices (see Fig. 5(a)). As frequency increases and the cluster tree becomes deeper, the near field matrices will contain less elements, resulting in smaller computation times. The jumps of the tree depth are very well shown by the times t sparse, and are also indicated by separators. The column #rhs contains the number of reduced right hand side excitation vectors ũ j. These data can be seen plotted in Fig. 7. Column #iter shows the total number of iterations needed to solve the reduced equations for all the right hand sides. The column t iter contains the time spent on one iteration step. These values increase with frequency due to the increasing truncation length L of the translation operator and the increasing number of integration points on the unit sphere. Column t sol shows the total time spent on the GMRES iterations to compute all responses y j of the reduced system of equations. The last column of Tab. 1 contains the mean relative error of the FMBEM solution compared to the BEM solutions. The error values are around 1% that is the prescribed tolerance of the FMBEM solver. We note here that a good agreement like this is not always guaranteed, as the convergence criterion is a prescribed backward error that can be different from the actual relative errors. Tab. 1 shows similar results for the refined mesh #2 and the fine mesh #3. Again, for the same reasons, the sparse matrix assembly needs less time if frequency increases, the number of right hand side vectors #rhs, the total number of iterations #iter and the total solution time increases with frequency.

12 2312 PROCEEDINGS OF ISMA2010 INCLUDING USD2010 Table 1: Computation times and accuracy of conventional BEM and FMBEM at (a) low frequencies (mesh #1), (b) mid frequencies (mesh #2) and (c) high frequencies (mesh #3) (b) (a) conv. BEM FM BEM f [Hz] t mat t inv t sparse #rhs #iter t iter t sol ɛ [%] % % % % % % % % f [Hz] t sparse #rhs #iter t iter t sol (c) f [Hz] t sparse #rhs #iter t iter t sol min min min min min min min min min min min min 6 The computed pressure fields The computed total pressure fields and the head related transfer functions obtained by the computations are presented in this section. The total sound pressure field p tot (x) around the dummy head is plotted in Fig. 12 for two selected frequencies: f = Hz and f = Hz. In the figures, the magnitude of the total wave field is displayed in a db scale. It is clear that the most intensive resonances are around the neck due to the reflections between the shoulders and the head and in the ear. At the high frequency, the shadow effect of the body is very strong, this can be seen for the case of the incidence φ = 90 o. The head related transfer functions were obtained by evaluating the total sound pressure p tot = p inc + p sca at the left and right ear positions, e.g. averaging the total sound pressure in the DOF related to the left and right ears. The transfer functions are plotted in Fig. 13. In each plot, the upper row contains the computed HRTF curves and the lower row contains the measured functions. The blue curves correspond to the left ear and the green correspond to the right ear. For the case of the measurements, only the data of the left ear is available, so the right ear curve has been generated symmetrically. For the case of the computations, the left and right ear curves may differ for the case of the fine mesh, as this is not symmetrically meshed. All transfer functions are normalized using the magnitude maxima. Comparing the modeled and measured HRTFs, we can conclude that the agreement is very good in the low frequency range (below 3 khz), but above 5 khz, the results start to deviate, even up to 5 10 db. 7 Conclusions The fast multipole boundary element method has been applied to compute the head related transfer functions of a dummy head.

13 MEDIUM AND HIGH FREQUENCY TECHNIQUES 2313 (a) (b) Figure 12: The total pressure wave field p tot around the dummy head for (a) f = Hz and (b) f = Hz at an angle of incidence φ = 90 f = Hz f = Hz f = Hz computed measured Figure 13: Computed and measured HRTF

14 2314 PROCEEDINGS OF ISMA2010 INCLUDING USD2010 It has been shown how the matrix-vector multiplications of a BEM system of equations can be accelerated by means of a multilevel fast multipole algorithm, yielding sufficient base for the development of an effective iterative algorithm to solve large scale acoustic problems by means of the BEM. A main development of the study is the optimal application of the singular value decomposition method to solve the system of equations with multiple right hand sides with prescribed accuracy. The numerical study showed in details how the methodology is applied to compute head related transfer functions of a dummy head model. The transfer functions were computed by splitting the frequency domain into three parts, and applying separate meshes for separate domains. The fast multipole method allowed to compute 360 HRTFs of a mesh with more than DOF in a few hours, using a single-core desktop PC with 4 GB RAM. We mention here, that in a similar application reported in [3], the conventional BEM computations of a element head model without torso required 77 hours CPU time on a parallel system. Comparing the results of computations and measurements, lots of differences can be found above Hz. The main error sources can be the (1) inaccurate choice of boundary conditions at the dummy head surface, (2) inaccurate representation of the loudspeaker source by an acoustic monopole, (3) wrong closed model of the eardrum, (4) measurement inaccuracies with respect to the positioning of the loudspeaker and (5) mainly the geometrical differences between the modeled and measured dummy heads. HRTFs are well-known to vary very rapidly with frequency and angle of incidence, especially for the higher frequencies, therefore, we can not expect from a numerical model to reproduce measurement results within 1-2 db error band. The methodology presented here should be rather applied in parametric studies to investigate the effect of parameter changes on the HRTFs. It has been demonstrated that FMBEM combined with SVD enables the computation of HRTFs of arbitrary head geometries and complex sources within just a couple of hours on a standard desktop computer. 8 Acknowledgements The work reported here was funded by the project IWT MIDAS Next generation numerical tools for mid-frequency acoustics. The financial support of the Flemish Community is gratefully acknowledged. Furthermore, the authors kindly acknowledge the European Commission for supporting the Collaborative Project CP MID-MOD mid-frequency vibro-acoustic modeling tools. References [1] C.P. Brown and R.O. Duda. An efficient HRTF model for 3-D sound. In 1997 IEEE ASSP Workshop on In Applications of Signal Processing to Audio and Acoustics, [2] Y. Kahana and P.A. Nelson. Numerical modelling of the spatial acoustic response of the human pinna. Journal of Sound and Vibration, 292: , [3] Y. Kahana and P.A. Nelson. Boundary element simulations of the transfer function of human heads and baffled pinnae using accurate geometric models. Journal of Sound and Vibration, 300: , [4] B. Gardner and K. Martin. HRTF measurements of a KEMAR dummy-head microphone. Technical Report 280, MIT Media Lab Perceptual Computing, May url: [5] J. Huijssen, P. Fiala, R. Hallez, and W. Desmet. Simulation of pass-by noise of automotive vehicles in the mid-frequency range using Fast Multipole BEM. In Proceedings of ISMA2010, International Congress on Sound and Vibration, Leuven, Belgium, 2010.

15 MEDIUM AND HIGH FREQUENCY TECHNIQUES 2315 [6] S. Marburg and S. Schneider. Performance of iterative solvers for acoustic problems. Part I. Solvers and effect of preconditioning. Engineering Analysis with Boundary Elements, 27: , [7] L. Greengard and V. Rokhlin. A fast algorithm for particle simulations. Journal of Computational Physics, 73(2): , [8] Y.J. Liu and N. Nishimura. The fast multipole boundary element method for potential problems: A tutorial. Engineering Analysis with Boundary Elements, 30: , [9] E. Darve. The fast multipole method I: Error analysis and asymptotic complexity. 160:98 128, [10] T. Sakuma and Y. Yasuda. Fast multipole boundary element method for large-scale steady-state sound field analysis. Part I: Setup and validation. Acustica-Acta Acustica, 88: , [11] Y. Yasuda and T. Sakuma. Fast multipole boundary element method for large-scale steady-state sound field analysis. Part II: Examination of numerical items. Acustica-Acta Acustica, 89:28 38, [12] M. Fischer. The Fast Multipole Boundary Element Method and its Application to Structure-Acoustic Field Interaction. PhD thesis, Universität Stuttgart, [13] J.A. Sanz, M. Bonnet, and J. Domingez. Fast multipole method applied to 3-D frequency domain elastodynamics. Engineering Analysis with Boundary Elements, 32: , [14] J. Langou. Solving large linear systems with multiple right-hand sides. PhD thesis, L Institut National des Sciences Appliquées de Toulouse, [15] Y. Saad and H. Schultz. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. 7(3): , [16] J. Lee, J Zhang, and C. Lu. Sparse inverse preconditioning of multilevel fast multipole algorithm for hybrid integral equations in electromagnetics. 52(9): , 2004.

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