19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007 HIGH FREQUENCY ACOUSTIC SIMULATIONS VIA FMM ACCELERATED BEM

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1 19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007 HIGH FREQUENCY ACOUSTIC SIMULATIONS VIA FMM ACCELERATED BEM PACS: Fn Gumerov, Nail A.; Duraiswami, Ramani; Fantalgo LLC, 7496 Merrymaker Way, Elkridge, MD 21075; ABSTRACT High frequency simulations of acoustics are among the most expensive problems to simulate. In practice 6 to 10 points per wavelength are required. Since the wavenumber k is inversely proportional to wavelength, if a numerical method is a surface based method (such as the BEM), then problem size scales as O(k 2 D 2 ),whered is the size of the domain. Dense matrices appear and typical iterative solution can be achieved for O(k 4 D 4 ) memory and O(k 4 D 4 ) per iteration cost. We employ an algorithm based on the fast multipole method (FMM) using coaxial and diagonal translation operators based on the frequency of simulation to reduce the memory requirements and per iteration cost to O(k 2 D 2 log kd) for larger kd (kd ~250). The number of iterations needed depend upon the condition of the matrix, and preconditioning chosen. Preconditioned Krylov methods such as the flexible generalized minimum residual method (FGMRES), with preconditioning based upon a lower accuracy FMM, usually ensure a convergent iteration. Example calculations are presented. 1 INTRODUCTION For many problems boundary element methods have long been considered as very promising. They can handle complex shapes, lead to problems in boundary variables alone, and lead to simpler meshes as the boundary alone must be discretized rather than the entire domain. Despite these advantages, one issue that has impeded their widespread adoption of these methods is that they lead to dense nonsymmetric complex valued linear systems, for which efficient iterative solvers may not be available. For N unknowns this requires O N 2 storage. The computation of the individual matrix elements is expensive requiring quadrature of singular or hypersingular functions. To reduce the singularity order and achieve symmetric matrices, many investigators employ Galerkin techniques, which lead to further O N 2 integral computations. Direct solution of the linear systems has an O(N 3 ) cost. Use of iterative methods can reduce the cost to O(N iter N 2 ) operations, where N iter is the number of iterations required, but this is still quite large. An iteration strategy that minimizes N iter is also needed. These expenses have meant that the BEM was not used for very large problems. For the Helmholtz equation the cost of computations grows with frequency. In a BEM calculation, the number of variables also grow with frequency. Any calculation must resolve the smallest wavelengths of interest, and in practice 6-10 points per wavelength are required. Since the wavenumber k is inversely proportional to wavelength, if a numerical method is a surface based method (such as the BEM), then problem N size scales as O k 2 D 2,whereD is the size of the domain. Thus the cost of a direct solver will be O(k 6 D 6 ), while an iterative BEM solver will cost O(N iter k 4 D 4 ) operations. The development of the fast multipole method (FMM) [2] and use of Krylov iterative methods presents a promising approach to improving the scalability of BEM. FMM allows the matrix vector product in an iterative method to be performed to a given precision ² in O(N log N) operations, and does not require the computation or storage of all N 2 matrix elements, reducing storage costs to O(N log N) as well. Incorporating the FMM in a quickly convergent iterative scheme allows rapid solved with O(N iter N log N) cost. Later this method was intensively studied and extended to solution of many other problems. While the literature and previous work is extensive, reasons of space do not permit us to discussion them. We refer the reader to the comprehensive review [6]. While this appears to be promising for the solution of BEM problems, for the Helmholtz equation the situations is not as nice as it may initially appear, as there are hidden factors which also grow 1

2 with the frequency increasing the cost of the solution. The FMM exploits factorized local and farfield (multipole) expansions of the Green s function obtained via the addition theorem. The basic idea of the FMM is to truncate the infinite series up to p 2 terms in 3D, and use translation operators for reexpansion of the solution about an arbitrary spatial point. The truncation number p depends on the frequency of the problem being simulated, and this is the cause of the trouble. In fact as the frequency (wavenumber k) and the domain size D increase, the truncation number must grow as O (kd) for kd > 40. In the original FMM the translation cost for a O p 2 representation was O p 4 which makes the FMM-BEM cost at high frequencies as O(N iter k 6 D 6 log kd) which is even larger than the direct solver! Improved translation methods with costs of O p 3, O p 2 log p and even O p 2 diagonal methods [7] are available. As is discussed in details in [5] by changing p in the FMM procedure with the level of the discretization we can achieve using the O(p 3 ) and O(p 2 ) translation methods O(N log N) matrix vector products. Contributions of the present paper: The FMM for both the Helmholtz equations at moderate frequencies can be handled well by O p 3 translation techniques based on a decomposition of the translation to rotation and coaxial translations (see [3]), and the switch to the O(p 2 ) diagonal forms at higher frequencies can be handled automatically. We must emphasize that several other authors (e.g., [9]) have developed FMM/BEM algorithms. Here our intention is to present some aspects of our implementations. In particular our approach is characterized by the use of the Burton Miller formulation, collocation techniques, preconditioned Krylov iterative methods, and an overall choice of techniques that have a consistent accuracy in all aspects of the algorithm. The results of solutions of the test problems and issues related to errors and performance of the methods employed are discussed. 2 PROBLEM FORMULATION We consider solutions of the Helmholtz equation with real wavenumber k 2 φ + k 2 φ =0, (1) inside or outside finite three dimensional domain V bounded by closed surface S, subject to mixed boundary conditions α (x) φ (x)+β(x) q (x) =γ (x), q(x) = φ (x), α + β 6= 0, x S. (2) n Here and below all normal derivatives are taken assuming that the normal to the surface is directed outward V. For external problems we also assume that the field satisfies the Sommerfeld radiation condition µ φ lim r r r ikφ =0, r = x. (3) This means that for scattering problems φ is treated as the scattered potential. Boundary integral equations: The boundary element method presumes that the problem will be formulated in terms of boundary integral equations, which solution together with the boundary conditions provides the values of φ (x) and q (x) on the boundary. Knowledge of these quantities enables determination of φ (y) for any domain point y. This can be done, e.g. using Greens identity, which is ±φ (y) =L [q] M [φ], y / S ± 1 φ (y) =L [q] M [φ], y S. (4) 2 Here the upper sign in the left hand side should be taken for the internal domain while the lower the lower sign is for the external domain (this convention is used everywhere below), and L and M denote the following boundary operators: Z L [q] = S Z q (x) G (x, y) ds(x), M [φ] = S φ (x) G (x, y) ds(x), (5) n (x) 19th INTERNATIONAL CONGRESS ON ACOUSTICS - ICA2007MADRID 2

3 where G is the free-space Green s function for the Helmholtz equation. G (x, y) = eikr, r = x y. (6) 4πr Burton Miller formulation: The well-known deficiency of this formulation is related to possible degeneration of operators L and M 2 1 at certain frequencies depending on S, which correspond to resonances of the internal problem for sound-soft and sound-hard boundaries. Despite the solution of the external problem is unique for these frequencies formulation (4) is deficient in these cases, since it provides non-unique solution. Moreover, despite the spurious values of resonance frequencies, for frequencies in the vicinity of the resonances the system becomes poorly conditioned. On the other hand, when solving the internal problems (e.g. room acoustics), the non-uniqueness of the solution for the internal problem, seems, have physical meaning, as there, in fact, can be resonances. In any case, boundary integral equation (4) can be modified to avoid the problem of degeneracy of boundary operators when solving correctly posed problem (1)-(3). In the present paper we use direct formulation based on the integral equation combining Green s and Maue s identities, which is the same trick as proposed by Burton and Miller (1971) for sound-hard boundaries. The latter identity is ± 1 2 q (y) =L0 [q] M 0 [φ]; L 0 [q] = Z q (x) G (x, y) ds, M 0 [φ] = Z G (x, y) φ (x) n (y) S n (y) S n (x) ds. (7) Multiplying Eq. (7) by some complex constant λ and summing with Eq. (4), we obtain ± 1 2 [φ (y)+λq(y)] = (L + λl0 )[q] (M + λm 0 )[φ]. (8) Burton & Miller (1971) proved that it is sufficient to have Im (λ) 6= 0to guarantee the uniqueness of the solution for external problem with sound-hard boundaries. Below we demonstrate a case where use of this formulation in our FMM BEM code resulted in proper computation of a spuriously resonant case of a sound hard sphere insonified by a plane wave. 3 BEM SPEED UP WITH FMM BEM and FMM data structures: Detailed description of the FMM for the Laplace and Helmholtz equations can be found elsewhere [2, 5]. The data structure of the FMM is based on the octree space subdivision up to some level l max. Selection of l max is dictated by an optimization problem for FMM costs, which balances the costs of the translation and direct summation operations. Deviations from optimal l max are not desirable, as they heavily influence the FMM performance [5]. On the other hand the split between near and far is dictated by the accuracy of the BEM and therefore Abs error 1.E+00 1.E-01 1.E-02 1.E-03 1.E-04 Preconditioned Unpreconditioned Iteration # Computational Cost Converged (10-4 ) Preconditioned Converged (10-4 ) Unpreconditioned Iteration # Figure 1: IfGMRES solve with a low-accurace FMM-BEM preconditioner. Left: No. of unpreconditioned GMRES loops vs. number of outer loops in the fgmres. Right: Since fgmres iterations are more expensive, the total computational cost is shown. Our preconditioning strategy is very effective. 19th INTERNATIONAL CONGRESS ON ACOUSTICS - ICA2007MADRID 3

4 the size of the neighborhood where we perform direct computations of integrals may not coincide with the sizes of neighborhoods at different levels of the FMM. Our numerical tests show that (fortunately) the size of the neighborhood at the optimal maximum space subdivision level is larger than the size required for direct integral evaluations. This means that all elements which intersect the neighborhood of an element i at the finest level, and whose contribution to the potential at the element center can be classified into two sets. First are the elements j that arecloserthansomedistancer min from the center of i to the closest corner of j. The contribution of these elements is performed using higher order quadrature or special integral treatment (for singular and hypersingular integrals) as in the conventional BEM. The second set of elements are located further than r min. Their contribution is taken into account using low order quadrature, as they are remote in the sense of BEM, but close in the sense of the FMM. Iterative methods: The basic iterative method we used is the Generalized Minimal Residual Method (GMRES) and its modifications (flexible GMRES, fgmres) [8], which allow the use of various FMM-based preconditioners [4]. In all cases for the Helmholtz equation at low frequencies unpreconditioned GMRES showed good results, while the convergence rate of this method for the Helmholtz equation substantially reduces at higher frequencies. Preconditioning: The best preconditioners we found were those that compute solution of the system Aψ j = c j on the jth step using unpreconditioned GMRES (inner loop), low accuracy FMM, and lower bounds for convergence of iterations. For example, if in the outer loop of the fgmres the prescribed accuracy for the FMM solution was 10 5, and the iterative process was terminated as the residual reaches 10 4, for the inner preconditioning loop we used FMM with prescribed accuracy 0.05 and the iterative process was terminated at residual value 0.5. Computation of normal derivatives in the FMM: Eq. (8) requires four matrix-vector products, and the use of Eq. (6) requires two matrix-vector products (in the case of mixed boundary conditions). These can be performed by the FMM in a single run, as for a single matrix-vector product. Indeed, the first step of the regular FMM is to build multipole expansions about the centers of the source boxes, x. We handle this step using expansions of Greens function over the multipole basis functions Sn m (r): G (y x) Xp 1 nx n=0 m= n C m n S m n (y x ), C m n = C m n (x, x ), (9) where p is the truncation number, and expressions for coefficients Cn m for the Laplace and Helmholtz equations can be found elsewhere [5]. We compute the normal derivative of arbitrary function F (x), which can be expanded inton a series over the basis functions with coefficients {Cn m } using sparsematrix differential operators Dnn o: mm0 0 Xp 1 n x F (x) nx n=0m= n Bn m Sn m (y x ), {Bn m }= nd mm0 nn 0 (n) on Cn m0 0 Expressions for Dnn mm0 0 for the Helmholtz equation can be found in [5]. 4 RESULTS o p 1 X, Bn m = nx Xp 1 Xn 0 D mm 0 nn 0 n=0m= nn 0 =0m 0 = n 0 The tests of the BEM for the Helmholtz equation were performed by solution of acoustic scattering problems off objects of different shape. The objects were sound-hard leading to an external Neumann problem for the scattered field. As a test we selected scattering of the plane incident wave from a single sphere, which has an analytical solution in the form of the infinite series (due to Lord Rayleigh, can be found elsewhere, e.g. [5]). These series can be appropriately truncated and the error of the solution can be accurately estimated. We performed tests using the same four methods as reported (10) Cm0 n. 0 19th INTERNATIONAL CONGRESS ON ACOUSTICS - ICA2007MADRID 4

5 Total CPU time (s) 1.E+04 1.E+03 1.E+02 1.E+01 kd=34.64 y=cx 3 y=bx 2 GMRES+FMM y=ax BEM Neumann Problem for Sphere (3D Helmholtz) GMRES+Low Mem Direct O(N 2 ) Memory GMRES+High Mem Direct Threshold LU-decomposition 1.E+00 1.E+03 1.E+04 1.E+05 1.E+06 Number of Vertices, N Single Matrix-Vector Multiplication CPU Time (s) 1.E+03 1.E+02 1.E+01 1.E+00 1.E-01 Number of GMRES Iterations Low Mem Direct y=bx 2 Multiplication of Stored Matrix y=ax FMM 3D Helmholtz, kd= E-02 1.E+03 1.E+04 1.E+05 1.E+06 Number of Vertices, N Figure 3: Left: CPU time for computation of scattering off a sphere in a domain of size kd =34.64 as a function of the degrees of freedom. Right: Dependence of the matrix vector multiplication time on N. Crosses show number of iterations. for the Laplace equation, plus we compared solutions obtained using boundary integral equations resulting from the layer potential and Green s identity. Also some tests using the fgmres with different preconditioners were performed. Fig. 2 shows scattering off a sphere. The numerical solution reproduces fine oscillating structure of the acoustic field near the point θ = 180 which happens at large ka, wherea is the sphere radius. Even the use of high frequency mesh with 480,000 elements for ka =30provided kδ =0.4, where δ is the size of the largest boundary element (about 16 elements per wavelength). Computations of this were performed with 6 levels of the octree with maximum truncation number p =22at level 2, and convergence to error ² =10 5 took 115 unpreconditioned iterations. The CPU time required for solution of the problem with ka =10using different db methods is shown in Fig. 3. As the kd for different N is fixed (kd =34.64) themeth- ods show scaling close to O(N 3 ), O(N 2 ), O(N 2 ), and O(N), respectively. Finally we applied the software for solution of some large scattering problems off a animal head, and from composite media for which the developed BEM/FMM works Plane wave scattering from a sphere, ka=30 (kd=103.9) Incident Wave well. Parameters such as BEM/FMM, vertices, elements sound pressure or head related transfer functions (HRTF) are computed. Fig. 4 provides one such case. (A dence angle is 0 for the front point and 180 o for the rear point. Figure 2: Scattering of a plane wave from a sound-hard sphere. The inci- color movie visualizing scattering can be viewed online on the web sites of the authors.) 19th INTERNATIONAL CONGRESS ON ACOUSTICS - ICA2007MADRID 5

6 Figure 4: An example of computation of sound scattering from bunny model using the BEM/FMM. The sound pressure for different frequencies at some moment of time is shown on the bottom pictures. Figure 5: A large case of dense object scattering run with our code. On the left is shown a case of plane-wave scattering from sound hard ellipsoids (kd = 29), while on the right is a case of plane wave scattering with a Robin boundary condition (kd = 144.5). There are vertices and elements in the mesh over 512 ellipsoids. References [1] Chen, L.H and Zhou, J. (1992) Boundary Element Methods. Academic Press. [2] Greengard, L. (1988). The Rapid Evaluation of Potential Fields in Particle Sytems. (MIT Press, MA). [3] Gumerov, N.A., Duraiswami, R. (2003). Recursions for the computation of multipole translation and rotation coefficients for the 3-D Helmholtz equation, SIAM J. Sci. Stat. Comput. 25(4), , [4] Gumerov, N.A. and Duraiswami, R. (2005). Computation of scattering from clusters of spheres using the fast multipole method, J. Acoust. Soc. Am., 117 (4), Pt. 1, [5] Gumerov, N.A. and Duraiswami, R. (2005) Fast Multipole Methods for the Helmholtz Equation in Three Dimensions. (Elsevier, Oxford, UK). [6] N. Nishimura, Fast multipole accelerated boundary integral equation methods, App Mech. 55: , [7] Rokhlin, V. (1993). Diagonal forms of translation operators for the Helmholtz equation in three dimensions, Appl. and Comp. Harmonic Analysis, 1: [8] Saad, Y.(1993). A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput. 14: [9] Sakuma, T. and Yasuda, Y. (2002). Fast multipole boundary element method for large scale steady state sound field analysis, part i: Setup and validation. Acustica/Acta Acustica, 88: th INTERNATIONAL CONGRESS ON ACOUSTICS - ICA2007MADRID 6