Abstract
Wave propagation and scattering problems in acoustics are often solved with boundary element methods. They lead to a discretization matrix that is typically dense and large: its size and condition number grow with increasing frequency. Yet, high frequency scattering problems are intrinsically local in nature, which is well represented by highly localized rays bouncing around. Asymptotic methods can be used to reduce the size of the linear system, even making it frequency independent, by explicitly extracting the oscillatory properties from the solution using ray tracing or analogous techniques. However, ray tracing becomes expensive or even intractable in the presence of (multiple) scattering obstacles with complicated geometries. In this paper, we start from the same discretization that constructs the fully resolved large and dense matrix, and achieve asymptotic compression by explicitly localizing the Green’s function instead. This results in a large but sparse matrix, with a faster associated matrix-vector product and, as numerical experiments indicate, a much improved condition number. Though an appropriate localisation of the Green’s function also depends on asymptotic information unavailable for general geometries, we can construct it adaptively in a frequency sweep from small to large frequencies in a way which automatically takes into account a general incident wave. We show that the approach is robust with respect to non-convex, multiple and even near-trapping domains, though the compression rate is clearly lower in the latter case. Furthermore, in spite of its asymptotic nature, the method is robust with respect to low-order discretizations such as piecewise constants, linears or cubics, commonly used in applications. On the other hand, we do not decrease the total number of degrees of freedom compared to a conventional classical discretization. The combination of the sparsifying modification of the Green’s function with other accelerating schemes, such as the fast multipole method, appears possible in principle and is a future research topic.
Similar content being viewed by others
References
Alouges, F., Aussal, M.: The sparse cardinal sine decomposition and its application for fast numerical convolution. Numer. Algorithms 70, 427–448 (2015)
Anand, A., Boubendir, Y., Ecevit, F., Reitich, F.: Analysis of multiple scattering iterations for high-frequency scattering problems. II: the three-dimensional scalar case. Numer. Math. 114, 373–427 (2010)
Asheim, A., Huybrechs, D.: Extraction of uniformly accurate phase functions across smooth shadow boundaries in high frequency scattering problems. SIAM J. Appl. Math. 74(2), 454–476 (2014)
Babich, V.M., Buldyrev, V.S.: Short-Wavelength Diffraction Theory. Springer, Berlin (1991)
Bebendorf, M.: Hierarchical LU decomposition-based preconditioners for BEM. Computing 74, 225–247 (2005)
Bleistein, N., Handelsman, R.A.: Asymptotic Expansions of Integrals. Dover Publications Inc, Mineola (1986)
Brakhage, H., Werner, P.: Über das Dirichletsche Außenraumproblem für die Helmholtzsche Schwingungsgleichung. Arch. Math. 16, 325–329 (1965)
Bruno, O., Geuzaine, C., Monro, J.J., Reitich, F.: Prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency: the convex case. Philos. Trans. R. Soc. Lond. A 362, 629–645 (2004)
Beylkin, G., Kurcz, C., Monzón, L.: Fast algorithms for Helmholtz Green’s functions. Proc. R. Soc. Ser. A 464, 3301–3326 (2008)
Chandler-Wilde, S., Graham, I., Langdon, S., Spence, E.: Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering. Acta Numer. 21, 89–305 (2012)
Chandler-Wilde, S.N., Hewett, D.P., Langdon, S., Twigger, A.: A high frequency boundary element method for scattering by a class of nonconvex obstacles. Numer. Math. 129(4), 647–689 (2015)
Cheng, H., Crutchfield, W.Y., Gimbutas, Z., Greengard, L.F., Ethridge, J.F., Huang, J., Rokhlin, V., Yarvin, N., Zhao, J.: A wideband fast multipole method for the helmholtz equation in three dimensions. J. Comput. Phys. 216(1), 300–325 (2006)
Colton, D.L., Kress, R.: Integral Equation Methods in Scattering Theory. Wiley, New York (1983)
Deaño, A., Huybrechs, D., Iserles, A.: Computing Highly Oscillatory Integrals, vol. 155. SIAM, Philadelphia (2018)
Domínguez, V., Graham, I.G., Smyshlyaev, V.: A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering. Numer. Math. 106, 471–510 (2007)
Ecevit, F., Reitich, F.: Analysis of multiple scattering iterations for high-frequency scattering problems. I: the two-dimensional case. Numer. Math. 114, 271–354 (2009)
Ganesh, M., Hawkins, S.: A fully discrete Galerkin method for high frequency exterior acoustic scattering in three dimensions. J. Comput. Phys. 230, 104–125 (2011)
Ganesh, M., Langdon, S., Sloan, I.: Efficient evaluation of highly oscillatory acoustic scattering surface integrals. J. Comput. Appl. Math. 204, 363–374 (2007)
Geuzaine, C., Bruno, O., Reitich, F.: On the O(1) solution of multiple-scattering problems. IEEE Trans. Magn. 41(5), 1488–1491 (2005)
Geuzaine, C., Remacle, J.F.: Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009)
Giladi, E.: Asymptotically derived boundary elements for the Helmholtz equation in high frequencies. J. Comput. Appl. Math. 198, 52–74 (2007)
Greengard, L., Rokhlin, V.: A fast algorithm for particle simulations. J. Comput. Phys. 73(2), 325–348 (1987)
Groth, S., Huybrechs, D., Opsomer, P.: High-order terms in the ray expansion for high frequency scattering by single and multiple obstacles (2018) (in preparation)
Groth, S.P., Hewett, D.P., Langdon, S.: Hybrid numerical-asymptotic approximation for high-frequency scattering by penetrable convex polygons. IMA J. Appl. Math. 80, 324–353 (2015)
Harrington, R.F.: Time-Harmonic Electromagnetic Fields. IEEE Press, Piscatawat (1961)
Huybrechs, D., Vandewalle, S.: A two-dimensional wavelet-packet transform for matrix compression of integral equations with highly oscillatory kernel. J. Comput. Appl. Math. 197, 218–232 (2006)
Huybrechs, D., Vandewalle, S.: A sparse discretization for integral equation formulations of high frequency scattering problems. SIAM J. Sci. Comput. 29(6), 2305–2328 (2007)
Huybrechs, D., Vandewalle, S.: An efficient implementation of boundary element methods for computationally expensive Green’s functions. Eng. Anal. Bound. Elem. 32(8), 621–632 (2008)
Khoromskij, B.N.: Tensor-structured preconditioners and approximate inverse of elliptic operators in \({\mathbb{R}}^d\). J. Constr. Approx. 30, 599–620 (2009)
Khoromskij, B.N., Veit, A.: Efficient computation of highly oscillatory integrals by using qtt tensor approximation. Comput. Methods Appl. Math. 16(1), 145–159 (2016)
Klöckner, A., Barnett, A., Greengard, L., O’Neil, M.: Quadrature by expansion: a new method for the evaluation of layer potentials. J. Comput. Phys. 252, 332–349 (2013)
Kress, R., Spassov, W.T.: On the condition number of boundary integral operators in acoustic and electromagnetic scattering. Numer. Math. 42, 77–95 (1983)
Melrose, R.B., Taylor, M.E.: Near peak Scattering and the corrected Kirchhoff approximation for a convex obstacle. Adv. Math. 55, 242–315 (1985)
Opsomer, P.: Release: Asymptotic compression version 3. https://github.com/popsomer/bempp.git (2016)
Rokhlin, V.: Rapid solution of integral equations of classic potential theory. J. Comput. Phys. 60, 187–207 (1985)
Śmigaj, W., Betcke, T., Arridge, S., Phillips, J., Schweiger, M.: Solving boundary integral problems with BEM++. ACM Trans. Math. Softw. 41(2), 6:1–6:40 (2015)
Sweldens, W., Piessens, R.: Quadrature formulae and asymptotic error expansions for wavelet approximations of smooth functions. SIAM J. Numer. Anal. 31(4), 1240–1264 (1994)
Wald, I., Mark, W.R., Gunther, J., Boulos, S., Thiago, I., Hunt, W., Parker, S.G., Shirley, P.: State of the Art in Ray Tracing Animated Scenes. Eurograph, Newport (2007)
Wong, R.S.: Asymptotic Approximations of Integrals. SIAM, Philadelphia (2001). (Republication of 1944)
Wu, T.: Boundary Element Acoustics. WIT Press (2000). (Reprint 2005)
Ying, L.: Fast directional computation of high frequency boundary integrals via local FFTs. SIAM Multiscale Model. Simul. 13(1), 423–439 (2015)
Acknowledgements
The authors would like to thank Samuel Groth, Stephen Langdon, Niels Billen, Philip Dutré, Karl Meerbergen, Laurent Jacques and Dave Hewett for interesting and helpful discussions on this paper and related topics. The authors were supported by FWO Flanders (Projects G.0617.10, G.0641.11 and G.A004.14).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Huybrechs, D., Opsomer, P. High-Frequency Asymptotic Compression of Dense BEM Matrices for General Geometries Without Ray Tracing. J Sci Comput 78, 710–745 (2019). https://doi.org/10.1007/s10915-018-0786-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-018-0786-7
Keywords
- Boundary element method
- Oscillatory integration
- High-frequency scattering
- Compression
- Condition number
- Smooth window functions