A fast solver for elastic scattering from axisymmetric objects by boundary integral equations. (English) Zbl 1490.35111
Summary: Fast and high-order accurate algorithms for three-dimensional elastic scattering are of great importance when modeling physical phenomena in mechanics, seismic imaging, and many other fields of applied science. In this paper, we develop a novel boundary integral formulation for the three-dimensional elastic scattering based on the Helmholtz decomposition of elastic fields, which converts the Navier equation to a coupled system consisted of Helmholtz and Maxwell equations. An FFT-accelerated separation of variables solver is proposed to efficiently invert boundary integral formulations of the coupled system for elastic scattering from axisymmetric rigid bodies. In particular, by combining the regularization properties of the singular boundary integral operators and the FFT-based fast evaluation of modal Green’s functions, our numerical solver can rapidly solve the resulting integral equations with a high-order accuracy. Several numerical examples are provided to demonstrate the efficiency and accuracy of the proposed algorithm, including geometries with corners at different wavenumbers.
MSC:
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35Q61 | Maxwell equations |
| 65R20 | Numerical methods for integral equations |
Keywords:
Helmholtz equation; Maxwell equations; boundary integral equations; elastic scattering; numerical solverReferences:
| [1] | Ahner, JF; Hsiao, GC, On the two-dimensional exterior boundary-value problems of elasticity, SIAM J. Appl. Math., 31, 677-685 (1976) · Zbl 0355.73019 · doi:10.1137/0131060 |
| [2] | Albella Martínez, J.; Imperiale, S.; Joly, P.; Rodríguez, J., Solving 2D linear isotropic elastodynamics by means of scalar potentials: a new challenge for finite elements, J. Sci. Comput., 77, 1832-1873 (2018) · Zbl 1407.65177 · doi:10.1007/s10915-018-0768-9 |
| [3] | Alpert, B., Hybrid Gauss-trapezoidal quadrature rules, SIAM J. Sci. Comput., 20, 5, 1551-1584 (1999) · Zbl 0933.41019 · doi:10.1137/S1064827597325141 |
| [4] | Ammari, H., Bretin, E., Garnier, J., Kang, H., Lee, H., Wahab, A.: Mathematical Methods in Elasticity Imaging. Princeton University Press, New Jersey (2015) · Zbl 1332.35002 |
| [5] | Bao, G.; Xu, L.; Yin, T., An accurate boundary element method for the exterior elastic scattering problem in two dimensions, J. Comput. Phys., 348, 343-363 (2017) · Zbl 1380.74059 · doi:10.1016/j.jcp.2017.07.032 |
| [6] | Borges, C.; Lai, J., Inverse scattering reconstruction of a three dimensional sound-soft axis-symmetric impenetrable object, Inverse Prob., 36, 10, 105005 (2020) · Zbl 1450.35288 · doi:10.1088/1361-6420/abac9b |
| [7] | Bremer, J., On the Nyström discretization of integral equations on planar curves with corners, Appl. Comput. Harm. Anal., 32, 45-64 (2012) · Zbl 1269.65131 · doi:10.1016/j.acha.2011.03.002 |
| [8] | Bremer, J.; Gimbutas, Z., A Nyström method for weakly singular integral operators on surfaces, J. Comput. Phys., 231, 14, 4885-4903 (2012) · Zbl 1245.65177 · doi:10.1016/j.jcp.2012.04.003 |
| [9] | Bremer, J.; Gimbutas, Z.; Rokhlin, V., A nonlinear optimization procedure for generalized Gaussian quadratures, SIAM J. Sci. Comput., 32, 4, 1761-1788 (2010) · Zbl 1215.65045 · doi:10.1137/080737046 |
| [10] | Bruno, OP; Yin, T., Regularized integral equation methods for elastic scatteringproblems in three dimensions, J. Comput. Phys., 410, 109350 (2020) · Zbl 1436.65211 · doi:10.1016/j.jcp.2020.109350 |
| [11] | Bu, F.; Lin, J.; Reitich, F., A fast and high-order method for the three-dimensional elastic wave scattering problem, J. Comput. Phys., 258, 856-870 (2014) · Zbl 1349.74348 · doi:10.1016/j.jcp.2013.11.009 |
| [12] | Cohl, HS; Tohline, JE, A compact cylindrical Green’s function expansion for the solution of potential problems, Astrophys. J., 527, 1, 86-101 (1999) · doi:10.1086/308062 |
| [13] | Conway, JT; Cohl, HS, Exact Fourier expansion in cylindrical coordinates for the three-dimensional Helmholtz Green function, Z. Angew. Math. Phys., 61, 425-442 (2010) · Zbl 1200.35057 · doi:10.1007/s00033-009-0039-6 |
| [14] | Colton, D.; Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, 3rd edn (2013), New York: Springer, New York · Zbl 1266.35121 · doi:10.1007/978-1-4614-4942-3 |
| [15] | Dong, H.; Lai, J.; Li, P., Inverse obstacle scattering for elastic waves with phased or phaseless far-field data, SIAM J. Imaging Sci., 12, 809-838 (2019) · Zbl 1524.78044 · doi:10.1137/18M1227263 |
| [16] | Dong, H.; Lai, J.; Li, P., An inverse acoustic-elastic interaction problem with phased or phaseless far-field data, Inverse Probl., 36, 035014 (2020) · Zbl 1437.35702 · doi:10.1088/1361-6420/ab693e |
| [17] | Dong, H.; Lai, J.; Li, P., A highly accurate boundary integral method for the elastic obstacle scattering problem, Math. Comput., 90, 2785-2814 (2021) · Zbl 1479.65035 · doi:10.1090/mcom/3660 |
| [18] | Epstein, CL; Greengard, L.; O’Neil, M., A high-order wideband direct solver for electromagnetic scattering from bodies of revolution, J. Comput. Phys., 387, 205-229 (2019) · Zbl 1452.78011 · doi:10.1016/j.jcp.2019.02.041 |
| [19] | Felipe, V.; Leslie, G.; Zydrunas, G., Boundary integral equation analysis on the sphere, Numer. Math., 128, 463-487 (2014) · Zbl 1308.65208 · doi:10.1007/s00211-014-0619-z |
| [20] | Geng, N.; Carin, L., Wide-band electromagnetic scattering from a dielectric BOR buried in a layered lossy dispersive medium, IEEE Trans. Antennas Propag., 47, 4, 610-619 (1999) · doi:10.1109/8.768799 |
| [21] | Gillman, A.; Young, PM; Martinsson, PG, A direct solver with O(N) complexity for integral equations on one-dimensional domains, Front. Math. China, 7, 2, 217-247 (2012) · Zbl 1262.65198 · doi:10.1007/s11464-012-0188-3 |
| [22] | Gimbutas, Z.; Greengard, L., Fast multi-particle scattering: a hybrid solver for the Maxwell equations in microstructured materials, J. Comput. Phys., 232, 22-32 (2013) · Zbl 1291.78058 · doi:10.1016/j.jcp.2012.01.041 |
| [23] | Greengard, L.; Rokhlin, V., A fast algorithm for particle simulations, J. Comput. Phys., 73, 325-348 (1987) · Zbl 0629.65005 · doi:10.1016/0021-9991(87)90140-9 |
| [24] | Greengard, L., O’Neil, M., Rachh, M., Vico, F.: Fast multipole methods for evaluation of layer potentials with locally-corrected quadratures. Submitted. arXiv:2006.02545 |
| [25] | Gustafsson, M., Accurate and efficient evaluation of modal green’s functions, J. Electromagnet. Wave., 24, 10, 1291-1301 (2010) · doi:10.1163/156939310791958752 |
| [26] | Hao, S.; Barnett, AH; Martinsson, PG; Young, P., High-order accurate Nyström discretization of integral equations with weakly singular kernels on smooth curves in the plane, Adv. Comput. Math., 40, 245-272 (2014) · Zbl 1300.65093 · doi:10.1007/s10444-013-9306-3 |
| [27] | Hao, S.; Martinsson, PG; Young, P., An efficient and highly accurate solver for multi-body acoustic scattering problems involving rotationally symmetric scatterers, Comput. Math. Appl., 69, 304-318 (2015) · Zbl 1360.76253 · doi:10.1016/j.camwa.2014.11.014 |
| [28] | Helsing, J.; Holst, A., Variants of an explicit kernel-split panel-based Nyström discretization scheme for Helmholtz boundary value problems, Adv. Comput. Math., 41, 691-708 (2015) · Zbl 1319.65117 · doi:10.1007/s10444-014-9383-y |
| [29] | Helsing, J.; Karlsson, A., An explicit kernel-split panel-based Nyström scheme for integral equations on axially symmetric surfaces, J. Comput. Phys., 272, 686-703 (2014) · Zbl 1349.65709 · doi:10.1016/j.jcp.2014.04.053 |
| [30] | Helsing, J.; Karlsson, A., Determination of normalized electric eigenfields in microwave cavities with sharp edges, J. Comput. Phys., 304, Supplement C, 465-486 (2016) · Zbl 1349.78113 · doi:10.1016/j.jcp.2015.09.054 |
| [31] | Helsing, J.; Karlsson, A., Resonances in axially symmetric dielectric objects, IEEE Trans. Microw. Theory Tech., 65, 7, 2214-2227 (2017) · doi:10.1109/TMTT.2017.2653773 |
| [32] | Helsing, J.; Ojala, R., Corner singularities for elliptic problems: integral equations, graded meshes, quadrature, and compressed inverse preconditioning, J. Comput. Phys., 227, 8820-8840 (2008) · Zbl 1152.65114 · doi:10.1016/j.jcp.2008.06.022 |
| [33] | Ho, KL; Greengard, L., A fast direct solver for structured linear systems by recursive skeletonization, SIAM J. Sci. Comput., 34, 5, 2507-2532 (2012) · Zbl 1259.65062 · doi:10.1137/120866683 |
| [34] | Hu, G.; Kirsch, A.; Sini, M., Some inverse problems arising from elastic scattering by rigid obstacles, Inverse Probl., 29, 015009 (2013) · Zbl 1334.74047 · doi:10.1088/0266-5611/29/1/015009 |
| [35] | Kolm, P.; Jiang, S.; Rokhlin, V., Quadruple and octuple layer potentials in two dimensions I: analytical apparatus, Appl. Comput. Harmon. Anal., 14, 47-74 (2003) · Zbl 1139.35397 · doi:10.1016/S1063-5203(03)00004-6 |
| [36] | Kress, R., Linear Integral Equations (1999), New York: Springer, New York · Zbl 0920.45001 · doi:10.1007/978-1-4612-0559-3 |
| [37] | Lai, J.; Li, P., A framework for simulation of multiple elastic scattering in two dimensions, SIAM J. Sci. Comput., 41, A3276-A3299 (2019) · Zbl 1439.35358 · doi:10.1137/18M1232814 |
| [38] | Lai, J.; O’Neil, M., An FFT-accelerated direct solver for electromagnetic scattering from penetrable axisymmetric objects, J. Comput. Phys., 390, 152-174 (2019) · Zbl 1452.65407 · doi:10.1016/j.jcp.2019.04.005 |
| [39] | Lai, J.; Ambikasaran, S.; Greengard, LF, A fast direct solver for high frequency scattering from a large cavity in two dimensions, SIAM J. Sci. Comput., 36, 6, B887-B903 (2014) · Zbl 1319.78008 · doi:10.1137/140964904 |
| [40] | Lai, J.; Greengard, L.; O’Neil, M., Robust integral formulations for electromagnetic scattering from three-dimensional cavities, J. Comput. Phys., 345, 1-16 (2017) · Zbl 1378.78020 · doi:10.1016/j.jcp.2017.05.008 |
| [41] | Liu, Y.; Barnett, AH, Efficient numerical solution of acoustic scattering from doubly-periodic arrays of axisymmetric objects, J. Comput. Phys., 324, 226-245 (2016) · Zbl 1360.65297 · doi:10.1016/j.jcp.2016.08.011 |
| [42] | Louër, FL, On the fréchet derivative in elastic obstacle scattering, SIAM J. Appl. Math., 72, 1493-1507 (2012) · Zbl 1259.35154 · doi:10.1137/110834160 |
| [43] | Louër, FL, A high order spectral algorithm for elastic obstacle scattering in three dimensions, J. Comput. Phys., 279, 1-17 (2014) · Zbl 1351.74113 · doi:10.1016/j.jcp.2014.08.047 |
| [44] | O’Neil, M.; Cerfon, AJ, An integral equation-based numerical solver for Taylor states in toroidal geometries, J. Comput. Phys., 359, 263-282 (2018) · Zbl 1383.76550 · doi:10.1016/j.jcp.2018.01.004 |
| [45] | Pao, YH; Varatharajulu, V., Huygens’ principle, radiation conditions, and integral formulas for the scattering of elastic waves, J. Acoust. Soc. Am., 59, 1361-1371 (1976) · Zbl 0348.73018 · doi:10.1121/1.381022 |
| [46] | Tong, MS; Chew, WC, Nyström method for elastic wave scattering by three-dimensional obstacles, J. Comput. Phys., 226, 1845-1858 (2007) · Zbl 1219.74046 · doi:10.1016/j.jcp.2007.06.013 |
| [47] | Young, P.; Hao, S.; Martinsson, PG, A high-order Nyström discretization scheme for boundary integral equations defined on rotationally symmetric surfaces, J. Comput. Phys., 231, 11, 4142-4159 (2012) · Zbl 1250.65146 · doi:10.1016/j.jcp.2012.02.008 |
| [48] | Yu, WM; Fang, DG; Cui, TJ, Closed form modal green’s functions for accelerated computation of bodies of revolution, IEEE Trans. Antennas Propag., 56, 11, 3452-3461 (2008) · Zbl 1369.78639 · doi:10.1109/TAP.2008.2005459 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
