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A wideband fast multipole accelerated boundary integral equation method for time-harmonic elastodynamics in two dimensions. (English) Zbl 1253.74122

Summary: This article presents a wideband fast multipole method (FMM) to accelerate the boundary integral equation method for two-dimensional elastodynamics in frequency domain. The present wideband FMM is established by coupling the low-frequency FMM and the high-frequency FMM that are formulated on the ingenious decomposition of the elastodynamic fundamental solution developed by Nishimura’s group. For each of the two FMMs, we estimated the approximation parameters, that is, the expansion order for the low-frequency FMM and the quadrature order for the high-frequency FMM according to the requested accuracy, considering the coexistence of the derivatives of the Helmholtz kernels for the longitudinal and transcendental waves in the Burton-Muller type boundary integral equation of interest. In the numerical tests, the error resulting from the fast multipole approximation was monotonically decreased as the requested accuracy level was raised. Also, the computational complexity of the present fast boundary integral equation method agreed with the theory, that is, \(N\log N\), where \(N\) is the number of boundary elements in a series of scattering problems. The present fast boundary integral equation method is promising for simulations of the elastic systems with subwavelength structures. As an example, the wave propagation along a waveguide fabricated in a finite-size phononic crystal was demonstrated.

MSC:

74S15 Boundary element methods applied to problems in solid mechanics
74J20 Wave scattering in solid mechanics
65N38 Boundary element methods for boundary value problems involving PDEs

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[1] Cheng, A wideband fast multipole method for the Helmholtz equation in three dimensions, J. Comp. Phys. 216 (1) pp 300– (2006) · Zbl 1093.65117
[2] Cheng, Contemporary Mathematics 408, in: Inverse Problems, Multi-Scale Analysis and Effective Medium Theory pp 99– (2006)
[3] Otani, A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals, J. Comp. Phys. 227 (9) pp 4630– (2008) · Zbl 1206.78084
[4] Gumerov N Duraiswami R Fast multipole accelerated boundary element method (FMBEM) for solution of 3D scattering problems Proc. Acoustics 2008 http://intellagence.eu.com/acoustics2008/acoustics2008/cd1/data/index.html
[5] Cho, A wideband fast multipole method for the two dimensional complex Helmholtz equation, Computer Physics Communications 181 pp 2086– (2010) · Zbl 1219.65140
[6] Wolf, Wideband fast multipole boundary element method: application to acoustic scattering from aerodynamic bodies, Int. J. Numer. Meth. Fluids 66 pp 2108– (2010)
[7] Rokhlin, Rapid solution of integral equations of scattering theory in two dimensions, J. Comp. Phys. 86 (2) pp 414– (1990) · Zbl 0686.65079
[8] Rokhlin, Diagonal forms of translation operators for the Helmholtz equation in three dimension, Applied and Computational Harmonic Analysis 1 pp 82– (1993) · Zbl 0795.35021
[9] Dembart, The accuracy of fast multipole methods for Maxwell’s equations, IEEE Computational Science & Engineering 5 (3) pp 48– (1998) · Zbl 05092191
[10] Greengard, Accelerating fast multipole methods for the Helmholtz equation at low frequencies, IEEE Computational Science & Engineering 5 (3) pp 32– (1998) · Zbl 05092188
[11] Jiang, Low-frequency fast inhomogeneous plane-wave algorithm (LF-FIPWA), Microw. Opt. Technol. Lett. 40 (2) pp 117– (2004)
[12] Wulf, An efficient implementation of the combined wideband MLFMA/LF-FIPWA, IEEE Antennas and propagation 57 (2) pp 467– (2009)
[13] Darve, Efficient fast multipole method for low-frequency scattering, J. Comp. Phys. 197 (1) pp 341– (2004) · Zbl 1073.65133
[14] Kobayashi, Mechanics and Mathematical Methods-Series of Handbooks, in: Boundary Element Methods in Mechanics pp 192– (1987)
[15] Bonnet, Boundary Integral Equation Methods for Solids and Fluids (1995)
[16] Nishimura, Fast multipole accelerated boundary integral equation methods, Applied Mechanics Reviews 55 pp 299– (2002)
[17] Eringen, Elastodynamics, Volume II Linear Theory (1975)
[18] Burton, The application of integral equation methods to the numerical solution of some exterior boundary-value problems, Proc. R. Soc. London Ser. A 323 pp 201– (1971) · Zbl 0235.65080
[19] Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (eighth Dover printing) (1972) · Zbl 0543.33001
[20] Chen, Fast multipole method as an efficient solver for 2D elastic wave surface integral equations, Computational Mechanics 20 (6) pp 495– (1997) · Zbl 0912.73077
[21] Fujiwara, The fast multipole method for integral equations of seismic scattering problems, Geophysical Journal International 133 pp 773– (1998)
[22] Fukui, Fast multipole boundary element method for 2D elastodynamics, J. Applied Mechanics JSCE 1 pp 373– (1998)
[23] Fujiwara, The fast multipole method for solving integral equations of three-dimensional topography and basin problems, Geophysical Journal International 140 pp 198– (2000)
[24] Yoshida, Analysis of three dimensional scattering of elastic waves by a crack with fast multipole boundary integral equation method, J. Applied Mechanics JSCE 3 pp 143– (2000)
[25] Yoshida K Application of fast multipole method to boundary integral equation method Doctoral dissertation 2001 http://gspsun1.gee.kyoto-u.ac.jp/yoshida/doctoral_thesis/index.html
[26] Yoshida, Application of a diagonal form fast multipole BIEM to the analysis of three dimensional scattering of elastic waves by cracks, Proc. Eighteenth Japan National Symposium on Boundary Element Methods 18 pp 77– (2001)
[27] Iritani, A new formulation of FMBIEM in elastodynamics, Proc. Conference on Computational Engineering and Science 5 (1) pp 301– (2000)
[28] Sakamoto, On the periodic fast multipole method in elastodynamics in 2D, Trans. Japan Society for Computational Methods in Engineering 8 pp 77– (2008)
[29] Isakari, A periodic FMM for three dimensional elastodynamics with considerations related to Wood’s anomaly, J. Applied Mechanics JSCE 12 pp 171– (2009)
[30] Isakari, A periodic FMM for elastodynamics in 3D and its applications to problems related to waves scattered by a doubly periodic layer of scatters, J. Applied Mechanics JSCE 13 pp 169– (2010)
[31] Isakari, Preconditioning based on Calderon’s formulae for the periodic FMM for elastodynamics in 3D, Trans. Japan Society for Computational Methods in Engineering 10 pp 45– (2010)
[32] Takahashi, A fast BIEM for three-dimensional elastodynamics in time domain, Eng. Anal. Bound. Elem. 27 pp 491– (2003) · Zbl 1047.74547
[33] Otani, Lecture Notes in Applied and Computational Mechanics 29, in: Boundary Element Analysis: Mathematical Aspects and Application pp 161– (2007) · Zbl 1298.74247
[34] Shidooka, A time domain fast multipole boundary integral equation method for anisotropic elastodynamics in 3D, J. of Applied Mechanics JSCE 11 pp 109– (2008)
[35] Saitoh, Application of a 2.5-D BEM and 3-D diagonal form fast multipole BEM to environmental vibration analysis with a moving load, Journal of Boundary Element Methods 21 pp 27– (2004)
[36] Chaillat, A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain, Comput. Methods Appl. Mech. Eng. 197 pp 4233– (2008) · Zbl 1194.74109
[37] Sanz, Fast multipole method applied to 3-D frequency domain elastodynamics, Eng. Anal. Bound. Elem. 32 pp 787– (2008) · Zbl 1244.74190
[38] Tong, Multilevel fast multipole algorithm for elastic wave scattering by large three-dimensional objects, J. Comp. Phys. 228 (3) pp 921– (2009) · Zbl 1259.74017
[39] Chew, Fast and Efficient Algorithms in Computational Electromagnetics (2001)
[40] Kobayashi, Wave Analysis and Boundary Element Methods (2000)
[41] Saad, GMRES; A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. on Scientific and Statistical Computing 7 pp 856– (1986) · Zbl 0599.65018
[42] Billings, American Institute of Physics Handbook (1972)
[43] Lua, Phononic crystals and acoustic metamaterials, Materials Today 12 (2) pp 34– (2009)
[44] Guenneau, Localised bending modes in split ring resonators, Physica B 394 pp 141– (2007)
[45] Pearson, Theoretical Elasticity (1959)
[46] Christensen, Collimation of sound assisted by acoustic surface waves, Nature Phys. 3 pp 851– (2007)
[47] Ooura’s Mathematical Software Packages Numerical Integration (Quadrature) - DE Formula (Almighty Quadrature) http://www.kurims.kyoto-u.ac.jp/ ooura/index.html
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