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A meshless generalized finite difference method for inverse Cauchy problems associated with three-dimensional inhomogeneous Helmholtz-type equations. (English) Zbl 1403.65118
Summary: The generalized finite difference method (GFDM) is a relatively new domain-type meshless method for the numerical solution of certain boundary value problems. The method involves a coupling between the Taylor series expansions and weighted moving least-squares method. The main idea here is to fully inherit the high-accuracy advantage of the former and the stability and meshless attributes of the latter. This paper makes the first attempt to apply the method for the numerical solution of inverse Cauchy problems associated with three-dimensional (3D) Helmholtz-type equations. Numerical results for three benchmark examples involving Helmholtz and modified Helmholtz equations in both smooth and piecewise smooth 3D geometries have been analyzed. The convergence, accuracy and stability of the method with respect to increasing the number of scatted nodes inside the whole domain and decreasing the amount of noise added into the input data, respectively, have been well-studied.

MSC:
65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
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[1] Karageorghis, A; Smyrlis, YS, Matrix decomposition MFS algorithms for elasticity and thermo-elasticity problems in axisymmetric domains, J Comput Appl Math, 206, 2, 774-795, (2007) · Zbl 1206.35067
[2] Chen, CS; Muleshkov, AS; Golberg, MA; Mattheij, RMM, A mesh-free approach to solving the axisymmetric Poisson’s equation, Numer Methods Partial Differ Equ, 21, 2, 349-367, (2005) · Zbl 1072.65154
[3] Benito, JJ; Urena, F; Gavete, L, Solving parabolic and hyperbolic equations by the generalized finite difference method, J Comput Appl Math, 209, 2, 208-233, (2007) · Zbl 1139.35007
[4] Cheng, AHD; Cheng, DT, Heritage and early history of the boundary element method, Eng Anal Bound Elem, 29, 3, 268-302, (2005) · Zbl 1182.65005
[5] Marin, L; Lesnic, D, The method of fundamental solutions for nonlinear functionally graded materials, Int J Solids Struct, 44, 21, 6878-6890, (2007) · Zbl 1166.80300
[6] Sarler, B., Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions, Eng Anal Bound Elem, 33, 12, 1374-1382, (2009) · Zbl 1244.76084
[7] Marin, L., An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation, Comput Mech, 45, 6, 665-677, (2010) · Zbl 1398.65278
[8] Liu, GR; Gu, YT, A meshfree method: meshfree weak-strong (MWS) form method, for 2-D solids, Comput Mech, 33, 1, 2-14, (2003) · Zbl 1063.74105
[9] Karageorghis, A; Fairweather, G, The method of fundamental solutions for axisymmetric elasticity problems, Comput Mech, 25, 6, 524-532, (2000) · Zbl 1011.74005
[10] Benito, JJ; Urena, F; Gavete, L, Influence of several factors in the generalized finite difference method, Appl Math Model, 25, 12, 1039-1053, (2001) · Zbl 0994.65111
[11] Liszka, T., An interpolation method for an irregular net of nodes, Int J Numer Methods Eng, 20, 9, 1599-1612, (1984) · Zbl 0544.65006
[12] Liszka, T; Orkisz, J, The finite difference method at arbitrary irregular grids and its application in applied mechanics, Comput Struct, 11, 1, 83-95, (1980) · Zbl 0427.73077
[13] Gavete, L; Gavete, ML; Benito, JJ, Improvements of generalized finite difference method and comparison with other meshless method, Appl Math Model, 27, 10, 831-847, (2003) · Zbl 1046.65085
[14] Benito, JJ; Urena, F; Gavete, L; Alvarez, R, An h-adaptive method in the generalized finite differences, Comput Methods Appl Mech Eng, 192, 5-6, 735-759, (2003) · Zbl 1024.65099
[15] Ureña, F; Salete, E; Benito, JJ; Gavete, L, Solving third- and fourth-order partial differential equations using GFDM: application to solve problems of plates, Int J Comput Math, 89, 3, 366-376, (2012) · Zbl 1242.65217
[16] Gavete, L; Ureña, F; Benito, JJ; García, A; Ureña, M; Salete, E, Solving second order non-linear elliptic partial differential equations using generalized finite difference method, J Comput Appl Math, 318, 378-387, (2017) · Zbl 1357.65232
[17] Fan, CM; Huang, YK; Li, PW; Chiu, CL, Application of the generalized finite-difference method to inverse biharmonic boundary-value problems, Numer Heat Transf B, 65, 2, 129-154, (2014)
[18] Fan, CM; Li, PW; Yeih, WC, Generalized finite difference method for solving two-dimensional inverse Cauchy problems, Inverse Probl Sci Eng, 23, 5, 737-759, (2015) · Zbl 1329.65257
[19] Yang, C; Tang, DL; Atluri, S, Three-dimensional carotid plaque progression simulation using mesh less generalized finite difference method based on multi-year MRI patient-tracking data, Comp Model Eng Sci, 57, 1, 51-76, (2010) · Zbl 1231.74474
[20] Gavete, L; Urena, F; Benito, JJ; Salete, E, A note on the dynamic analysis using the generalized finite difference method, J Comput Appl Math, 252, 132-147, (2013) · Zbl 1290.74043
[21] Salete, E; Benito, JJ; Ureña, F; Gavete, L; Ureña, M; García, A, Stability of perfectly matched layer regions in generalized finite difference method for wave problems, J Comput Appl Math, 312, 231-239, (2017) · Zbl 1351.65060
[22] Hosseini, SM., Shock-induced two dimensional coupled non-Fickian diffusion-elasticity analysis using meshless generalized finite difference (GFD) method, Eng Anal Bound Elem, 61, 232-240, (2015) · Zbl 1403.74278
[23] Hosseini, SM., Application of a hybrid mesh-free method for shock-induced thermoelastic wave propagation analysis in a layered functionally graded thick hollow cylinder with nonlinear grading patterns, Eng Anal Bound Elem, 43, 56-66, (2014) · Zbl 1297.74164
[24] Hosseini, SM, Application of a hybrid mesh-free method based on generalized finite difference (GFD) method for natural frequency analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotubes, Comp Model Eng Sci, 95, 1, 1-29, (2013) · Zbl 1356.74010
[25] Bai, MR., Application of BEM (boundary element method)‐based acoustic holography to radiation analysis of sound sources with arbitrarily shaped geometries, J Acoust Soc Am, 92, 1, 533-549, (1992)
[26] Gu, Y; Chen, W; Zhang, C; He, X, A meshless singular boundary method for three-dimensional inverse heat conduction problems in general anisotropic media, Int J Heat Mass Transf, 84, 0, 91-102, (2015)
[27] Marin, L., A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations, Appl Math Comput, 165, 2, 355-374, (2005) · Zbl 1070.65115
[28] Karageorghis, A; Lesnic, D; Marin, L, The method of fundamental solutions for solving direct and inverse Signorini problems, Comput Struct, 151, 11-19, (2015)
[29] Liu, GR; Chen, XL, A mesh-free method for static and free vibration analyses of thin plates of complicated shape, J Sound Vib, 241, 5, 839-855, (2001)
[30] Liu, GR; Wang, XJ; Xi, ZC, Elastodynamic responses of an immersed to a Gaussian beam pressure, J Sound Vib, 233, 5, 813-833, (2000)
[31] Beskos, DE, Boundary element methods in dynamic analysis, Appl Mech Rev, 40, 1, 1-23, (1987)
[32] Chen, JT; Chang, MH; Chen, KH; Lin, SR, The boundary collocation method with meshless concept for acoustic eigenanalysis of two-dimensional cavities using radial basis function, J Sound Vib, 257, 4, 667-711, (2002)
[33] Harari, I; Barbone, E.; Slavutin, M; Shalom, R., Boundary infinite elements for the Helmholtz equation in exterior domains, Int J Numer Methods Eng, 41, 1105-1131, (1998) · Zbl 0911.76035
[34] Hall, WS; Mao, XQ, A boundary element investigation of irregular frequencies in electromagnetic scattering, Eng Anal Bound Elem, 16, 3, 245-252, (1995)
[35] Jeon, I-Y; Ih, J-G, On the holographic reconstruction of vibroacoustic fields using equivalent sources and inverse boundary element method, J Acoust Soc Am, 118, 6, 3473-3482, (2005)
[36] Chen, W; Fu, ZJ, Boundary particle method for inverse Cauchy problems of inhomogeneous Helmholtz equations, J Mar Sci Technol, 17, 3, 157-163, (2009)
[37] Benito, JJ; Urena, F; Salete, E; Muelas, A; Gavete, L; Galindo, R, Wave propagation in soils problems using the generalized finite difference method, Soil Dyn Earthq Eng, 79, 190-198, (2015)
[38] Chan, HF; Fan, CM; Kuo, CW, Generalized finite difference method for solving two-dimensional non-linear obstacle problems, Eng Anal Bound Elem, 37, 9, 1189-1196, (2013) · Zbl 1287.74056
[39] Marin, L., Regularized method of fundamental solutions for boundary identification in two-dimensional isotropic linear elasticity, Int J Solids Struct, 47, 24, 3326-3340, (2010) · Zbl 1203.74056
[40] Karageorghis, A; Lesnic, D, Application of the MFS to inverse obstacle scattering problems, Eng Anal Bound Elem, 35, 4, 631-638, (2011) · Zbl 1259.76046
[41] Qin, HH; Wen, DW, Tikhonov type regularization method for the Cauchy problem of the modified Helmholtz equation, Appl Math Comput, 203, 2, 617-628, (2008) · Zbl 1158.65070
[42] Fu, Z-J; Chen, W; Gu, Y, Burton-Miller-type singular boundary method for acoustic radiation and scattering, J Sound Vib, 333, 16, 3776-3793, (2014)
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