A hybrid LBKM/Semi-analytical approach for SH-Wave scattering in an elastic half-space

Shuwei Zhang; Shuihuai Yang; Xing Wei; Haowen Wu; Linlin Sun; Shuwei Zhang; Shuihuai Yang; Xing Wei; Haowen Wu; Linlin Sun

doi:10.3934/math.2026877

AIMS Mathematics

2026, Volume 11, Issue 7: 21674-21703. doi: 10.3934/math.2026877

Previous Article Next Article

Research article

A hybrid LBKM/Semi-analytical approach for SH-Wave scattering in an elastic half-space

1.
School of mathematics and statistics, Nantong University, Nantong, Jiangsu 226019, China
2.
School of Civil Engineering & Architecture, East China Jiaotong University, Nanchang 330013, China

Received: 12 March 2026 Revised: 11 June 2026 Accepted: 29 June 2026 Published: 22 July 2026
MSC : 65N85, 74J20

In this paper, a novel method combining the localized boundary knot method (LBKM) and a semi-analytical formulation is developed for analyzing SH wave scattering problems in half spaces. To solve the problem effectively, the computational domain is divided into a finite interior domain containing all scatterers and an infinite exterior region by a semicircle. In the finite domain, the LBKM, which is accurate and yields a sparse coefficient matrix, is employed. For the infinite exterior region, the solution is expressed by a semi-analytical formulation that exactly satisfies both the governing equation and the Sommerfeld radiation condition. The two formulations are then coupled through interface conditions. Three numerical examples involving the scattering of SH waves by canyons, underground tunnels, and inclusions were conducted to validate the proposed method. Its superior performance is illustrated through a comparison with the generalized finite difference method, localized method of fundamental solution, and radial point interpolation method. For the cases without analytical solutions, the effectiveness of the proposed approach is validated via the results from the finite element method.
- localized boundary knot method,
- Meshless method,
- SH wave,
- semi-analytical formulation,
- half spaces
Citation: Shuwei Zhang, Shuihuai Yang, Xing Wei, Haowen Wu, Linlin Sun. A hybrid LBKM/Semi-analytical approach for SH-Wave scattering in an elastic half-space[J]. AIMS Mathematics, 2026, 11(7): 21674-21703. doi: 10.3934/math.2026877

Related Papers:

Abstract

In this paper, a novel method combining the localized boundary knot method (LBKM) and a semi-analytical formulation is developed for analyzing SH wave scattering problems in half spaces. To solve the problem effectively, the computational domain is divided into a finite interior domain containing all scatterers and an infinite exterior region by a semicircle. In the finite domain, the LBKM, which is accurate and yields a sparse coefficient matrix, is employed. For the infinite exterior region, the solution is expressed by a semi-analytical formulation that exactly satisfies both the governing equation and the Sommerfeld radiation condition. The two formulations are then coupled through interface conditions. Three numerical examples involving the scattering of SH waves by canyons, underground tunnels, and inclusions were conducted to validate the proposed method. Its superior performance is illustrated through a comparison with the generalized finite difference method, localized method of fundamental solution, and radial point interpolation method. For the cases without analytical solutions, the effectiveness of the proposed approach is validated via the results from the finite element method.

References

[1]	M. I. Todorovska, V. W. Lee, Surface motion of shallow circular alluvial valleys for incident plane SH waves-analytical solution, Soil Dyn. Earthquake Eng., 10 (1991), 192–200. https://doi.org/10.1016/0267-7261(91)90033-v doi: 10.1016/0267-7261(91)90033-v
[2]	C. Smerzini, J. Avilés, F. J. Sánchez-Sesma, R. Paolucci, A. Santini, N. Moraci, Analytical solutions for the seismic response of underground structures under SH wave propagation, AIP Conf. Proc., 1020 (2008), 674–683. https://doi.org/10.1063/1.2963900 doi: 10.1063/1.2963900
[3]	N. Zhang, X. Chen, Y. Gao, D. Dai, Analytical solution to scattering of SH waves by a circular lined tunnel embedded in a semi–circular alluvial valley in an elastic half–space, Tunn. Undergr. Space Technol., 106 (2020), 103615. https://doi.org/10.1016/j.tust.2020.103615 doi: 10.1016/j.tust.2020.103615
[4]	T. Sun, P. Wang, G. Zhang, Y. Chai, Transient analyses of wave propagations in nonhomogeneous media employing the novel finite element method with the appropriate enrichment function, Comput. Math. Appl., 129 (2023), 90–112. https://doi.org/10.1016/j.camwa.2022.10.004 doi: 10.1016/j.camwa.2022.10.004
[5]	C. Xu, Q. Liu, J. Wu, P. Deng, P. Liu, H. Zhang, Numerical study on P-wave propagation across the jointed rock masses by the combined finite-discrete element method, Comput. Geotech., 142 (2022), 104554. https://doi.org/10.1016/j.compgeo.2021.104554 doi: 10.1016/j.compgeo.2021.104554
[6]	P. Destuynder, L. Hervella-Nieto, P. M. López-Pérez, J. Orellana, A. Prieto, A modal-based Partition of Unity Finite Element Method for elastic wave propagation problems in layered media, Comput. Struct., 265 (2022), 106759. https://doi.org/10.1016/j.compstruc.2022.106759 doi: 10.1016/j.compstruc.2022.106759
[7]	L. Huang, X. Liu, Z. Liu, T. Ai, Scattering of seismic waves by reef sites in saturated layered sea areas-simulated using the indirect boundary element method, Soil Dyn. Earthquake Eng., 183 (2024), 108804. https://doi.org/10.1016/j.soildyn.2024.108804 doi: 10.1016/j.soildyn.2024.108804
[8]	L. Huang, Z. Liu, C. Wu, J. Liang, Q. Pei, A three-dimensional indirect boundary integral equation method for the scattering of seismic waves in a poroelastic layered half-space, Eng. Anal. Boundary Elem., 135 (2022), 167–181. https://doi.org/10.1016/j.enganabound.2021.11.012 doi: 10.1016/j.enganabound.2021.11.012
[9]	L. Chen, H. Lian, Y. Xu, S. Li, Z. Liu, E. Atroshchenko, et al., Generalized isogeometric boundary element method for uncertainty analysis of time-harmonic wave propagation in infinite domains, Appl. Math. Model., 114 (2023), 360–378. https://doi.org/10.1016/j.apm.2022.09.030 doi: 10.1016/j.apm.2022.09.030
[10]	M. Rasht-Behesht, C. Huber, K. Shukla, G. E. Karniadakis, Physics-informed neural networks (PINNs) for wave propagation and full waveform inversions, J. Geophys. Res.: Solid Earth, 127 (2022), e2021JB023120. https://doi.org/10.1029/2021JB023120 doi: 10.1029/2021JB023120
[11]	L. Qiu, Y. Wang, Y. Gu, Q. H. Qin, F. Wang, Adaptive physics-informed neural networks for dynamic coupled thermo-mechanical problems in large-size-ratio functionally graded materials, Appl. Math. Model., 140 (2025), 115906. https://doi.org/10.1016/j.apm.2024.115906 doi: 10.1016/j.apm.2024.115906
[12]	J. Li, L. Zhang, S. Cai, N. Li, Regularized singular boundary method for calculating wave forces on three-dimensional large offshore structure, Appl. Math. Lett., 149 (2024), 108931. https://doi.org/10.1016/j.aml.2023.108931 doi: 10.1016/j.aml.2023.108931
[13]	L. Sun, Z. Fu, Z. Chen, A localized collocation solver based on fundamental solutions for 3D time harmonic elastic wave propagation analysis, Appl. Math. Comput., 439 (2023), 127600. https://doi.org/10.1016/j.amc.2022.127600 doi: 10.1016/j.amc.2022.127600
[14]	X. Wei, W. Luo, 2.5D singular boundary method for acoustic wave propagation, Appl. Math. Lett., 112 (2021), 106760. https://doi.org/10.1016/j.aml.2020.106760 doi: 10.1016/j.aml.2020.106760
[15]	M. Xiang, H. Qi, J. Guo, Technical note: the anti-plane scattering of SH waves by the non-circular cavity in an infinite strip, J. Earthquake Eng., 26 (2022), 3228–3245. https://doi.org/10.1080/13632469.2020.1793843 doi: 10.1080/13632469.2020.1793843
[16]	Z. Ge, Simulation of the seismic response of sedimentary basins with constant-gradient velocity along arbitrary direction using boundary element method: SH case, Earthquake Sci., 23 (2010), 149–155. https://doi.org/10.1007/s11589-009-0060-z doi: 10.1007/s11589-009-0060-z
[17]	S. Mojtabazadeh-Hasanlouei, M. Panji, M. Kamalian, Attenuated orthotropic time-domain half-space BEM for SH-wave scattering problems, Geophys. J. Int., 229 (2022), 1881–1913. https://doi.org/10.1093/gji/ggac032 doi: 10.1093/gji/ggac032
[18]	C. Song, H. Geng, Y. Wang, U. B. Waheed, C. Liu, Simultaneous P- and S-wave seismic traveltime tomography using physics-informed neural networks, Geophys. Prospect., 73 (2025), e70034. https://doi.org/10.1111/1365-2478.70034 doi: 10.1111/1365-2478.70034
[19]	K. Shaker, M. Eskandari-Ghadi, S. Mohammadi, Meshless method for wave propagation in poroelastic transversely isotropic half-space with the use of perfectly matched layer, Int. J. Numer. Anal. Methods Geomech., 48 (2024), 3751–3779. https://doi.org/10.1002/nag.3797 doi: 10.1002/nag.3797
[20]	S. Hou, J. Li, L. Sun, A GFDM with radial PML for SH-wave scattering problems in an orthotropic half plane, Appl. Math. Lett., 171 (2025), 109661. https://doi.org/10.1016/j.aml.2025.109661 doi: 10.1016/j.aml.2025.109661
[21]	H. Xue, X. Zhang, J. Cheng, Transient analyses of underwater acoustic propagation with the modified meshfree radial point interpolation method and Newmark time integration techniques, Ocean Eng., 304 (2024), 117842. https://doi.org/10.1016/j.oceaneng.2024.117842 doi: 10.1016/j.oceaneng.2024.117842
[22]	C. Z. Shi, H. Zheng, P. H. Wen, Y. C. Hon, The local radial basis function collocation method for elastic wave propagation analysis in 2D composite plate, Eng. Anal. Boundary Elem., 150 (2023), 571–582. https://doi.org/10.1016/j.enganabound.2023.02.021 doi: 10.1016/j.enganabound.2023.02.021
[23]	Z. Liu, Z. Wang, L. Guo, D. Wang, F. Wu, A fast-multi-pole accelerated method of fundamental solutions for 2-D broadband scattering of SH waves in an infinite half space, J. Vibroeng., 21 (2019), 250–264. https://doi.org/10.21595/jve.2018.19515 doi: 10.21595/jve.2018.19515
[24]	C. Yang, Efficient forward and inverse approaches for guided-wave scattering, Eng. Anal. Boundary Elem., 186 (2026), 106680. https://doi.org/10.1016/j.enganabound.2026.106680 doi: 10.1016/j.enganabound.2026.106680
[25]	C. Yang, J. Nakaoka, S. Hirose, Semi-analytical FEM for modelling 3D guided waves scattering in an infinite plate, Appl. Math. Model., 148 (2025), 116236. https://doi.org/10.1016/j.apm.2025.116236 doi: 10.1016/j.apm.2025.116236
[26]	C. Yang, R. Qin, S. Hirose, Semi-analytical FEM for modeling 2D guided wave scattering by interface delamination in composite laminates, Eng. Anal. Boundary Elem., 179 (2025), 106427. https://doi.org/10.1016/j.enganabound.2025.106427 doi: 10.1016/j.enganabound.2025.106427
[27]	C. S. Chen, A. Karageorghis, Y. Li, On choosing the location of the sources in the MFS, Numer. Algors., 72 (2016), 107–130. https://doi.org/10.1007/s11075-015-0036-0 doi: 10.1007/s11075-015-0036-0
[28]	W. Chen, M. Tanaka, A meshless, integration-free, and boundary-only RBF technique, Comput. Math. Appl., 43 (2002), 379–391. https://doi.org/10.1016/s0898-1221(01)00293-0 doi: 10.1016/s0898-1221(01)00293-0
[29]	A. Canelas, B. Sensale, A boundary knot method for harmonic elastic and viscoelastic problems using single-domain approach, Eng. Anal. Boundary Elem., 34 (2010), 845–855. https://doi.org/10.1016/j.enganabound.2010.05.010 doi: 10.1016/j.enganabound.2010.05.010
[30]	L. Sun, C. Zhang, Y. Yu, A boundary knot method for 3D time harmonic elastic wave problems, Appl. Math. Lett., 104 (2020), 106210. https://doi.org/10.1016/j.aml.2020.106210 doi: 10.1016/j.aml.2020.106210
[31]	M. Lei, T. Li, H. Meng, Comparative analysis of the improved boundary knot and fundamental solutions methods for complex multi-connected Helmholtz-type equations, Appl. Math. Model., 142 (2025) 115971. https://doi.org/10.1016/j.apm.2025.115971 doi: 10.1016/j.apm.2025.115971
[32]	F. Wang, Y. Gu, W. Qu, C. Zhang, Localized boundary knot method and its application to large-scale acoustic problems, Comput. Methods Appl. Mech. Eng., 361 (2020), 112729. https://doi.org/10.1016/j.cma.2019.112729 doi: 10.1016/j.cma.2019.112729
[33]	X. Yue, F. Wang, P. W. Li, C. M. Fan, Local non-singular knot method for large-scale computation of acoustic problems in complicated geometries, Comput. Math. Appl., 84 (2021), 128–143. https://doi.org/10.1016/j.camwa.2020.12.014 doi: 10.1016/j.camwa.2020.12.014
[34]	Q. Zhang, Z. Ji, L. Sun, An improved localized boundary knot method for 3D acoustic problems, Appl. Math. Lett., 149 (2024), 108900. https://doi.org/10.1016/j.aml.2023.108900 doi: 10.1016/j.aml.2023.108900
[35]	F. Wang, C. Wang, Z. Chen, Local knot method for 2D and 3D convection–diffusion–reaction equations in arbitrary domains, Appl. Math. Lett., 105 (2020), 106308. https://doi.org/10.1016/j.aml.2020.106308 doi: 10.1016/j.aml.2020.106308
[36]	D. L. Young, K. H. Chen, C. W. Lee, Singular meshless method using double layer potentials for exterior acoustics, J. Acoust. Soc. Am., 119 (2006), 96–107. https://doi.org/10.1121/1.2141130 doi: 10.1121/1.2141130
[37]	J. Huang, A. Wood, Numerical simulation of electromagnetic scattering induced by an overfilled cavity in the ground plane, IEEE Antennas Wireless Propag. Lett., 4 (2005), 224–228. https://doi.org/10.1109/lawp.2005.851101 doi: 10.1109/lawp.2005.851101
[38]	Z. Tang, Z. Fu, D. Zheng, J. Huang, Singular boundary method to simulate scattering of SH wave by the canyon topography, Adv. Appl. Math. Mech., 10 (2018), 912–924. https://doi.org/10.4208/aamm.oa-2017-0301 doi: 10.4208/aamm.oa-2017-0301
[39]	W. Qu, C. M. Fan, Y. Gu, Localized method of fundamental solutions for interior Helmholtz problems with high wave number, Eng. Anal. Boundary Elem., 107 (2019), 25–32. https://doi.org/10.1016/j.enganabound.2019.06.018 doi: 10.1016/j.enganabound.2019.06.018
[40]	L. Sun, Z. Ji, Q. Zhang, X. Wei, A localized meshless collocation method based on fundamental solutions for 3D uncoupled transient thermoelastic problems, Int. J. Heat Mass Transfer, 232 (2024), 125945. https://doi.org/10.1016/j.ijheatmasstransfer.2024.125945 doi: 10.1016/j.ijheatmasstransfer.2024.125945
[41]	Ö Oruç, A local meshfree radial point interpolation method for Berger equation arising in modelling of thin plates, Appl. Math. Model., 122 (2023), 555–571. https://doi.org/10.1016/j.apm.2023.03.014 doi: 10.1016/j.apm.2023.03.014
[42]	L. Chen, R. Huo, H. Lian, B. Yu, M. Zhang, S. Natarajan, et al., Uncertainty quantification of 3D acoustic shape sensitivities with generalized nth-order perturbation boundary element methods, Comput. Methods Appl. Mech. Eng., 433 (2025), 117464. https://doi.org/10.1016/j.cma.2024.117464 doi: 10.1016/j.cma.2024.117464
[43]	H. Jin, X. Wei, L. Sun, Z. Fu, Dynamic analysis of 3D saturated poroelastic media with singular boundary method, Comput. Math. Appl., 217 (2026), 115–137. https://doi.org/10.1016/j.camwa.2026.05.024 doi: 10.1016/j.camwa.2026.05.024

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)