在本論文中提出以無網格法與無限元素法的耦合作為一種全新的混合方法來解決半無限域中的問題。對於半無限域可以劃分出受到邊界限制的近域以及延伸至無窮遠的遠域，而在本研究中，對於近域將以無網格法中的再生核質點法進行分析，遠域則使用動態無限元素法模擬波源消散至無窮遠的情形。其中，再生核質點法直接在定義域中以離散點建構出近似函數，可以避免傳統上使用網格為分析元素時發生的扭曲以及糾纏問題。對於動態無限元素法，透過適當的調整欲解決問題的波數和消散係數，在動態和靜態的半無限域問題上都可以得到良好且穩定的分析結果。此外，針對無網格法上有許多種數值積分方法可以使用，例如:高斯積分法、直接節點積分法、變分一致積分法、自然穩定節點積分法等。透過此研究成果可以得知，在使用直接節點積分法加上自然穩定節點積分法修正後，可以獲得最佳的分析結果。此外，針對非均勻離散化的分析之下，透過加入變分一致積分法可以獲得更加準確的分析結果。

This paper proposes a new coupled method of RKPM-DIEM. This efficient and stable coupled method can tackle both dynamic and static half-space problems.
For the half-space problem, the near field and far field can be defined as bounded and unbounded, respectively, and analyzed using the reproducing kernel particle method (RKPM) and the dynamic infinite element method (DIEM). In contrast to the element-based method, RKPM constructs the shape function in the domain using nodes, thereby avoiding mesh distortion and entanglement.
By adjusting the wave number and decay factor properly for DIEM, dynamic and static problems can achieve stable results. In addition, several meshfree integration techniques, such as the Gaussian integration, the direct nodal integration, the natural stabilized nodal integration, and the variationally consistent integration, are evaluated for their accuracy and stability and can obtain good results in both dynamic and static problems.

[1] L.B. Lucy, “A numerical approach to the testing of the fission hypothesis,”Astron. J., vol. 82, no. 12, pp. 1013–1024, 1977.
[2] R.A. Gingold and J.J. Monaghan, “Smoothed particle hydrodynamics-theory and application to non-spherical stars,” Mon. Not. R. Astron. Soc., vol. 181, pp. 375–389, 1977.
[3] B. Nayroles, G. Touzot, and P. Villon, “Generalizing the finite element method: Diffuse approximation and diffuse elements,” Comput. Mech., vol. 10, no. 5, pp. 307–318, 1992.
[4] T. Belytschko, Y.Y. LU, and L. Gu, “Element-free galerkin methods,” vol. 37, no. April 1993, pp. 229–256, 1994.
[5] W.K. Liu, S. Jun, and Y.F. Zhang, “Reproducing kernel particle methods,” Int. J. Numer. Methods Fluids, vol. 20, no. 8–9, pp. 1081–1106, 1995.
[6] J.S. Chen, M. Hillman, and S.W. Chi, “Meshfree Methods: Progress Made after 20 Years,” J. Eng. Mech., vol. 143, no. 4, 2017.
[7] J.S. Chen, C.T.Wu, S. Yoon, and Y. You, “A stabilized conforming nodal integration for Galerkin mesh-free methods,” Int. J. Numer. Methods Eng., vol. 50, no. 2, pp. 435–466, 2001.
[8] J.S. Chen, S. Yoon, and C.T. Wu, “Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods,” Int. J. Numer. Methods Eng., vol. 53, no. 12, pp. 2587–2615, 2002.
[9] P.C. Guan et al., “Semi-Lagrangian reproducing kernel formulation and application to modeling earth moving operations,” Mech. Mater., vol. 41, no. 6, pp. 670–683, 2009.
[10] P.C. Guan, S.W. Chi, J.S. Chen, T.R. Slawson, and M.J. Roth, “Semi-Lagrangian reproducing kernel particle method for fragment-impact problems,” Int. J. Impact Eng., vol. 38, no. 12, pp. 1033–1047, 2011.
[11] J.S. Chen, M. Hillman, and M. Rüter, “An arbitrary order variationally consistent integration for Galerkin meshfree methods,” Int. J. Numer. Methods Eng., vol. 95, pp. 387–418, 2013.
[12] M. Hillman and J.S. Chen, “An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics,” Int. J. Numer. Methods Eng., pp. 603–630, 2016.
[13] M. Hillman, J.S. Chen, and Y. Bazilevs, “Variationally consistent domain integration for isogeometric analysis,” Comput. Methods Appl. Mech. Eng., vol. 284, pp. 521–540, 2015.
[14] B. Engquist and A. Majda, “Absorbing boundary conditions for numerical simulation of waves,” Proc. Natl. Acad. Sci., vol. 74, no. 5, pp. 1765–1766, 1977.
[15] Z.S. Sacks, D.M. Kingsland, R. Lee, and J.F. Lee, “A Perfectly Matched Anisotropic Absorber for Use as an Absorbing Boundary Condition,” IEEE Trans. Antennas Propag., vol. 43, no. 12, pp. 1460–1463, 1995.
[16] C. Song and J.P. Wolf, “Finite-element modelling of unbounded media,” Simulation Practice and Theory, vol. 5, no. 6. pp. p27–p28, 1997.
[17] Y. He, H. Yang, and A.J. Deeks, “An element-free Galerkin (EFG) scaled boundary method,” Finite Elem. Anal. Des., vol. 62, pp. 28–36, 2012.
[18] Y. He, H. Yang, and A.J. Deeks, “An element-free galerkin scaled boundary method for steady-state heat transfer problems,” Numer. Heat Transf. Part B Fundam., vol. 64, no. 3, pp. 199–217, 2013.
[19] O.C. Zienkiewicz and P. Bettess, “Infinite elements in the study of fluid-structure interaction problems.,” pp. 133–172, 1976.
[20] J.A. Bettess and P. Bettess, “Infinite elements for static problems,” Eng. Comput., vol. 1, no. 1, pp. 4–16, 1984.
[21] A. Curnier, “A static infinite element,” Int. J. Numer. Methods Eng., vol. 19, no. 10, pp. 1479–1488, 1983.
[22] Y.B. Yang and H.H. Hung, “A 2.5D finite/infinite element approach for modelling visco-elastic bodies subjected to moving loads,” Int. J. Numer. Methods Eng., vol. 51, no. 11, pp. 1317–1336, 2001.
[23] Y.B. Yang, H.H. Hung, and D.W. Chang, “Train-induced wave propagation in layered soils using finite/infinite element simulation,” Soil Dyn. Earthq. Eng., vol. 23, no. 4, pp. 263–278, 2003.
[24] K.C. Lin, H.H. Hung, J.P. Yang, and Y.B. Yang, “Seismic analysis of underground tunnels by the 2.5D finite/infinite element approach,” Soil Dyn. Earthq. Eng., vol. 85, pp. 31–43, 2016.
[25] Y.B. Yang, H.H. Hung, K.C. Lin, and K.W. Cheng, “Dynamic Response of Elastic Half-Space with Cavity Subjected to P and SV Waves by Finite/Infinite Element Approach,” Int. J. Struct. Stab. Dyn., vol. 15, no. 7, 2015.
[26] R.F. Ungless, “An infinite finite element,” vol. 21, no. July, 1973.
[27] J.S. Chen, C. Pan, and C.T. Wu, “Large deformation analysis of rubber based on a reproducing kernel particle method,” Comput. Mech., vol. 19, no. 3, pp. 211–227, 1997.
[28] J.S. Chen, C. Pan, C.T. Wu, and W.K. Liu, “Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures,” Comput. Methods Appl. Mech. Eng., vol. 139, no. 1–4, pp. 195–227, 1996.
[29] K.M. Liew, H.K. Lim, M.J. Tan, and X.Q. He, “Analysis of laminated composite beams and plates with piezoelectric patches using the element-free Galerkin method,” Comput. Mech., vol. 29, no. 6, pp. 486–497, 2002.
[30] S. Fernández-Méndez and A. Huerta, “Imposing essential boundary conditions in mesh-free methods,” Comput. Methods Appl. Mech. Eng., vol. 193, no. 12–14, pp. 1257–1275, 2004.
[31] P. Bettess and O. Zienkiewicz, “Diffraction and refraction of surface waves using finite and infinite elements,” Int. J. Numer. Methods Eng., vol. 11, no. July 1976, pp. 1271–1290, 1977.
[32] J.W. Brown and R.V. Churchill, Complex Variables and Applications, 9th ed. McGraw-Hill Education, 2013.