無元素法是用來解決偏微分方程式的一種新興數值方法，且已成功地應用在不同的領域上。無元素法中常用的形狀函數包括了移動最小二乘法(MLS),、點插值法(PIM)、再生核近似法(RKPM)、徑向基底法(RPIM)等。克利金法為地理統計上常使用的內插方法，但很少被應用於無元素法的分析。
本研究實現並評估各種不同組合的克利金法與半變異圖於無元素法並用來解算固體力學問題。透過這些評估，克利金法於無元素分析時之可用性與可能發生的問題可以被瞭解與探討。
除了克利金內插於無元素法上的使用外，本研究同時利用克利金法去分析領域的耦合，且此領域耦合有一些實際的應用：1)在有限元素法中可簡化網格的產生、2)減少無元素法的計算時間、3)可使用採用非均質材料於無元素法計算上。本研究施加位移諧和的條件在領域耦合的邊界上以達到領域耦合的目的，而所使用的程序則可同時應用於接合網格及非接合網格，而耦合的兩領域可為有限元素法與無元素法的任意組合，並在耦合項次的計算上亦採用克利金內插。此程序經驗證證實可行，並與精確解比較時顯示其可產生合理的解答。

Meshfree methods are new class of numerical methods for solving partial differential equations, and they have found many applications in different fields. Commonly used shape functions in meshfree methods are moving least-square method (MLS), point-interpolation method (PIM), reproducing kernel particle method (RKPM), radial-basis point interpolation method (RPIM), etc. Kriging interpolant, a widely used interpolation scheme in geostatistics, is rarely applied in the context of meshfree methods.
In this work, various combinations of Kriging methods and semivariograms are implemented, evaluated, and incorporated in meshfree methods for solving solid mechanics problems. Through these evaluations, the applicability and issues on using Kriging interpolants in meshfree methods can be studied and identified.
In addition to the study of using Kriging interpolants as shape functions in meshfree methods, this work also studies the use of Kriging interpolants as domain couplers. Domain coupling has many practical applications: 1) simplifications of mesh generation in finite element methods, 2) reduce computation time of meshfree methods, and 3) introducing heterogeneous material properties in meshfree methods. In this work, a procedure for coupling two domains with matching or nonmatching meshes by imposing displacement continuity condition across the boundary is implemented. The procedure can coupled domains formulated using any combination of finite element method or meshfree method, and the coupling terms are computed using Kriging interpolants. It is shown that the proposed procedure for coupling can work, and produce reasonable results when compare to exact solutions.

【1】. 賴銘峰(2010)，「應用GIS於集水區降雨空間特性之研究－以陳有蘭溪為例」，朝陽科技大學營建工程系碩士論文。
【2】. 林宛儒(2010) ，「應用地理統計方法於雨量站網調整之研究－以臺北水源特定區為例」，逢甲大學水利工程與資源保育學系碩士論文。
【3】. Dorsel , D. & Breche , T.L. (1997), “Environmental Sampling & Monitoring Primer”, http://www.cee.vt.edu/ewr/environmental/teach/smprimer/kriging/kriging.html#TheoryEqns
【4】. “Wikipedia”, http://en.wikipedia.org/wiki/Kriging
【5】. Bohlin, G. (2005), “KRIGING”, http://people.ku.edu/~gbohling/cpe940/
【6】. 李天浩(2009)，「應用克利金法建立高解析度網格點氣象數據之研究」，交通部中央氣象局委託研究計畫成果報告
【7】. “Universal Kriging Algorithm”http://spatial-analyst.net/ILWIS/htm/ilwisapp/universal_kriging_algorithm.htm
【8】. 吳育霖(2000) ，「苗栗地區之重力研究」，國立中央大學地球物理研究所碩士論文。
【9】. Lucy, L.B. (1977 ), “Smoothed Particle Hydrodynamics”, Astrophysical Journal, 82,1013-1024.
【10】. 蔡松栩(2010) ，「基於無元素分析之迭代設計軟體系統」，國立台灣科技大學營建工程系碩士論文。
【11】. Liu, G. R. and Gu, Y. T. ( 2005 ), “An introduction to meshfree methods and their programming” , Springer, ISBN-13: 978-1402032288
【12】. Li, S. and Liu, W. K. ( 2002 ), “Meshfree and particle methods and their applications” , ASME Applied Mechanics Review, 54, 1-34.
【13】. Wong, F. T. and W. Kanok-Nukulchai (2009). "ON THE CONVERGENCE OF THE KRIGING-BASED FINITE ELEMENT METHOD." International Journal of Computational Methods 6(1): 93-118.
【14】. Dai, K. Y., G. R. Liu, et al. (2003). "Comparison between the radial point interpolation and the Kriging interpolation used in meshfree methods." Computational Mechanics 32(1-2): 60-70.
【15】. Bui, T. Q., T. N. Nguyen, et al. (2009). "A moving Kriging interpolation-based meshless method for numerical simulation of Kirchhoff plate problems." International Journal for Numerical Methods in Engineering 77(10): 1371-1395.
【16】. Gu, L. (2003). "Moving kriging interpolation and element-free Galerkin method." International Journal for Numerical Methods in Engineering 56(1): 1-11.
【17】. Gu, Y. T., Wang, Q. X., et al. (2007). "A meshless local Kriging method for large deformation analyses." Computer Methods in Applied Mechanics and Engineering 196(9-12): 1673-1684.
【18】. Belytschko, T., Organ, D. and Krongauz,Y. (1995). “A coupled finite element-element-free Galerkin method.” Computational Mechanics, 17, 186-195.
【19】. Puso, M. A. and T. A. Laursen (2003). "Mesh tying on curved interfaces in 3D." Engineering Computations 20(3): 305-319
【20】. Tian, R. and G. Yagawa (2007). "Non-matching mesh gluing by meshless interpolation—an alternative to Lagrange multipliers." International Journal for Numerical Methods in Engineering 71(4): 473-503.
【21】. Karutz, H., R. Chudoba, et al. (2002). "Automatic adaptive generation of a coupled finite element/element-free Galerkin discretization." Finite Elements in Analysis and Design 38(11): 1075-1091.
【22】. Li, G. (2003). "Coupling of the mesh-free finite cloud method with the boundary element method: a collocation approach." Computer Methods in Applied Mechanics and Engineering 192(20-21): 2355-2375.
【23】. TSAI, H.C., et al. (2004). Element free formulation used for connecting domain boundaries. Taipei, TAIWAN, PROVINCE DE CHINE, Chinese Institute of Engineers.
【24】. Tian, R. and G. Yagawa (2007). "Non-matching mesh gluing by meshless interpolation—an alternative to Lagrange multipliers." International Journal for Numerical Methods in Engineering 71(4): 473-503.
【25】. 林建勳(2006)，「元素釋放法與有限元素法之聯合應用」，國立台灣科技大學營建工程系博士論文。
【26】. Huerta, A., S. Fernandez-Mendez, et al. (2004). "A comparison of two formulations to blend finite elements and mesh-free methods." Computer Methods in Applied Mechanics and Engineering 193(12-14): 1105-1117.
【27】. Miles, R., Hamilton, K. (2006). “Learning UML 2.0”, Sebastopol, CA : O'Reilly, 2006. - 269 p.