不均勻佈點之無網格法主要用於解決徑向基底函數配置方法(RBFCM)中因為使用大量均勻佈點所造成之非條件化問題。在解析解可獲得的前提下，穩態及暫態的地下水流問題將用於測試各種型態之不均勻佈點方式之效果，並在相同條件下與均勻佈點方式所得到之結果進行比較。研究中所呈現的地下水流問題，目的為了解座標系統和邊界條件對於數值結果的影響。本研究於徑向基底函數配置方法中所使用的徑向基底函數為含有形狀參數(c)的multiquardrics (MQ)，其中最佳形狀參數之選擇在本研究也運用回歸分析進行探討。研究中也針對形狀參數，以及使用有限差分法所引進之時間參數進行在MQ-RBFCM模式之敏感性分析。結果顯示，比起均勻佈點法，使用不均勻佈點法可有效節省總佈點數至上千甚至更多，且達到與均勻佈點法相同之數值誤差。研究中所提出之最佳不均勻佈點型態在數值模式受到邊界效應影響前與Theis’s解進行比較，其平均相對誤差可減至0.033%。迴歸分析中，得到一些節點相距之特定距離與形狀參數之關係式，有助於使用不均勻佈點之無網格法於地下水流之模擬。

Non-uniformly-distributed nodes in meshless method were proposed to overcome the ill-conditioned problem caused by the large number of uniformly-distributed nodes use in global radial basis function collocation method. Both steady and transient groundwater flow problems with analytical solutions in literature are proposed to demonstrate the efficiency of the proposed configurations of non-uniformly-distributed nodes. The results are compared to the one uses uniformly-distributed nodes with the same condition. The groundwater flow problems were designed to explore the effect of coordinate system and boundary conditions. The radial basis function used in RBFCM is multiquardrics (MQ) which contains shape parameters (c). The way of choosing optimal c was explored by performing regressions. Sensitivity analysis of the parameters in MQ-RBFCM focuses on the shape parameter and time parameters for finite difference method. Results showed that by applying the configurations of non-uniformly-distributed nodes, the total number of nodes can be effectively reduced thousands and more to reach the same accuracy compared with uniformly-distributed nodes. For the best configuration of non-uniformly-distributed nodes, the average relative error is down to 0.033 % compared with Theis’s solution before the effect of boundary occuring. In regression analysis, several useful relations between specific distances of source nodes and shape parameter have been derived when applying non-uniformly-node-distributed meshless method in groundwater flow modeling.

1.Amaziane, B., Naji, A. and Ouazar, D., Radial basis function and genetic algorithms for parameter identification to some groundwater flow problems, CMC, Vol. 1, No.2, pp.117-128, (2004).
2.Atluri, S.N. and Shen, S., The meshless local Petrov-Galerkin (MLPG) method : a simple and less-costly alternative to the finite element and boundary element method, Computer Modelling in Engineering and Sciences, Vol. 3, pp. 11-52, (2002).
3.Bergman, S., Functions satisfying certain partial differential equations of elliptic type and their representation, Duke Math. J., Vol. 14, pp. 349–366, (1947).
4.Belytschko, T., Lu, Y.Y. and Gu, L., Element-free Galerkin methods. International Journal for Numerical Methods in Engineering, Vol. 37, pp. 229–256, (1994)
5.Brebbia, C. A., Telles, J. C. F. and Wrobel, L. C., Boundary element techniques : theory and applications in engineering, Berlin, Heidelberg, Springer-Verlag.
6.Domenico, P. A. and Schwartz, F. W., Physical and Chemical Hydrogeology, John Wiley & Sons, New York, (1998).
7.Foley, T. A., Interpolation and approximation of 3-D and 4-D scattered data, Comput. Math. Applic., Vol. 13, No. 8, pp. 711-740, (1987).
8.Franke, R., Scattered data interpolation : tests of some methods, Maths Comput., Vol. 38, No. 157, pp. 181-200, (1982).
9.Franke, R. and Schaback, R., Solving partial differential equations by collocation using radial functions, Applied Mathematics and Computation, Vol. 93, pp. 73-82, (1998).
10.Hardy, R. L., Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., Vol. 76, pp. 1905-1915, (1971).
11.Kansa, E. J., Multiquadrics-¬a scattered data approximation scheme with applications to computational fluid-dynamics-I. Surface approximations and partial derivatives, Computers and Mathematics with Applications, Vol. 19, No. 8/9, pp. 127-145, (1990a).
12.Kansa, E. J., Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics-II. Solutions to parabolic, hyperbolic and elliptic partial differential equations, Computers and Mathematics with Applications, Vol. 19, No. 8/9, pp. 147-161, (1990b).
13.Kupradze, V.D. and Aleksidze, M.A., The method of functional equations for the approximate solution of certain boundary value problem, Zh. Vych. Mat., Vol. 4, No. 4, pp. 683, (1964).
14.Liu, G. R. and Gu, Y. T., An Introduction to Meshfree Methods and their Programming, Spriger Dordrecht., Berlin, Heidelberg, New York, (2005).
15.Mai-Duy, N. and Tran-Cong, T., Numerical solution of differential equations using multiquadric radial basis function networks, Neural Networks, Vol. 14, pp185-199, (2001).
16.Mai-Duy, N. and Tran-Cong, T., Indirect RBFN method with thin plate splines for numerical solution of differential equations, CMES, Vol. 4, No. 1, pp. 85-102, (2003).
17.Meenal, M. and Eldho, T. I., Simulation of groundwater flow in unconfined aquifer using meshfree pointcollocation method, Engineering Analysis with Boundary Elements, Vol. 35, pp. 700-707, (2011).
18.Meenal, M. and Eldho, T. I., Meshless point collocation method for 1D and 2D groundwater flow simulation, ISH Journal of Hydraulic Engineering, Vol. 17, No. 1, pp. 71-87, (2011).
19.Moody, J. and Darken, C. J., Fast learning in networks of locally-tuned processing units, Neural Computations, Vol. 1, pp. 281-294, (1989).
20.Patankar, S. V., Numerical heat transfer and fluid flow, New York, McGraw-Hill, (1980).
21.Park, Y. C. and Leap, D. I., Modeling groundwater flow by the element-free Galerkin (EFG) method, Geosciences Journal, Vol. 4, No. 3, pp.231-241, (2000).
22.Schwartz, F. W. and Zhang, H., Fundamentals of Groundwater, Wiley, (2003).
23.Smith, G.D., Numerical solutions of partial differential equations : finite difference methods, Oxford, Clarendon Press, (1978).
24.Tarwater, A. E., A parameter study of Hardy’s multiquadric method for scattered data interpolation, Lawerence Livermore National Laboratory, Technical Report UCRL-563670, (1985).
25.Theis, C. V., The relation between the lowering of the piezometric surface and rate and duration of discharge of a well using groundwater storage, Trans. Am. Geophys. Union, Vol. 2, pp. 519-524, (1935).
26.Thiem, G., Hydrologische Methode, Gebhardt, Leipzig, (1906).
27.Thomas, G. B., Calculus and Analytic Geometry, 3rd edition, Addison-Wesley Publishing Company, Reading, Massachusetts, (1972).
28.Trefftz, E., Ein Gegenstu ̈ck zum ritzschen Verfahren. Proc. 2nd Int. Cong. Appl. Mech., Zurich, pp. 131-37, (1926).
29.Vekua, N., Novye metody res ̌enija elliptic ̌kikh uravnenij (New Methods for Solving Elliptic Equations), OGIZ, Moskow and Leningrad, (1948).
30.Wang, J. G. and Liu, G. R., On the optimal shape parameters of radial basis functions used for 2-D meshless methods, Comput. Methods Appl. Mech. Engrg., Vol. 191, pp. 2611-2630, (2002).
31.Wu, Z. and Hon, Y. C., Convergence error estimate in solving free boundary diffusion problem by radial basis functions method, Eng. Anal. Bound. Elem., Vol. 27, pp. 73-79, (2003).
32.Young, D. L., Jane, S. C., Lin, C. Y., Chiu, C. L. and Chen, C. S., Solutions of 2D and 3D Stokes laws using multiquardrics method, Engineering Analysis with Boundary Elements, Vol. 28, pp. 1233-1243, (2004).
33.Young, D. L., Chen, C. S. and Wong, T. K., Solution of Maxwell’s equations using MQ method, CMC, Vol. 2, No. 4, pp. 267-276, (2005).
34.Zerroukat, M., Power, H. and Chen, C. S., A numerical method for heat transfer problems using collocation and radial basis functions, International Journal for Numerical Methods in Engineering, Vol. 42, pp. 1263-1278, (1998).
35.Zienkiewicz, O. C., The finite element method in engineering science, McGraw-Hill, New York, (1971).