本文發展了無網格徑向點插值法(RPIM)應用在二維平板破裂之位移、應力分析，作為一種新興的數值方法，在處理不連續材料問題時，能避免傳統有限元素法難以解決的問題。徑向基函數為無網格法的一種，由於收斂速度快、形式簡單、形狀函數具備無窮階可微且連續等優點備受學者重視。為了驗證RPIM算法，模擬懸臂樑於受力後的位移、應力情況，與文獻結果一致，並討論不同形狀函數、節點數對此方法的影響。接著本文模擬具有凹槽的樑，透過不同形式的凹槽了解材料不連續對結構的影響，並討論不同深度的凹槽，使其在RPIM都有相當好的結果。最後藉由不連續結構的經驗及可視法模擬裂紋對結構的影響，可以發現其位移、應力在不同形式下的裂紋有不同的分佈方式，並討論不同深度的裂紋在RPIM方法求解都相當連續且平滑。

This thesis presents a meshless radial point interpolation method (RPIM) model for simulation of two dimensional displacement and stress of fracture analysis in the plate. As a kind of newly developed numerical methods, meshfree method can overcome discontinuous material problems which trouble the conventional finite element method. As a kind of meshfree method, radial basis function received many researchers’ interest due to its characteristics of rapid convergence, simple expression, shape function are infinitely differentiable and continuous,etc. In order to verify the model, the displacement and stress condition of cantilever beam in force, results is well-agreement of previously literature. And discuss the impact of different shape functions, different nodes by this method. Then this paper simulates a beam with notch, understand the discontinuous material and different depths impact on the linear elastic structure , making it through different forms of notch has very good results in RPIM. Finally, the effect of the crack on the structure by notch and visibility criterion was present. Different distribution of displacement and stress can be discussed in different forms of cracks, and the different depth of crack also had well-agreement by RPIM.

[1] Asprone, D., Auricchio, F., & Reali, A. (2014). Modified Finite Particle Method: applications to elasticity and plasticity problems. International Journal of Computational Methods, 11(01).
[2] Alan T. Zehnder.(2012). Fracture Mechanics. ,Springer Netherlands.
[3] Belytschko, T., Lu, Y. Y., & Gu, L. (1995). Crack propagation by element-free Galerkin methods. Engineering Fracture Mechanics, 51(2), 295-315.
[4] Cao,Y.,Yao,L.Q., & Yi,S.C.(2014). A weighted nodal-radial point interpolation meshless method for 2D solid problems.Engineering Analysis with Boundary Elements. 88–100
[5] Chen, Y., Lee, J. D., & Eskandarian, A. (2006). Meshless methods in solid mechanics. Heidelberg: Springer.
[6] Duflot, M., & Nguyen-Dang, H. (2004). Fatigue crack growth analysis by an enriched meshless method. Journal of Computational and Applied Mathematics, 168(1), 155-164.
[7] Engle Jr, R. M. (1970). CRACKS, a FORTRAN IV digital computer program for crack propagation analysis (No. AFFDL-TR-70-107). AIR FORCE FLIGHT DYNAMICS LAB WRIGHT-PATTERSON AFB OH.
[8] Fleming, M., Chu, Y. A., Moran, B., Belytschko, T., Lu, Y. Y., & Gu, L. (1997). Enriched element-free Galerkin methods for crack tip fields. International Journal for Numerical Methods in Engineering, 40(8), 1483-1504.
[9] Graça, A., Cardoso, R. P., & Yoon, J. W. (2013). Subspace analysis to alleviate the volumetric locking in the 3D solid-shell EFG method. Journal of Computational and Applied Mathematics, 246, 185-194.
[10] Hong, W. T. (2009). A meshless method with enriched basis functions for singularity problems ,Doctoral dissertation, THE UNIVERSITY OF NORTH CAROLINA AT CHARLOTTE.
[11] Kolodziej, J. A., Jankowska, M. A., & Mierzwiczak, M. (2013). Meshless methods for the inverse problem related to the determination of elastoplastic properties from the torsional experiment. International Journal of Solids and Structures, 50(25), 4217-4225.
[12] Li, S. C., Li, S. C., & Cheng, Y. M. (2005). Enriched meshless manifold method for two-dimensional crack modeling. Theoretical and applied fracture mechanics,44(3), 234-248.
[13] Liu, G. R., & Gu, Y. T. (2005). An introduction to meshfree methods and their programming. Springer.
[14] Monaghan, J.J.,(1992). Smoothed particle hydrodynamics. Annual Review of Astronomical and Astrophysics, 30, 543-574.
[15] Moosavi, M. R., Delfanian, F., & Khelil, A. (2012). Orthogonal meshless finite volume method applied to crack problems. Thin-Walled Structures, 52, 61-65.
[16] Moosavi, M. R., Delfanian, F., & Khelil, A. (2012). Performance of orthogonal meshless finite volume method applied to elastodynamic crack problems. Thin-Walled Structures, 53, 156-160.
[17] Pant, M., Singh, I. V., & Mishra, B. K. (2010). Numerical simulation of thermo-elastic fracture problems using element free Galerkin method. International Journal of Mechanical Sciences, 52(12), 1745-1755.
[18] Rao, B. N., & Rahman,S. (2001) ,A coupled meshless-finite element method for fracture analysis of cracks.,International Journal of Pressure Vessels and Piping. 78(9),647-657.
[19] Rice, J. (1988). Elastic fracture mechanics concepts for interfacial cracks. Journal of applied mechanics, 55(1), 98-103.
[20] Suliman, R., Oxtoby, O. F., Malan, A. G., & Kok, S. (2013). An enhanced finite volume method to model 2D linear elastic structures. Applied Mathematical Modelling.
[21] Su, R. K. L., & Feng, W. J. (2005). Accurate determination of mode I and II leading coefficients of the Williams expansion by finite element analysis. Finite elements in Analysis and Design, 41(11), 1175-1186.
[22] Sladek, J., Sladek, V., Krahulec, S., Zhang, C., & Wünsche, M. (2013). Crack analysis in decagonal quasicrystals by the MLPG. International Journal of Fracture,181(1), 115-126.
[23] Shi, J., Ma, W., & Li, N. (2013). Extended meshless method based on partition of unity for solving multiple crack problems. Meccanica, 48(9), 2263-2270.
[24] Sagaresan, N. (2013). A SIMPLIFIED MESHLESS METHODS FOR BRITTLE FRACTURE OF CONCRETE. International Journal of Computational Methods,10(03).
[25] Yuqiu, L., & Yiqiang, Z. (1985). Technical note: Calculation of stress intensity factors in plane problems by the sub-region mixed finite element method. Advances in Engineering Software (1978), 7(1), 32-35.
[26] Zhuang, X., Cai, Y., & Augarde, C. (2013). A meshless sub-region radial point interpolation method for accurate calculation of crack tip fields. Theoretical and Applied Fracture Mechanics.
[27] Zhuang, X., Augarde, C., & Bordas, S. (2011). Accurate fracture modelling using meshless methods, the visibility criterion and level sets: Formulation and 2D modelling. International Journal for Numerical Methods in Engineering, 86(2), 249-268.
[28] Zehnder, A. T. (2012). Linear Elastic Stress Analysis of 2D Cracks. In Fracture Mechanics (pp. 7-32). Springer Netherlands.
[29] Zhu, T. (1998). Meshless methods in computational mechanics.
[30] 沈國瑞(2002)，「無元素分析之積分權值調整法」，國立中央大學土木工程研究所博士論文。
[31] 吳柏壽(2001)，「元素釋放法在非均質材料之應用」，國立中央大學土木工程研究所碩士論文。
[32] 張鈺翎(2013)，「光滑粒子流體動力學在微流道中紅血球變形之模擬與應用」，國立成功大學機械工程研究所碩士論文。
[33] 張正群(2012)，「懸臂樑之平面應力分析」，國立成功大學土木工程研究所碩士論文。
[34] 詹奇峰(2003)，「應力集中效應對表面疲勞裂縫之影響」，國立成功大學造船暨船舶機械工程研究所碩士論文。
[35] 羅漢中(2013)， 「徑向基函數無網格配點法及其在岩石力學中的應用研究」，上海交通大學船舶海洋與建築工程學院博士論文。
[36] 秦伶俐(2005)，「無網格法在結構計算上的改進」，中國農業大學車輛工程所學位論文。