二维各向异性位势问题的改进插值型边界无单元法
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摘要
【目的】把边界积分方程方法和基于非奇异权函数的改进移动最小二乘插值法相结合,建立数值求解二维各向异性位势问题的改进插值型边界无单元法。【方法】在改进移动最小二乘插值法的基础上,讨论了非奇异权函数的改进移动最小二乘插值法,它的形函数满足Kroneckerδ函数的性质,因此可以直接施加边界条件。【结果】数值算例表明该方法求解二维各向异性位势问题是有效和可行的。【结论】与边界元方法相比,该方法精度和收敛性更好。
[Purposes]Combining the boundary integral equation method with the improved interpolating moving least-square method(IIMLS)based on nonsingular weight function,an improved interpolating boundary element-free method(IIBEFM)is developed for solving two-dimensional anisotropic potential problems.[Methods]On the basis of the improved moving least-square interpolation method,the improved interpolating moving least-square interpolation method with nonsingular weight function is discussed.Its shape function satisfies the property of Kronecker delta function,so boundary conditions can be applied directly.[Findings]Numerical examples show that the method is effective and feasible for solving two-dimensional anisotropic potential problems.[Conclusions]Compared with the boundary element method,this method has better accuracy and convergence.
[Purposes]Combining the boundary integral equation method with the improved interpolating moving least-square method(IIMLS)based on nonsingular weight function,an improved interpolating boundary element-free method(IIBEFM)is developed for solving two-dimensional anisotropic potential problems.[Methods]On the basis of the improved moving least-square interpolation method,the improved interpolating moving least-square interpolation method with nonsingular weight function is discussed.Its shape function satisfies the property of Kronecker delta function,so boundary conditions can be applied directly.[Findings]Numerical examples show that the method is effective and feasible for solving two-dimensional anisotropic potential problems.[Conclusions]Compared with the boundary element method,this method has better accuracy and convergence.
引文
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[9]LANCASTER P,SALkAUSKAS K.Surfaces generated by moving least squares methods[J].Mathematics of Computation,1981,37(155):141-158.
[10]LIEW K M,CHENG Y M,KITIPORNCHAI S.Boundary element-free method(BEFM)and its application to twodimensional elasticity problems[J].International Journal for Numerical Methods in Engineering,2006,65(8):1310-1332.
[11]PENG M J,CHENG Y M.A boundary element-free method(BEFM)for two-dimensional potential problems[J].Engineering Analysis with Boundary Elements,2009,33(1):77-82.
[12]REN H P,CHENG Y M,ZHANG W.An improved boundary element-free method(IBEFM)for two-dimensional potential problems[J].Chinese Physics B,2009,18(10):4065-4073.
[13]REN H P,CHENG Y M,ZHANG W.An improved boundary element-free method(IBEFM)for two-dimensional elasticity problems[J].Science China Physics,Mechanics&Astronomy,2010,53(4):758-766.
[14]WANG J F,WANG J F,SUN F X,et al.An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems[J].International Journal of computational Methods,2013,10(6):1350043.
[15]孙凤欣.基于非奇异权的改进的插值型无网格方法研究[D].上海:上海大学,2014.SUN X F.Research on improved interpolating meshless methods based on nonsingular weight functions[D].Shanghai:Shanghai University,2014.
[16]LI X L.An interpolating boundary element free method for three-dimensional potential problems[J].Applied Mathematical Modelling,2015,39(10/11):3116-3134.
[17]TELUKUNTA S,MUKHERJEE S.An extended boundary node method for modeling normal derivative discontinuties in potential theory across edges and corners[J].Engineering Analysis with Boundary Elements,2004,28(9):1099-1110.
[2]BREBBIA C A,TELLES J C F,WROBEL L C.Boundary element techniques[M].Berlin,Heidelberg:Springer Verlag,1984.
[3]周焕林,牛忠荣,程长征,等.各向异性位势问题边界元法中几乎奇异积分的解析算法[J].计算力学学报,2008,25(3):333-339.ZHOU H L,LIU Z R,CHENG C Z,et al.Analytical algorithm of the nearly singular integrals in element method to anisotropic potential problems[J].Chinese Journal of Computational Mechanics,2008,25(3):333-339.
[4]周焕林,王剑,牛忠荣.薄体各向异性位势边界条件识别反问题正则化边界元方法[J].固体力学学报,2008,29(S1):45-48.ZHOU H L,WANG J,LIU Z R.Inverse identification of boundary conditions for anisotropic potential problems of thin body using the regularized boundary element method[J].Chinese Journal of Solid Mechanics,2008,29(S1):45-48.
[5]陈关忠,周爱华,张耀明.二维各向异性位势薄体问题的虚边界元法[J].山东理工大学学报(自然科学版),2013,27(2):14-19.CHEN G Z,ZHOU A H,ZHANG Y M.Virtual boundary element method for 2Danisotropic thin-body structure in potential problems[J].Journal of Shandong University of Technology(Natural Science Edition),2013,27(2):14-19.
[6]程玉民.无网格方法[M].北京:科学出版社,2015.CHENG Y M.Meshless method[M].Beijing:Science Press,2015.
[7]BELYTSCHKO T,KRONGAUZ Y,ORGAN D,et al.Meshless methods:an overview and recent developments[J].Computer Methods in Applied Mechanics and Engineering,1996,139(1/2/3/4):3-47.
[8]陈林冲.二维Helmholtz方程的边界点解法[J].重庆理工大学学报(自然科学),2017,31(7):188-194.CHEN L C.Boundary node method for the 2-D Helmholtz equation[J].Journal of Chongqing University of Technology(Natural Science),2017,31(7):188-194.
[9]LANCASTER P,SALkAUSKAS K.Surfaces generated by moving least squares methods[J].Mathematics of Computation,1981,37(155):141-158.
[10]LIEW K M,CHENG Y M,KITIPORNCHAI S.Boundary element-free method(BEFM)and its application to twodimensional elasticity problems[J].International Journal for Numerical Methods in Engineering,2006,65(8):1310-1332.
[11]PENG M J,CHENG Y M.A boundary element-free method(BEFM)for two-dimensional potential problems[J].Engineering Analysis with Boundary Elements,2009,33(1):77-82.
[12]REN H P,CHENG Y M,ZHANG W.An improved boundary element-free method(IBEFM)for two-dimensional potential problems[J].Chinese Physics B,2009,18(10):4065-4073.
[13]REN H P,CHENG Y M,ZHANG W.An improved boundary element-free method(IBEFM)for two-dimensional elasticity problems[J].Science China Physics,Mechanics&Astronomy,2010,53(4):758-766.
[14]WANG J F,WANG J F,SUN F X,et al.An interpolating boundary element-free method with nonsingular weight function for two-dimensional potential problems[J].International Journal of computational Methods,2013,10(6):1350043.
[15]孙凤欣.基于非奇异权的改进的插值型无网格方法研究[D].上海:上海大学,2014.SUN X F.Research on improved interpolating meshless methods based on nonsingular weight functions[D].Shanghai:Shanghai University,2014.
[16]LI X L.An interpolating boundary element free method for three-dimensional potential problems[J].Applied Mathematical Modelling,2015,39(10/11):3116-3134.
[17]TELUKUNTA S,MUKHERJEE S.An extended boundary node method for modeling normal derivative discontinuties in potential theory across edges and corners[J].Engineering Analysis with Boundary Elements,2004,28(9):1099-1110.