自然元方法的分析及其在偏微分方程中的应用
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摘要
自然邻接插值是一种广泛应用于多元数据拟合的插值方法.自然元方法就是以自然邻接插值(形函数)作为试探函数和检验函数应用在Galerkin过程的数值计算方法,其依赖于Voronoi图和Delaunay三角几何结构.自然邻接插值只与节点的分布有关,而与网格形状无关.因此在处理不规则区域的问题时,自然元方法具有天然的优势.形函数满足插值性质；在凸包边界的两相邻点之间是严格线性连续等性质.自然元方法既具有经典有限元方法、无网格方法的优点,又克服了两者的一些缺陷.
本文讨论以Sibson插值为形函数的自然单元方法,介绍了自然元形函数的性质、构造以及自然单元方法的应用.我们首先给出了Sibson插值的定义和光滑性、插值性质等各项性质的证明,并构造了自然元方法Galerkin过程的核心内容一形函数；其次利用标准的Galerkin离散方法,介绍了自然元程序的设计思想和主要结构；最后将自然元方法用于求解平板问题和圆孔问题的数值试验,并在圆孔问题中比较了自然元方法和有限元方法的数值结果.由于自然元形函数在凸包边界上是线性的,在数值实验中可以很便利地强加本质边界条件.以上算例也说明了自然元方法应用于求解不规则区域的偏微分方程是非常有吸引力的.
Natural neighbor interpolation is a widely used multivariate scattered data interpolation method.Natural element method(NEM),which depends on the Voronoi diagram and Delaunay triangulation,is a numerical methods that natural neighbor interpolation(shape function)is used as trial and test approximations in a Galerkin method.Natural neighbor interpolation is only related to the distribution of nodes,nothing with the grid shape. Therefore, for the issue of irregular region, NEM has a natural advantage. Furthermore the shape function meets the interpolation property; and on the boundary of convex hull it is strictly linear continuous between two neighbor points, etc. so NEM not only has the advantages of the classical finite element(FEM) method and meshless method, but also overcomes some shortcomings of both.
In this paper,we discuss NEM in which Sibson interpolation is used as shape function, and describe the properties,construction of shape function and the ap-plication of the NEM.First,we illustrate the definition of Sibson interpolation and give out some proofs involving smoothness,interpolation property, etc.Mean-while, we construct the key of the Galerkin method-shape function.Second, using a standard Galerkin method,we introduce the design and main structure of the NEM.Finally, we use NEM to solve the plane problem and hole plate prob-lem.In the experiment of hole plate problem,we compare the numerical results of the NEM and FEM.Since shape function is linear on the boundary of con-vex hull,we can conveniently impose the essential conditions on the boundary in the numerical experiment.At the same time,these examples also illustrate that NEM is very attractive,which is applied to solve partial differential equations of the irregular region.
本文讨论以Sibson插值为形函数的自然单元方法,介绍了自然元形函数的性质、构造以及自然单元方法的应用.我们首先给出了Sibson插值的定义和光滑性、插值性质等各项性质的证明,并构造了自然元方法Galerkin过程的核心内容一形函数；其次利用标准的Galerkin离散方法,介绍了自然元程序的设计思想和主要结构；最后将自然元方法用于求解平板问题和圆孔问题的数值试验,并在圆孔问题中比较了自然元方法和有限元方法的数值结果.由于自然元形函数在凸包边界上是线性的,在数值实验中可以很便利地强加本质边界条件.以上算例也说明了自然元方法应用于求解不规则区域的偏微分方程是非常有吸引力的.
Natural neighbor interpolation is a widely used multivariate scattered data interpolation method.Natural element method(NEM),which depends on the Voronoi diagram and Delaunay triangulation,is a numerical methods that natural neighbor interpolation(shape function)is used as trial and test approximations in a Galerkin method.Natural neighbor interpolation is only related to the distribution of nodes,nothing with the grid shape. Therefore, for the issue of irregular region, NEM has a natural advantage. Furthermore the shape function meets the interpolation property; and on the boundary of convex hull it is strictly linear continuous between two neighbor points, etc. so NEM not only has the advantages of the classical finite element(FEM) method and meshless method, but also overcomes some shortcomings of both.
In this paper,we discuss NEM in which Sibson interpolation is used as shape function, and describe the properties,construction of shape function and the ap-plication of the NEM.First,we illustrate the definition of Sibson interpolation and give out some proofs involving smoothness,interpolation property, etc.Mean-while, we construct the key of the Galerkin method-shape function.Second, using a standard Galerkin method,we introduce the design and main structure of the NEM.Finally, we use NEM to solve the plane problem and hole plate prob-lem.In the experiment of hole plate problem,we compare the numerical results of the NEM and FEM.Since shape function is linear on the boundary of con-vex hull,we can conveniently impose the essential conditions on the boundary in the numerical experiment.At the same time,these examples also illustrate that NEM is very attractive,which is applied to solve partial differential equations of the irregular region.
引文
[1]李开泰,黄艾香,黄庆怀.有限元方法及其应用,西安交通大学出版社,1984.
[2]陈传淼,黄云清.有限元高精度理论,湖南科学出版社,1995.
[3]Randall,J.LeVeque.Finite Difference Methods for Ordinary and Partial Dif-ferential Equations,SIAM,2007.
[4]J.Braun and M.Sambridge. A numerical method for solving partial differential equations on highly irregular evolving grids,Nature,376:655-660,1995.
[5]R. Sibson. A vector identity for the Dirichlet tesselation, Mathematical Pro-ceedingsof the Cambridge Philosophical Society,87:151-155,1980.
[6]R. Sibson. A brief description of natural neighbor interpolation,in V. Bar-nett(ed.),Interpreting Multivariate Data, Wiley, Chichester, Chapter 2:21-36, 1981.
[7]G.Farin. Surfaces over Dirichlet tessellations, Computer Aided Geometric De-sign,7:281-292,1990.
[8]L.Traversoni. Natural neighbour finite elements, In International Conference on Hydraulic Enginnering Software, Hydrosoft Proceedings 2:291-297, Com-putational Mechanics Publications,1994.
[9]P. Piper. Properties of local coordinates based on Dirichlet tessellations,in G. Farin, H. Hagen and H. Noltemeier(eds.),Geometric Modelling, Springer-verlag,Wien,Computing Supplementun 8 227-239,1993.
[10]E. Cueto, N.Sukumar.Overview and recent advances in natural neighbour Galerkin methods, Archives of Computational Methods in Engineering,10(4): 307-384,2003.
[11]D.F.Watson.Contouring:A Guide to the Analysis and Display of Spatial Data, Pergamon, Oxford,1992.
[12]C.L.Lawson.Software for C1 surface interpolation, in Mathematical Software, 3:161-193,Rice J.(ed.),Academic Press,New York,1977.
[13]湘潭大学偏微分数值解讲义,2008.
[14]F.P.普雷帕拉塔,M.I.沙莫斯著，庄心谷译.计算几何导论,科学出版社,1990.
[15]周培德.计算几何：算法设计与分析(第二版),清华大学出版社,2005.
[16]Atsuyuki Okabe, Barry Boots,Kokichi Sugihara. Spatial Tessellations:Con-cepts and Applications of Voronoi Diagrams,John Wiley sons,2000.
[17]姜礼尚,庞之垣.有限元方法及其理论基础,人民教育出版社,1979.
[18]斯特朗,菲克斯著,崔俊芝,宫著铭译.有限元法分析,科学出版社,1983.
[19]O.C.Zienkiewicz,R. L.Taylor.The Finite Element Method,世界图书出版公司,2005.
[20]P. G.Ciarlet,J.L.Lions.Handbook of Numerical Analysis V1 Part1:Finite Difference Methods,Elsevier Science,North-holland,1990.
[21]蔡大用.数值代数,清华大学出版社,1987.
[22]徐芝纶.弹性力学(第二版),人民教育出版社,1981.
[23]铁摩辛柯,古地尔著,徐芝纶译.弹性理论,高等教育出版社,1990.
[24]Bojan Niceno. Easymesh:A Two-Dimensional Quality Mesh Generator, http://www-dinma.univ.trieste.it/nirftc/research/easymesh/
[25]N.Sukumar, B Moran. T Belytschko, The natural element method in solid mechanics, Int.J.Numer.Meth. Eng,43:839-887,1998.
[26]J.Braun, M. Sambridge and H. McQueen.Geophysical parameterization and interpolation of irregular data using natural neighbors,Geophys. J.Int.,122: 837-857 1995.
[27]T.Belytschko, Y. Krongauz.Meshless methods:An overview and recent devel-opments, Comput Methods.Appl.Mech.Engr g.,139:3-47,1996.
[28]T.Belytschko,Y. Y. Lu,L.Gu. Element-free Galerkin methods.International Journal for Numerical methods in Engineering,37:229-256,1994.
[29]Y. krongauz, T.Belytschko. Enforcement of essential boundary conditions in meshless approximations using finite elements.Comput.Methods. Appl.Mech. Engrg.,131:133-145,1996.
[30]M.Griebel,M.A.Schweitzer. A particle-partition of unity method for the so-lution of elliptic,parabolic and hyperbolic PDEs,SIAM.J.Sci.Comput.,22: 853-890,2001.
[31]E.Onate, S.Idelsohn. A mesh-free finite point method for advective-diffusive transport and fluid flow problems,Comput.Mech.,21:283-292,1998.
[32]吴宗敏.散乱数据拟合的模型、方法和理论,科学出版社,2007.
[33]王兆清,冯伟.自然单元法研究进展,力学进展[J]，34(4)：437-445,2004.
[34]卢波,葛修润,孔祥礼.有限元法、无单元法及自然单元法之比较研究,岩石力学与工程学报,24(5)：780-786,2005.
[35]朱合华,杨宝红,蔡永昌,徐斌.无网格自然单元法在弹塑性分析中的应用,岩土力学,4：671-674,2004.
[36]张雄,刘岩.无网格法,清华大学出版社,2004.
[37]B.Nayroles,G.Touzot.Generalizing the finite element method:diffuse approx-imation and diffuse elements,Comput.Mech.,10:307-318,1992.
[38]I.Babuska, G.Caloz,J.Osborn. Special finite element methods for a class of second order elliptic problems with rough coefficients.SIAM.J.Numer.Anal., 31:954-981,1994.
[39]V.N.Parthasarathya, S.Kodiyalamb. A constrained optimization approach to finite element mesh smoothing, Finite Elements in Analysis and Design 9: 309-320,1991.
[40]曹国金,姜弘道.无单元法研究和应用现状及动态,力学进展,32(4)：526-534,2002.
[2]陈传淼,黄云清.有限元高精度理论,湖南科学出版社,1995.
[3]Randall,J.LeVeque.Finite Difference Methods for Ordinary and Partial Dif-ferential Equations,SIAM,2007.
[4]J.Braun and M.Sambridge. A numerical method for solving partial differential equations on highly irregular evolving grids,Nature,376:655-660,1995.
[5]R. Sibson. A vector identity for the Dirichlet tesselation, Mathematical Pro-ceedingsof the Cambridge Philosophical Society,87:151-155,1980.
[6]R. Sibson. A brief description of natural neighbor interpolation,in V. Bar-nett(ed.),Interpreting Multivariate Data, Wiley, Chichester, Chapter 2:21-36, 1981.
[7]G.Farin. Surfaces over Dirichlet tessellations, Computer Aided Geometric De-sign,7:281-292,1990.
[8]L.Traversoni. Natural neighbour finite elements, In International Conference on Hydraulic Enginnering Software, Hydrosoft Proceedings 2:291-297, Com-putational Mechanics Publications,1994.
[9]P. Piper. Properties of local coordinates based on Dirichlet tessellations,in G. Farin, H. Hagen and H. Noltemeier(eds.),Geometric Modelling, Springer-verlag,Wien,Computing Supplementun 8 227-239,1993.
[10]E. Cueto, N.Sukumar.Overview and recent advances in natural neighbour Galerkin methods, Archives of Computational Methods in Engineering,10(4): 307-384,2003.
[11]D.F.Watson.Contouring:A Guide to the Analysis and Display of Spatial Data, Pergamon, Oxford,1992.
[12]C.L.Lawson.Software for C1 surface interpolation, in Mathematical Software, 3:161-193,Rice J.(ed.),Academic Press,New York,1977.
[13]湘潭大学偏微分数值解讲义,2008.
[14]F.P.普雷帕拉塔,M.I.沙莫斯著，庄心谷译.计算几何导论,科学出版社,1990.
[15]周培德.计算几何：算法设计与分析(第二版),清华大学出版社,2005.
[16]Atsuyuki Okabe, Barry Boots,Kokichi Sugihara. Spatial Tessellations:Con-cepts and Applications of Voronoi Diagrams,John Wiley sons,2000.
[17]姜礼尚,庞之垣.有限元方法及其理论基础,人民教育出版社,1979.
[18]斯特朗,菲克斯著,崔俊芝,宫著铭译.有限元法分析,科学出版社,1983.
[19]O.C.Zienkiewicz,R. L.Taylor.The Finite Element Method,世界图书出版公司,2005.
[20]P. G.Ciarlet,J.L.Lions.Handbook of Numerical Analysis V1 Part1:Finite Difference Methods,Elsevier Science,North-holland,1990.
[21]蔡大用.数值代数,清华大学出版社,1987.
[22]徐芝纶.弹性力学(第二版),人民教育出版社,1981.
[23]铁摩辛柯,古地尔著,徐芝纶译.弹性理论,高等教育出版社,1990.
[24]Bojan Niceno. Easymesh:A Two-Dimensional Quality Mesh Generator, http://www-dinma.univ.trieste.it/nirftc/research/easymesh/
[25]N.Sukumar, B Moran. T Belytschko, The natural element method in solid mechanics, Int.J.Numer.Meth. Eng,43:839-887,1998.
[26]J.Braun, M. Sambridge and H. McQueen.Geophysical parameterization and interpolation of irregular data using natural neighbors,Geophys. J.Int.,122: 837-857 1995.
[27]T.Belytschko, Y. Krongauz.Meshless methods:An overview and recent devel-opments, Comput Methods.Appl.Mech.Engr g.,139:3-47,1996.
[28]T.Belytschko,Y. Y. Lu,L.Gu. Element-free Galerkin methods.International Journal for Numerical methods in Engineering,37:229-256,1994.
[29]Y. krongauz, T.Belytschko. Enforcement of essential boundary conditions in meshless approximations using finite elements.Comput.Methods. Appl.Mech. Engrg.,131:133-145,1996.
[30]M.Griebel,M.A.Schweitzer. A particle-partition of unity method for the so-lution of elliptic,parabolic and hyperbolic PDEs,SIAM.J.Sci.Comput.,22: 853-890,2001.
[31]E.Onate, S.Idelsohn. A mesh-free finite point method for advective-diffusive transport and fluid flow problems,Comput.Mech.,21:283-292,1998.
[32]吴宗敏.散乱数据拟合的模型、方法和理论,科学出版社,2007.
[33]王兆清,冯伟.自然单元法研究进展,力学进展[J]，34(4)：437-445,2004.
[34]卢波,葛修润,孔祥礼.有限元法、无单元法及自然单元法之比较研究,岩石力学与工程学报,24(5)：780-786,2005.
[35]朱合华,杨宝红,蔡永昌,徐斌.无网格自然单元法在弹塑性分析中的应用,岩土力学,4：671-674,2004.
[36]张雄,刘岩.无网格法,清华大学出版社,2004.
[37]B.Nayroles,G.Touzot.Generalizing the finite element method:diffuse approx-imation and diffuse elements,Comput.Mech.,10:307-318,1992.
[38]I.Babuska, G.Caloz,J.Osborn. Special finite element methods for a class of second order elliptic problems with rough coefficients.SIAM.J.Numer.Anal., 31:954-981,1994.
[39]V.N.Parthasarathya, S.Kodiyalamb. A constrained optimization approach to finite element mesh smoothing, Finite Elements in Analysis and Design 9: 309-320,1991.
[40]曹国金,姜弘道.无单元法研究和应用现状及动态,力学进展,32(4)：526-534,2002.