偏微分方程的径向基无网格配置法
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摘要
偏微分方程在工程实际和科学技术中有广泛的应用背景,研究其数值解对处理在电磁学、声学等领域中的很多物理问题具有很重要的意义。传统求解偏微分方程的数值方法都基于网格划分,如有限差分法、有限元法等。针对网格划分的有效算法,研究人员进行了大量的工作,目前仍没有一种针对各种复杂区域问题的通用算法。无网格法是过去十多年兴起的一种新的数值方法,该方法基于点的近似,不需要预先定义或生成网格,为计算力学研究者回避网格划分难题提供了一条新的途径。到目前在不同领域已产生了十余种无网格方法,每种方法都是不同的近似方案和离散方案的结合。用径向基函数插值求偏微分方程定解问题的数值解,是近年来国际上比较流行的一种无网格方法,其原因是径向基函数具有形式简单、各向同性等优点。如果采用径向基函数进行近似,离散方案一般采用最小二乘方法、伽辽金方法、配置法等。用最小二乘法、伽辽金法逼近时,已得到不错的收敛性结果,而对于配置法在计算方面具有简单实用的优点,但进行的相关分析与使用还不是很多。
本文首先引入了一种径向基函数,对其性质进行了具体的研究。然后将这种径向基函数与配置法相结合,构造了求解椭圆型方程和抛物型方程的径向基无网格配置法,并讨论了数值解的存在唯一性。在进行具体数值计算时,给出了径向基函数中自由参数在不同维空间中取值的经验公式,并且讨论了这些经验公式的适用范围。通过与经典有限元方法的比较,表明本文构造的径向基无网格配置法有效、实用；再通过与其它径向基函数构造的径向基无网格配置法相比较,说明本文引入的径向基函数对于偏微分方程求解效果更好。
Partial differential equations have been widely applied in physics, engineering and scientific computation. It is significant to research efficient numerical methods for solving partial differential equation in dealing with many physical equations in some fields, such as electromagnetics, acoustics and so on. The traditional numerical methods are based on mesh discretization mostly such as FEM and FDM. For the efficient algorithm of mesh discretization, the researchers carried out a lot of work. There is still not a general algorithm for the complex regional problems. Meshless methods have been proposed and achieved remarkable progress in numerical computations in last decade. It is based on the point approximation and needn't to pre-defined or generated grid, so that a new way is provided for the computational mechanics researchers to avoid the mesh discretization. Now more than ten kinds of meshless methods have been produced in different fields, each method is the combination of different approximation and discrete programs. RBF interpolation for partial differential equation is a very popular meshless method in recent years in the international aspects because of simple form, isotropic and so on. For the radial basis function approximation, discrete programs generally use the least squares method, Galerkin method, collocation method and so on. There are good convergence results of the least square and Galerkin approximation, but the collocation method has not much application and relational analysis although simple and practial in the calculation.
First, we introduce a radial basis function and give a specific research for its naturein this paper. Then combining this radial basis function with collocation method, we constructed meshless methods for elliptic and parabolic equations by collocation and radial basis function. We also give theoratical proof of existence and uniqueness of sol-utions when using meshless method to solve partial differential equation by collocation with radial basis functions. Through the numerical examples,we give the empirical form-ula of the free parameter in the radial basis function in different values of dimensional space and their scope of application. We illustrate the effectiveness and practicality of the method through compared with the classic finite element method. We also illustrate the superiority of the method for partial differential equation through compared with the others.
本文首先引入了一种径向基函数,对其性质进行了具体的研究。然后将这种径向基函数与配置法相结合,构造了求解椭圆型方程和抛物型方程的径向基无网格配置法,并讨论了数值解的存在唯一性。在进行具体数值计算时,给出了径向基函数中自由参数在不同维空间中取值的经验公式,并且讨论了这些经验公式的适用范围。通过与经典有限元方法的比较,表明本文构造的径向基无网格配置法有效、实用；再通过与其它径向基函数构造的径向基无网格配置法相比较,说明本文引入的径向基函数对于偏微分方程求解效果更好。
Partial differential equations have been widely applied in physics, engineering and scientific computation. It is significant to research efficient numerical methods for solving partial differential equation in dealing with many physical equations in some fields, such as electromagnetics, acoustics and so on. The traditional numerical methods are based on mesh discretization mostly such as FEM and FDM. For the efficient algorithm of mesh discretization, the researchers carried out a lot of work. There is still not a general algorithm for the complex regional problems. Meshless methods have been proposed and achieved remarkable progress in numerical computations in last decade. It is based on the point approximation and needn't to pre-defined or generated grid, so that a new way is provided for the computational mechanics researchers to avoid the mesh discretization. Now more than ten kinds of meshless methods have been produced in different fields, each method is the combination of different approximation and discrete programs. RBF interpolation for partial differential equation is a very popular meshless method in recent years in the international aspects because of simple form, isotropic and so on. For the radial basis function approximation, discrete programs generally use the least squares method, Galerkin method, collocation method and so on. There are good convergence results of the least square and Galerkin approximation, but the collocation method has not much application and relational analysis although simple and practial in the calculation.
First, we introduce a radial basis function and give a specific research for its naturein this paper. Then combining this radial basis function with collocation method, we constructed meshless methods for elliptic and parabolic equations by collocation and radial basis function. We also give theoratical proof of existence and uniqueness of sol-utions when using meshless method to solve partial differential equation by collocation with radial basis functions. Through the numerical examples,we give the empirical form-ula of the free parameter in the radial basis function in different values of dimensional space and their scope of application. We illustrate the effectiveness and practicality of the method through compared with the classic finite element method. We also illustrate the superiority of the method for partial differential equation through compared with the others.
引文
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[2]G.Evans, J.Blackledge and P.Yardley. Numerical Methods for Partial Differential Equations[M]. London:Springer,2000:72-73.
[3]A.R. Mitchell and D.F.Griffiths. The Finite Difference Method in Partial Differential Equations [M].New York:Wiley,1980:164-166.
[4]应隆安.有限元方法讲义[M].北京：北京大学出版社,1988：1-153.
[5]曾余庚,刘京生,张雪阳.有限元方法与边界元方法[M].西安：西安电子科技大学出版社：1991：1-265.
[6]朱幼兰,钟锡昌,陈炳木等.初边值问题差分方法及绕流[M].北京：科学出版社,1980：1-380.
[7]R.K.Beatson. Fast fitting of radial basis functions:Methods based on precondition GMRES ite-ration, Adv. Comput. Math.,[J].1999,11:253-270.
[8]R.K.Beatson and M.J.D. Powell. Univariate multiquadric approximation:quasi-interpolation to sca-ttered data[J]. Constr. Approx.,1992,8:275-288.
[9]R.K.Beatson and N.Dyn. Multiquadric B-splines[J]. J.Approx.theory,1996,87,1-24.
[10]M.D. Buhmann and C.A. Micchelli. Multiquadric interpolation improved[J]. Computers Math. Aplic.,1992,24:21-25.
[11]M.S. Floater and A.Iske.Multistep scattered data interpolation using compactly supported radial basis functions[J]. J.Comput. Appl. Math.,1996,73:65-78.
[12]H.M. Gutmann. On the semi-normof radial basis function interpolants[J].J.Approx.Theory,2001, 111:315-328.
[13]F.J. Hickerenll and Y.C.Hon.Radial basis function approximations as smoothing splines[J]. Appl. Math. Comput.,1999,102:1-24.
[14]F.J. Narcowich and J.D.Ward. Scattered data interpolation on spheres:Error estimates and loc-ally support basis functions[J]. SIAM J.Numer.Anal.,2002,33:1393-1410.
[15]R.Schaback. A unified theory of radial basis functions, Native Hilbert spaces for radial basis functions Ⅱ [J]. J. Comput. Appl. Math.,2000,121:165-177.
[16]R. Schaback and H. Wendland. Adaptive greedy techniques for approximate solution of large RBF systems[J].Numer. Algorithms,2000,24:239-254.
[17]R. Schaback and Z.M. Wu. Operators on radial functions[J]. J. Comput.Appl. Math.,1996,73:2 57-270.
[18]Z.M. Wu. Compactly supported radial functions and the Strang-Fix condition[J]. Appl. Math. Comput.,1997,84:115-124.
[19]Z.M. Wu. Generalized Bochner;s theorem for radial function[J].Approx. Theory Appl.,1997,13 (3):47-57.
[20]Z.M. Wu and Y.C. Hon.Convergence error estimate in solving free boundary diffusion problem by radial basis functions method[J]. Engrg. Anal. Bound. Elem.,2003,27:73-79.
[21]Z.M. Wu and R. Schaback. Local error estimates for radial basis function interpolation of sca-ttered data[J]. IMA J. Numer. Anal.,1993,13:13-27.
[22]Z.M. Wu and R. Schaback. Shape preserving properties and convergence of univariate multi-quadric quasi-interpolation[J]. ACTA Math. Appl. Sinica,1994,10(4):441-446.
[23]Zhang Weixiang and Wu Zongmin. Some shape-preserving quasi-interpolants to non-uniformly distributed data by MQ-B-splines[J]. Appl. Math. J. Chinese Univ. Ser. B,2004,19:191-202.
[24]T.P.Chen and H.Chen.Universal approximation capability of RBF neural networks with arbitrary activation functions[J]. Circuits-Systema-Signal-Process,1996,15:671-683.
[25]X. Li.On simultaneous approximations by radial basis function neural networks[J]. Appl. Math. Comput.,1998,95:75-89.
[26]X. Li and C.A.Micchelli. Approximation by radial basis and neural networks[J]. Numer. Algori-thms,2000,25:241-262.
[27]N. Mai-Duy and T. Tran-Cong. Numerical solution of differential equations using multiquadric radial basis function networks[J]. Neural Networks,2001,14:185-199.
[28]N. Mai-Duy and T. Tran-Cong.Approximation of function and its derivatives using radial basis function networks[J].Appl. Math. Model.,2003,27:197-220.
[29]A.J.M.Ferreira. A formulation of the multiquadric radial basis function method for the analysis of laminated composite plates[J]. Composite Structures,2003,59:385-392.
[30]A.J.M.F.ERREIRA, c.m.c.Roque and P.A.L.S. Martins. Analysis of composite plates using high-er-order shear deformation theory and a finite point formulation based on the multiquadric ra-dial basis function method[J].Composites:Part B,2003,34:627-636.
[31]张雄,刘岩.无网格方法[M].北京：清华大学出版社,2004：1-124.
[32]Y.C.Hon and Z.M.Wu. A quasi-interpolation method for solving stiff ordinary differential equa-tions[J]. Int. J.Numer.Meth. Engng.,2000,48:1187-1197.
[33]A.I.Fedoseyev, M.J.Friedman and E.J.Kansa. Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary[J]. Computers Math. Appl.,2002,43:4 39-455.
[34]E.J.Kansa and Y.C.Hon.Circumventing the ill-conditioning problemwith multiquadric radial basis functions:applications to ellipitic partial differential aquations[J].Comput.Math.Appl.,2000,39:123-137.
[35]Z.M. Wu and Y.C. Hon. Numerical intergration of harmonic functions with restricted sampling
data[J]. J. Complexity.2001,17:898-909.
[36]Y. C. Hon and X. Z. Mao.An efficient numerical scheme foe Burgers'equation[J]. Appl. Math. Comput.,1998,95:37-50.
[37]Z.M. Wu. Dynamically knots setting in meshless method for solving time dependent propaga-tions equation[J]. Comput. Methods Appl. Mech. Engrg.2004,193:1221-1229.
[38]Z.M. Wu. Hermite-Brikhoff interpolation of scattered data by radial basis functions[J]. Approx. Theory Appl.,1992,8(2):1-10.
[39]H. Wendland. Meshless Galerkin methods using radial basis functions[J]. Math. Comput.,1999, 68:1521-1531.
[40]吴宗敏.径向基函数、散乱数据拟合与无网格偏微分方程数值解[J].工程数学学报,2002,19(2)：1-12.
[41]E.M. Stein and G Weiss. Introduction to Fourier Analysis on Euclidean Spaces[M]. Princeton, New Jersey:Princeton University Press,1971,14:133-137.
[42]H. Wendland. Piecewise polynomial, positive definiteand compactly supported radial basis func-tions of minimal degree[J]. Adv. Comput. Math.,1995,4:389-396.
[43]Z.M. Wu. Compactly supported positive definite radial functions[J]. Adv. Comput. Math.,1995, 4:282-292.
[44]K. Miller. Moving finite elements I [J]. SIAM J. Numer. Anal.,1981,18:1033-1057.
[45]M.D.Buhmann. Radial basis functions[M]. in:Acta Numerica Vol.9,2000,Cambridge:Cambridge University Press,2000:1-38.
[46]吴宗敏.散乱数据拟合-模型、方法及其理论[M].待出版137-202.
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