不规则区域上抛物问题的配置有限元法
|
摘要
配置法是近几十年发展起来的一种数值求解方法，它是以满足纯插值约束条件的方式，寻求算子方程近似解的方法．配置法不必计算数值积分，逼近方程容易形成，计算简便而且收敛精度高，因此在数值求解椭圆型方程、抛物型方程、双曲型方程中得到广泛应用，但是主要研究工作大都针对规则区域展开．
本文建立了解一类不规则区域上抛物方程初边值问题的配置有限元法．该方法结合了正交配置法的简单性和可变有限元的多样性．在每个结点使用了具有4个自由度的双三次Hermite元．
本文共分两章：
第一章简单介绍了Frind和Pinder在[1]中建立的一种解一类边值问题的潜力巨大的数值方法．该方法结合了正交配置法的简单性和可变有限元的多样性，在每个结点使用了具有4个自由度的双三次Hermite元．其中使用的亚参数变换允许最高精度配置点的准确定位以及不规则边界的唯一表示．
本章分为三节：第一节是引言，简单介绍了配置法的发展概况；第二节详细介绍了配置方程的推导．该方法对一类不规则区域上的势能问题：提出了一种配置有限元解法，并给出了与Galerkin有限元法计算效率的比较．第三节对这种方法的优势做了简单总结．
第二章针对初边值问题展开讨论．对抛物型方程应用第一章的方法，给出了在扇环形区域上一类热传导方程初边值问题的相应的配置解法．
本章分为四节：第一节是引言，简单介绍了抛物问题的配置法发展；第二节给出了在扇环区域上的热传导问题：并应用第一章介绍的方法得出配置方程；第三节把前面的结论推广到更一般的情形，指出本文所介绍方法具有重要的应用价值；第四节指出了本文存在的不足．
The collocation method is a numerical method which has developed for several decades. It is a method which search for the approximate solutions of the operator functions by satisfying pure interpolation condition. The collocation method has many advantages , for example, it needn't calculate numerical integral; it can form the appproximate equation easily; its compute is simple and convenient, and it has high-order accuracy. Hence, the collocation is widely used for solving elliptie equations , prarbolic equations and hyperbolic equations. But most study was ristricted in regular domains.
In this paper, a collocation finite element method for solving certain initial boundary value problems in irregular domains. The method combines the simplicity of orthogonal collocation with the versatility of deformable finite elements. Bicubic Hermite elements with four degrees-of-freedom per node are used. Analysis of this paper show that by taking advantage of the boundary conditions, a minimum number of collocation points can be used.
This article is divided into two chapters.
Chapter 1 introduces a potentially powerful numerical method for solving certain boundary value problems which is developed in [1]. The method combines the simplicity of orthogonal collocation with the versatility of deformable finite elements. Bicubic Hermite elements with four degrees-of-freedom per node are used. A sub- parametric transformation permits the precise positioning of the collocation points for maximum accuracy as well as a unique representation of irregular boundaries.
This chapter is devided into three sections. The first section is introduction, which introduces the general situation of the collocation method simply. The second section gives the introduction of the arithmetic. This method sets up a collocation finite method for potential problems in irregular domains: and gives the compare with the Galerkin finite method in the computational efficiency . The third section concludes the method simply.
Chapter 2 applies the method to a parabolic equation, and gives the corresponding collocation method of certain heat exchanging problem in irregular domains.
This chapter is devided into four sections. The first section is the introductions which introduces the development of the collocation method for parabolic problems simply, and it also gives the main problems of this article as well. The second section gives a heat exchanging problem in sector-cirque-shaped domain:Using the method introduced in the Chapter 1 , we gain the collocation method . The third section extends the method of this article, and indicates that the method has very important value; The fourth section gives the shortage of this paper.
本文建立了解一类不规则区域上抛物方程初边值问题的配置有限元法．该方法结合了正交配置法的简单性和可变有限元的多样性．在每个结点使用了具有4个自由度的双三次Hermite元．
本文共分两章：
第一章简单介绍了Frind和Pinder在[1]中建立的一种解一类边值问题的潜力巨大的数值方法．该方法结合了正交配置法的简单性和可变有限元的多样性，在每个结点使用了具有4个自由度的双三次Hermite元．其中使用的亚参数变换允许最高精度配置点的准确定位以及不规则边界的唯一表示．
本章分为三节：第一节是引言，简单介绍了配置法的发展概况；第二节详细介绍了配置方程的推导．该方法对一类不规则区域上的势能问题：提出了一种配置有限元解法，并给出了与Galerkin有限元法计算效率的比较．第三节对这种方法的优势做了简单总结．
第二章针对初边值问题展开讨论．对抛物型方程应用第一章的方法，给出了在扇环形区域上一类热传导方程初边值问题的相应的配置解法．
本章分为四节：第一节是引言，简单介绍了抛物问题的配置法发展；第二节给出了在扇环区域上的热传导问题：并应用第一章介绍的方法得出配置方程；第三节把前面的结论推广到更一般的情形，指出本文所介绍方法具有重要的应用价值；第四节指出了本文存在的不足．
The collocation method is a numerical method which has developed for several decades. It is a method which search for the approximate solutions of the operator functions by satisfying pure interpolation condition. The collocation method has many advantages , for example, it needn't calculate numerical integral; it can form the appproximate equation easily; its compute is simple and convenient, and it has high-order accuracy. Hence, the collocation is widely used for solving elliptie equations , prarbolic equations and hyperbolic equations. But most study was ristricted in regular domains.
In this paper, a collocation finite element method for solving certain initial boundary value problems in irregular domains. The method combines the simplicity of orthogonal collocation with the versatility of deformable finite elements. Bicubic Hermite elements with four degrees-of-freedom per node are used. Analysis of this paper show that by taking advantage of the boundary conditions, a minimum number of collocation points can be used.
This article is divided into two chapters.
Chapter 1 introduces a potentially powerful numerical method for solving certain boundary value problems which is developed in [1]. The method combines the simplicity of orthogonal collocation with the versatility of deformable finite elements. Bicubic Hermite elements with four degrees-of-freedom per node are used. A sub- parametric transformation permits the precise positioning of the collocation points for maximum accuracy as well as a unique representation of irregular boundaries.
This chapter is devided into three sections. The first section is introduction, which introduces the general situation of the collocation method simply. The second section gives the introduction of the arithmetic. This method sets up a collocation finite method for potential problems in irregular domains: and gives the compare with the Galerkin finite method in the computational efficiency . The third section concludes the method simply.
Chapter 2 applies the method to a parabolic equation, and gives the corresponding collocation method of certain heat exchanging problem in irregular domains.
This chapter is devided into four sections. The first section is the introductions which introduces the development of the collocation method for parabolic problems simply, and it also gives the main problems of this article as well. The second section gives a heat exchanging problem in sector-cirque-shaped domain:Using the method introduced in the Chapter 1 , we gain the collocation method . The third section extends the method of this article, and indicates that the method has very important value; The fourth section gives the shortage of this paper.
引文
[1] Emil O.Frind and George F.Pinder, A collocation finite element method for potential problems in irregvlar domains, International Journal for Numerical Methods in Engineering,1979,14:681-701.
[2] J.Douglas Jr.,T.Dupont, Collocation methods for parabolic equations in a single space variable, Lecture Notes in Mathematics,Vol.385,Springer-Verlag, Heidelberg, 1974.
[3] B.Bialecki and R.I.Pernandes, Orthogonal spline collocation Laplace-modified and alternating-direction methods for parabolic problems on rectangles, Math.Comp.,1993,60:545-573.
[4] P.Percell and M.F.Wheeler, A C~1 finite element collocation method of elliptic equations[J].SlAM J.Numer.Anal., 1980,17:605-622.
[5] D.S.Dillery, High order orthogonal spline collocation schemes for elliptic and parabolic problems,Ph.D thesis.University of Kentuck, Lexington,KY,1994.
[6] J.Douglas Jr.,T.Dupont, A finite element collocation method for quasilinearparabolic equations[J]. ,Math.Comp., 1973,27:17-28.
[7] L.Hoyes, An alternating-direction collocation mrthodfor finite element approximations on rectangles, Math.Comp., 1980,6: 45-50.
[8] M. T. van Genuchten,G.F.Pinder and E.O.Frind, Simulation of two-dimensional contaminant transport with isoparametric Hermitian finite elements , Water Resources Research,1977,13:451-458.
[9] E.Isaacson and H.B.Keller, Analysis of Numerical Methods Wiley,New York, 1966.
[10] P.M.Prenter and R.D.Russell, Orthogonal collocation for elliptic partial differential equations, SIAM J.Numer.Anal., 1976,13:923-939.
[11] P．M．Prenter，样条函数与变分方法，上海科学技术出版社，上海，1980．
[2] J.Douglas Jr.,T.Dupont, Collocation methods for parabolic equations in a single space variable, Lecture Notes in Mathematics,Vol.385,Springer-Verlag, Heidelberg, 1974.
[3] B.Bialecki and R.I.Pernandes, Orthogonal spline collocation Laplace-modified and alternating-direction methods for parabolic problems on rectangles, Math.Comp.,1993,60:545-573.
[4] P.Percell and M.F.Wheeler, A C~1 finite element collocation method of elliptic equations[J].SlAM J.Numer.Anal., 1980,17:605-622.
[5] D.S.Dillery, High order orthogonal spline collocation schemes for elliptic and parabolic problems,Ph.D thesis.University of Kentuck, Lexington,KY,1994.
[6] J.Douglas Jr.,T.Dupont, A finite element collocation method for quasilinearparabolic equations[J]. ,Math.Comp., 1973,27:17-28.
[7] L.Hoyes, An alternating-direction collocation mrthodfor finite element approximations on rectangles, Math.Comp., 1980,6: 45-50.
[8] M. T. van Genuchten,G.F.Pinder and E.O.Frind, Simulation of two-dimensional contaminant transport with isoparametric Hermitian finite elements , Water Resources Research,1977,13:451-458.
[9] E.Isaacson and H.B.Keller, Analysis of Numerical Methods Wiley,New York, 1966.
[10] P.M.Prenter and R.D.Russell, Orthogonal collocation for elliptic partial differential equations, SIAM J.Numer.Anal., 1976,13:923-939.
[11] P．M．Prenter，样条函数与变分方法，上海科学技术出版社，上海，1980．