含开边界二维Stokes问题的Galerkin边界元解法
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摘要
不可压缩粘性流体动力学方程组的Navier-Stokes方程在粘性很大,即雷诺数很小的情况下,可线性化为Stokes方程。考虑到边界的粘附条件,形成Stokes问题。鉴于边界元方法在处理Stokes问题中的连续性方程或者不可压缩条件时明显优于其它类型的方法,而且可以把计算速度场和计算压力分别开,因而是求Stokes问题数值解的理想方法。本文用边界元方法求解平面有界区域内带有非闭合直线段或曲线段的开边界的Stokes问题,即带有屏障的二维流体流动问题。求解这类问题的难点在于在数值逼近时需要妥善处理解在边界的端点附近存在的奇异性,本文对于含有开边界端点的边界单元,采用特别的插值函数,以模拟其固有的奇异性。
我们利用对应于Stokes算子的Green公式和基本解所推导出来的第一类Fredholm向量积分方程。建立起与之等价的边界变分方程,采用Galerkin边界元求解,得出单层位势的向量密度,进而根据速度的边界积分表达式的离散形式计算流场中任意点的流速值。
本文首先阐述了复连通区域二维Stokes问题的Galerkin边界元解法的思路,利用Fortran语言编制程序计算速度场,用Matlab画出相应的流线图,以检验程序的可靠性。然后主要针对有界区域内含有开边界的问题进行数值模拟。由于Galerkin边界元方法需要计算二重积分,本文推导出了带普通插值形函数和奇异形函数的积分公式,对存在奇异性的单元上的第一重积分中采用解析积分,针对第二重积分推导出了带奇异积分权函数的Gauss积分公式,对不存在奇异性的单元所有积分都采用数值积分。论文用若干数值算例模拟了复连通区域上以及含有开边界的有界区域上不可压缩粘性流体的绕流。
Stokes equation is the linearization of Navier-Stokes equation, which is the governing equation of incompressible viscous flow, with small Reynolds number. It forms Stokes problem with the viscous boundary conditions attached. Among so many numerical methods to solve this problem, the boundary element method is an ideal method, since it is easy to handle the continuity equation, or the incompressibility condition,moreover, computing the velocity and computing the pressure can be separated in the solution procedure. In this paper,we solved the Stokes problem in a bounded plane region with non-closed line or curve segment as an open boundary,which is a two-Dimensional screen problem in fluid flow. The difficulty in solving this kind of problems is how to deal with the singularity near the endpoints on the open boundary. We have established the singular element with singular shape function while the element contains endpoint in order to simulate the inherent singularity.
We establish the variational formulation based on the Fredholm vector integral equations of the first kind, which are derived by the Green’s formula and the fundamental solution corresponding to Stokes operator. Then, we solve the boundary variational equation by Galerkin Boundary Element Method, to get vector density function of the simple layer. The velocity at any point in the flow field is calculated by the discrete form of boundary integral expression.
We first set forth the formulation of the Galerkin Boundary Element Method to solve the 2-D Stokes problem in a complex connected region, and devise the computing codes by Fortran program for calculate the velocity field of the flow. The streamlines are drawn by a Matlab program to verify the reliability of the program. Then we focus on the numerical simulation of the problem containing open boundary in a bounded domain. Because by Galerkin boundary element method we need to calculate the double integrals in computing matrix coefficients, we derived the analytical integral formula with shape functions contained singularity or not, and special Gauss integral formula with singular weight function. For the double integrations on singular boundary element, we carry out the first integration by analytical integral formula and the second integration by a Gauss integral formula with a special singular weight function, while the usual Gauss integral formulas are used when there is no singularity on the element. In this paper, several numerical tests simulated the flow of the viscous incompressible fluid in a complex connected region and a bounded region with an open boundary.
我们利用对应于Stokes算子的Green公式和基本解所推导出来的第一类Fredholm向量积分方程。建立起与之等价的边界变分方程,采用Galerkin边界元求解,得出单层位势的向量密度,进而根据速度的边界积分表达式的离散形式计算流场中任意点的流速值。
本文首先阐述了复连通区域二维Stokes问题的Galerkin边界元解法的思路,利用Fortran语言编制程序计算速度场,用Matlab画出相应的流线图,以检验程序的可靠性。然后主要针对有界区域内含有开边界的问题进行数值模拟。由于Galerkin边界元方法需要计算二重积分,本文推导出了带普通插值形函数和奇异形函数的积分公式,对存在奇异性的单元上的第一重积分中采用解析积分,针对第二重积分推导出了带奇异积分权函数的Gauss积分公式,对不存在奇异性的单元所有积分都采用数值积分。论文用若干数值算例模拟了复连通区域上以及含有开边界的有界区域上不可压缩粘性流体的绕流。
Stokes equation is the linearization of Navier-Stokes equation, which is the governing equation of incompressible viscous flow, with small Reynolds number. It forms Stokes problem with the viscous boundary conditions attached. Among so many numerical methods to solve this problem, the boundary element method is an ideal method, since it is easy to handle the continuity equation, or the incompressibility condition,moreover, computing the velocity and computing the pressure can be separated in the solution procedure. In this paper,we solved the Stokes problem in a bounded plane region with non-closed line or curve segment as an open boundary,which is a two-Dimensional screen problem in fluid flow. The difficulty in solving this kind of problems is how to deal with the singularity near the endpoints on the open boundary. We have established the singular element with singular shape function while the element contains endpoint in order to simulate the inherent singularity.
We establish the variational formulation based on the Fredholm vector integral equations of the first kind, which are derived by the Green’s formula and the fundamental solution corresponding to Stokes operator. Then, we solve the boundary variational equation by Galerkin Boundary Element Method, to get vector density function of the simple layer. The velocity at any point in the flow field is calculated by the discrete form of boundary integral expression.
We first set forth the formulation of the Galerkin Boundary Element Method to solve the 2-D Stokes problem in a complex connected region, and devise the computing codes by Fortran program for calculate the velocity field of the flow. The streamlines are drawn by a Matlab program to verify the reliability of the program. Then we focus on the numerical simulation of the problem containing open boundary in a bounded domain. Because by Galerkin boundary element method we need to calculate the double integrals in computing matrix coefficients, we derived the analytical integral formula with shape functions contained singularity or not, and special Gauss integral formula with singular weight function. For the double integrations on singular boundary element, we carry out the first integration by analytical integral formula and the second integration by a Gauss integral formula with a special singular weight function, while the usual Gauss integral formulas are used when there is no singularity on the element. In this paper, several numerical tests simulated the flow of the viscous incompressible fluid in a complex connected region and a bounded region with an open boundary.
引文
[1] John T.Katsikadelis Boundary elements:Theory and Applications[J], Department of Civil Engineering, National Technical University of Athens, Athens, Greece. 2002.
[2]冯康著.数值计算方法[M].北京:国防工业出版社,1978.
[3]谷超豪等.数学物理方程[M].北京:人民教育出版社,1979.
[4]祝家麟著.椭圆边值问题的边界元分析[M].北京:科学出版社,1991.
[5]杜庆华著.边界积分方程方法-边界元法[M].北京:高等教育出版社,1989.
[6]王元淳著.边界元法基础[M].上海:上海交通大学出版社,1988.
[7]杨德全,赵忠生著.边界元理论及应用[M].北京:北京理工大学出版社,2002.
[8]申光宪等著.边界元法[M].北京:机械工业出版社,1998.
[9]孙炳楠等著.工程中边界元法及其应用[M].浙江:浙江大学出版社,1991.
[10]布瑞比亚C.A.Brebbia,泰勒斯J.C.F,诺贝尔L.C著,龙述尧,刘腾喜,蔡松柏译[M].边界单元法的理论和工程应用.国防工业出版社,1988.
[11]导向科技编著. MATLAB6.0程序设计与实例应用[M].北京:中国铁道出版社,2001.
[12]余德浩,自然边界元方法的数学理论[M],科学出版社,1993.
[13]祝家麟.边界元分析[M],科学出版社,2009.
[14]姚寿广著.边界元数值方法及其工程应用[M].北京:国防工业出版社,1991.
[15] Brebbia C.A. The Boundary Element Method for Engineers[M]. London: Pentech Press. 1978.
[16] C.A.Brebbia.J.C.F.Telles.L.C.Wrobel. Boudary Element techniques–Theory and Applications in Engineering[M]. Springer-Verlag,Berlin Heidelberg New York Tokyo,1984.
[17]祝家麟.用边界积分方程法解平面双调和方程的Dirichlet问题[J].计算数学, 1984(3):278-286.
[18] G.C. Hsiao, P. Kopp and W.L. Wendland, A Galerkin collocation method for some integral equations of the first kind[J], Computing 25, 89-130 (1980).
[19] I.H. Sloan and A. Spence, The Galerkin method for integral equations of the first kind with logarithmic kerne[J]l: Theory and applications, IMA J. Numer. Anal. 8, 105-140 (1988).
[20] Hsiao G C. A modified Galerkin scheme for equations with natural boundary conditions[J]. In: Vichnevetsky R, Vigness J. Numerical Mathematics and Applications. B. V: Elsevier Science Publishers, 1986: 193-197.
[21]董海云,祝家麟.二维Laplace方程的直接边界积分方程的Galerkin边界元解法[D].重庆大学硕士论文集,2005.
[22]李荣华.解边值问题的Galerkin方法[M].上海:上海科学技术出版社.1988.12.
[23]张守贵,用双层位势求解二维Laplace方程的Neumann问题的Galerkin边界元解法[D].重庆大学硕士论文集,2005.
[24] J.T.Katsikadelis. Boundary elements theory and applications[M]. Elsevier,2002.
[25]祝家麟.边界元方法――它的产生、内容和意义.第一届工程中的边界元法会议论文集[C].杜庆华,1985.
[26]祝家麟.边界元方法—一个老而新的发展中的数值方法[J].数学的实践与认识, 1983(3):74-79.
[27]张雄,刘岩著.无网格法[M].北京:清华大学出版社,2004.
[28]张耀明,孙焕纯,杨家新.虚边界元法的理论分析[J].计算力学学报,2000,17(1):56-62.
[29]邹广德,孙焕纯等.求解位势问题的虚边界元法[J].计算结构力学及其应用,1994.11.
[30]祝家麟.边界元方法中的奇异性[J].重庆建筑工程学院学报,1985,2(13):90—102.
[31]陈懋章,粘性流体动力学基础[M].北京:高等教育出版社,2002.
[32]王献孚,熊鳌魁.高等流体力学[M].武汉:华中科技大学出版社,2003
[33]许贤良,王传礼,张军,朱增宝.流体力学[M].北京:国防工业出版社,2006.
[34]祝家麟.用边界积分方程法通过流函数解二维Stokes问题[J].重庆建筑大学学报,1982.
[35]祝家麟.三维定常流Stokes问题的边界积分方程法[J],计算数学,第1期,1985.2.
[36] Hsiao.G.C (1982). Integral representations of solutions for 2-D viscous flow problems[J]. Integral Eqs and Oper. Theory,5.533-547.
[37] Zhu.Jialin(1985),The boundary integral equation method for solving stationary Stokes problems[J];(Ed Feng kang),Procof beijing Symposium on differential Geometry and Differential Equations Science Press.374-377.
[38] Jialin Zhu. Asymptotic Error Estimates of Bem FOR 2-Dimensional Flow Problems. Theory and applications of boundary element methods[J]. In proceedings of the 1st Japan-China Symposium on Boundary Elemnet Mthods,(Edited by Mastaka Tanaka and QingHua Du) pergamon press.Oxford 295-305.
[39] J.E.Osborn,Regularity of solutions of the Stokes problem in a polygonal domain[J], In Numerical Solution of PDEⅢ,(Edited by B.Hubbard),pp.393-411,Synspade,Academic Press, New York,(1976).
[40] P.Grisvard,Elliptic Problems in Nonsmooth Domains[M], Pitman, London,(1985).
[41] M.Costabel and E.P.Stephan,An improved boundary element Galerkin method for three-dimensional crack problems[J],integral Equs. and Operator Theory. 10,467-504(1987).
[42] M.Costabel and W.L.Wendlend,Strong ellipticity of boundary integral operators[J],j.fur die reine and angewandte Mathematik 372,34-63(1986).
[43] M.Costabel, E.Stephan and W.L.Wendland, On boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain[J],Ann.Scuola Norm.Sup. Pisa, ser.IV 10,197-242(1983).
[44] G.C.Hsiao,E.P.Stephan and W.L.Wendland, on the dirichlet problem in elasticity for a domain exterior to an arc[J]. Math.Inst.A.University of Stuttgart, Preprint 15,(1989)
[45] M.Dauge,Operateur de stokes dans des espaces de sobolev a poids sur des domaines anguleux[J], Can. J.Math.34,853-882(1982).
[46] Hsiao.G.C kress R (1985); on an integral equation for the 2-Dexterior Stokes problem[J]. Appl.Num.Math.,1,pp,73-93.
[47] W.L.Wendland,J.Zhu the boundary element method for three-dimensional stokes flows exterior to an open surface[J], Mathematical Compution Modelling Vol.15,No.6.19-41,1991
[48]向瑞银.平面定常STOKES问题的Galerkin线性边界元解法[D].重庆大学硕士论文集,2005.
[49]赵翔龙著. Fortron90学习教程[M].北京:北京大学出版社,2000.
[50] Zhu Jialin, The boundary integral equation method for solving stationary Stokes problems[J], In: Feng Kang. Proc. of the 1984 Beijing Symp. On Diff. Geometry and Diff. Equations. Beijing, Science Press, 1985.
[51] George C. Hsiao, Ernst P. Stephan and Wolfgang L. Wendland. an integral equation formulation for a boundary value problem of elasticity in the domain exterior to an arc[J]. Mathematical methods in the applied sciences. 1993(16:22): 87-114.
[52] W. L. Wendland E.P.Stephan A Hypersingular Boundary Integral Method for Two-Dimensional Screen and Crack Problems[J]. Arch. Rational Mech. Anal. 112 (1990) 363-390.
[53] D.L.Young, S.J.Jane, C.M.Fan, K.Murugesan, C.C.Tsai. The method of fundamental solutions for 2D and 3D Stokes problems[J].Journal of Computational Physics,2006,211,1-8.
[54]向瑞银等.平面定常Stokes方程的Galerkin边界元解法[J],重庆大学学报,2006 (2).29-2.
[55]张洁等.开弧段外Laplace方程单层位势解法探讨[J],后勤工程学院学报,2005 (1).69-71.
[56]张耀明,温卫东.平面定常Stokes问题的无奇异第一类边界积分方程[J],计算数学,2005 ,2(1):1-10.
[57] A.Zed,L.Elliott,D.B.Ingham,D.Lesnic.The boundary element method for the solution of Stokes equations in two-dimensional domains[J], Engineering Analysis with Boundary Elements 22(1998)317-326.
[58] D.H.Yu Canonical equations of Stokes problem, J.Comput.Math.,4:1(1986),62-73.
[59] B.I.Yun, S.Lee and U.J.Choi.A Modified Boundary Integral Method on Open Arts in the Plane[J],Computers & Mathematics with ApplicationsVolume 31, Issue 11, June 1996, Pages 37-43.
[60] Johannes Elschnera, Ernst P. Stephan A discrete collocation method for Symm's integral equation on curves with comers[J], Journal of Computational and Applied Mathematics 75 (1996) 131-146.
[61] F.-J. SAYAS,The Numerical Solution of Symm's Equation on Smooth Open Arcs by Spline Galerkin Methods[J], Computers and Mathematics with Applications 38 (1999) 87-99.
[62] Y.Yan,Cosine change of variable for Symm's integral equation on open arcs[J], IMA J.Numer.Anal.10, 521-535 (1990)
[63]余德浩.无界区域上Stokes问题的自然边界元与有限元耦合法[J],计算数学,14:3(1992), 371-378.
[64] D.H.Yu,The Natural Boundary Integral Method and Its Applications[J],Kluwer Academic Public,2002.
[65]李瑞遐.边界是光滑开弧Helmholtz方程的边界积分法[J],华东理工大学学报,1999-08 ,25(4).
[66]祝家麟,定常Stokes问题的边界积分方程法[J],计算数学,1986(8):281-289.
[67]张洁.开边界外调和方程的迦辽金解法[D].重庆大学硕士论文集,2003.
[68]张耀明,温卫东等.平面定常Stokes问题的无奇异第一类边界积分方程[J].计算数学.Vol.27(1),2005.2.
[69] J.T.Katsikadelis. Boundary elements theory and applications[M]. Elsevier,2002.
[70]何银年,李开泰外部非定常Navier-Stokes方程的非线性Galerkin算法[J],应用数学和力学, Vol.23(11),2002.11.
[71]吕涛,黄晋定常Stokes问题的边界积分方程的高精度求积方法与外推[J],计算物理, Vol.16(6),1999.11.
[72] George C. Hsiao, M.D.Marcozzi and S.Zhang an efficient computional method for the flow pastan airfoil[J]. Mathematics Subject Classification.65N30,65F10,1991.
[73] KENDALL E.ATKINSON AND IAN H.SLOAN the numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs[J], Mathematics Subject Classification 1980.
[74] Xiaolin Li,Jialin Zhu A meshless Galerkin method for Stokes problems using boundary integral equations[J], Comput.Methods Appl. Mech.Engrg,2009.
[75]稽醒,臧跃龙,程玉民著.边界元法进展及通用程序[M].上海:同济大学出版社,1997.
[76]谭浩强,田淑清. Fortran语言—Fortran77结构化程序设计[M].北京:清华大学出版社.1999.
[77]谭世语,范幸义. Fortran程序设计[M].重庆:重庆大学出版社,2002.
[78]徐明编. Fortran power station 4.0基础教程[M].北京:清华大学出版社,2000.
[79]梅志红,杨万铨. MATLAB程序设计基础及其应用[M].清华大学出版社,2005.
[80]张德喜,周予生. MATLAB语言程序设计教程[M].中国铁道出版社,2006.
[81]张瑞丰等著.精通MATLAB6.5[M].北京:中国水利水电出版社,2003.
[2]冯康著.数值计算方法[M].北京:国防工业出版社,1978.
[3]谷超豪等.数学物理方程[M].北京:人民教育出版社,1979.
[4]祝家麟著.椭圆边值问题的边界元分析[M].北京:科学出版社,1991.
[5]杜庆华著.边界积分方程方法-边界元法[M].北京:高等教育出版社,1989.
[6]王元淳著.边界元法基础[M].上海:上海交通大学出版社,1988.
[7]杨德全,赵忠生著.边界元理论及应用[M].北京:北京理工大学出版社,2002.
[8]申光宪等著.边界元法[M].北京:机械工业出版社,1998.
[9]孙炳楠等著.工程中边界元法及其应用[M].浙江:浙江大学出版社,1991.
[10]布瑞比亚C.A.Brebbia,泰勒斯J.C.F,诺贝尔L.C著,龙述尧,刘腾喜,蔡松柏译[M].边界单元法的理论和工程应用.国防工业出版社,1988.
[11]导向科技编著. MATLAB6.0程序设计与实例应用[M].北京:中国铁道出版社,2001.
[12]余德浩,自然边界元方法的数学理论[M],科学出版社,1993.
[13]祝家麟.边界元分析[M],科学出版社,2009.
[14]姚寿广著.边界元数值方法及其工程应用[M].北京:国防工业出版社,1991.
[15] Brebbia C.A. The Boundary Element Method for Engineers[M]. London: Pentech Press. 1978.
[16] C.A.Brebbia.J.C.F.Telles.L.C.Wrobel. Boudary Element techniques–Theory and Applications in Engineering[M]. Springer-Verlag,Berlin Heidelberg New York Tokyo,1984.
[17]祝家麟.用边界积分方程法解平面双调和方程的Dirichlet问题[J].计算数学, 1984(3):278-286.
[18] G.C. Hsiao, P. Kopp and W.L. Wendland, A Galerkin collocation method for some integral equations of the first kind[J], Computing 25, 89-130 (1980).
[19] I.H. Sloan and A. Spence, The Galerkin method for integral equations of the first kind with logarithmic kerne[J]l: Theory and applications, IMA J. Numer. Anal. 8, 105-140 (1988).
[20] Hsiao G C. A modified Galerkin scheme for equations with natural boundary conditions[J]. In: Vichnevetsky R, Vigness J. Numerical Mathematics and Applications. B. V: Elsevier Science Publishers, 1986: 193-197.
[21]董海云,祝家麟.二维Laplace方程的直接边界积分方程的Galerkin边界元解法[D].重庆大学硕士论文集,2005.
[22]李荣华.解边值问题的Galerkin方法[M].上海:上海科学技术出版社.1988.12.
[23]张守贵,用双层位势求解二维Laplace方程的Neumann问题的Galerkin边界元解法[D].重庆大学硕士论文集,2005.
[24] J.T.Katsikadelis. Boundary elements theory and applications[M]. Elsevier,2002.
[25]祝家麟.边界元方法――它的产生、内容和意义.第一届工程中的边界元法会议论文集[C].杜庆华,1985.
[26]祝家麟.边界元方法—一个老而新的发展中的数值方法[J].数学的实践与认识, 1983(3):74-79.
[27]张雄,刘岩著.无网格法[M].北京:清华大学出版社,2004.
[28]张耀明,孙焕纯,杨家新.虚边界元法的理论分析[J].计算力学学报,2000,17(1):56-62.
[29]邹广德,孙焕纯等.求解位势问题的虚边界元法[J].计算结构力学及其应用,1994.11.
[30]祝家麟.边界元方法中的奇异性[J].重庆建筑工程学院学报,1985,2(13):90—102.
[31]陈懋章,粘性流体动力学基础[M].北京:高等教育出版社,2002.
[32]王献孚,熊鳌魁.高等流体力学[M].武汉:华中科技大学出版社,2003
[33]许贤良,王传礼,张军,朱增宝.流体力学[M].北京:国防工业出版社,2006.
[34]祝家麟.用边界积分方程法通过流函数解二维Stokes问题[J].重庆建筑大学学报,1982.
[35]祝家麟.三维定常流Stokes问题的边界积分方程法[J],计算数学,第1期,1985.2.
[36] Hsiao.G.C (1982). Integral representations of solutions for 2-D viscous flow problems[J]. Integral Eqs and Oper. Theory,5.533-547.
[37] Zhu.Jialin(1985),The boundary integral equation method for solving stationary Stokes problems[J];(Ed Feng kang),Procof beijing Symposium on differential Geometry and Differential Equations Science Press.374-377.
[38] Jialin Zhu. Asymptotic Error Estimates of Bem FOR 2-Dimensional Flow Problems. Theory and applications of boundary element methods[J]. In proceedings of the 1st Japan-China Symposium on Boundary Elemnet Mthods,(Edited by Mastaka Tanaka and QingHua Du) pergamon press.Oxford 295-305.
[39] J.E.Osborn,Regularity of solutions of the Stokes problem in a polygonal domain[J], In Numerical Solution of PDEⅢ,(Edited by B.Hubbard),pp.393-411,Synspade,Academic Press, New York,(1976).
[40] P.Grisvard,Elliptic Problems in Nonsmooth Domains[M], Pitman, London,(1985).
[41] M.Costabel and E.P.Stephan,An improved boundary element Galerkin method for three-dimensional crack problems[J],integral Equs. and Operator Theory. 10,467-504(1987).
[42] M.Costabel and W.L.Wendlend,Strong ellipticity of boundary integral operators[J],j.fur die reine and angewandte Mathematik 372,34-63(1986).
[43] M.Costabel, E.Stephan and W.L.Wendland, On boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain[J],Ann.Scuola Norm.Sup. Pisa, ser.IV 10,197-242(1983).
[44] G.C.Hsiao,E.P.Stephan and W.L.Wendland, on the dirichlet problem in elasticity for a domain exterior to an arc[J]. Math.Inst.A.University of Stuttgart, Preprint 15,(1989)
[45] M.Dauge,Operateur de stokes dans des espaces de sobolev a poids sur des domaines anguleux[J], Can. J.Math.34,853-882(1982).
[46] Hsiao.G.C kress R (1985); on an integral equation for the 2-Dexterior Stokes problem[J]. Appl.Num.Math.,1,pp,73-93.
[47] W.L.Wendland,J.Zhu the boundary element method for three-dimensional stokes flows exterior to an open surface[J], Mathematical Compution Modelling Vol.15,No.6.19-41,1991
[48]向瑞银.平面定常STOKES问题的Galerkin线性边界元解法[D].重庆大学硕士论文集,2005.
[49]赵翔龙著. Fortron90学习教程[M].北京:北京大学出版社,2000.
[50] Zhu Jialin, The boundary integral equation method for solving stationary Stokes problems[J], In: Feng Kang. Proc. of the 1984 Beijing Symp. On Diff. Geometry and Diff. Equations. Beijing, Science Press, 1985.
[51] George C. Hsiao, Ernst P. Stephan and Wolfgang L. Wendland. an integral equation formulation for a boundary value problem of elasticity in the domain exterior to an arc[J]. Mathematical methods in the applied sciences. 1993(16:22): 87-114.
[52] W. L. Wendland E.P.Stephan A Hypersingular Boundary Integral Method for Two-Dimensional Screen and Crack Problems[J]. Arch. Rational Mech. Anal. 112 (1990) 363-390.
[53] D.L.Young, S.J.Jane, C.M.Fan, K.Murugesan, C.C.Tsai. The method of fundamental solutions for 2D and 3D Stokes problems[J].Journal of Computational Physics,2006,211,1-8.
[54]向瑞银等.平面定常Stokes方程的Galerkin边界元解法[J],重庆大学学报,2006 (2).29-2.
[55]张洁等.开弧段外Laplace方程单层位势解法探讨[J],后勤工程学院学报,2005 (1).69-71.
[56]张耀明,温卫东.平面定常Stokes问题的无奇异第一类边界积分方程[J],计算数学,2005 ,2(1):1-10.
[57] A.Zed,L.Elliott,D.B.Ingham,D.Lesnic.The boundary element method for the solution of Stokes equations in two-dimensional domains[J], Engineering Analysis with Boundary Elements 22(1998)317-326.
[58] D.H.Yu Canonical equations of Stokes problem, J.Comput.Math.,4:1(1986),62-73.
[59] B.I.Yun, S.Lee and U.J.Choi.A Modified Boundary Integral Method on Open Arts in the Plane[J],Computers & Mathematics with ApplicationsVolume 31, Issue 11, June 1996, Pages 37-43.
[60] Johannes Elschnera, Ernst P. Stephan A discrete collocation method for Symm's integral equation on curves with comers[J], Journal of Computational and Applied Mathematics 75 (1996) 131-146.
[61] F.-J. SAYAS,The Numerical Solution of Symm's Equation on Smooth Open Arcs by Spline Galerkin Methods[J], Computers and Mathematics with Applications 38 (1999) 87-99.
[62] Y.Yan,Cosine change of variable for Symm's integral equation on open arcs[J], IMA J.Numer.Anal.10, 521-535 (1990)
[63]余德浩.无界区域上Stokes问题的自然边界元与有限元耦合法[J],计算数学,14:3(1992), 371-378.
[64] D.H.Yu,The Natural Boundary Integral Method and Its Applications[J],Kluwer Academic Public,2002.
[65]李瑞遐.边界是光滑开弧Helmholtz方程的边界积分法[J],华东理工大学学报,1999-08 ,25(4).
[66]祝家麟,定常Stokes问题的边界积分方程法[J],计算数学,1986(8):281-289.
[67]张洁.开边界外调和方程的迦辽金解法[D].重庆大学硕士论文集,2003.
[68]张耀明,温卫东等.平面定常Stokes问题的无奇异第一类边界积分方程[J].计算数学.Vol.27(1),2005.2.
[69] J.T.Katsikadelis. Boundary elements theory and applications[M]. Elsevier,2002.
[70]何银年,李开泰外部非定常Navier-Stokes方程的非线性Galerkin算法[J],应用数学和力学, Vol.23(11),2002.11.
[71]吕涛,黄晋定常Stokes问题的边界积分方程的高精度求积方法与外推[J],计算物理, Vol.16(6),1999.11.
[72] George C. Hsiao, M.D.Marcozzi and S.Zhang an efficient computional method for the flow pastan airfoil[J]. Mathematics Subject Classification.65N30,65F10,1991.
[73] KENDALL E.ATKINSON AND IAN H.SLOAN the numerical solution of first-kind logarithmic-kernel integral equations on smooth open arcs[J], Mathematics Subject Classification 1980.
[74] Xiaolin Li,Jialin Zhu A meshless Galerkin method for Stokes problems using boundary integral equations[J], Comput.Methods Appl. Mech.Engrg,2009.
[75]稽醒,臧跃龙,程玉民著.边界元法进展及通用程序[M].上海:同济大学出版社,1997.
[76]谭浩强,田淑清. Fortran语言—Fortran77结构化程序设计[M].北京:清华大学出版社.1999.
[77]谭世语,范幸义. Fortran程序设计[M].重庆:重庆大学出版社,2002.
[78]徐明编. Fortran power station 4.0基础教程[M].北京:清华大学出版社,2000.
[79]梅志红,杨万铨. MATLAB程序设计基础及其应用[M].清华大学出版社,2005.
[80]张德喜,周予生. MATLAB语言程序设计教程[M].中国铁道出版社,2006.
[81]张瑞丰等著.精通MATLAB6.5[M].北京:中国水利水电出版社,2003.