拟协调有限元弱形式的辛算法
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摘要
基于弱形式的力学方程,阐述了弱形式广义方程是拟协调有限元的内在本质,用弱形式给出的微分方程和边界条件根本上是降低了函数光滑性,不过对工程问题而言,给出的有限元解比原始方程更接近真实解,其数值解就是广义协调方程的直接解,同时满足平衡和几何方程弱连续条件。进而就导出的对偶体系弱形式哈密尔顿方程,采用辛相似变换,利用平方约化法求解哈密尔顿矩阵特征值问题,使其哈密尔顿结构得到了保证。辛算法具有较强的有效性,可以解决常规有限元难以适应的领域,对计算力学发展有着重要的作用。
The paper explains that the generalized weak formulation is of the inherent nature for the quasi-conforming element based on weak formulation elasticity equations. It points out that the existing equilibrium differential equations with boundary conditions also can be seen as a starting point for variational formulation, the weak formulation is of more fundamental and original. Formally, through the weak form, the continuity of the function is reduced, as generalized direct solutions, more accurate solutions to the differential equations are obtained. The equilibrium equations and geometric equations, then their weak forms, are also satisfied. Weak formulation of the dual system of Hamilton equations can be solved by means of symplectic similarity transformation which reflects the structure of the Hamiltonian matrices. This algorithm has preferable validity, and therefore the quasi-conforming element can be used in the field in which the common finite element is not feasible, so it is a landmark in computational mechanics.
The paper explains that the generalized weak formulation is of the inherent nature for the quasi-conforming element based on weak formulation elasticity equations. It points out that the existing equilibrium differential equations with boundary conditions also can be seen as a starting point for variational formulation, the weak formulation is of more fundamental and original. Formally, through the weak form, the continuity of the function is reduced, as generalized direct solutions, more accurate solutions to the differential equations are obtained. The equilibrium equations and geometric equations, then their weak forms, are also satisfied. Weak formulation of the dual system of Hamilton equations can be solved by means of symplectic similarity transformation which reflects the structure of the Hamiltonian matrices. This algorithm has preferable validity, and therefore the quasi-conforming element can be used in the field in which the common finite element is not feasible, so it is a landmark in computational mechanics.
引文
[1]Zienkiewicz R L,Taylor.The finite element method[M].New York:Mcgraw-Hill Book Company,1988.
[2]Pian T H H,Chen D P.On the suppression of zero energy deformation modes[J].International Journal for Numerical Methods in Engineering,1983,19(12):1741-1752.
[3]唐立民.有限元分析的若干基本问题[J].大连工学院学报,1979,18(2):1-15.(Tang Limin.Some basic problems of finite element method[J].Journal of Dalian Institute of Technology,1979,18(2):1-15(in Chinese)).
[4]唐立民,陈万吉,刘迎曦.有限元分析中的拟协调元[J].大连工学院学报,1980,19(2):19-35.(Tang Limin,Chen Wanji,Liu Yingxi.Quasi-conforming elements for finite element analysis[J].Journal of Dalian Institute of Technology,1980,19(2):19-35(in Chinese)).
[5]唐立民,刘迎曦.多变量拟协调元解流函数形式的两维N-S方程[J].大连工学院学报,1987,26(1):1-7.(Tang Limin,Liu Yingxi.Two dimensional navier-stokes equations of stream-function formulation solved by multivariate quasi-conforming techniques[J].Journal of Dalian Institute of Technology,1987,26(1):1-7(in Chinese)).
[6]唐立民,陈万吉.多变量拟协调平面四边元[J].计算结构力学及其应用,1988,5(1):1-6.(Tang Limin,Chen Wanji.A multi-variable quasi-conforming quadrilateral element[J].Computational Structural Mechanics and Applications,1988,5(1):1-6(in Chinese)).
[7]Tang L,Chen W,Liu Y.String net function approximation and quasi-conforming technique,hybid and mixed finite element methods[M].Atluri S N,Gallagher R H,Zienkiewicz O C,translated.New York:John Wiley&Sons,1983.
[8]Tang L,Chen W,Liu Y.Formulation of quasi-conforming element and Hu-Washizu principle[J].Computers&Structures,1984,19(1):247-250.
[9]陈万吉.再论拟协调元与广义变分原理[J].固体力学学报,1983(1):129-135.(Chen Wanji.More on the quasi-conforming element methods and the generalized variational principles[J].Acta Mechanica Solida Sinica,1983(1):129-135(in Chinese)).
[10]陈万吉,刘迎曦.拟协调元与广义变分原理[J].大连工学院学报,1980,19(3):71-80.(Chen Wanji,Liu Yingxi.The quasi-compatible element models and the generalized variational principle[J].Journal of Dalian Institute of Technology,1980,19(3):71-80(in Chinese)).
[11]张鸿庆,王鸣.多套函数有限元逼近与拟协调元[J].应用数学和力学,1985(1):41-52.(Zhang Hongqing,Wang Ming.Finite element approximations with multiple sets of functions and quasi-conforming elements for plate bending problems[J].Applied Mathematics and Mechanics,1985(1):41-52(in Chinese).
[12]张鸿庆.多套函数的广义分片试验与十二参拟协调元[J].大连工学院学报,1982,21(3):11-19.(Zhang Hongqing.The generalized patch test of multiple sets of functions and the 12-parameter quasi-conforming element[J].Journal of Dalian Institute of Technology,1982,21(3):11-19(in Chinese)).
[13]龙驭球,辛克贵.广义协调元[J].土木工程学报,1987(1):1-14.(Long Yuqiu,Xin Kegui.Generalized conforming element[J].China Civil Engineering Journal,1987(1):1-14(in Chinese)).
[14]陈绍春,石钟慈.构造单元刚度矩阵的双参数法[J].计算数学,1991(3):286-296.(Chen Shaochun,Shi Zhongci.Double set parameter method of constructing stiffness matrices[J].Mathematica Numerica Sinica,1991(3):286-296(in Chinese)).
[15]唐立民.弹性力学的混合方程和Hamilton正则方程[J].计算结构力学及其应用,1991,8(4):343-350.(Tang Limin.Mixed formulation and Hamilton canonical equations of theory of elasticity[J].Computational Structural Mechanics and Applications,1991,8(4):343-350(in Chinese)).
[16]唐立民,褚致中,邹贵平,等.混合状态Hamilton元的半解析解和叠层板的计算[J].计算结构力学及其应用,1992,9(4):347-360.(Tang Limin,Chu Zhizhong,Zou Guiping,et al.The semi-analytical solution of mixed state Hamiltonian element and the computation of laminated plates[J].Computational Structural Mechanics and Applications,1992,9(4):347-360(in Chinese)).
[17]唐立民,齐朝晖,丁克伟,等.弹性力学弱形式广义基本方程的建立和应用[J].大连理工大学学报,2001,41(1):1-8.(Tang Limin,Qi Zhaohui,Ding Kewei,et al.Establishment and applications of weak form generalized basic equations in elasticity[J].Journal of Dalian University of Technology,2001,41(1):1-8(in Chinese)).
[18]Van Loan C.A symplectic method for approximating all the eigenvalue of a Hamiltonian matrix[J].Linear Algebra and Its Applications,1984,61(c):233-251.
[19]Bunse A G,Mehrmann V.A symplectic QR-like algorithm for the solution of the real algebraic Riccati equation[J].IEEE Transactions on Automatic Control,1986,31(12):1104-1113.
[20]Benner P,Fabbender H.A implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem[J].Linear Algebra and Its Applications,1997,263,75-111.
[21]Benner P,Fabbender H.An implicitly restarted symplectic Lanczos method for the symplectic eigenvalue problem[J].SIAM Journal on Matrix Analysis and Applications,2000,22(3):682-713.
[22]Mehrmann V,Watkins D.Structure preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/Hamiltonian pencils[J].SIAM Journal on Scientific Computing,2002,22(6):1905-1925.
[23]Fabbender H.Error analysis of the symplectic Lanczos method for symplectic eigenvalue problem[J].BIT Numerical Mathematics,2000,40(3):471-496.
[24]闫庆友,魏小鹏.求解大规模Hamilton矩阵特征问题的辛Lanczos算法的误差分析[J].数学研究与评论,2004,24(1):91-106.(Yan Qingyou,Wei Xiaopeng.Error analysis of symplectic Lanczos method for Hamiltonian eigenvalue problem[J].Journal of Mathematical Research and Exposition,2004,24(1):91-106(in Chinese)).
[25]丁克伟,唐立民.弹性力学混合状态方程的弱形式及其边值问题[J].力学学报,1998,30(5):580-586.(Ding Kewei,Tang Limin.Weak formulation of mixed state equation and boundary value problem of theory of elasticity[J].Acta Mechanica Sinica,1998,30(5):580-586(in Chinese)).
[26]丁克伟,唐立民.轴对称层合圆柱壳混合状态方程的弱形式及其边值问题[J].应用力学学报,1998,15(4):32-36.(Ding Kewei,Tang Limin.Weak formulation of mixed state equation and boundary value problem for axisymmetric problem of laminated cylindrical shel[J].Chinese Journal of Applied Mechanics,1998,15(4):32-36(in Chinese)).
[27]Ding Kewei,Tang Limin.Exact solution for axisymmetric thick laminated shells[J].Composite Structures,1999,46(2):125-129.
[28]丁克伟.测试Hamiltonian矩阵结构问题的辛算法[J].合肥工业大学学报,2006,29(2):189-192.(Ding Kewei.Symplectic algorithm for testing the Hamiltonian matrix[J].Journal of Hefei University of Technology,2006,29(2):189-192(in Chinese)).
[29]Ding Kewei.Weak formulation study for thermoelastic analysis of thick open laminated shell[J].Mechanics of Advanced Materials and Structures,2008,15(15):33-39.
[30]丁克伟.拟协调有限元与弱形式广义方程[J].合肥工业大学学报,2009,32(12):1875-1879.(Ding Kewei.Quasi-conforming element methods and generalized weak formulation equations in elasticity[J].Journal of Hefei University of Technology,2009,32(12):1875-1879(in Chinese)).
[2]Pian T H H,Chen D P.On the suppression of zero energy deformation modes[J].International Journal for Numerical Methods in Engineering,1983,19(12):1741-1752.
[3]唐立民.有限元分析的若干基本问题[J].大连工学院学报,1979,18(2):1-15.(Tang Limin.Some basic problems of finite element method[J].Journal of Dalian Institute of Technology,1979,18(2):1-15(in Chinese)).
[4]唐立民,陈万吉,刘迎曦.有限元分析中的拟协调元[J].大连工学院学报,1980,19(2):19-35.(Tang Limin,Chen Wanji,Liu Yingxi.Quasi-conforming elements for finite element analysis[J].Journal of Dalian Institute of Technology,1980,19(2):19-35(in Chinese)).
[5]唐立民,刘迎曦.多变量拟协调元解流函数形式的两维N-S方程[J].大连工学院学报,1987,26(1):1-7.(Tang Limin,Liu Yingxi.Two dimensional navier-stokes equations of stream-function formulation solved by multivariate quasi-conforming techniques[J].Journal of Dalian Institute of Technology,1987,26(1):1-7(in Chinese)).
[6]唐立民,陈万吉.多变量拟协调平面四边元[J].计算结构力学及其应用,1988,5(1):1-6.(Tang Limin,Chen Wanji.A multi-variable quasi-conforming quadrilateral element[J].Computational Structural Mechanics and Applications,1988,5(1):1-6(in Chinese)).
[7]Tang L,Chen W,Liu Y.String net function approximation and quasi-conforming technique,hybid and mixed finite element methods[M].Atluri S N,Gallagher R H,Zienkiewicz O C,translated.New York:John Wiley&Sons,1983.
[8]Tang L,Chen W,Liu Y.Formulation of quasi-conforming element and Hu-Washizu principle[J].Computers&Structures,1984,19(1):247-250.
[9]陈万吉.再论拟协调元与广义变分原理[J].固体力学学报,1983(1):129-135.(Chen Wanji.More on the quasi-conforming element methods and the generalized variational principles[J].Acta Mechanica Solida Sinica,1983(1):129-135(in Chinese)).
[10]陈万吉,刘迎曦.拟协调元与广义变分原理[J].大连工学院学报,1980,19(3):71-80.(Chen Wanji,Liu Yingxi.The quasi-compatible element models and the generalized variational principle[J].Journal of Dalian Institute of Technology,1980,19(3):71-80(in Chinese)).
[11]张鸿庆,王鸣.多套函数有限元逼近与拟协调元[J].应用数学和力学,1985(1):41-52.(Zhang Hongqing,Wang Ming.Finite element approximations with multiple sets of functions and quasi-conforming elements for plate bending problems[J].Applied Mathematics and Mechanics,1985(1):41-52(in Chinese).
[12]张鸿庆.多套函数的广义分片试验与十二参拟协调元[J].大连工学院学报,1982,21(3):11-19.(Zhang Hongqing.The generalized patch test of multiple sets of functions and the 12-parameter quasi-conforming element[J].Journal of Dalian Institute of Technology,1982,21(3):11-19(in Chinese)).
[13]龙驭球,辛克贵.广义协调元[J].土木工程学报,1987(1):1-14.(Long Yuqiu,Xin Kegui.Generalized conforming element[J].China Civil Engineering Journal,1987(1):1-14(in Chinese)).
[14]陈绍春,石钟慈.构造单元刚度矩阵的双参数法[J].计算数学,1991(3):286-296.(Chen Shaochun,Shi Zhongci.Double set parameter method of constructing stiffness matrices[J].Mathematica Numerica Sinica,1991(3):286-296(in Chinese)).
[15]唐立民.弹性力学的混合方程和Hamilton正则方程[J].计算结构力学及其应用,1991,8(4):343-350.(Tang Limin.Mixed formulation and Hamilton canonical equations of theory of elasticity[J].Computational Structural Mechanics and Applications,1991,8(4):343-350(in Chinese)).
[16]唐立民,褚致中,邹贵平,等.混合状态Hamilton元的半解析解和叠层板的计算[J].计算结构力学及其应用,1992,9(4):347-360.(Tang Limin,Chu Zhizhong,Zou Guiping,et al.The semi-analytical solution of mixed state Hamiltonian element and the computation of laminated plates[J].Computational Structural Mechanics and Applications,1992,9(4):347-360(in Chinese)).
[17]唐立民,齐朝晖,丁克伟,等.弹性力学弱形式广义基本方程的建立和应用[J].大连理工大学学报,2001,41(1):1-8.(Tang Limin,Qi Zhaohui,Ding Kewei,et al.Establishment and applications of weak form generalized basic equations in elasticity[J].Journal of Dalian University of Technology,2001,41(1):1-8(in Chinese)).
[18]Van Loan C.A symplectic method for approximating all the eigenvalue of a Hamiltonian matrix[J].Linear Algebra and Its Applications,1984,61(c):233-251.
[19]Bunse A G,Mehrmann V.A symplectic QR-like algorithm for the solution of the real algebraic Riccati equation[J].IEEE Transactions on Automatic Control,1986,31(12):1104-1113.
[20]Benner P,Fabbender H.A implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem[J].Linear Algebra and Its Applications,1997,263,75-111.
[21]Benner P,Fabbender H.An implicitly restarted symplectic Lanczos method for the symplectic eigenvalue problem[J].SIAM Journal on Matrix Analysis and Applications,2000,22(3):682-713.
[22]Mehrmann V,Watkins D.Structure preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/Hamiltonian pencils[J].SIAM Journal on Scientific Computing,2002,22(6):1905-1925.
[23]Fabbender H.Error analysis of the symplectic Lanczos method for symplectic eigenvalue problem[J].BIT Numerical Mathematics,2000,40(3):471-496.
[24]闫庆友,魏小鹏.求解大规模Hamilton矩阵特征问题的辛Lanczos算法的误差分析[J].数学研究与评论,2004,24(1):91-106.(Yan Qingyou,Wei Xiaopeng.Error analysis of symplectic Lanczos method for Hamiltonian eigenvalue problem[J].Journal of Mathematical Research and Exposition,2004,24(1):91-106(in Chinese)).
[25]丁克伟,唐立民.弹性力学混合状态方程的弱形式及其边值问题[J].力学学报,1998,30(5):580-586.(Ding Kewei,Tang Limin.Weak formulation of mixed state equation and boundary value problem of theory of elasticity[J].Acta Mechanica Sinica,1998,30(5):580-586(in Chinese)).
[26]丁克伟,唐立民.轴对称层合圆柱壳混合状态方程的弱形式及其边值问题[J].应用力学学报,1998,15(4):32-36.(Ding Kewei,Tang Limin.Weak formulation of mixed state equation and boundary value problem for axisymmetric problem of laminated cylindrical shel[J].Chinese Journal of Applied Mechanics,1998,15(4):32-36(in Chinese)).
[27]Ding Kewei,Tang Limin.Exact solution for axisymmetric thick laminated shells[J].Composite Structures,1999,46(2):125-129.
[28]丁克伟.测试Hamiltonian矩阵结构问题的辛算法[J].合肥工业大学学报,2006,29(2):189-192.(Ding Kewei.Symplectic algorithm for testing the Hamiltonian matrix[J].Journal of Hefei University of Technology,2006,29(2):189-192(in Chinese)).
[29]Ding Kewei.Weak formulation study for thermoelastic analysis of thick open laminated shell[J].Mechanics of Advanced Materials and Structures,2008,15(15):33-39.
[30]丁克伟.拟协调有限元与弱形式广义方程[J].合肥工业大学学报,2009,32(12):1875-1879.(Ding Kewei.Quasi-conforming element methods and generalized weak formulation equations in elasticity[J].Journal of Hefei University of Technology,2009,32(12):1875-1879(in Chinese)).