层合结构Hamiltonian元弱形式的辛方法
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摘要
文章基于力学平衡方程,在柱坐标系下,导出正交各向异性层合柱壳混合方程和边界条件算子的弱形式,继而给出层合结构的Hamilton正则方程,建立半离散半解析的Hamiltonian元的微分方程,用弱形式给出的微分方程和边界条件对函数的连续性要求降低了,用于解决实际的工程问题常常比原始的微分方程更逼近真正解;针对其Hamiltonian元的矩阵结构,构建分析计算Hamiltonian元弱形式的辛方法,Hamilton结构在辛约化过程中得到充分保证,文中提出的辛方法简易可行,具有较强的有效性和稳定性。
It was presented of weak formulations of mixed equations including boundary conditions of laminated cylindrical shell in a cylindrical coordinate system based upon the mechanical equilibrium equation.Hamilton canonical equation of laminated structure was established,and then semi-discrete and semi-analytical differential equation of Hamiltonian element was set up.The differential equations and boundary conditions given in weak formulation make the requirements for the continuity of functions relaxed,and for the actual physical problem,it often more approaches the true solution than the original differential equations.The symplectic algorithm for testing Hamiltonian matrix structure was established in the process of reducing Hamiltonian matrix.This method works with symplectic similarity transformation which reflects the structure of the Hamiltonian matrices.This algorithm is simple and feasible,and has preferable validity and stability.
It was presented of weak formulations of mixed equations including boundary conditions of laminated cylindrical shell in a cylindrical coordinate system based upon the mechanical equilibrium equation.Hamilton canonical equation of laminated structure was established,and then semi-discrete and semi-analytical differential equation of Hamiltonian element was set up.The differential equations and boundary conditions given in weak formulation make the requirements for the continuity of functions relaxed,and for the actual physical problem,it often more approaches the true solution than the original differential equations.The symplectic algorithm for testing Hamiltonian matrix structure was established in the process of reducing Hamiltonian matrix.This method works with symplectic similarity transformation which reflects the structure of the Hamiltonian matrices.This algorithm is simple and feasible,and has preferable validity and stability.
引文
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[7]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.
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[9]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.
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[12]丁克伟.Hamiltonian矩阵平方约化求解特征问题的辛算法[J].安徽理工大学学报(自然科学版),2005,25(2):24-28.
[13]DING Kewei,Thermal stresses of weak formulation study for thick laminated shell[J].Journal of Thermal Stresses,2008,31(4):389-400.
[14]MOLER C,VAN LOAN C.Nineteen dubious ways to computer the exponential of a matrix[J].SIAM Review,1978,20(4):89-108.
[15]LAUB A J.Numerical linear algebra aspects of control design computations[J].IEEE Transactions on Automatic Control,1985,30(2):97-108.
[16]YAO G,LI F M.The stability analysis and active control of a composite laminated open cylindrical shell in subsonic airflow[J].Journal of Intelligent Material Systems and Structures,2014,25(3):259-270.
[17]MAGHAMI S A,TAHANI M.Thermal bending analysis of doubly curved composite laminated shell panels with general boundary conditions and laminations[J].Journal of Thermal Stresses,2015,38(2):250-270.
[18]BURGUENO R,HU N,HEERINGA A,et al.Tailoring the elastic postbuckling response of thin-walled cylindrical composite shells under axial compression[J].Thin-Walled Structures,2014,84:14-25.
[19]ELYSEEVA J.Comparison theorems for conjoined bases of linear Hamiltonian differential systems and the comparative index[J].Journal of Mathematical Analysis and Applications,2016,444(2):1260-1273.
[20]LERMAN L M,YAKOVLEV E I.Geometry of slow-fast Hamiltonian systems and Painleve equations[J].Indagationes Mathematicae:New Series,2016,27(5):1219-1244.
[21]GRILLO S,PADRON E.A Hamilton-Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds[J].Journal of Geometry and Physics,2015,110:101-129.
[2]REDDY J N.Exact solution of moderately thick laminated shells[J].Journal of Engineering Mechanics Division,1984,111:794-809.
[3]何陵辉,刘人怀.一种考虑层间位移和横向剪应力连续条件的层合板理论[J].固体力学学报,1994,15(4):319-326.
[4]DING H J,CHEN W Q.Exact shell theory analysis of free vibration of submerged thin spherical shells[J].International Journal of Solids and Structures,1998,35(33):4381-4389.
[5]冯康,秦孟兆.Hamilton动力体系的Hamilton算法[J].自然科学进展国家重点实验室通讯,1991,1(2):102-112.
[6]FENG Kang.The step-transition operator for multi-step methods of ODEs[J].Journal of Computation Mathematics,1998,16(3):193-202.
[7]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.
[8]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.
[9]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.
[10]VAN LOAN C.A symplectic method for approximating all the eigenvalue of a Hamiltonian matrix[J].Linear Algebra and Its Applications,1984,61:233-251.
[11]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.
[12]丁克伟.Hamiltonian矩阵平方约化求解特征问题的辛算法[J].安徽理工大学学报(自然科学版),2005,25(2):24-28.
[13]DING Kewei,Thermal stresses of weak formulation study for thick laminated shell[J].Journal of Thermal Stresses,2008,31(4):389-400.
[14]MOLER C,VAN LOAN C.Nineteen dubious ways to computer the exponential of a matrix[J].SIAM Review,1978,20(4):89-108.
[15]LAUB A J.Numerical linear algebra aspects of control design computations[J].IEEE Transactions on Automatic Control,1985,30(2):97-108.
[16]YAO G,LI F M.The stability analysis and active control of a composite laminated open cylindrical shell in subsonic airflow[J].Journal of Intelligent Material Systems and Structures,2014,25(3):259-270.
[17]MAGHAMI S A,TAHANI M.Thermal bending analysis of doubly curved composite laminated shell panels with general boundary conditions and laminations[J].Journal of Thermal Stresses,2015,38(2):250-270.
[18]BURGUENO R,HU N,HEERINGA A,et al.Tailoring the elastic postbuckling response of thin-walled cylindrical composite shells under axial compression[J].Thin-Walled Structures,2014,84:14-25.
[19]ELYSEEVA J.Comparison theorems for conjoined bases of linear Hamiltonian differential systems and the comparative index[J].Journal of Mathematical Analysis and Applications,2016,444(2):1260-1273.
[20]LERMAN L M,YAKOVLEV E I.Geometry of slow-fast Hamiltonian systems and Painleve equations[J].Indagationes Mathematicae:New Series,2016,27(5):1219-1244.
[21]GRILLO S,PADRON E.A Hamilton-Jacobi Theory for general dynamical systems and integrability by quadratures in symplectic and Poisson manifolds[J].Journal of Geometry and Physics,2015,110:101-129.