Symplectic mathematics and problems of 1D, 2D, 3Dstructural mechanics under multi-symplectic framework

• Main Authors: Hsiao-Wei Lee, 李孝威
• Other Authors: 洪宏基
• Format: Others
• Language: en_US
• Published: 2016
• Online Access: http://ndltd.ncl.edu.tw/handle/70022315899351052000

碩士 === 國立臺灣大學 === 土木工程學研究所 === 104 === This thesis focuses on the multi-symplectic analysis in structural and solids mechanics, including 1D, 2D and 3D problems. In linear mechanics, we propose matrix exponential solutions to multi-symplectic governing equations based on matrix algebra and matrix functions. The matrix exponential solution requires the commutator of matrices equal zero. To seek for the condition under which the commutator of matrices equals zero, we utilize the method of Jordan decomposition to classify all possible patterns of the commutative matrices. This classification provides us a way to formulate different patterns of solutions, and establishes a method to tackle the initial-boundary value problems with a variety of initial and boundary conditions. In non-linear mechanics, we successfully derive extended multi-symplectic governing equations and their related conservation laws by the variational principle. The dual relation between compatibility and equilibrium when nonlinear terms exist is demonstrated. Furthermore, the conservation properties including the conservation of area, local and global properties in linear and non-linear structural mechanics are investigated by giving appropriate physical meanings.