Symplectic mathematics and problems of 1D, 2D, 3D structural mechanics under multi-symplectic framework
Date Issued
2016
Date
2016
Author(s)
Lee, Hsiao-Wei
Abstract
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.
Subjects
Hamiltonian system
symplectic group
Lie algebra
matrix exponential functions
Jordan decomposition
eigenmode expansion
multi-symplectic structure
structural mechanics
SDGs
Type
thesis
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