辛數學與一維、二維、三維結構力學問題之多辛架構

Abstract

本論文旨在探討結構和固體力學的多辛架構，包括一維、二維及三 維問題。在線性力學，基於矩陣代數及矩陣函數的基礎，我們提出一 套對於多辛控制方程式的矩陣指數解。該矩陣指數解要求矩陣指數函 數的微分連鎖律與交換性，從而定義矩陣交換子為零的條件。為了找 出滿足此矩陣交換子為零的情況，我們利用約旦分解法來分類所有可 交換矩陣的可能樣式。這種分類方式提供我們方法建構不同模式的解 及處理複雜初邊值問題的能力。 在非線性結構力學中，我們透過變分法成功推導出一套延伸的多辛 偏微方程以及其對應之守恆式。清楚呈現當非線性項同時存在於相容 式與力平衡式時，兩者間的對偶關係。我們同時更進一步針對線性及 非線性結構力學之多辛守恒式，包括局部特性、全域特性及面積守恆 等特性，賦予適切之物理意義。

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.