發展基於微分再生核(Differential Reproducing Kernel, DRK)適點法於彈性體問題中一維及二維之常微分及偏微分控制方程的求解。在傳統的再生核(Reproducing Kernel, RK)近似法中，再生核近似函數之導數的形狀函數須對再生核近似函數(Liu, Jun and Zhang, 1995) 直接微分以進行求取，尤其是對高階導數的微分計算相當耗時。相對於的直接微分處理，我們提出一組微分再生狀態於求解再生核近似函數之導數的形狀函數。根據本文中發展的無網格適點法，應用於分析一維的彈性桿問題、二維的勢能問題及彈性體的平面問題，並驗證其準確性及發現其收斂率。接著，使用DRK內插法發展無網格置點法於求解功能性梯度(Functionally Graded, FG)彈性梁及板承載側向力學載重平面問題之靜態分析，其中DRK內插法是由隨機分佈的節點建構而成。發展以DRK內插法為基礎之適點法求解齊次FG彈性梁及板的平面應力及應變問題。結果展示，基於DRK近似-及基於DRK內插-適點法，此二種真正的無網格方法確實皆具備較優良的準確度及較快的收斂率。
本文以Reissner混合變分理論(the Reissner Mixed Variational Theorem, RMVT)基礎，結合先前提出的DRK近似函數與無網格適點 (Meshless Collocation, MC) 法及無元素Galerkin (Element-free Galerkin, EFG)法，發展近似三維靜態及自由振動三維彈性力學問題分析，求解圍繞簡支撐、承受力學載重之多層複合材料及功能梯度材料(Functionally Graded Material, FGM)中空圓柱殼之物理問題。此三維彈性力學問題的強形式及弱形式在靜態問題中皆被使用，前者由物理問題配合相關邊界條件組成尤拉-拉格朗日方程式(Euler－Lagrange equations)，後者為加權殘餘積分式，其差異均勻分佈於主變數量及其變異之中。使用DRK內插法來建構主變數，以滿足Kronecker Delta之性質，且與主變數相關的邊界及連續條件可以直接應用。採用強形式及弱形式結合DRK內插函數可獲得基於RMVT之無網格法及無元素Galerkin法的系統方程。其中，各獨立之功能性梯度材料層之材料性質沿該層之厚度座標方向以體積分數的冪級數律分佈組成，基於RMVT可求導出此三維動態問題之弱形式。

A differential reproducing kernel (DRK) approximation-based collocation method is developed for solving ordinary and partial differential equations governing the one- and two-dimensional problems of elastic bodies. In the conventional reproducing kernel (RK) approximation, the shape functions for the derivatives of RK approximants are determined by directly differentiating the RK approximants (Liu, Jun and Zhang, 1995), and this is very time-consuming, especially for the calculations of their higher-order derivatives. Contrary to the previous differentiation manipulation, we construct a set of differential reproducing conditions to determine the shape functions for the derivatives of RK approximants. A meshless collocation method based on the present DRK approximation is developed and applied to the analysis of one-dimensional problems of elastic bars, two-dimensional potential problems, and plane elasticity problems of elastic solids to validate its accuracy and find the rate of convergence. Subsequently, a meshless collocation method is developed for the static analysis of plane problems of functionally graded (FG) elastic beams and plates under transverse mechanical loads using the DRK interpolation, in which the DRK interpolant is constructed by the randomly-distributed nodes. A point collocation method based on this DRK interpolation is developed for the plane stress and strain problems of homogeneous and FG elastic beams and plates. It is shown that the present method, both the DRK approximation- and the DRK interpolation- based collocation method, is indeed a fully meshless approach with excellent accuracy and fast convergence rate.
In this dissertation, a meshless collocation (MC) and an element-free Galerkin (EFG) method in conjunction with an earlier proposed DRK interpolation are developed for the approximate three-dimensional (3D) static and free vibration analysis of the 3D elasticity problem, centering on simply supported, multilayered composite and functionally graded material (FGM) circular hollow cylinders under mechanical loads, derived on the basis of the Reissner mixed variational theorem (RMVT). Both the strong and weak formulations of this 3D elasticity problem are used in static problems, the former consists of the Euler－Lagrange equations of this problem and its associated boundary conditions, while the latter represents a weighted-residual integral in which the differentiation is equally distributed among the primary field variables and their variations. An earlier proposed DRK interpolation is used to construct the primary field variables where the Kronecker delta properties are satisfied, and the boundary and continuity conditions related to the primary variables themselves can be directly applied. The system equations of both the RMVT-based MC and EFG methods are obtained using these strong and weak formulations in combination with the DRK interpolation. Based on the RMVT, the weak formulation of this 3D dynamic problem is derived, in which the material properties of each individual FGM layer are assumed to obey the power-law distributions of the volume fractions of the constituents through the thickness coordinate of the layer.

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