本文發展基於Reissner混合變分原理(Reissner’s Mixed Variational Theorem, RMVT)之無網格適點與無元素Galerkin方法，其中各場量變數相應之形狀函數則引用微分再生核(Differential Reproducing Kernel, DRK)函數予以建構，文中該二方法亦應用於功能性梯度材料板之三維靜、動態行為分析。此三維彈性力學問題經由Reissner混合變分原理獲得強形式問題相應之Euler-Lagrange方程式和可能的邊界條件，與弱形式問題相應之加權殘餘積分式，其中各獨立層之材料性質考慮沿厚度方向為冪級數律分佈。DRK內插法中，各主變數相應之形狀函數是由具有Kronecker Delta性質的原始函數(The Primitive Function)與滿足再生條件的改善函數(The Enrichment Function)所組成，可方便於幾何邊界條件的代入，與有限元素法的用法相似。以DRK內插函數代入RMVT無網格適點與無元素Galerkin方法中，求得該二方法相關之系統方程，進而求得相關之三維數值解。文中對無網格方法應用時相關之關鍵參數，諸如：最佳影響範圍(a)、節點數(Np)與基底函數階數(n)，均有討論，並作出合宜之建議。數值範例的結果顯示，該二方法求得之解與文獻中既有之三維解有相當一致的結果且收斂迅速。

A meshless collocation (MC) and an element-free Galerkin (EFG) method, using the differential reproducing kernel (DRK) interpolation, are developed for the three-dimensional (3D) static and dynamic analyses of simply supported, multilayered composite and functionally graded material (FGM) plates. Based on the Reissner Mixed Variational Theorem (RMVT), the strong and weak formulations of these 3D problems are derived, in which the material properties of each individual FGM layer, constituting the plate, are assumed to obey the power-law distributions of the volume fractions of the constituents. The system equations of both the RMVT-based MC and EFG methods are obtained using these strong and weak formulations, respectively, in combination with the DRK interpolation, in which the shape functions of the unknown functions satisfy the Kronecker delta properties, and the essential boundary conditions can be readily applied, exactly like the implementation in the finite element method. In the illustrative examples, the primary field variables and their variations, the natural frequencies and their corresponding modal field variables varying along the thickness coordinate of the plate are studied. It is shown that the solutions obtained using these methods are in excellent agreement with the available 3D solutions, and their convergence rates are rapid.

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