本文提出無網格伽遼金法(Element free Galerkin method；EFGM)和四叉樹無網格伽遼金法(Quadtree Element free Galerkin method；QEFGM)模擬在熱環境下之電磁彈性材料之靜態和破壞問題。藉由結合最小勢能原理和馬克士威方程組推導電磁彈性材料之統御方程組，使用最小移動二乘法(moving Least Squares；MLS)建構形狀函數。由於此形狀函數不滿足克羅內克(Kronecker delta)函數性質，因此在邊界條件上施加罰函數(penalty method)來修正。探討裂縫問題時，由於裂紋造成位移場的不連續而引入繞射法(diffraction method)對形狀函數進行修正和利用背景分解法(the background decomposition method；BDM) 取代傳統背景網格生成來提高數值解之精確度，最後透過M-integral方法計算應力強度因子。首先，透過對電磁彈性結構柱進行驗證並觀察在熱環境下熱電/熱磁效應對材料物理特性之影響。更進一步探討在不同熱環境下，電磁彈性結構樑之物理特性變化。最後，藉由中間裂紋電磁彈性平板進行驗證並觀察熱環境對應力強度因子的影響。在這些數值模擬問題中，顯示EFGM/QEFGM對電磁彈性材料之多物理場耦合和破裂問題都具有計算高效率和準確性。

In this dissertation report, the element-free Galerkin method (EFGM) and quadtree element-free Galerkin method (QEFGM) are proposed for the static analysis and fracture behaviour of a magnetoelectroelastic (MEE) structure in thermal environments. The system governing equations of MEE structures are derived from the principle of minimum potential energy and Maxwell's equations. In EFGM, the moving least-squares (MLS) method is utilized to generate shape functions, however, because it does not satisfy the principle of the Kronecker delta, the penalty method is implemented to impose approximate boundary conditions. In the fracture problem, the crack caused a discontinuous displacement field; therefore, the shape function is corrected using the diffraction method. The background decomposition method is used to replace the background cell generation to improve the accuracy of the numerical solution and to calculate the stress intensity factors using the M-integral.
First, a plane strain MEE material with the z-axis polarization direction is assumed. And then, the results are verified and applied to reveal the effects of thermo-magnetic and thermo-electric coupling using the MEE column and a comparative study is conducted to evaluate the variations in the physical characteristics along the longitudinal direction of the MEE beam in different thermal environments. Finally, the SIFs of the internal cracked plate are calculated to verify and observe them effectively in a thermal environment. The numerical results demonstrate the efficiency and accuracy of the EFG/QEFGM formulation for multi-physics or fracture simulation of MEE structures.

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