摘要
本文對有限元素法提出改善元素之內插函數的想法，在元素內部增加高階函數項以提高精確度，產生一系列四邊形和六面體強化型元素，並對其操作及效能進行討論。強化型元素內部選擇拉格朗日元素內部節點對應的內插函數，元素邊上節點對應的內插函數則由巧湊邊點元素的內插函數經克羅內克函數修正而得。所以強化型元素除了邊上節點外，內部插入節點以加入高階多項式。這些內部節點與其他元素節點不相連結的特性，所以可在產生元素係數矩陣時將其分離出來，不會造成全域聯立方程組維度增大的問題。
在強化型元素的運用方面可利用靜態縮減法以及次參數單元形式簡化計算。靜態縮減的過程是對係數矩陣進行部分分解，因此能夠視為一個預加條件。而且這個預加條件矩陣是直接計算到每個係數矩陣的行列上，所以採用疊代法解聯立方程組時，可以再採用另一個預加條件矩陣。使用次參數單元形式來表示元素幾何不需要內部節點的座標，所以在網格化的過程中也不需要產生這些節點，因此強化型元素可以直接運用現有的網格化軟體。
本文分別採用平面應力問題和三維彈力問題來驗證四邊形和六面體強化型元素的效能。由數值實驗結果發現強化型元素比起傳統的巧湊邊點元素更為準確，並且具有與巧湊邊點元素相同的收斂速率。在一階強化型元素的實驗中，一階強化型元素能夠有效降低誤差值，但是增加些微的計算時間。而二階和三階強化型元素不但能夠大幅地降低誤差值，更能使疊代法所需要疊代次數下降，進而節省總計算時間。

Abstract
The concept that adds the high order terms to the shape function of finite element method is presented to enrich the quadrilateral and hexahedral elements and the performance of the enriched elements is discussed in this thesis. The enriched elements combine the shape functions of interior nodes of the Langrage elements and the shape functions of the serendipity elements which are corrected by Kronecker delta function. Since the interior nodes of enriched elements don’t connect with any other element, their degree of freedom can be separated from the linear system.
In order to simplify the use of enriched elements, both the static condensation and subparametric formulation are employed in finite element analysis. By the use of the static condensation technique at the element level, the extra computation time in using these elements can be ignored. The procedure can be seen that the coefficient matrix is applied a partial factorization. Therefore, static condensation can be regarded as a precondition. Since this precondition is applied directly to the entities of the coefficient matrix, the iterative method can use another precondition to solve the linear system. By the use of the subparametric formulation, the coordinates of the interior nodes are not necessary in the finite element analysis and the existing programs can generate the mesh for enriched elements.
The plane stress problems and three dimensional elastic problems are used to evaluate the performance of enriched quadrilateral and hexahedral elements, respectively. It shows that the results obtained by using the enriched elements are more accurate than those of the traditional serendipity elements. The convergence rate of the proposed elements is the same as that of the traditional serendipity elements. In the numerical examples, the error norm of the first order enriched elements can be reduced when compared with the use of the traditional serendipity element, but the computation time is increased slightly. The use of the second and third order proposed elements not only give an improvement in element accuracy but also save computation time, when the precondition conjugate gradient method is used to solve the linear system. The saving of computation time is due to the decrease of iteration number.

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