本文利用B-spline函數為基底函數，解二維雙材料不規則邊界形狀的平面應力問題。最初的作法是將不同材料共用相同的基底函數，因此不同材料間仍具有Ck-2 的連續性，但是不同的彈性模數且應變連續的狀況下會使應力被迫呈現不連續的現象，導致應力值的不準確。處理方式是將不同材料使用不同的基底函數並在交界處施加相合條件，使交界處的位移一致。因此本文除了針對B-spline有限元素法於雙材料問題進行處理，同時使用二至六階的B-spline函數並改變元素大小，對不同的雙材料不規則邊界形狀做詳細的探討。

首先對正方形平板圓孔邊界形狀問題作研究，發現當元素逐漸縮小時，可顯現三階以上B-spline函數不錯的h收斂性；而在適當的元素大小且當元素大小保持不變，增加B-spline函數階數時，則可顯現不錯的p收斂性。

為了使B-spline有限元素法能更廣泛應用到其他不規則邊界形狀，因此本論文也研究將雙材料正方形平板之橢圓及圓角矩形邊界形狀的應力問題。對於細小積分面積而引起的勁度矩陣條件數變大的影響，本論文也有所探討。經由雙材料矩形邊界形狀的研究，發現B-spline函數為了維持Ck-2連續性，在最大應力點附近區域準確度不如二階有限元素法的結果，但最大應力點的準確度比二階有限元素法佳。另外，相合條件也可能會影響B-spline函數的收斂性。

本文提出雙材料問題的細切方法，此方法在應力集中的區域可任意增加小元素B-spline函數，可用較少的自由度得到更準確的分析，提升B-spline有限元素法的效率。

綜合以上各分析，本研究顯示B-spline有限元素法可以廣泛應用在二維雙材料不規則邊界形狀平面應力問題。

We used the B-spline functions as the basis functions to solve the two dimensional plane stress problems of bi-material with irregular boundary shapes. Originally, we used the same basis function in different material. But stress was forced to be discontinues due to the continuity of strain and different modulus of elasticity. Finally, this problem induced by inaccuracy of stress. Therefore we used different basis functions in different material and applied compatibility condition in the border between two materials, producing the consistency of displacement in the border between two materials. We used B-spline functions from second order to sixth order simultaneously and different element size on the problems with different irregular boundary shapes.

The first irregular boundary shape in these studies is a square plane with a circular shape. The h convergence of high order B-spline functions is excellent when the element size decreases, and the p convergence is also excellent when increases the order of B-spline functions in the suitable and the same element size.

In order to use the B-spline finite element method widely on other irregular boundary shapes, we also study stress problems of bi-material on a square plate with a elliptic boundary shape, and a rectangle boundary shape with round edges. We also study the effect of the stiffness matrix with a large condition number, which is caused by small integration area.According to the study of a rectangle boundary shape with round edges in a square plate, for maintaining the property of Ck-2 continuity, the accuracy of the B-spline functions on areas around the maximum stress point is not as good as the results of the second-order finite element method, but the accuracy of maximum stress point is better than the results of the second-order finite element method. Specially, compatibility condition may affect convergence of B-spline function.

We proposed a method of refinement for bi-material. With this mehod, we can add small B-spline functions on small elements at will, and we can use lesser degree of freedom to get a more accurate result in the analysis. The uses of refinement will increase the efficiency of the B-spline finite element method.

In summary, the studies in this thesis show that the B-spline finite element method can be used widely in the analysis on the two dimensional plane stress problem of bi-material with irregular boundary shapes.

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