本文利用B-spline函數為基底函數，解二維不規則形狀的平面應力問題。本文同時使用二至六階的B-spline函數並改變元素大小，針對不同的不規則形狀做詳細的探討。
首先對正方形平板挖圓孔的不規則形狀作研究，發現幾何模擬誤差對高階B-spline函數的收斂性影響很大。藉由降低幾何模擬誤差，當元素逐漸縮小時，可顯現三階以上B-spline函數相當佳的h收斂性，而當元素大小不變，增加B-spline函數階數時，則可顯現相當佳的p收斂性。
為了使B-spline有限元素法能更廣泛應用到其他幾何形狀，因此本論文也研究正方形平板挖多圓孔、橢圓孔及圓角矩形孔的應力問題。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，大條件數對正方形平板挖圓孔問題的應力誤差約為0.001%，而對正方形平板挖橢圓孔的應力誤差約為0.1%~0.3%。經由正方形平板挖圓角矩形孔的研究，發現高階B-spline函數為了維持Ck-1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。
本文提出細切方法，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不規則形狀平面應力問題。

We used the B-spline functions as the basis functions to solve the two dimensional plane stress problems with irregular shapes. We used B-spline functions from second order to sixth order simultaneously and different element size on the problems with different irregular shapes.
The first irregular shape in these studies is a square plate with a circular hole. The geometry simulation error has a great effect upon the convergence rate of high order B-spline functions. By decreasing the simulation error of geometry, 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 same element size.
In order to use the B-spline finite element method widely on other irregular shapes, we also study stress problems on a square plate with multiple circular holes, a elliptic hole and a rectangle hole with round edges. We also study the effect of the stiffness matrix with a large condition number, which is caused by small integration area. The stress error induced by large condition number on a square plate with a circular hole is about 0.001%, and the stress error on a square plate with a elliptic hole is about 0.1%~0.3%. According to the study of a rectangle hole with round edges in a square plate, for maintaining the property of Ck-1 continuity, the accuracy of the high order 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.
We proposed a method of refinement. With this method, 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 problems with irregular shapes.

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