以邊界積分方程法正算尤拉梁問題

Abstract

橋梁的振動在土木工程中是一個非常重要的議題，而橋梁可以簡化為尤拉梁模型來作為分析。本文提供的分析方法為邊界積分方程法(BIEM)，其搭配了伴隨Trefftz測試函數為基底作係數的展開，而伴隨Trefftz測試函數本身是滿足齊性控制方程式和邊界條件的，因此能夠消除吉布斯現象和避免矩陣運算，也就是說能夠在誤差極小的情況下得到數值解。邊界積分方程法能將難以求得解析解的微分方程式問題轉換成依靠邊界條件來描述整個場的等效積分方程式問題。最後由數值算例可以知道邊界積分方程法在追求高精度、高效率的情況下是可行的。

In this thesis we numerically solve the direct Euler-Bernoulli beam problems by using a boundary integral equation method(BIEM) which is based on the generalized Green’s second identity and the self-adjoint operators. In the BIEM, we choose a set of adjoint Trefftz test functions which can be obtained by the method of separation of variables. In the numerical algorithm, we can expand a trial solution by using the bases satisfying the homogeneous governing equation and the boundary conditions simultaneously. To satisfy the above two properties of the bases, we use the adjoint Trefftz test functions as the bases and impose the specified boundary condition. By using these bases, moreover, we can eliminate the Gibbs phenomenon and avoid the matrix computations. Finally, there are several numerical examples to validate the effectiveness of the proposed scheme in this thesis and the results show that the BIEM is a highly accurate numerical method.