無網格邊界積分方程法求解內域勢能問題及外域反平面力場問題

Numerical analysis of interior potential and exterior anti-plane problems by using the mesh-free boundary integral equation method

鄧適婕(Shih-Chieh Teng)

指導教授 ： 李家瑋

淡江大學/工學院/土木工程學系碩士班/碩士(2021年)

https://doi.org/10.6846/TKU.2021.00557

摘要

本文提出了一種求解滿足拉普拉斯(Laplace)方程二維邊界問題的無網格邊界積分方程方法(Meshfree Boundary Integral Equation Method)來解決內域勢能問題和反平面力場外域問題，此方法與需要生成元素網格的傳統邊界元方法不同，本方法只需要邊界節點。在處理柯西主值奇異積分和固體角的計算時，引入了邊界點局部正確解的邊界積分方程。該局部正確解必須滿足三個條件，其中包括1.場解要滿足拉普拉斯方程式 2.邊界點的原始場量 3.其對邊界點場量的法向導數，因此，局部正確解是邊界物理量與滿足二維拉普拉斯方程對應形狀函數的線性組合。透過減去原問題的邊界積分方程式和局部正確解的邊界積分方程式，可技巧性地計算柯西主值奇異積分，也可以免去計算邊界點上的固體角。計算上述的邊界積分方程式只使用了一次高斯積分，所以邊界積分方程式只不過是一個代數方程式，這些邊界節點就是高斯積分點，這就是為什麼在本方法中只需要邊界節點的原因，同時它們也是獲得聯立方程式的配置點，這個想法還可以保留數值方法的靈活性，因此它適用於任何幾何形狀。總之，本方法有兩個優點，一種是無網格化，另一種是不用主值計算奇異積分。最後，本文考慮了一些例子，如壩基滲流問題、穩態熱傳導問題和受反平面剪切應力中的包含孔洞/剛性夾雜物的無限域問題，以檢驗此無網格邊界積分方程方法的有效性。

關鍵字

無網格邊界積分方程法 ； 勢能問題 ； 反平面力場問題 ； 高斯積分 ； 局部正確解

並列摘要

In this thesis, a meshfree boundary integral equation method is proposed to solve 2D boundary value problems for the Laplace equation. Both interior potential problems and exterior problems under anti-plane shear are considered. Only boundary nodes are required for the present method different from the conventional boundary element method that needs to generate the mesh. To technically deal with the Cauchy principal value and the solid angle, a boundary integral equation of a local exact solution for a boundary point is introduced. This local exact solution must satisfy three conditions, including Laplace equation for the domain point, original boundary datum and its normal derivative on a boundary point. Therefore, a local exact solution is a linear combination of boundary data with corresponding shape functions which satisfy the 2D Laplace equation. By subtracting the boundary integral equation of original problem and boundary integral equation of a local exact solution, the singular integral in the sense of the Cauchy principal value can be technically determined. Free of calculating the solid angle for the boundary point is also gained. Then, the Gaussian quadrature is employed only once for the above boundary integral equation, the boundary integral equation is nothing but an algebraic equation. This is the reason why only boundary nodes are required in the present method. Those boundary nodes are just the Gaussian quadrature points. Simultaneously, they are also the collocation points to obtain the simultaneous equation. This idea can also preserve the flexibility of numerical method, hence it is suitable for any geometry shape. In a word, there are two advantages in the present method. One is meshfree. The other is free of calculating singular integral by using the sense of principal value. Finally, some examples such as a seepage problem with a dam foundation, steady heat conduction problems and infinite plane problems containing a hole/rigid inclusion in anti-plane shear are considered to examine the validity of the present meshfree boundary integral equation method.

並列關鍵字

Meshfree boundary integral equation method ； Potential problems ； External anti-plane field problems ； Gauss integral ； Local exact solution

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