以加權高階梯度再生核函數配置法求解反算柯西問題

Solving Inverse Cauchy Problems by Weighted High-Order Gradient Reproducing Kernel Collocation Method

林漢峰(Hon-Fung-Samuel Lam)

指導教授 ： 楊子儀

國立交通大學/工學院/土木工程系所/碩士(2020年)

摘要

反算問題乃指不適定與不穩定之邊界值問題，通常具備不完整之邊界條件，也因此往往無法用解析方式分析，需仰賴數值近似。由於文獻上具有高效率、高精準度之數值求解方法仍有待探討，故本研究率先以加權高階梯度再生核函數配置法求解反算柯西問題。此數值方法之優點為簡潔高效，引入高階梯度再生核形狀函數近似未知之導數，透過高階梯度再生核形狀函數滿足高階梯度再生條件以構建形狀函數，免去計算反矩陣與核函數之導數，大幅提升計算效率。本研究透過四個數值算例驗證數值方法之精度、穩定性與運算效率，考慮之反算柯西問題包含單連通域與多重連通域，在多重連通域中進一步考慮不同邊界未知之情況。由數值結果可知此數值方法之性能良好，而其運算效能更比梯度再生核函數配置法快25~50%。

關鍵字

反算柯西問題 ； 再生核形狀函數 ； 高階梯度近似 ； 加權強形式配點法 ； 多重連通域

並列摘要

The inverse problems refer to ill-posed and unstable boundary value problems, and they often have incomplete boundary conditions. Thus, the inverse problems cannot be solved analytically, and they need to be approximated numerically. As there are limited numerical methods with high efficiency and accuracy reported in the literature, this study first analyzes inverse Cauchy problems by using the weighted high-order gradient reproducing kernel collocation method. The advantages of the method are neat and effective. By introducing the high-order gradient reproducing kernel shape functions to approximate the derivatives of unknown function and using the high-order gradient reproducing conditions, the high-order gradient shape functions can be established. Thus, the calculation of derivatives of inverse matrix and kernel function is avoided, thereby significantly improving the efficiency. Four numerical examples are used to verify the performance of the method, including simply connected and multiply connected domains; especially, different missing boundaries are considered in multiply connected domains. The numerical results show that the method is effective, and its computational efficiency is 25~50% faster than gradient reproducing kernel collocation method.

並列關鍵字

inverse Cauchy problem ； reproducing kernel shape function ； high-order gradient reproducing kernel approximation ； weighted strong form collocation method ； multiply connected domain

參考文獻

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