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A Study of the Moving Modified Trefftz Method for Solving the Inversed Laplace Problem with the Multiply Connected Domain
|Contributors: ||NTOU:Department of Systems Engineering and Naval Architecture|
|Keywords: ||反算拉普拉斯問題;修正Trefftz 法;移動式Trefftz 法;病態問題|
Inverse Laplace problem;modified Trefftz method;moving Trefftz method;ill-posed problem
|Issue Date: ||2011-06-28T08:19:59Z
|Abstract: ||本研究計畫擬發展出移動式的修正Trefftz 法求解由拉普拉斯方程所統御的多連通區域反算問題。傳統的Trefftz 法求解正算問題時，會導致病態的線性方程，這是因為問題的解以發散級數展開的緣故，而在求解反算問題時，由於其解並不連續依賴所給定之邊界條件，使得求解困難，病態問題更加嚴重。雖有其他學者提出許多改善此種病態問題之方式，卻未能有效改善。本研究首先擬針對此病態問題，考慮引進特徵尺度的概念，將原本的發散級數轉為收斂級數，並進一步結合移動式的Trefftz法，解決多連通區域的問題。此方法主要的重點是將解作級數展開，再藉由邊界條件的量測，求解每個基底的係數。而本法與其它數值方法對該問題解的準確性將進行比較。最後，吾人將設計數個數值算例，特別是針對帶有邊界資料量測誤差以及量測位置不同造成干擾的情況對解的穩定性及精確度之影響，進行深入之探討與解析，藉此來驗證我們所提出方法是否可有效地解決上述諸多的問題及其適用性。|
In this research proposal, the inverse problem in a multiply connected domain governed by the Laplace equation will be investigated numerically by the developed moving modified Trefftz method. When solving the direct Laplace problem with the conventional Trefftz method, one may treat the ill-posed linear algebraic equations because the solution is obtained by expanding the diverging series; while when the inverse Laplace problem is encountered, it is more difficult to treat the more seriously ill-posed behaviors because the incomplete boundary data, and its solution, if exists, does not depend on the given boundary data continuously. Even many researchers have proposed lots of methods to overcome the ill-posed problem; however, an effective numerical scheme to tackle the problem is still not available. To begin with, this ill-posed problem will be analyzed by introducing the characteristic length concept and then, the diverging series can be transferred to a converging one. Thereafter, the developed moving modified Trefftz method will be combined to solve the multiply connected domain problem. The key point is that the solution can be represented by the series expansion and further be combined with boundary measurements to compute the coefficients of bases functions. Comparisons of the proposed method with other available numerical methods will be conducted. Several numerical examples for discussing how measurement errors or different measuring positions influence the accuracy of the numerical solution will be designed. Finally, the numerical instability of the proposed method will also be investigated such that it can further verify the wideness and effectiveness of the present method.
|Appears in Collections:||[系統工程暨造船學系] 研究計畫|
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