以加權再生核函數配置法與誤差分析求解反算彈性力學問題

Weighted Reproducing Kernel Collocation Method and Error Analysis for Solving Inverse Elasticity Problems

辛玟錡

指導教授 ： 楊子儀

國立交通大學/工學院/土木工程系所/碩士(2018年)

摘要

在工程應用與科學領域中， 在工程應用與科學領域中， 在工程應用與科學領域中， 在工程應用與科學領域中， 在工程應用與科學領域中， 在工程應用與科學領域中， 在工程應用與科學領域中， 在工程應用與科學領域中， 在工程應用與科學領域中， 在工程應用與科學領域中， 反算問題由於本質上具有不完整之邊界條件，使其成為邊界值問題求解中較為困難之問題，隨著計算力學之蓬勃發展，近二十年來無網格法之相關研究亦日趨成熟，應用強形式無網格法求解反算問題為近年研究之趨勢。有鑑於先前之研究中所採用求解反算問題之數值方法，其所得之離散系統往往為滿矩陣，容易導致病態系統，增加數值求解之困難，本論文研究提出以最小二乘泛函建立之加權再生核函數配置法（W-RKCM）求解反算彈性力學問題，透過誤差分析以推得適當之權重，進而提高近似解之精度，其中再生核形狀函數為局部函數，雖然僅具有代數收斂率，其所建立之線性離散系統卻較為穩定，本論文將以數個數值例題來驗證W-RKCM之精確度與穩定性。

關鍵字

無網格法 ； 再生核函數配置法 ； 反算問題 ； 彈性力學 ； 誤差分析

並列摘要

In the area of engineering aplicaiton and science, the inverse problems have become one of tough boundary value problems due to their incomplete boundary conditions. With the rapid development of computational mechanics, the meshless methods are well developed in recent twenty years has become. The application of strong-form meshless methods in solving inverse problems has become a popular trend. It is noted that the meshless methods proposed and adopted in the former research often lead to full matrices in solving inverse problems, thereby making systems ill-conditioned and hard to be solved. As such, this study proposes a weighted reproducing kernel collocation method (W-RKCM), which is formulated on the basis of a least-squares functional defined for inverse elasticity problems. The weights on the boundary are determined by the error analysis, from which the desired accuracy can be reached. Particularly, the RK shape function is a local function, and it has algebraic convergence rate. The resulting system is more stable compared to the one obtained by global approximation. Several numerical examples are provided to demonstrate the efficacy and stability of the proposed method.

並列關鍵字

meshfree method ； reproducing kernel collocation method ； inverse problems ； elasticity ； error analysis

參考文獻

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