變量形狀參數於虛擬點為基礎之徑向基函數選點法與兩步特解基礎解法之應用

Ghost-point Based Radial Basis Function Collocation and Two-step MPS-MFS Methods with Variable Shape Parameters

林欣瑞(Shin-Ruei Lin)

指導教授 ： 陳俊杉

國立臺灣大學/工學院/土木工程學研究所/博士(2022年)

https://doi.org/10.6342/NTU202200229

摘要

無網格法最早在1970 年代開始發展，但是一直到1990 年代才開始有大量投入無網格領域的研究。在所有無網格法中，徑向基底函數的選點法一直是最熱門的主題之一。除此之外還有與徑向基底函數選點法非常類似的基本解法。概念上，基本解法可以看成將徑向基底函數選點法的徑向基底函數，以微分運算子的基本解來取代，使基底函數更有能力去近似對應的微分方程。到了1996 年，基於基本解法做出改進的特解法誕生了。特解法因為引入徑向基底函數的來近似其特解，簡化了求得特解的計算流程。
然而，在傳統的徑向函數選點法中，用來標示資料位置的資料點。而另一個在徑向基底函數研究領域常見的爭議則是形狀參數的選擇。形狀參數在徑向基底函數內，是個可自由調整的參數，由於這個參數也會大幅影響近似出來的結果，其選擇過程又往往與問題的相依性非常大，且在許多研究都可以發現到，計算的誤差對於形狀參數的變化非常敏感，而且前面提到，因為特解法也用徑向基底函數求得特解，也無法避免源自徑向基底函數的這些問題。本研究的目的是提出演算法，該演算法能夠將虛擬點法以及變量形狀參數法整合應用至徑向基底函數選點法和基本解特解法的兩者的求解過程。以期望能夠減少形狀參數對於問題的相依性以及更進一步的降低近似誤差。
在本論文中會描述到該演算法如何透過虛擬點法，將基底函數中的中心點獨立成新的一組點，以此提升徑向基底函數的建構彈性，讓建構出來的基底函數可以更有能力去近似目標函數。變量形狀參數則是為了降低形狀參數本身對問題的相依性而提出。而當前變量形狀函數的變量範圍卻是同樣有著難以確定的問題。本研究會提出全新的方法，用來預測變量形狀參數的變動範圍。該方法會以Franke 提出的形狀參數計算式微基本，求得出新的變量形狀參數的任意變動範圍。
在研究過程，測試了各式各樣的偏微分方程，包括二階與四階，還有二維跟三維，基本解特解法的案例都涵蓋在內。有部分的測試案例也會透過有效狀態數來驗證，以了解本研究提出的方法是否真的有改進近似誤差。因為引入了虛擬點法，徑向基底函數可以有效的建構更好的基底函數，測試案例的結果也發現到近似誤差有非常明顯的減少。變量函數的部分，則是減少了計算流程中，變量形參數對於問題的依賴性，不用再針對問題去調整形狀參數。可以用單一的流程求得正確的形狀函數，且也從案例的計算結果得到了更準確的近似解。

關鍵字

徑向函數法 ； 變量形狀參數 ； 多元二次曲面法 ； 虛擬點法 ； 特解法 ； 基本解法 ； 有效條件數 ； 時空問題

並列摘要

Meshless methods were first created in the 1970s, while significant developments were made after the 1990s. Among meshless methods, the radial basis function (RBF) collocation method (RBFCM) used for solving partial differential equations (PDE) is one of the most popular research topics. Another similar numerical scheme is the method of fundamental solutions (MFS), where the basis function is replaced with a fundamental solution of a differential operator. This replacement dramatically boosts the approximation accuracy. In 1996, the method of particular solutions (MPS) was proposed to extend the ability of MFS to solve inhomogeneous problems. One notable feature of MPS is the ability to approximate a particular solution with RBFs; the treatment simplifies the procedure in MPS.
However, in conventional RBFCM, the centers are usually taken as being the same as collocation points. Another issue in dispute is the selection of shape parameters. The shape parameter in an RBF is a free parameter that controls the smoothness of the basis function. Unfortunately, selection of this parameter strongly depends on the problem. Usually, studies on the shape parameter find an optimal value with certain criteria, but the strategies for selection are varied and the approximation error has been found to be too sensitive for varying the shape parameter. In MPS, the particular solution is approximated with RBFs. Hence the same problems also occur in the MPS method. The objective of this research is to diminish the dependency of variables on problems and reduce the approximation error.
This dissertation proposes an algorithm for the solution procedure of RBFCM and MPS-MFS. The proposed algorithm includes the ghost point method and variable shape parameters. In the ghost point method, the centers and collocation points are different point sets. The construction of the RBFs is therefore decoupled from the placement of collocation points. The variable shape parameters are introduced. A new strategy is proposed to determine the interval of the variable shape parameters for the ghost point method using RBFs and for MPS. However, determination of a suitable interval for the variable shape parameter remains a challenge. The modified Franke formula is used as an initial predictor of the center of the interval of the variable shape parameter in this dissertation.
Several numerical examples are presented, including second- and fourth-order partial differential equations in 2D and 3D with steady and space-time transients. Some of the examples are tested with the effective condition number for validating the effectiveness of the proposed method.

並列關鍵字

radial basis functions ； variable shape parameter ； Franke's formula ； multiquadrics ； ghost point method ； The Kansa method ； method of particular solutions ； method of fundamental solutions ； effective condition number ； space-time

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