In this study, the effectiveness of the particle swarm optimization in determining the optimal source locations in the method of fundamental solutions (MFS) is investigated and verified. The MFS, truly free from mesh and numerical quadrature, is one of the classical meshless methods and only boundary nodes and sources are required in the implementation of the MFS. Although the MFS is very powerful in dealing with homogeneous partial differential equations with known fundamental solutions, to determine the optimal locations of sources during the simulations are still a challenging task for practical engineering applications. In this study, we proposed to use the particle swarm optimization (PSO) to identify the optimal locations of sources in the MFS. The PSO is one of the metaheuristic algorithms in mathematical optimization, so the optimal locations of sources can be efficiently determined without calculating Jacobian matrices. Several numerical examples, governed by different linear partial differential equations, are provided to demonstrated the accuracy of the proposed combination of the MFS and the PSO. Furthermore, the efficiency of the proposed numerical scheme is clearly verified by providing the CPU time of numerical tests. For the future study, some possible engineering applications of the proposed approach will be also discussed in this talk.

In this paper, we present a space–time boundary-type meshfree method based on the collocation Trefftz method for solving two-dimensional transient subsurface flow problem. The Trefftz basis functions were derived for the transient flow in heterogeneous geological media. The domain decomposition method was adopted in which flow in consecutive layers at the interface satisfies the continuity conditions. Numerical solutions were approximated using initial and boundary conditions assigned on the space–time boundaries. Consequentially, the time-marching technique is not required for solving two-dimensional transient subsurface flow problem. Numerical verifications were carried out. It is found that high accurate results were obtained.

本研究提出了一種使用等參幾何分析方法(IGA)的新方法，該方法具有權重優化的非均勻有理B樣條(NURBS)函數。IGA方法的主要優點是使用NURBS 函數去近似 PDE 的幾何和物理量。在處理高階幾何時，引入權重函數來擴展解空間並準確描述問題域。通常，PDE近似解將繼承相同的NURBS和權重，然而問題域的幾何形狀通常會影響 PDE 的物理量，所以PDE的解可能比幾何更複雜，需要進一步調整近似值以獲得更準確的解。因此，在本研究中，我們引入了優化的方法，將控制值和權重都視為變數，以獲得更好的物理解。此次使用最小二乘法和非線性迭代方法計算權重，以便更完整地重建物理場。此外，IGA 方法中通常使用懲罰法來處理一類邊界條件。但是，如果一類邊界條件是一個無理函數，邊界誤差會大於內部誤差。出現這種現象的原因是形函數的逼近階數不夠。為了解決這個問題，我們使用曲線擬合來擬合無理邊界條件並計算邊界權重以最小化邊界誤差。當按照Galerkin弱形式求解內部解時，優化後的權重不會給我們最好的解，這是因為更新後的權重必須對應新的控制值，因此控制值也須同步更新。統整以上原因，我們提出了一種交替迭代方案，利用控制方程的Galerkin弱形式求解控制值，然後使用最小二乘法和牛頓法更新權重，以更新後的權重重新計算新的控制值，重複此步驟求得最佳解。
最後，使用二維泊松方程和穩態熱傳問題對所提出方法的結果進行了測試。優化結果表明，可以用更少的離散點和更少的計算成本獲得更準確的近似解。

關鍵詞：等參幾何法(IGA)；NURBS；最佳化；熱傳問題

In this study, a collocation method utilizing the multiquadric (MQ) and inverse multiquadric (IMQ) radial basis functions (RBFs) without adopting the shape parameter for dealing with boundary value problems is presented. The center points were separated from the interior points where the center points are regarded as source points placed outside the domain. We then place the boundary, interior, and source points on, within, and outside the domain, respectively. Because the distance between the source and the interior point is always greater than zero, the MQ, IMQ RBFs and their derivatives in the governing equation remain continuous. The shape parameter is therefore no longer needed in the MQ and IMQ RBFs. Several verifications are carried out to validate the accuracy of the proposed approach. Results illustrate that the proposed approach may provide more accurate solutions than the traditional RBF collocation method with the optimal shape parameter. Additionally, it appears that the positions of the source points are of low sensitivity with respect to the numerical results.

本研究使用最新開發的幽靈點法(ghost point method)求解二維柯西反算問題。柯西反算問題在工程應用問題中非常常見，因其組建之線性代數系統具有嚴重的病態問題，故數值模擬對於量測獲得之邊界條件非常敏感，在有噪音(noise)的影響下會嚴重影響數值解的穩定性與準確性，故在有噪音的影響下獲得良好且準確的數值解是此問題所關心的重點。幽靈點法是以傳統之徑向基底函數配點法(radial basis function collocation method, RBFCM)為基礎，搭配幽靈點的概念加以改良完成之新型態無網格法，故其無須建置網格與進行數值積分，是非常簡單且準確的數值模擬方法之一。徑向基底函數配點法原本佈在計算域內之計算源點，幽靈點法將其延伸拓展至計算域外之區域，並採用修正的弗蘭卡公式(Franke’s formula)與留一交叉驗證法(leave-one-out cross-validation, LOOCV)獲得徑向基底函數之最佳形狀參數，幽靈點法改善徑向基底函數配點法未知形狀參數的選擇問題，同時也大幅提升了數值模擬結果的準確性與穩定性，是一個具有高度發展潛力之數值方法。本研究將提供數個數值算例，以驗證幽靈點法在求解二維柯西反算問題之可行性、準確性與穩定性。

退化邊界廣泛存在於地下水板樁結構、薄板裂縫等問題。其退化边界奇異性問題對傳統的數值方法，如有限元素法、有限差分法和有限體積法提出了挑戰。本文提出了一種區域選點法（Domain Selection Method, DSM）搭配區域型無網格廣義有限差分法（Generalized Finite Difference Method, GFDM）來分析具有退化邊界的拉普拉斯問題。我們將計算域分解為若干子域，根據各自特性進行廣義有限差分法(GFDM)區域點位互選，避免退化邊界存在引起的不連續性問題，然后結合所需的邊界條件以及控制方程式即可進行求解。本文利用該數值方案開展了防滲壩、半圓形土層、電磁場中帶狀導體等拉普拉斯問題的研究，將結果與傳統區域分解法（Domain Decomposition method, DDM）結果進行比對，吻合度較高。案例說明該數值方案可以有效克服奇異點問題，保證了整個域的平滑解，是一種有效而簡單的獲得退化邊界問題求解的數值工具。