奇異邊界法

奇異邊界法（Singular boundary method）是一種與邊界元法相對應的邊界型數值計算方法，同基本解法^[1]、邊界節點法^[2]、去奇異無網格法^[3]、邊界粒子法^[4]和改進的基本解法^[5]等同屬於邊界型無網格數值離散方法。該類方法的共同特點是通過節點信息建立插值基函數，不需要對區域或者其邊界作網格劃分，克服了傳統基於網格的方法（如有限元法、有限差分法和邊界元法）對網格的依賴性，特別適合於求解大變形問題、移動邊界問題、反問題、薄體結構問題、和複雜-高維幾何區域等問題。

發展歷史：[編輯]

奇異邊界法最初於2009年^[6]提出。該方法使用基本解作為插值基函數，並直接將插值源點佈置在問題的真實邊界上，克服了傳統基本解方法中最複雜最頭疼的虛擬邊界問題。為避免配置點與插值源點重合時帶來的基本解源點奇異性，該方法首次提出了源點強度因子的概念，從而將邊界型強格式方法的核心歸結為求解源點強度因子。目前，該方法已成功應用於求解位勢問題^[7]^[8]、無限域問題^[9]、彈性力學問題^[10]等。

目前發展了兩種方法來計算源點強度因子。第一種方法採用一種反插值的數值處理技術間接求解源點強度因子。該方法首先需要在物理域內佈置一組與邊界源點不重合的內部點(稱之為樣本點)，然後通過求解一個代數方程組來間接計算源點強度因子；第二種方法基於邊界元法中處理奇異積分的技術與方法，直接導出了源點強度因子的解析表達式^[11]。數值算例表明，第二種方法更為穩定，精度更高。

邊界層效應[編輯]

當配置點接近但又不在邊界單元時，配置點與源點的距離不等於零但又十分趨近於零。從數學的意義上，基本解此時是非奇異的。但是從計算的角度講，當配置點接近邊界時，基本解表現出劇烈的震盪特性，產生所謂的「幾乎奇異性」，反應了邊界型方法中的邊界層效應（Boundary layer effect）問題^[12]。

針對奇異邊界法中的邊界層效應問題，文^[13]採用一類非線性變量替換法，有效地改善了近邊界點基本解的震盪特性，消除了基本解的幾乎奇異性。在不增加計算量的情況下，極大地改進了幾乎奇異核函數的計算精度，成功求解了近邊界點上的力學參量，克服了邊界層效應問題。數值試驗結果表明，即使配置點和邊界的距離達到1.0E-10的數量級，依然能取得理想的數值結果。

大規模問題[編輯]

同基本解法和邊界元法類似，奇異邊界法形成的係數矩陣通常是非對稱的稠密矩陣。如果邊界總的自由度是N的量級，計算和存儲這樣的係數矩陣則需要O(N²) 的量級，使得常規求解技術效率較低。快速多極算法能將計算量和存儲量都降至O(NlogN)甚至O(N)。從而使奇異邊界法快速、準確的求解大規模複雜工程問題成為可能。

參考文獻[編輯]

^ Fairweather G, Karageorghis A. The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 1998;9(1): 69-95
^ Chen W, Tanaka M. A meshless, exponential convergence, integration-free, and boundary-only RBF technique. Comput Math Appl 2002;43: 379-91.
^ Young DL, Chen KH, Lee CW. Novel meshless method for solving the potential problems with arbitrary domain. J Comput Phys 2005; 209(1): 290-321.
^ Chen W. High-order fundamental and general solutions of convection-diffusion equation and their applications with boundary particle method. Eng Anal Bound Elem 2002;26(7): 571-75.
^ Sarler B. Solution of potential flow problems by the modified method of fundamental solutions: Formulations with the single layer and the double layer fundamental solutions. Eng Anal Bound Elem 2009;33(12): 1374-82.
^ 陳文. 奇異邊界法：一個新的、簡單、無網格、邊界配點數值方法. 固體力學學報 2009;30(6): 592-99.
^ 谷岩, 陳文. 改進的奇異邊界法模擬三維位勢問題. 力學學報 2012;42(2): 351-60.
^ Chen W, Wang FZ. A method of fundamental solutions without fictitious boundary. Eng Anal Bound Elem 2010;34(5): 530-32.
^ Chen W, Fu Z. A novel numerical method for infinite domain potential problems. Chin Sci Bull 2010;55(16): 1598-603.
^ Gu Y, Chen W, Zhang C-Z. Singular boundary method for solving plane strain elastostatic problems. Int J Solids Struct 2011;48(18): 2549-56.
^ Chen W, Gu Y. Recent advances on singular boundary method. Joint International Workshop on Trefftz Method VI and Method of Fundamental Solution II, Taiwan 2011.
^ Johnston BM, Johnston PR, Elliott D. A sinh transformation for evaluating two-dimensional nearly singular boundary element integrals. Int J Numer Methods Eng 2007;69(7): 1460-79.
^ Gu Y, Chen W, Zhang J. Investigation on near-boundary solutions by singular boundary method. Eng Anal Bound Elem 2012;36(8): 1173-82 .

[1] Fairweather G, Karageorghis A. The method of fundamental solutions for elliptic boundary value problems. Adv Comput Math 1998;9(1): 69-95

[2] Chen W, Tanaka M. A meshless, exponential convergence, integration-free, and boundary-only RBF technique. Comput Math Appl 2002;43: 379-91.

[3] Young DL, Chen KH, Lee CW. Novel meshless method for solving the potential problems with arbitrary domain. J Comput Phys 2005; 209(1): 290-321.

[4] Chen W. High-order fundamental and general solutions of convection-diffusion equation and their applications with boundary particle method. Eng Anal Bound Elem 2002;26(7): 571-75.

[5] Sarler B. Solution of potential flow problems by the modified method of fundamental solutions: Formulations with the single layer and the double layer fundamental solutions. Eng Anal Bound Elem 2009;33(12): 1374-82.

[6] 陳文. 奇異邊界法：一個新的、簡單、無網格、邊界配點數值方法. 固體力學學報 2009;30(6): 592-99.

[7] 谷岩, 陳文. 改進的奇異邊界法模擬三維位勢問題. 力學學報 2012;42(2): 351-60.

[8] Chen W, Wang FZ. A method of fundamental solutions without fictitious boundary. Eng Anal Bound Elem 2010;34(5): 530-32.

[9] Chen W, Fu Z. A novel numerical method for infinite domain potential problems. Chin Sci Bull 2010;55(16): 1598-603.

[10] Gu Y, Chen W, Zhang C-Z. Singular boundary method for solving plane strain elastostatic problems. Int J Solids Struct 2011;48(18): 2549-56.

[11] Chen W, Gu Y. Recent advances on singular boundary method. Joint International Workshop on Trefftz Method VI and Method of Fundamental Solution II, Taiwan 2011.

[12] Johnston BM, Johnston PR, Elliott D. A sinh transformation for evaluating two-dimensional nearly singular boundary element integrals. Int J Numer Methods Eng 2007;69(7): 1460-79.

[13] Gu Y, Chen W, Zhang J. Investigation on near-boundary solutions by singular boundary method. Eng Anal Bound Elem 2012;36(8): 1173-82 .

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發展歷史：[編輯]

最新進展[編輯]

邊界層效應[編輯]

大規模問題[編輯]

參考文獻[編輯]