SH-wave scattering by a semi-elliptical hill using a null-field boundary integral equation method and a hybrid method

Abstract

Following the success of seismic analysis of a canyon [J.T. Chen, P.Y. Chen, C.T. Chen, Surface motion of multiple alluvial valleys for incident plane SH-waves by using a semi-analytical approach, Soil Dyn. Earthq. Eng. 28 (2008) 58-72.], the problem of SH-wave scattering by a semi-elliptical hill is revisited by using the null-field boundary integral equation method (BIEM). To fully utilize the analytical property in the null-field boundary integral equation approach in conjunction with degenerate kernels for solving the semi-elliptical hill scattering problem, the problem is decomposed to two regions to produce elliptical boundaries by using the technique of taking free body. One is the half-plane problem containing a semi-elliptical boundary. This semi-infinite problem is imbedded in an infinite plane with an artificial full elliptical boundary such that degenerate kernel can be fully applied. The other is an interior problem bounded by an elliptical boundary. The degenerate kernel in the elliptic coordinates for two subdomains is utilized to expand the closed-form fundamental solution. The semi-analytical formulation in companion with matching boundary conditions yields six constraint equations. Instead of finding admissible wave expansion bases, our null-field BIEM approach in conjunction with degenerate kernels have the five features over the conventional BIEM/BEM, (1) free of calculating principal values, (2) exponential convergence, (3) elimination of boundary-layer effect, (4) meshless, and (5) well-posed system. All the numerical results are compared well with those of using the hybrid method which is also described in this paper. It is the interesting to find that a focusing phenomenon is also observed in this study. 基於零場邊界積分方程法求解半圓形山谷SH波散射問題的成功經驗，本文利用零場邊界積分方程法搭配退化核的解析特性來求解半橢圓山峰SH波的散射問題。此問題可藉由取自由體的技巧來獲得兩個含橢圓邊界的次領域。其中一個是含一個半橢圓邊界的半平面問題，可將此問題埋藏在一個全平面裡，並製造一個人工的橢圓邊界，如此一來退化核就可被充分的應用。另一個則是含橢圓邊界的內域問題。這時閉合型基本解可在橢圓座標下展開成退化核的形式，並充分分別應用在這兩個次領域。此半解析法結合邊界條件的滿足可得到六條束制方程式。我們用零場邊界積分方程法搭配退化核來取代尋找是當基底的波函數展開法，並且它有比傳統的邊界積分方程法與邊界元素法好的五大優點，包含(1) 免除主值的計算 (2) 指數收斂 (3) 消除邊界層效應 (4) 無須建構網格 (5) 良態代數系統。所得的數值結果與複合法的結果都一致吻合，並且針對複合法也在文中做一描述。最後，有趣的是在這研究裡發現到了SH波散射問題的聚焦現象。