A semi-analytical approach for solving surface motion of multiple alluvial valleys for incident plane SH-waves

Abstract

In this paper, the degenerate kernels and Fourier series expansions are adopted in the null-field integral equation to solve the exterior Helmholtz problems with alluvial valleys. The main gain of using degenerate kernels in integral equations is free of calculating the principal values for singular integrals when the null-field point exactly locates on the real boundary. An adaptive observer system is addressed to fully employ the property of degenerate kernels for circular boundaries in the polar coordinate. Image concept and technique of decomposition are utilized for half-plane problems. After moving the null-field point to the boundary and matching the boundary conditions, a linear algebraic system is obtained without boundary discretization. The unknown coefficients in the algebraic system can be easily determined. The present method is treated as a"semi-analytical" solution since error only attributes to the truncation of Fourier series. Earthquake analysis for the site response of alluvial valley or canyon subject to the incident SH-wave is the main concern. Numerical examples including single and successive alluvial valleys are given to test our program. Limiting cases of a single canyon and two successive canyons are also addressed. The validity of the semi-analytical method is verified. Our advantages, well-posed model, principal value free, elimination of boundary layer effect and exponential convergence and mesh-free, by using the present method are achieved.本文係使用零場積分方程搭配分離核函數與傅立葉級數求解含沉積山谷的外域赫姆茲問題。文中，利用退化核函數之特性，可解析求得當零場點直接佈在真實邊界上時的所有的奇異積分並免除主值計算的困擾。採用自適性觀察座標系統來充分掌握分離核函數的特性。本文中，半平面的問題使用映射法以及疊加的技 巧來求解。透過零場積分方程推向邊界且均勻佈點，滿足邊界條件後可以得到一線性代數方程式，其中的未知傅立葉係數均可輕易地求得。由於誤差僅來自於擷取有限項的傅立葉級數，故本方法可視為"半解析法"。SH 坡入射沉積土或山谷的地震反應分析是本文探討的重點。在數值算例中，利用單個或連續沉積土以及特例的山谷與剛性夾雜問題來驗證此半解析法的正確性。本方法同時兼具五種優點:有良態模式、免於計算主值、無邊界曾效應以及不需佈網格的優點。