浅海波导赫姆霍茨方程外问题多尺度有限元法

A multi-scale finite element method for the exterior problem of the Helmholtz equation in an oceanic waveguide

研究了二维多尺度海洋波导中声波与散射体相互作用的问题.将海洋波导中散射体的声散射问题分解为两个外问题和一个内问题进行求解,外问题和内问题的解通过在人工边界上的DtN映射耦合,建立了相应的DtN映射.通过DtN映射,把赫姆霍茨方程在无限长带状区域中的定解问题,转化为在一个有限区域中的定解问题.基于伽辽金变分和DtN映射方法,发展了数值求解这类声散射问题的一种多尺度有限元求解方法,提出了有效的数值实现手段.数值计算结果表明了所建立的有限元理论方法是有效和实用的,而且它的数值实现是很快速的.

The interaction of acoustic waves with the scattering object in an multiscale oceanic waveguide was examined. The acoustic scattered problem is divided and solved into two interior and a exterior problems. DtN （Dirichlet to Neumann） mapping was set up by coupling the solution of the interior and exterior problems in the artificial boundary. The problem of the infinite zonal region was transformed into the one of finite region for Helmholtz equation. A multi-scale finite element method is developed to solve numerically the acoustic scattering problem based on the Galerkin variational principle and the DtN mapping method. The results show that the proposed method is valid and its numerical implementation is very fast.