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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

二维非光滑散射体声波散射场数值解

赵海峰，刘继军

东南大学数学系，南京, 210096

摘要：在实际应用领域, 给定入射波和不可穿透散射体, 声波散射问题的研究有着重要意义.

如果散射体的边界不光滑, 采用标准的算法, 如有限元 (FEM) 或边界元 (BEM), 边界角点会影

响算法的精度. 考虑散射体边界带有角点的声学散射问题. 在二维情况, 此模型可以用

Helmholtz 方程外问题描述. 散射体角点的奇异性会影响算法的数值结果. 本文通过引入人工

边界条件避免角点附近散射场的计算, 并引入完全匹配层 (PML) 将散射问题截断为有界域问

题, 其中 PML 层的外边界赋予零边值 Dirichlet 条件. 最后用间断 Galerkin 有限元求解此边值

问题. 数值结果表明算法是有效的.

关键词：散射波, 非光滑散射体, 间断 Galerkin 有限元, 完全匹配层, 数值算法.

中图分类号： O241.8

On numerical solution to acoustic wave

scattering by a 2-dimensional non-smooth

obstacle

Zhao Haifeng, Liu Jijun

Department of Mathematics, Southeast University, Nanjing, 210096

Abstract: For given incident wave and an impenetrable obstacle, the wave scattering

problems are of great importance in applied area. In case of the obstacle boundary being

nonsmooth, the numerical solution of scattered wave outside of the obstacle by standard

schemes such as FEM and BEM will be contaminated due to the singularity of scattered wave

at the obstacle corner. Consider the acoustic wave scattering by an obstacle with a corner on

its boundary. In 2-dimensional case, this problem is governed by an exterior problem for the

Helmholtz equation, which is much complicated for stable numerical computation due to the

existence of obstacle corner. We introduce an artiﬁcial boundary condition (ABC) to

eliminate the singularity of scattered wave at the corner and introduce a perfectly matched

layer (PML) to truncate the unbounded domain where the zero Dirichlet boundary condition

is setup on the outer boundary of PML. Then the discontinuous Galerkin method is applied

to solve this boundary value problem. Finally, some numerical examples are presented to

show the validity of the proposed scheme.

基金项目： This work is supported by Research Found for the Doctoral Program of Higher Education of China

(No.20110092110018)

作者简介： Correspondence author：Liu Jijun（1965-），male，Professor，major research direction：inverse scatterings,

medical imaging, industry mathematics. Email: jjliu@seu.edu.cn.

- 1 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

Key words: Wave scattering, non-smooth obstacle, discontinuous Galerkin method,

perfectly matched layer, numerics.

0 Introduction

For given incident acoustic waves, the wave scattering problems by an obstacle D are of

great importance. Under some assumptions, the scattering process can be described by an

exterior problem for the Helmholtz equation, where the Sommerfeld radiation condition for

scattered wave is imposed to guarantee the uniqueness of the solution ([1]). Numerically this

requirement is the boundary condition at inﬁnite place for the exterior problem. To solve

the boundary value problem in an unbounded domain, there are two diﬀerent schemes. One

scheme is to introduce some artiﬁcial boundary replacing the Sommerfeld radiation condition

to truncate the unbounded domain so that the numerical schemes such as FEM and FDM

can be carried out in a bounded domain ([2, 3]). For this scheme, the key point is some

suitable artiﬁcial boundary condition such that there is no reﬂection in this artiﬁcial boundary

to guarantee the approximation accuracy. The other scheme is the boundary element method

which ﬁrstly expresses the scattered wave outside of the obstacle as the boundary integral

such as single- or double layer-potentials in terms of some density function to be determined

([4, 5, 6]). In this representation, the radiation condition is satisﬁed automatically. Once the

density is determined from the boundary condition based on the potential theory, the scattered

wave can be computed by computing an integral deﬁned on the obstacle boundary.

However, the application of the boundary integral equation method for the computation of

scattered wave requires the smoothness of obstacle boundary, the reason is that the solvability

of the integral equation for the density function from the boundary condition requires the

compactness of the integral operator in the obstacle boundary, which leads to the requirement

of the smoothness of the obstacle boundary ([1]). In other words, the boundary integral equation

method can not be directly applied to solve the scattering problem for non-smooth obstacle. On

the other hand, the scattered waves have singularities for the scatterer with corners ([4, 5, 7]),

such a singular behavior at the corners has unavoidable inﬂuence on the accuracy of the ﬁnite

element method throughout the whole domain. There are much work on the numerical solutions

of Laplace equation in a domain with boundary corners ([8, 9]). However, the numerical

solutions to the Helmholtz equation outside of the non-smooth obstacle are still at the initial

stage. For this problem, we need also to consider the inﬂuence of wave number k. Theoretically,

we should keep the numerical scheme for computing the scattered wave eﬃcient even if for large

wave number.

The purpose of this paper is to solve the acoustic wave scattering problem for an obstacle

with corner. For simplicity, we consider an impenetrable D ⊂ R

with non-smooth boundary

- 2 -

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