散射体边界辨识问题中散射场的数值计算

考虑带有混合边界条件的散射体利用反射波信息进行边界识别的反问题.该类问题在利用优化技术迭代求解时的关键一步是散射波及其远场数据的数值求解.由于是混合边界条件,经典的只利用单层位势或双层位势建立在整个边界上一个统一的积分方程的求解的方法不再适用.提出了综合利用单双层位势求解该问题的数值方案,得到的是边界上不同部分的线性积分方程组.利用势函数的阶跃理论数学上证明了所提出方案的可解性,进而给出了方程中奇性积分的计算方法及方程组在有限维空间的离散方案,最后给出了数值例子,证明了该求解方法的有效性.

Computation of scattered wave in boundary identification problem

The boundary identification for an obstacle with mixed boundary condition is considered using the scattered wave information.The key step to this problem applying optimization technique is the numerical solution of the scattered wave as well as its far-field pattern.Since the boundary is of mixed type,the classic layer-potential method can not be applied any more.An inversion scheme is proposed using the combined single and double-layer potential.Using the jump relation for the potential function,the solvability of the proposed inversion scheme is proved and the computation scheme for the improper integrals is given.The numerical results are presented to show the validity of the proposed inversion scheme.