弹性波正、反障碍物和开腔体散射问题的理论及数值研究

In recent years, the elastic wave scattering has received ever-increasing attention in the fields of military and medical science because it is beneficial to the inversion of higher-precision solutions. However, due to the shortcomings of the direct and inverse wave scattering problems (in particular time-domain scattering problem), such as the high computational complexity, the low efficiency, the nonlinear and ill-posedness, there are still many issues in mathematical theory and numerical algorithm of the scattering problems. In order to solve these problems, this project will establish mathematical models of time-harmonic and time-domain elastic wave scattering in the unbounded domain, and study the theory and numerical algorithm of the direct and inverse scattering by the obstacle or open cavity. Based on the existing foundation, the numerical algorithm for the direct scattering of time-harmonic elastic waves is further studied, and the smoothed finite element method and the adaptive finite element method based on the transparent boundary condition (TBC) are proposed to obtain the solution with high accuracy. Then, for the time-domain elastic wave scattering problem, the well-posedness of the direct problem including Navier equations and Helmholtz equations with coupled boundary will be established, and an effective numerical method for the time-domain inverse problem is developed to improve the accuracy of the reconstruction algorithm. The project is expected to provide theoretical foundation and technical support for military and medical detection, and has important theoretical and engineering practical value.

近年来，由于弹性波散射有利于反演出更高精度的解，故而在军事和医学等科学领域中有着越来越重要的应用。然而由于正、反弹性波散射问题(特别是时域散射问题)存在一些缺陷，如计算复杂度高、效率低下，以及具有非线性和不适定性等，使其数学理论和数值算法仍有诸多问题亟待解决。针对此问题，本项目将建立无界区域上的时谐、时域弹性波散射问题的数学模型，并研究正、反障碍物散射和开腔体散射的理论及数值算法。基于现有基础，首先对时谐弹性波正散射问题的数值算法进行进一步研究，拟采用光滑有限元和基于透明边界条件的自适应有限元来获得较高精度的解。然后，研究时域弹性波散射问题的适定性，包括其满足的时域Navier方程和具有耦合边界条件的时域Helmholtz方程组，并为时域反散射问题研发一类新的高效数值算法来提高重构算法的精度。本项目有望为军事和医学检测等领域提供理论基础和技术支撑，具有重要的理论及工程实用价值。

弹性波散射问题广泛存在于工业和医学等领域。本项目主要研究无界区域上的时谐和时域弹性波散射问题。针对时谐弹性波散射问题，基于完美匹配层（PML）和透明边界条件（TBC）构建了有界区域上的Navier方程和耦合边界条件的Helmholtz方程组的边值问题，并采用多种类型的光滑有限元方法（基于边的光滑有限元、稳定的基于节点的光滑有限元方法和具有线性梯度的基于节点的光滑有限元方法）、基于加权最小二乘的配置方法、修正的边界节点方法和改进的有限积分法对弹性波散射问题进行了深入研究。针对时域弹性波散射问题，利用Helmholtz分解，基于时域TBC构建了具有耦合边界条件的初边值问题，并采用Newmark和基于边的光滑有限元方法对该问题进行了深入研究。同时，针对时域弹性波散射的反界面问题，通过变分方法，推导了总场势满足的具有耦合边界条件的初边值问题的区域导数，并证明了该区域导数是原问题的伴随问题的唯一弱解，进而构建了基于区域导数的重构算法。针对三维无限大薄板的弯曲波散射问题，分别构建了障碍物和周期空腔结构弯曲波散射的TBC用来将无界域截断为有界区域，并通过引入辅助变量将四阶波动方程降阶为两个二阶方程。然后分别提出和采用连续内罚的线性有限元和自适应有限元进行数值求解，获得了较好的结果。本项目有望为工业和医学等领域提供理论基础和技术支撑，具有重要的理论及工程实用价值。