论文标题
通过Helmholtz分解的弹性散射问题解决方案的强大边界积分方程
Robust boundary integral equations for the solution of elastic scattering problems via Helmholtz decompositions
论文作者
论文摘要
弹性场的Helmholtz分解为通过边界积分方程（Bie）解决线性弹性问题的新途径[Bie）[Dong，Lai，Li，Li，计算数学，2021年]。这种方法的主要吸引力是，随之而来的BIE系统功能仅与Helmholtz方程相关的积分运算符。但是，这些bie涉及非标准边界积分运算符，在将dirichlet或neumann Trace应用于Helmholtz单层和双层电位之后，它们不会产生。相反，Helmholtz的分解方法导致弹性散射问题与Neumann边界条件的弹性散射问题，涉及Helmholtz层潜力的Hessians的边界痕迹。结果，在Helmholtz分解的框架中应用的经典组合野外方法导致了BIE配方，尽管强大，但并非第二类。在[Boubendir，Dominguez，Levadoux，Turc，Siam Applied Mathematics 2015]中引入的正规化方法之后，我们设计和分析了新颖的鲁棒Helmholtz分解bie，以解决弹性散射的解决方案，这些弹性在两个尺寸中是第二种。我们基于NyStrom离散化提供了多种数值结果，这些结果说明了第二种正则配方在与迭代求解器的连接中的良好性能。
Helmholtz decompositions of the elastic fields open up new avenues for the solution of linear elastic scattering problems via boundary integral equations (BIE) [Dong, Lai, Li, Mathematics of Computation,2021]. The main appeal of this approach is that the ensuing systems of BIE feature only integral operators associated with the Helmholtz equation. However, these BIE involve non standard boundary integral operators that do not result after the application of either the Dirichlet or the Neumann trace to Helmholtz single and double layer potentials. Rather, the Helmholtz decomposition approach leads to BIE formulations of elastic scattering problems with Neumann boundary conditions that involve boundary traces of the Hessians of Helmholtz layer potential. As a consequence, the classical combined field approach applied in the framework of the Helmholtz decompositions leads to BIE formulations which, although robust, are not of the second kind. Following the regularizing methodology introduced in [Boubendir, Dominguez, Levadoux, Turc, SIAM Journal on Applied Mathematics 2015] we design and analyze novel robust Helmholtz decomposition BIE for the solution of elastic scattering that are of the second kind in the case of smooth scatterers in two dimensions. We present a variety of numerical results based on Nystrom discretizations that illustrate the good performance of the second kind regularized formulations in connections to iterative solvers.
