论文标题
边界积分方程的NyStrom离散化解决方程式二维弹性散射问题的解决方案
Nystrom discretizations of boundary integral equations for the solution of 2D elastic scattering problems
论文作者
论文摘要
我们介绍了在二维中无法穿透的时谐波纳维尔方程的各种边界积分方程公式的三种高阶Nystrom离散策略。一类此类公式基于与Navier Operator的绿色功能相关的四个经典边界积分运算符（BIOS）。我们考虑了这些运算符的两种类型的Nystrom离散化，一种依赖于Kussmaul-Martensen对数分裂，另一种依赖于Alpert四倍。此外，我们考虑了基于弹性场的Helmholtz分解的Navier散射问题的替代公式，可以通过边界积分方程系统求解，该系统具有与Helmholtz方程相关的积分运算符。由于这些配方中的某些BIOS是非标准的事实，因此我们使用扩展（QBX）方法来进行高阶NyStrom离散化。另外，我们使用零件技术的MAUE集成来根据单层和双层Helmholtz Bios来重塑这些非标准操作员，其NyStrom离散化适用于Kussmaul-Martensen方法论。我们提出了各种数值结果，这些结果涉及我们的NyStrom离散化弹性散射求解器为光滑和Lipschitz边界实现的高阶精度。我们还对基于高频制度中不同积分方程的求解行为进行了广泛的比较。最后，我们说明了我们认为我们认为的一些NyStrom离散化是如何无缝地纳入卷积正交（CQ）方法的，以提供时域弹性散射问题的高阶解。
We present three high-order Nystrom discretization strategies of various boundary integral equation formulations of the impenetrable time-harmonic Navier equations in two dimensions. One class of such formulations is based on the four classical Boundary Integral Operators (BIOs) associated with the Green's function of the Navier operator. We consider two types of Nystrom discretizations of these operators, one that relies on Kussmaul-Martensen logarithmic splittings and the other on Alpert quadratures. In addition, we consider an alternative formulation of Navier scattering problems based on Helmholtz decompositions of the elastic fields, which can be solved via a system of boundary integral equations that feature integral operators associated with the Helmholtz equation. Owing to the fact that some of the BIOs that are featured in those formulations are non-standard, we use Quadrature by Expansion (QBX) methods for their high order Nystrom discretization. Alternatively, we use Maue integration by parts techniques to recast those non-standard operators in terms of single and double layer Helmholtz BIOs whose Nystrom discretizations is amenable to the Kussmaul-Martensen methodology. We present a variety of numerical results concerning the high order accuracy that our Nystrom discretization elastic scattering solvers achieve for both smooth and Lipschitz boundaries. We also present extensive comparisons regarding the iterative behavior of solvers based on different integral equations in the high frequency regime. Finally, we illustrate how some of the Nystrom discretizations we considered can be incorporated seamlessly into the Convolution Quadrature (CQ) methodology to deliver high-order solutions of the time domain elastic scattering problems.
