Novel Methods for Numerical Simulation of Wave Propagation in Inhomogeneous Media

许多类型的波在非均匀介质中传播，例如在空气中传播的光波，其中颗粒密度取决于大气层的高度；在水中传播的声波，其中盐度取决于海洋深度；或在地球中传播的地震波，其中地质层的密度根据其沉积历史而变化。该项目的科学主旨是致力于开发各向异性和非均匀介质中波传播的数值模拟工具。主要的应用是空气声学中的波传播，以研究涡流反应堆在流动中产生的噪声，其中不均匀和各向异性的来源是发动机周围的非均匀气流。然而，所考虑的方法也适用于其他可变的材料性质，如介电常数或声速。本项目致力于一类Trefftz方法的进一步发展。广义地说，Trefftz方法依赖于使用局部基函数逼近偏微分方程解的思想，局部基函数是控制方程的精确解，显式地利用了关于环境介质的信息。相反，新方法依赖于作为控制方程的近似解的基函数，而不是精确解。因此，PI称为准Trefftz方法。这个项目的目标有两个：(1)研究使用高阶局部基时准Trefftz方法的性质；(2)研究相应基函数的数值积分技术。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Many types of waves propagate through inhomogeneous media, such as light waves propagating in the air, where the particle density depends on the height of the atmospheric layers, acoustic waves propagating in the water, where the salinity depends on the ocean depth, or seismic waves propagating in the earth, where the density of geological layers varies depending on their sedimentation history. The scientific thrust of this project is devoted to the development of numerical simulation tools for wave propagation in anisotropic and inhomogeneous media. The principal application targeted is wave propagation in aeroacoustics to study the noise generated by a turbo-reactor in a flow, where the source of inhomogeneity and anisotropy is the non-uniform air flow around the engine. Yet the methods considered will also apply to other variable material properties such as permittivity or sound speed.This project is concerned with the further development of a class of Trefftz-like methods. Trefftz methods rely, in broad terms, on the idea of approximating solutions to partial differential equations using local basis functions, which are exact solutions of the governing equation, making explicit use of information about the ambient medium. Instead, the new methods rely on basis functions that are approximate solutions of the governing equation rather than exact solutions. The PI therefore refers to such methods as quasi-Trefftz methods. The goal of this project is two-fold: (1) investigate the properties of the quasi-Trefftz methods when using a high-order local basis; (2) investigate numerical integration techniques for the corresponding basis functions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.