Advances in Numerical Methods for Wave Propagation in Inhomogeneous Media

本项目的科学重点是致力于创建具有可变材料特性的复杂材料中波传播的数值模拟方法。广义平面波函数，由PI在博士期间引入，已经被证明是解决简单问题的有效工具。分析相应的方法将是我们工作的中心焦点，并应用于涡轮反应堆的降噪。所提出的工作有望使用于研究涡轮反应器周围气流声学效应的数值方法得到显著改进。正如《华盛顿邮报》最近报道的那样，飞机对噪音污染有巨大的影响。在设计未来的飞机时考虑到降低噪音是非常具有挑战性的，并且会影响公共健康和政策。虽然基于gpw的方案是一种非常有前途的数值工具，但对其性能的分析却知之甚少。然而，为了增加它们对社区的影响，有必要进行更清晰、更详细的估计。本文主要研究二维和三维变系数非均匀介质中波传播问题的数值模拟。与空客SAS合作的主要应用目标是空气声学中的波传播，其中不均匀性的来源是不均匀流动，但该项目中考虑的方法也将适用于其他可变材料特性，如介电常数或声速。新的数学和计算挑战需要解决，以避免由系数的分段常数近似先验地引入的数值误差。从广义上讲，Trefftz方法依赖于使用精确解的基函数近似偏微分方程解的思想，明确地利用有关环境介质的信息。本课题研究了基于广义平面波基函数的变系数数值方法的设计、数学分析和计算机实现。提出了以下研究方向：(1)构造卷积Helmholtz方程的gpw； (2) h版收敛分析，对应于网格的细化；(3)p版收敛分析，对应于固定网格增加基函数的数量；(4)实现GPW-Trefftz代码原型，与其他方法进行性能比较。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The scientific thrust of this project is devoted to the creation of methods for the numerical simulation of wave propagation in complicated materials with variable material properties. Generalized Plane Wave functions, introduced by the PI during her PhD, have already been proven to lead to be efficient tools for simple problems. Analyzing the corresponding methods will be a central focus of our work, with application to noise reduction in turboreactors. The proposed work is expected to enable noticeable improvements in the numerical methods used to study acoustic effects in an air flow around a turboreactor. As recently reported by the Washington Post, airplanes have a huge impact on noise pollution. Taking into account noise reduction in the design of future aircrafts is very challenging, and will impact public health and policy.Although GPW-based schemes represent a very promising numerical tool, little is known analytically about their performance. Sharper and more detailed estimates are necessary, however, to increase their impact on the community. This proposal focuses on the numerical simulation of wave propagation problems in inhomogeneous media, modeled by variable coefficients, in two and three dimensions. The principal application targeted is wave propagation in aeroacoustics in collaboration with Airbus SAS, where the source of inhomogeneity is the non-uniform flow, but the methods considered in this project will also apply to other variable material properties such as permittivity or sound speed. Novel mathematical and computational challenges need to be addressed in order to avoid the numerical error introduced a priori by a piece-wise constant approximation of the coefficients. Trefftz methods rely, in broad terms, on the idea of approximating solutions to PDEs using basis functions which are exact solutions, making explicit use of information about the ambient medium. This project is concerned with the design, mathematical analysis and computer implementation of numerical methods adapted to variable coefficients via Generalized Plane Wave (GPW) basis functions. The following research directions are proposed: (1) construction of GPWs for the convected Helmholtz equation, (2) h-version of convergence analysis, corresponding to refining the mesh, (3) p-version of convergence analysis, corresponding to increasing the number of basis functions with a fixed mesh, (4) implementation of a prototype GPW-Trefftz code for performance comparison with other methods.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.