Advances in Numerical Methods for Wave Propagation in Inhomogeneous Media

这个项目的科学主旨是致力于创造方法来数值模拟具有不同材料性质的复杂材料中的波传播。PI在博士期间引入的广义平面波函数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，并应用于涡轮反应堆的降噪。这项拟议的工作有望使用于研究涡轮反应堆周围气流中声学效应的数值方法得到显着改进。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。在未来飞机的设计中考虑降噪是非常具有挑战性的，并将影响公众健康和政策。尽管基于GPW的方案代表着一种非常有前途的数值工具，但人们对其性能的分析知之甚少。然而，为了增加它们对社会的影响，有必要进行更准确和更详细的估计。本文主要对变系数非均匀介质中的波传播问题进行了二维和三维的数值模拟。主要目标是与空中客车SAS合作在空气声学中的波传播，其中不均匀的来源是非均匀流动，但本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。需要解决新的数学和计算挑战，以避免系数的分段常数近似先验地引入数值误差。广义地说，Trefftz方法依赖于使用基函数来近似偏微分方程组的解的想法，基函数是精确的解，显式地利用关于环境介质的信息。该项目涉及通过广义平面波基函数适应变系数的数值方法的设计、数学分析和计算机实现。提出了以下研究方向：(1)构造对流Helmholtz方程的GPWS，(2)h版本的收敛分析，对应于网格的细化，(3)p版本的收敛分析，对应于增加固定网格的基函数的数量，(4)实现一个原型GPW-Trefftz代码，用于与其他方法的性能比较。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。

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.