Advances in Numerical Methods for Wave Propagation in Inhomogeneous Media

非均匀介质中波传播数值方法的进展

基本信息

批准号：
2105487
负责人：
Lise-Marie Imbert-Gerard
金额：
$ 12.23万
依托单位：
University of Arizona
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2105487&HistoricalAwards=false
关键词：
Advances Numerical Methods Wave Propagation

项目摘要

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.

该项目的科学推力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。 PI在其博士学位期间引入的广义平面波函数已经被证明是用于简单问题的有效工具。分析相应的方法将是我们工作的主要重点，并应用了降低涡轮反应器的降噪功能。预计拟议的工作将在用于研究涡轮反应器周围空气流中的声流效应的数值方法上有明显的改进。正如《华盛顿邮报》最近报道的那样，飞机对噪声污染产生了巨大影响。考虑到未来飞机的设计降低噪音非常具有挑战性，并且会影响公共卫生和政策。尽管基于GPW的方案代表了一种非常有希望的数值工具，但在分析上几乎不知道其性能。但是，需要更加敏锐，更详细的估计，以增加其对社区的影响。该提案的重点是对不均匀介质的波传播问题的数值模拟，以两个和三个维度为模型。针对性的主要应用是与空中客车SAS合作的航空声中的波传播，在该空气声中，不均匀性的来源是不均匀的流量，但是该项目中考虑的方法也将适用于其他可变材料属性，例如介电常数或声音速度。为了避免数值错误，需要解决新颖的数学和计算挑战，以避免通过系数的零件恒定近似来提出的数值误差。 TREFFTZ方法以广义依赖于将解决方案近似于PDE的概念，这些函数是确切的解决方案，从而明确使用有关环境介质的信息。该项目涉及通过广义平面波（GPW）基础函数适应可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）为对流的Helmholtz方程的GPW构建，（2）收敛分析的H反复，对应于精炼网格，（3）收敛分析的p反复分析，对应于增加数量具有固定网格的基础功能，（4）实施原型GPW-Trefftz代码，以进行性能与其他方法进行比较。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛的影响评估来获得支持的标准。