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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.

该项目的科学推动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在她的博士学位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焦点，并将其应用于涡轮反应器的降噪。所提出的工作，预计将使显着的改进，用于研究在涡轮增压器周围的空气流的声学效应的数值方法。正如《华盛顿邮报》最近报道的那样，飞机对噪音污染有着巨大的影响。在未来飞机的设计中考虑噪声降低是非常具有挑战性的，并且将影响公共健康和政策。虽然基于GPW的方案代表了非常有前途的数值工具，但很少有人知道它们的性能分析。然而，需要更精确和更详细的估计，以增加其对社区的影响。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。主要应用目标是与Airbus SAS合作的航空声学中的波传播，其中不均匀性的来源是非均匀流，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。需要解决新的数学和计算挑战，以避免由系数的分段常数近似先验引入的数值误差。Trefftz方法依赖于，在广义上，对偏微分方程的近似解决方案的想法，使用基函数，这是精确的解决方案，明确使用有关环境介质的信息。本计画系关于以广义平面波（GPW）基函数为基础之变系数数值方法之设计、数学分析与电脑执行。提出以下研究方向：（1）构造对流Helmholtz方程的GPW，（2）h-版本的收敛分析，对应于细化网格，（3）p-版本的收敛分析，对应于增加具有固定网格的基函数的数目，（4）GPW原型的实现-与其他方法进行性能比较的Trefftz代码。该奖项反映了NSF的法定使命，并通过评估被认为值得支持使用基金会的知识价值和更广泛的影响审查标准。