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.
