Efficient Methods for Electromagnetic and Acoustic Problems

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The investigator proposes a family of new algorithms to respond to the increasing demand for computational efficiency and/or accuracy for acoustic and electromagnetic problems. The main idea of this project consists in adequately combining several numerical techniques such as finite elements, integral equations, and domain decomposition methods. The investigator plans to develop a quasi-optimal non-overlapping domain decomposition method for Maxwell's equations using an appropriate approximation of the Dirirchlet to Neumann operator. In the case of partially coated dielectric objects, the following are proposed: (1) design well-conditioned integral equations; and (2) use these new integral formulations to introduce a novel robust domain decomposition method, where the iteration operator is only defined on the aperture interface. It is also proposed to explore hybrid algorithms for large and complex bodies such as aircraft and satellites. In particular, the investigator plans to couple finite elements, localization techniques of the Dirirchlet to Neumann map, and substructuring methods to deal with scatterers with large platforms where dielectric objects and deep cavities are attached. Parallel computing and mathematical analysis will be used to help achieve these goals. The proposed project is concerned with the improvement of computational tools required to face rapidly increasing engineering and industrial needs. Indeed, the computation of acoustic and electromagnetic waves is a vast area of research. This is largely due to breadth of applications, many of which have imposed technological requirements in systems, such as noise reduction, oceanic scattering, optical fibers, stealth technology, radar design, remote sensing, and many others. In developing modern aircraft, which consist of many very different components, engineers use high performance computing and innovative mathematical algorithms to enhance performance and optimize passenger safety. This procedure reduces the cost of the design, and allows to rapid response to new technological advances as well as minimization energy consumption. The results obtained through this research plan will be made readily available to engineers and scientists in the aerospace industry, which will contribute to enhancing U.S leadership in this field. In addition, this work can be used in the areas of underground water flow in hydrology, oil recovery in petroleum engineering and fluid flow through body tissues. Several aspects in this project will benefit the education of both undergraduate and graduate students, and will train them in state-of-art scientific computing and mathematical analysis. This will reinforce their preparation to face future challenges in science and technology.