This project is focused on the development of innovative and efficient algorithms dedicated to solving problems of acoustic and electromagnetic wave propagation. The strategy consists of using domain decomposition to design advanced numerical techniques for obtaining high computational efficiency, and improved convergence and accuracy properties. The proposed approach also allows for suitable utilization of parallel computing. The investigator is concerned with two classes of problems. The first consists of using domain decomposition methods to suitably combine, (1) finite element methods with boundary element methods, and (2) finite element methods with asymptotic techniques. In the second class, the investigator proposes to couple domain decomposition methods with a specific integral equation method for problems concerning multiple scatterers in the high frequency regime. The resulting algorithm bypasses the need to resolve at the wavelength scale while retaining error-controllability. A new Krylov-subspace method that significantly improves convergence of the iterative procedure will be investigated. This approach will decrease the computational time required to obtain a given accuracy. A careful mathematical analysis will be conducted to help achieve these goals.
The main focus of this proposal is the development of innovative and efficient algorithms dedicated to problems of acoustic and electromagnetic wave propagation. The strategy consists of careful mathematical analysis and design of high efficiency algorithms. The work is highly interdisciplinary and has impact on advanced technological applications including problems arising in areas such as telecommunications, aircraft design, and oceanography. For example, the response of large, geometrically complex structures to incoming electromagnetic radiation is a topic of great interest to the aerospace industry, which seeks to improve stealth capabilities of airborne vehicles. Engineers regularly employ computational tools to predict the radar signature of aircraft, which helps to minimize design costs. Clearly, such efforts will be greatly aided by the development of computationally efficient and rigorous numerical methods. Some aspects of this project will be supported by an active collaboration with engineers.
The increasing demand from the Aerospace industry in terms of computational efficiency, and the complexity of large bodies such as aircraft or satellites require the development of novel simulation techniques. This kind of industry seeks to both improve stealth capabilities of airborne vehicles and reduce the cost incurred during the design cycle. Indeed, standard techniques dealing with Electromagnetic, Acoustic, and Elastic wave propagation problems are limited to objects of large size. Under this project, novel algorithms related to some of these extremely challenging problems were produced. These new methods consists of combining Modern Numerical techniques and Mathematical concepts to develop solvers that can be used in the design of different products, including circuit devices, stealth technology, antennas, and medical equipment. The obtained results were validated on structures given by a submarine and a satellite. To support these new methods, a Mathematical theory was developed and published in top scientific research journals. It was also successfully used for problems related to porous media flows. These problems are of great interest in physiology when studying cancer growth and filtration of blood through arterial vessel walls, and in numerous industrial applications of air and oil filter design, oil exploration, chemical reactor simulations. All these findings contributed in enhancing the training of students involved in this project. Indeed, they were exposed to High Performance Computing techniques as well as to these sophisticated Mathematical algorithms.
