Optimized Schwarz methods for time-harmonic wave problems in resonating cavities
Final Report Abstract
Numerical simulations of time-harmonic wave problems are nowadays of paramount importance in the development of many key technologies, such as medical imaging, particle accelerators or photonics, just to cite a few. Moreover, the need for the numerical treatment of large-scale structures (with respect to the considered wavelength) is increasing. This requires therefore an appropriate numerical solver, capable of fully exploiting the computational power offered by super-computing facilities. Unfortunately, in the case of time-harmonic wave problems, only a few well-suited methods are available. They have been proven very efficient in different engineering disciplines, such as antenna arrays, photonic waveguides or medical images reconstruction. Nonetheless, when treating cavity structures, their performance frequently drops, pushing thus very often the computational effort beyond an acceptable limit. In this research project, we focused on a particular family of methods, namely the optimized Schwarz approach, where the computational domain is divided into smaller subdomains which are easier to handle numerically. The final solution is then iteratively computed by exchanging boundary information between the subdomains via the so-called transmission operator, which is the keystone of the optimized Schwarz method. This operator is basically an estimation of the behavior of the wave at the interface between two subdomains, when taking into account the entire computational domain. Consequently, the better this estimation is, the fewer iterations are required. Within the framework of this work, novel transmission operators have been designed for the case of time-harmonic acoustic (Helmholtz) cavity problems. When applied to configurations that deviates only slightly from a rectangular cavity, they outperform the state-of-the-art operators, which were developed for unbounded problems. When considering more general configurations, their performance is, in the worst case, comparable with the state-of-the-art, at least from the iteration count point of view. Indeed, the novel operators are associated with a higher computation cost per iteration, an aspect to be improved by future researches.
Publications
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Transmission operators for the non-overlapping Schwarz method for solving Helmholtz problems in rectangular cavities. Computers & Mathematics with Applications, 138, 37-64.
Marsic, Nicolas; Geuzaine, Christophe & De Gersem, Herbert
