Efficient spectral-element methods for acoustic scattering and related problems

Ying He, Purdue University


This dissertation focuses on the development of high-order numerical methods for acoustic and electromagnetic scattering problems, and nonlinear fluid-structure interaction problems. For the scattering problems, two cases are considered: 1) the scattering from a doubly layered periodic structure; and 2) the scattering from doubly layered, unbounded rough surface. For both cases, we first apply the transformed field expansion (TFE) method to reduce the two-dimensional Helmholtz equation with complex scattering surface into a successive sequence of the transmission problems with a plane interface. Then, we use Fourier-Spectral method in the periodic structure problem and Hermite-Spectral method in the unbounded rough surface problem to reduce the two-dimensional problems into a sequence of one-dimensional problems, which can then be efficiently solved by a Legendre-Galerkin method. In order for TFE method to work well, the scattering surface has to be a sufficiently small and smooth deformation of a plane surface. To deal with scattering problems from a non-smooth surface, we also develop a high-order spectral-element method which is more robust than the TFE method, but is computationally more expensive. We also consider the non-linear fluid-structure interaction problem, and develop a class of monolithic pressure-correction schemes, based on the standard pressure-correction and rotational pressure-correction schemes. The main advantage of these schemes is that they only require solving a pressure Poisson equation and a linear coupled elliptic equation at each time step. Hence, they are computationally very efficient. Furthermore, we prove that the proposed schemes are unconditionally stable.




Shen, Purdue University.

Subject Area

Applied Mathematics

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