Optimal PML transformations for the Helmholtz equation

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Abstract

The Helmholtz equation arises in a range of applications, from aircraft design to geophysical exploration, which often involve problems set in infinite domains. Simulating these infinite domains is a major challenge when using the finite element method (FEM). One way to overcome this challenge is by surrounding the finite computational domain with perfectly matched layers (PMLs), which use a coordinate transformation to simulate the effect of an infinite domain. The choice of transformation is key to creating an efficient method and is the central focus of the thesis. Prior work on optimising PML transformations by Bermudez et al. [Journal of Computational Physics, 223, (2007), 469–488] showed that transformations should be unbounded. Cimpeanu et al. [Journal of Computational Physics, 296, (2015), 329-347] then showed that the layer thickness should be as close to zero as practically possible. Our first novel contribution is finding a transformation which has the same effect as a PML with zero thickness. The transformation has the special property that it transforms a 1D planar wave to vary linearly through the PML. This property allows the PML to be resolved with a single linear element and therefore without introducing additional degrees of freedom. We use this property as our definition of an optimal transformation and show that in 1D the optimal transformation is unique. We then extend the idea of optimal transformations to 2D, which apart from the case of a planar wave at an angle of incidence, prove challenging to compute. Our first method is a series solution. It suffers from slow convergence, is expensive to compute, but provides useful insight into the character of the transformations. Our second method formulates the optimal transformation as a root of a non-linear equation, which we solve using Newton's method. We then use the optimal transformation with the FEM to simulate infinite domains in several examples. However, our basic method is not effective when the transformation exhibits discontinuities which we call rips. We then go on to study rips. We show that optimal transformations for fields with a single angular Fourier mode cannot have such rips, but optimal transformations for fields with multiple modes can. We then analyse the behaviour of the transformation around the critical point of the rip. By improving our method to compute the transformation, reformulating the weak form and improving our integration scheme we show that we can use discontinuous optimal transformations with the FEM. The FEM requires solving a system of linear equations. If the system is large, we may have to use an iterative method. One iterative method which is effective for the Helmholtz equation is GMRES with a complex shifted Laplacian preconditioner. We show that the performance of this solver typically deteriorates when used with the PML method, however, we also show that this is not the case when used with our optimal transformation procedure. Finally, we develop a computational model for a specific challenging fluid-structure interaction problem using conventional PMLs. The problem is an infinite vibrating submerged plate with an incomplete elastic coating. The primary challenges we discuss are the fluid-elastic interaction in the PML and the choice of a transformation which is effective in both the fluid and elastic domains. We demonstrate the accuracy of our model by comparing the solution on two domain sizes for an example problem.

Keyword(s)

Acoustics; FEM; Finite element method; Fluid-structure interaction; Helmholtz equation; Infinite domain; Iterative solvers; PML; Perfectly matched layers; Transformation

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