[Thesis]. Manchester, UK: The University of Manchester; 2020.
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), 469Ă˘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.