|dc.description.abstract||Conventional finite element methods (FEM) have been used for many years for the
solution of harmonic wave problems. To ensure accurate simulation, each wavelength
is discretised into around ten nodal points, with the finite element mesh being updated
for each frequency to maintain adequate resolution of the wave pattern. This
technique works well when the wavelength is long or the model domain is small.
However, when the converse applies and the wavelength is small or the domain of
interest is large, the finite element mesh requires a large number of elements, and
the procedure becomes computationally expensive and impractical.
The principal objective of this work is to accurately model two-dimensional Helmholtz
problems with the Partition of Unity Finite Element Method (PUFEM). This will
be achieved by applying the plane wave basis decomposition to the wave field. These
elements allow us to relax the traditional requirement of around ten nodal points per
wavelength and therefore solve Helmholtz wave problems without refining the mesh
of the computational domain at each frequency.
Various numerical aspects affecting the efficiency of the PUFEM are analysed in
order to improve its potential. The accuracy and effectiveness of the method are
investigated by comparing solutions for selected problems with available analytical
solutions or to high resolution numerical solutions using conventional finite elements.
First, the use of plane waves or cylindrical waves in the enrichment process is assessed
for wave scattering problems involving a rigid circular cylinder in both near field and
far field. In the far field, the cylindrical waves proved to be more effective in reducing
the computational effort. But given that the plane waves are simpler to analytically
integrate for straight edge elements, during the finite element assembling process,
they are retained for the remaining of the thesis.
The analysed numerical aspects, which may affect the PUFEM performance, include
the conjugated and unconjugated weighting, the geometry description, the use of
non-reflecting boundary conditions, and the h-, p- and q-convergence. To speed up
the element assembling process at high wave numbers, an exact integration procedure
The PUFEM is also assessed on multiple scattering problems, involving sets of circular
cylinders, and on exterior wave problems presenting singularities in the geometry
of the scatterer. Large and small elements, in comparison to the wavelength, are
used with both constant and variable numbers of enriching plane waves.
Last, the PUFEM resulting system is iteratively solved by using an incomplete lower
and upper based preconditioner. To further enhance the efficiency of the iterative
solution, the resulting system is solved into the wavelet domain.
Overall, compared to the FEM, the PUFEM leads to drastic reduction of the total
number of degrees of freedom required to solve a wave problem. It also leads to very
good performance when large elements, compared to the wavelength, are used with
high numbers of enriching plane waves, rather than small elements with low numbers
of plane waves. Due to geometry detail description, it is practical to use both large
and small elements. In this case, to keep the conditioning within acceptable limits
it is necessary to vary the number of enriching plane waves with the element size.||en_US