This dissertation presents the Generalized Finite Element Method (GFEM) for the scalar
Helmholtz equation, which describes the time harmonic acoustic wave propagation problem.
We introduce several handbook functions for the Helmholtz equation, namely the planewave,
wave-band, and Vekua functions, and we use these handbook functions to enrich the
Finite Element space via the Partition of Unity Method to create the GFEM space. The
enrichment of the approximation space by these handbook functions reduces the pollution
effect due to wave number and we are able to obtain a highly accurate solution with a
much smaller number of degrees-of-freedom compared with the classical Finite Element
Method. The q-convergence of the handbook functions is investigated, where q is the order
of the handbook function, and it is shown that asymptotically the handbook functions
exhibit the same rate of exponential convergence. Hence we can conclude that the selection
of the handbook functions from an admissible set should be dictated only by the ease of
implementation and computational costs.
Another issue addressed in this dissertation is the error coming from the artificial truncation
boundary condition, which is necessary to model the Helmholtz problem set in the
unbounded domain. We observe that for high q, the most significant component of the error
is the one due to the artificial truncation boundary condition. Here we propose a method
to assess this error by performing an additional computation on the extended domain using
GFEM with high q.
| Identifer | oai:union.ndltd.org:tamu.edu/oai:repository.tamu.edu:1969.1/ETD-TAMU-1221 |
| Date | 15 May 2009 |
| Creators | Hidajat, Realino Lulie |
| Contributors | Strouboulis, Theofanis |
| Source Sets | Texas A and M University |
| Language | en_US |
| Detected Language | English |
| Type | Book, Thesis, Electronic Dissertation, text |
| Format | electronic, application/pdf, born digital |
Page generated in 0.0025 seconds