Enriched finite elements for the solution of hyperbolic PDEs
Abstract
This doctoral research endeavors to reduce the computational cost involved in the
solution of initial boundary value problems for the hyperbolic partial differential
equation, with special functions used to enrich the solution basis for highly oscillatory solutions. The motivation for enrichment functions is derived from the fact that
the typical solutions of the hyperbolic partial differential equations are wave-like in
nature. To this end, the nodal coefficients of the standard finite element method
are decomposed into plane waves of variable amplitudes. These plane waves form
the basis for the proposed enrichment method, that are used for interpolating the
solution over the elements, and thus allow for a coarse computational mesh without
jeopardizing the numerical accuracy.
In this research, the time dependant wave problem is established into a semi-discrete
finite element formulation. Both implicit as well as explicit discretization schemes
are employed for temporal integration. In either approach, the assembled system
matrix needs to be inverted only at the first time step. This inverted matrix is
then reused in the subsequent time steps to update the numerical solution with
evolution of time. The implicit approach provides unconditional stability, whereas
the explicit scheme allows lumping the mass matrix into blocks that are cheaper
to invert as opposed to the consistent mass matrix. These methods are validated
with several numerical examples. A comparison of the performances of the implicit
and the explicit schemes, in conjunction with the enriched finite element basis, is
presented. Numerical results are also compared to gauge the performance of the enriched approach against the standard polynomial based finite element approaches.
Industrially relevant numerical examples are also studied to illustrate the utility of
the numerical methods developed through this research.