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.