Explicit time integration with lumped mass matrix for enriched finite elements solution of time domain wave problems

We present a partition of unity finite element method for wave propagation problems in the time domain {using} an explicit time integration scheme. Plane wave enrichment functions are introduced at the finite elements nodes which allows for a coarse mesh at low order polynomial shape functions even at high wavenumbers. The initial condition is formulated as a Galerkin approximation in the enriched function space. We also show the possibility of lumping the mass matrix which is approximated as a block diagonal system. The proposed method, with and without lumping, is validated using three test cases and compared to an implicit time integration approach. The stability of the proposed approach against different factors such as the choice of wavenumber for the enrichment functions, the spatial discretization, the distortions in mesh elements or the timestep size, is tested in the numerical studies. The method performance is measured for the solution accuracy and the CPU processing times. The results show significant advantages for the proposed lumping approach which outperforms other considered approaches in terms of stability. Furthermore, the resulting block diagonal system only requires a fraction of the CPU time needed to solve the full system associated with the non-lumped approaches.