An efficient time-stepping solution for transient elastodynamic problems using hybrid-Trefftz finite elements

An efficient time-stepping formulation for transient elastodynamic problems based on hybrid-Trefftz finite elements is presented. The proposed approach combines classical time-stepping algorithms with a novel Trefftz-based Dual Reciprocity Method to handle the non-homogeneous spectral problems that arise from time discretization. In contrast to time-series expansion techniques, the formulation allows transient analyses to be performed over arbitrarily long durations and introduces controllable numerical damping through standard time-stepping parameters. The key feature of the method lies in the construction of the particular solution using Trefftz-type trial functions posed with artificial frequencies, which are formally similar to the complementary solution basis. This strategy avoids the cumbersome and potentially singular expressions typically associated with Dual Reciprocity formulations in elastodynamics, while preserving completeness and enhancing computational efficiency. As a result, all domain equations are satisfied implicitly and the final solving system involves boundary integrals only, with a coefficient matrix that remains unchanged throughout the time-stepping process. The formulation is implemented in the open-source hybrid-Trefftz platform FreeHyTE and validated on benchmark problems involving shock wave propagation in bounded and semi-unbounded elastic media. Numerical results demonstrate excellent agreement with previously published wavelet-based solutions, while avoiding spurious high-frequency noise and featuring improved computational efficiency.