Topology optimization under mixed transient dynamic loads: Explicit–implicit time integration

Most existing topology optimization formulations for elastodynamic structures in the time domain adopt implicit integration schemes and treat either force or displacement (velocity) boundary conditions in isolation. Such approaches are not well suited to structures that are simultaneously subjected to different types of transient loads. In this paper we develop a density-based topology optimization framework for linear elastodynamics under mixed transient dynamic loads, in which time-dependent prescribed traction and prescribed loading rate act concurrently on different parts of the boundary. To balance solution accuracy and computational cost, the structural dynamics are solved using both the implicit HHT-α method and the explicit central difference method (CDM), and consistent adjoint sensitivities are derived for each scheme, with verification by the finite difference method. Structural performance is measured by the total mechanical work, which provides a unified metric that couples the work of external forces and the work associated with prescribed velocities over the entire loading history. Through a series of two-dimensional benchmark problems (half MBB beam, cantilever beam and clamped beam) we systematically investigate the influence of loading type, loading duration and time-integration scheme on the optimized topologies, dynamic responses and computational cost. The numerical results show that (i) mixed traction loading produces layouts that differ markedly from those obtained under single-type loading or static optimization, and (ii) the choice between HHT-α and CDM is crucial for short-duration, high-frequency excitations, where numerical damping in HHT-α significantly reduces the total mechanical work. These findings complement existing time-domain topology optimization studies and provide practical guidance for selecting appropriate loading representations and time-integration schemes in transient dynamic design.