A domain embedding strategy for acoustic wave scattering by moving obstacles

We propose a novel domain embedding method for simulating acoustic wave scattering by moving obstacles. The key innovation lies in reformulating the sound-soft boundary condition on the moving obstacle as a space-time-dependent source term in the governing acoustic wave equation. This is achieved through a weak formulation that embeds the obstacle domain into a larger, fixed computational domain. Within the embedded obstacle domain, we introduce an effective internal damping equation that enforces the sound-soft condition dynamically. In practical computations, the characteristic function of the embedded domain is replaced by a smoothed phase field variable. This formulation generalizes the diffuse domain method by incorporating enhanced damping dynamics to better impose boundary behavior in both space and time. The extended model is discretized using finite element methods in space and a hybrid Crank–Nicolson and backward Euler (BDF-1) scheme in time. The resulting fully discrete scheme is shown to satisfy a discrete energy dissipation law, ensuring numerical stability and physical fidelity. Benchmark comparisons with classical finite element solutions for fixed obstacle domains validate the accuracy of the proposed method. We further demonstrate the robustness and versatility of the approach by simulating acoustic wave scattering by moving obstacles of varying geometries and trajectories. This domain embedding strategy transforms a challenging initial-boundary value problem with moving interior boundaries into a fixed-domain problem augmented by a time-dependent source term. The method offers a promising framework for addressing more complex wave scattering phenomena involving moving objects such as electromagnetic wave scattering.