Validation of a 3-D high-order discontinuous Galerkin solver with transducer boundary conditions through comparisons with the fast nearfield method

Abstract

A 3-D discontinuous Galerkin (DG) solver is presented for ultrasound propagation in heterogeneous media. The formulation advances particle velocity and density perturbation fields using explicit time integration, while numerical fluxes couple elements and enforce material interfaces, enabling layered configurations and spatially varying material properties. DG is ideal for these calculations due to high-order accuracy, element-local operators, and a flux-based boundary-condition model for finite-aperture sources. Implementation choices governing accuracy and computational cost are considered, comparing consistent and diagonal (lumped) formulations. Transducer boundary conditions are implemented for single-element transducers and linear arrays. Two enforcement strategies are investigated: a pointwise imposition of the boundary drive at face evaluation points, and an L2 projection of the drive onto the face polynomial space. The projected approach reduces aliasing for coarse meshes and high-order discretizations by representing boundary data consistently within the face polynomial basis. Linear validation in homogeneous media is performed via comparisons to the fast nearfield method for single-element and multi-element configurations, including focused sub-apertures with transmit delays. Transient pressure fields and centerline waveforms are compared in order to demonstrate accurate boundary-driven excitation across various aperture and focusing conditions.