Polyhedral elements based on the scaled boundary finite element method with mass lumping for explicit analysis

An efficient and versatile numerical approach is critical for transient structural analyses in diverse engineering contexts. Explicit time integration schemes, such as the central difference method, benefit substantially from lumped (diagonal) mass matrices, which eliminate matrix inversions and enable efficient computation of nodal displacements. In wave propagation simulation in soil (e.g., seismic waves), domains often exhibit heterogeneity. Hierarchical meshes, which enable abrupt transitions in mesh size near material interfaces, efficiently capture such geometries but pose challenges for standard finite elements due to hanging nodes. The scaled boundary finite element method (SBFEM) offers an attractive alternative by facilitating the derivation of shape functions for polyhedral elements with arbitrary numbers of faces while discretizing only the subdomain boundary. In the static case, solutions are obtained numerically in the circumferential direction and analytically in the radial direction using power functions. When extended to general loading conditions, these semi-analytical formulations yield shape functions that are fully complete circumferentially but only linearly complete radially. While the use of higher-order elements in the surface discretization enhances static stiffness accuracy, their limited radial completeness restricts the accurate representation of inertial effects and convergence rates in linear dynamics. To address this, the shape functions are enriched with bubble functions that introduce higher-order terms in the interior, which vanish at boundaries, thus improving solution accuracy. The inherent Kronecker-delta property and partition-of-unity of the SBFEM formulation enable efficient mass matrix lumping. The computational efficiency and accuracy of the proposed approach are evaluated using transient simulations in heterogeneous domains.