Numerical Anisotropy and Band‐Gap Formation in Finite Element Wave Propagation: Analysis and Mitigation Using Spline‐Based Elements

ABSTRACT Numerical anisotropy and dispersion are inherent consequences of spatial discretization in finite element modeling of wave propagation. In two‐dimensional problems, these effects manifest not only as direction‐dependent phase and group velocities but also as angularly dependent numerical band gaps that may entirely suppress wave propagation. Such phenomena are often overlooked when dispersion characteristics are examined only along a limited set of symmetry directions. This paper presents a systematic analysis of numerical anisotropy in two‐dimensional wave propagation models, focusing on the spectral consequences of discretization‐induced discontinuities. Dispersion relations are examined using Bloch‐type formulations combined with mode assignment, extended Brillouin zone representations, and angular mapping, enabling a consistent interpretation of multimodal and directional effects. A spline‐based finite element formulation (spFEM) is proposed and extended from 1‐D to 2‐D problems. The method employs piecewise polynomial shape functions with enforced higher‐order global continuity, increasing the information carried by each node and reducing inter‐element derivative discontinuities. The performance of spFEM is assessed against classical FEM and a time‐domain spectral finite element method (TD‐SFEM) using Reissner–Mindlin plate models with comparable numbers of degrees of freedom. The results show that classical FEM and TD‐SFEM formulations exhibit pronounced numerical anisotropy and direction‐dependent band gaps, severely limiting their usable frequency ranges. In contrast, the proposed spFEM formulation significantly suppresses these artefacts and provides a substantially wider usable frequency range at comparable computational cost. The study demonstrates that analyzing numerical behavior beyond nominal applicability limits is essential for understanding error mechanisms and for the systematic design of improved finite element formulations.