A modal-based Partition of Unity Finite Element Method for layered wave propagation problems

The time-harmonic propagation of waves in layered media is simulated numerically by means of a modal-based Partition of Unity Finite Element Method (PUFEM). Instead of using typical plane wave or Bessel solutions of the Helmholtz equation to design the discretization basis, the proposed modal-based PUFEM uses explicitly the tensor-product expressions of the eigenmodes (the so-called Love and interior modes) of a spectral transverse problem, which fulfil the coupling conditions among layers. This modal-based PUFEM approach does not introduce quadrature errors since the coefficients of the discrete matrices are computed in closed-form. An preliminary analysis of the high condition number suffered by the proposed method is also analysed in terms of the mesh size and the number of eigenmodes involved in the discretization. The numerical methodology is validated through a number of test scenarios, where the reliability of the proposed PUFEM method is discussed by considering different modal basis and source terms. Finally, some indicators are introduced to select a convenient discrete PUFEM basis taking into account the observability of cracks placed on a coupling boundary between two adjacent layers.