Spectral formulation of finite element methods using Daubechies compactly-supported wavelets for elastic wave propagation simulation

Abstract

A novel and generic formulation of the wavelet-based spectral finite element approach, which is applicable to linear transient dynamics and elastic wave propagation problems, is presented in this paper. In order to spectrally formulate the variational form of governing equations in a temporally-decoupled manner, the wavelet-Galerkin discretization based on Daubechies compactly-supported wavelets is employed. The spectral formulation reduces the problem to a set of temporally independent equations which can be solved in parallel. It is demonstrated that for the spatial discretization, any class of standard finite element method can be adopted to facilitate capturing the complex geometries and boundary conditions. It is demonstrated that the wavelet-based temporal discretization is not influenced by the finite element mesh. Also frequency-dependent damping models are straight-forward to apply. These, along with the fact that the method is directly amenable to parallel computation make the approach very promising for wave propagation problems.To attain hp-refinement capability and spectral convergence properties for the spatial discretization, a higher-order Lagrangian FEM on the Gauss–Lobatto–Legendre grid, i.e. spectral element method, is adopted. For the sake of numerical investigation, a number of 2D and 3D examples including anisotropic and non-homogeneous structures are studied. The accuracy and the convergence rate of the method are evaluated and found to be superior to the explicit Newmark time integration. Possible approaches for speeding up the method are also discussed.