A wavelet multi-resolution enabled interpolation Galerkin method for two-dimensional solids

Abstract

A novel wavelet multi-resolution formulation in which arbitrary specified nodal coefficients can be exactly the function values at the corresponding nodes is proposed to approximate field functions for solid mechanics problems. This wavelet approximation is explicitly constructed based only on properly scattered nodes without any matrix inversion or ad-hoc parameters. It allows easy and automatic node generation and further refinement at any targeted zones, with a loose and explicit criterion. Using this wavelet multi-resolution approximation to create trial and weighted functions, a wavelet multi-resolution enabled interpolation Galerkin method (WMEIGM) is developed. In the proposed WMEIGM, the stiffness matrix can be efficiently evaluated through semi-analytical method. The essential boundary conditions can be imposed with ease as in the finite element method (FEM). The accuracy and convergence of the proposed WMEIGM are examined theoretically and numerically. Numerical results show that the present WMEIGM has an excellent stability against irregular nodal distribution even with an extremely large ratio of the maximum grid size to the minimum, and a strong capacity for handling irregular problem domains with complicated shape. Moreover, a robust multi-resolution enrichment technique is developed for the present WMEIGM for improving local accuracy and for capturing localized steep gradients.