Adaptive wavelet-enriched hierarchical finite element model for polycrystalline microstructures

This paper proposes an adaptive, wavelet-enriched hierarchical finite element model (FEM) to enhance computational efficiency and solution accuracy for heterogeneous domains of anisotropic elastic materials. It is motivated by the need to overcome shortcomings of conventional methods like the standard FEM or fast Fourier transformation (FFT) based method for analyzing deformation in large polycrystalline microstructures. The multi-resolution wavelet functions are advantageous for selection of an optimal set of functions that can adaptively enrich the solution space with a prescribed level of accuracy. The proposed method introduces a discretization space that conforms to the profile of the solution sought. The method engages a second generation family of wavelets with a lifting scheme to generate hierarchical interpolation functions. An iterative algorithm efficiently calculates estimates of the solution from the previous iterate using a modified Jacobi method. The adaptive FE method performs significantly better than uniformly refined FEM in validation tests with focus on the convergence rate and error mitigation. A polycrystalline microstructure with elastically anisotropic grains is simulated by the adaptive wavelet-enriched hierarchical method, showing high convergence rates.