An Adaptive Wavelet Method for Multiscale Modeling with Applications in Dynamic Damage

Multiscale and multiphysics problems need novel numerical methods in order to provide practical predictive results. To that end, this dissertation develops a wavelet based technique to solve a coupled system of nonlinear partial differential equations (PDEs) while resolving features on a wide range of spatial and temporal scales. The novel N-dimensional algorithm exploits the multiresolution nature of wavelet basis functions to solve initial-boundary value problems on finite domains. A sparse multiresolution spatial discretization is constructed by projecting fields and their spatial derivatives onto the wavelet basis. By leveraging wavelet theory and embedding a predictor-corrector procedure within the time advancement loop, the algorithm dynamically adapts the computational grid and maintains accuracy of the solutions of the PDEs as they evolve. Consequently, this new method provides high fidelity simulations with significant data compression. The implementation of this algorithm is verified, establishing mathematical correctness with spatial convergence in agreement with the theoretical estimates. The multiscale capabilities are demonstrated by modeling high-strain rate damage nucleation and propagation in nonlinear solids using a novel Eulerian-Lagrangian continuum framework.