An adaptive wavelet method for nonlinear partial differential equations with applications to dynamic damage modeling

TL;DR

This paper introduces an adaptive wavelet-based numerical method for solving complex nonlinear PDEs, enabling accurate, high-fidelity simulations of dynamic damage processes with efficient multiscale resolution.

Contribution

The paper presents a novel adaptive wavelet technique with a predictor-corrector scheme for efficiently solving coupled nonlinear PDEs in dynamic damage modeling.

Findings

Successfully models damage nucleation and propagation in nonlinear solids.

Achieves high accuracy with data compression through multiresolution discretization.

Demonstrates effectiveness in multiscale, multiphysics problems.

Abstract

Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop 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 algorithm exploits the multiresolution nature of wavelet basis functions to solve initial-boundary value problems on finite domains with a sparse multiresolution spatial discretization. By leveraging wavelet theory and embedding a predictor-corrector procedure within the time advancement loop, we dynamically adapt the computational grid and maintain accuracy of the solutions of the PDEs as they evolve. Consequently, our method provides high fidelity simulations with significant data compression. We present verification of the algorithm and demonstrate its capabilities by modeling high-strain rate damage nucleation and propagation in nonlinear solids using a novel Eulerian-Lagrangian continuum framework.