An explicit predictor/multicorrector time marching with automatic adaptivity for finite-strain elastodynamics

TL;DR

This paper introduces a stable, efficient, and adaptive predictor/multicorrector time integration scheme for finite-strain elastodynamics, capable of handling complex wave phenomena and energy conservation over long simulations.

Contribution

It develops a novel explicit predictor/multicorrector method with automatic adaptivity based on error estimation, improving stability and accuracy in hyperbolic PDE simulations.

Findings

Successfully applied to linear and nonlinear wave problems

Demonstrates dynamic adaptivity to energy releases and shocks

Conserves momentum and energy over long simulations

Abstract

We propose a time-adaptive predictor/multi-corrector method to solve hyperbolic partial differential equations, based on the generalized-$\alpha$ scheme that provides user-control on the numerical dissipation and second-order accuracy in time. Our time adaptivity uses an error estimation that exploits the recursive structure of the variable updates. The predictor/multicorrector method explicitly updates the equation system but computes the residual of the system implicitly. We analyze the method's stability and describe how to determine the parameters that ensure high-frequency dissipation and accurate low-frequency approximation. Subsequently, we solve a linear wave equation, followed by non-linear finite strain deformation problems with different boundary conditions. Thus, our method is a straightforward, stable and computationally efficient approach to simulate real-world engineering problems. Finally, to show the performance of our method, we provide several numerical examples in two and three dimensions. These challenging tests demonstrate that our predictor/multicorrector scheme dynamically adapts to sudden energy releases in the system, capturing impacts and boundary shocks. The method efficiently and stably solves dynamic equations with consistent and under-integrated mass matrices conserving the linear and angular momenta as well as the system's energy for long-integration times.