Optimization Methods for One Dimensional Elastodynamics

TL;DR

This paper introduces an optimization-based numerical scheme for one-dimensional elastodynamics that leverages variational formulations and constrained gradient descent, improving shock handling at the expense of higher computational cost.

Contribution

It presents a novel variational and optimization-based approach for solving elastodynamics systems, integrating constrained gradient descent with discontinuous Galerkin methods.

Findings

The scheme effectively manages oscillations near shocks.

It demonstrates good performance in solving elastodynamics equations.

Computational cost is higher but can be reduced with step selection techniques.

Abstract

We propose a new approach for solving systems of conservation laws that admit a variational formulation of the time-discretized form, and encompasses the p-system or the system of elastodynamics. The approach consists of using constrained gradient descent for solving an implicit scheme with variational formulation, while discontinuous Galerkin finite element methods is used for the spatial discretization. The resulting optimization scheme performs well, it has an advantage on how it handles oscillations near shocks, and a disadvantage in computational cost, which can be partly alleviated by using techniques on step selection from optimization methods.