Energy-conserving time-discretisation of abstract dynamic problems with applications in continuum mechanics of solids

TL;DR

This paper introduces an energy-conserving time discretisation method for abstract second-order evolution equations, applicable to continuum mechanics problems involving vibrations, waves, and inelastic processes, with demonstrated numerical simulations.

Contribution

It develops a novel energy-preserving time discretisation scheme for second-order evolution problems, especially effective for quadratic or near-quadratic energy systems.

Findings

Energy conservation in discretised systems under specific conditions

Application to continuum mechanics problems with internal variables

Numerical simulations demonstrating method effectiveness

Abstract

An abstract 2nd-order evolution equation or inclusion is discretised in time in such a way that the energy is conserved at least in qualified cases, typically in the cases when the governing energy is component-wise quadratic or "slightly-perturbed" quadratic. Specific applications in continuum mechanics of solids possibly with various internal variables cover vibrations or waves in linear viscoelastic materials at small strains, coupled with some inelastic processes as plasticity, damage, or phase transformations, and also some surface variants related to contact mechanics. The applicability is illustrated by numerical simulations of vibrations interacting with a frictional contact or waves emitted by an adhesive contact of a 2-dimensional viscoelastic body.