Energy-conserving time-discretisation of abstract dynamic problems with applications in continuum mechanics of solids
For researchers in computational solid mechanics, this provides a time-stepping scheme that preserves energy in qualified cases, improving long-term stability and accuracy for coupled inelastic processes.
The paper develops an energy-conserving time-discretization method for abstract 2nd-order evolution equations, applicable to continuum mechanics problems such as vibrations in viscoelastic materials with inelastic processes. Numerical simulations demonstrate the method's effectiveness for frictional and adhesive contact problems.
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.