Exponential integrators for stiff elastodynamic problems

Abstract

We investigate the application of exponential integrators to stiff elastodynamic problems governed by second-order differential equations. Classical explicit numerical integration schemes have the shortcoming that the stepsizes are limited by the highest frequency that occurs within the solution spectrum of the governing equations, while implicit methods suffer from an inevitable and mostly uncontrollable artificial viscosity that often leads to a nonphysical behavior. In order to overcome these specific detriments, we devise an appropriate class of exponential integrators that solve the stiff part of the governing equations of motion by employing a closed-form solution. As a consequence, we are able to handle up to three orders of magnitude larger time-steps as with conventional implicit integrators and at the same time achieve a tremendous increase in the overall long-term stability due to a strict energy conservation. The advantageous behavior of our approach is demonstrated on a broad spectrum of complex deformable models like fibers, textiles, and solids, including collision response, friction, and damping.