Energy conservation and accuracy of some MPM formulations

The success of the material point method (MPM) in solving many challenging problems nevertheless raises some open questions regarding the fundamental properties of the method such as time integration accuracy and energy conservation. The traditional MPM time integration methods are often based upon the symplectic Euler method or staggered central differences. This raises the question of how to best ensure energy conservation in explicit time integration for MPM. Two approaches are used here, one is to extend the symplectic Euler method (Cromer Euler) to provide better energy conservation and the second is to use a potentially more accurate symplectic methods, namely the widely used Stormer-Verlet method. The Stormer-Verlet method is shown to have locally third-order time accuracy of energy conservation in time, in contrast to the second order accuracy in energy conservation of the symplectic Euler methods that are used in many MPM calculations. It is shown that there is an extension to the symplectic Euler stress-last method that provides better energy conservation that is comparable with the Stormer-Verlet method. This extension is referred to as TRGIMP and also has third-order accuracy in energy conservation. When the interactions between space and time errors are studied it is seen that spatial errors may dominate in computed quantities such as displacement and velocity. This connection between the local errors in space and time is made explicit mathematically and explains the observed results that displacement and velocity errors are very similar for both methods. The observed and theoretically predicted third-order energy conservation accuracy and computational costs are demonstrated on a standard MPM test example.