Computational error estimation for the Material Point Method

A common feature of many methods in computational mechanics is that there is often a way of estimating the error in the computed solution. The situation for computational mechanics codes based upon the Material Point Method is very different in that there has been comparatively little work on computable error estimates for these methods. This work is concerned with introducing such an approach for the Material Point Method. Although it has been observed that spatial errors may dominate temporal ones at stable time steps, recent work has made more precise the sources and forms of different MPM errors. There is then a need to estimate these errors through computable estimates of different errors in the Material Point Method. The approach used involves linearity-preserving extensions of existing methods, which allow estimates of different spatial errors in the Material Point Method to be derived based upon nodal derivatives of different physical variables in MPM. These derivatives are then estimated using standard difference approximations calculated on the background mesh. The use of these estimates of the spatial error makes it possible to measure the growth of errors over time. A number of computational experiments are used to illustrate the performance of the computed error estimates for both the original MPM method and the GIMP method, when modified to preserve linearity. Finally, the form of the computed estimates also makes it possible to identify the order of the accuracy of the methods in space and time. For these methods, including the linearity preservation is clearly beneficial, as regards accuracy while not changing the preference for GIMP over MPM