# Improving accuracy in the MPM method using a null space filter

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## Abstract

The material point method (MPM) has been very successful in providing solutions to many challenging problems involving large deformations. Nevertheless there are some important issues that remain to be resolved with regard to its analysis. One key challenge applies to both MPM and particle-in-cell (PIC) methods and arises from the difference between the number of particles and the number of the nodal grid points to which the particles are mapped. This difference between the number of particles and the number of grid points gives rise to a non-trivial null space of the linear operator that maps particle values onto nodal grid point values. In other words, there are non-zero particle values that when mapped to the grid point nodes result in a zero value there. Moreover, when the nodal values at the grid points are mapped back to particles, part of those particle values may be in that same null space. Given positive mapping weights from particles to nodes such null space values are oscillatory in nature. While this problem has been observed almost since the beginning of PIC methods there are still elements of it that are problematical today as well as methods that transcend it. The null space may be viewed as being connected to the ringing instability identified by Brackbill for PIC methods. It will be shown that it is possible to remove these null space values from the solution using a null space filter. This filter improves the accuracy of the MPM methods using an approach that is based upon a local singular value decomposition (SVD) calculation. This local SVD approach is compared against the global SVD approach previously considered by the authors and to a recent MPM method by Zhang and colleagues.

## Keywords

MPM Particles Null space Instability## Notes

### Acknowledgments

We would like to thank the referees for their thoughtful and helpful comments that have helped to greatly improve this paper. This research was primarily sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-12-2-0023. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The authors would like to thank ARL for their support and their colleague Mike Kirby for suggesting the use of the global SVD approach.

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