Ghost stabilisation of the material point method for stable quasi‐static and dynamic analysis of large deformation problems

The unstable nature of the material point method (MPM) is widely documented and is a barrier to the method being used for routine engineering analyses of large deformation problems. The vast majority of articles concerning this issue are focused on the instabilities that manifest when a material point crosses between background grid cells. However, there are other issues related to the stability of MPMs. This article focuses on the issue of the conditioning of the global system of equations caused by the arbitrary nature of the position of the physical domain relative to the background computational grid. The issue is remedied here via the use of a ghost stabilisation technique that penalises variations in the gradient of the solution field near the boundaries of the physical domain. This technique transforms the stability of the MPM, providing a robust computational framework for large deformation explicit dynamic and implicit quasi ‐static analysis.