Improving the Material Point Method

The material point method (MPM) was designed to solve problems in solid mechanics, and it has been used widely in research and industry. In MPM, the equations of motion are solved on a background grid and Lagrangian material points represent the geometry of the body, carrying history-dependent material properties. The focus of this work is on the convergence properties of MPM in terms of order of accuracy and stability. It has been shown numerically that MPM loses convergence and suffers cell crossing errors for large deformation problems. Two remedies that have been proposed are the the generalized interpolation material point method (GIMP) and the convected particle domain interpolation method (CPDI). Both GIMP and CPDI try to improve MPM by altering the geometry of the evolved material-point domain. Such changes lead to improvement of the quadrature part of the MPM algorithm. In this work, we give a different approach to improving MPM by combining ideas from meshfree particle methods and finite element methods. Such an approach provides a general framework for improving MPM. Not only has the framework produced the improved material point method (IMPM), but it can also be used to improve existing variations of MPM, such as CPDI. In addition, a Neumann stability analysis of MPM on the linearized equations of motion is provided. However, this analysis did not provide insight into the observed behavior of MPM. Limitations of the analysis are given that perhaps account for the lack of correlation between the analysis and observations.MathematicsDoctoralUniversity of New Mexico. Dept. of Mathematics and StatisticsSulsky, DeborahSchreyer, HowardLau, StephenAtzberger, PaulAppelo, Danie