A Mathematical Introduction to the Material Point in a Finite Element Framework: Observations, Theory and Experiments

In its 30 or so years existence the Material Point Method (MPM) has proved to be outstanding in solving challenging modeling problems in engineering and in animation. The method models objects as a collection of particles that move on a fixed background grid. The method is used in a range of applications from modeling snow, food, landslides and explosions, and in animating moves such as Disney’s Frozen. However the theoretical background of the method continues to improve but is still lacking somewhat.

In this talk the method is introduced through the GIMP method of Bardenhagen and Kober whose Petrov-Galerkin type approach allows for an understanding of the Eulerian mapping from particles to grid and the mapping back of grid-advanced variables to the particles in defining a Lagrangian integration of particles and their velocities, accelerations, stresses and deformation gradients.

The use of finite element approaches for both these parts of MPM may be used to clarify and provide insight into many aspects of MPM such as mass-lumping , the grid crossing error and linearity preservation as well as allowing existing results to be improved.

The theoretical insights obtained by this approach are illustrated with a number of experiments that show the underlying GIMP accuracy, and how this accuracy translates into observed results. The results obtained show the importance of using the linearity preservation approach and error estimation in both improving the accuracy of MPM and showing what that accuracy is for a range of problems.