A number of physical problems involve localized phenomena that move in time, such as crack propagation and granular flows. In the finite difference and finite element methods approaches, a convenient treatment of these issues is problematic. Recently, a new method well suited for these problems was introduced. It is called the Material Point Method (MPM). This method splits the object into many Lagrangean particles, which are defined from material particles in the body under consideration. The particles are mapped to a Eulerian mesh to solve for the momentum equation. The mesh maps material properties back to the particles. Advantages of both the Lagrangean and Eulerian approaches are realized in the MPM. Also, dissipation normally found in Eulerian approaches is reduced. Because this MPM is new, many problems have yet to still be solved. Problems that were found from this MPM during the research of this project included discontinuities and instabilities in total energy. The problem of finding a time step that was reasonable to use but did not cause instabilities in the energy was examined. In the course of this project, Matlab was used to create from scratch, a working implementation of the Material Point Method, an algorithm created to model elastic materials on a particle by particle basis. The code was made more efficient by deleting loops. Visualization methods were added for the ability to see the deformations occurring in an elastic collision and to see the changes in energy that are occurring. Methods to correct for the instabilities were implemented including re-ordering of the steps of the algorithm, using a partial mass matrix instead of a lumped diagonal matrix, adding an iterative loop to fine tune a[n] intermediate velocity, and the addition of code to account for volume changes. Results include finding the maximum time step that can be taken for different methods while preserving stability. Also it was found that the addition of a partial mass matrix seemed to deter this method in the implementation applied in this paper. Lastly, the improvements which created the best improvements to the code were a re-ordering of the original steps of the method, application of an iterative loop to fine tune the intermediate velocity, and the addition of code to account for particle volume change during collision.
Worcester Polytechnic Institute
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