## The Material-Point Method

The material-point method (MPM) [2-4] is an extension of the hydrodynamics, particle-in-cell code, FLIP [1], to problems in solid mechanics. MPM was developed at UNM, in collaboration with Professor Howard Schreyer (Mechanical Engineering Department) and Professor Zhen Chen (Civil and Environmental Engineering, University of Missouri) with support from Sandia National Laboratories. Since then, many students and colleagues have been involved in using or developing MPM.

In MPM, a fluid or solid body is discretized using a collection of material points that are followed throughout the deformation history. Information from these points is transferred to a background computational grid, where the momentum equation is solved. The grid solution is then used to move the material points and update their properties. In contrast to FLIP, the full stress tensor is carried by the material points so that history-dependent constitutive models are easily implemented.

The method has several advantages, the Lagrangian description provided by the material points naturally tracks the position of bodies and is capable of representing large deformations without the problem of mesh tangling. Referring material-point data to a grid allows an efficient solution of the interactions (linear in the number of material points). Objects of any shape are defined by filling a region with material points, and the computational mesh does not have to conform to the object - greatly simplifying mesh construction. Since material points move in a single-valued velocity field determined from the grid solution, interpenetration of bodies is not possible. This feature makes MPM particularly suited to problems involving many bodies in contact, since no-slip contact can be enforced without a special algorithm.

MPM has been applied to impact, rebound and penetration or perforation problems; and to manufacturing problems such as metal rolling, cutting, extrusion and upsetting. It has also been modified to allow simulations of thin membranes, and the contact algorithm has been extended to allow frictional contact. Simulations of granular material have particularly benefited from the addition of frictional contact.

# References:

• [1] J. U. Brackbill, D. B. Kothe and H. M. Ruppel, FLIP: A Low-Dissipation, Particle-in-Cell Method for Fluid Flow, Comput. Phys. Comm. 48: 25-38 (1988).
• [2] D. Sulsky, Z. Chen and H. L. Schreyer, A Particle Method for History-Dependent Materials, Comput. Meths. Appl. Mech. Engrg. 118: 179-196 (1994).
• [3] D. Sulsky, S.-J. Zhou, H. L. Schreyer, Application of a Particle-in-Cell Method to Solid Mechanics, Comput. Phys. Commun. 87: 236-252 (1995).
• [4] D. Sulsky, H. L. Schreyer, Axisymmetric Form of the Material Point Method with Applications to Upsetting and Taylor Impact Problems, Comput. Meths. Appl. Mech. Engrg. 139: 409-429 (1996).