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Material Point Method (MPM)

The material point method (MPM) is a particle method for simulations in computational fluid and solid mechanics.The method uses a regular structured grid as a computational scratchpad for computing spatial gradients of field variables. The grid is convected with the particles during deformations that occur over a time step, eliminating the diffusion problems associated with advection on an Eulerian grid. The grid is restored to its original location at the end of a time step. In addition to avoiding the Eulerian diffusion problem, this approach also circumvents problems with mesh entanglement that can plague fully Lagrangian-based techniques when large deformations are encountered. MPM has also been successful in solving problems involving contact, having an advantage over traditional finite element (FE) methods in that the use of the regular grid eliminates the need for doing costly searches for contact surfaces.

Coupling of the MPM to computational fluid dynamics simulations is readily achieved because a regular grid is used for gradient calculations.The grid then serves as both an Eulerian reference frame for CFD calculations and an updated Lagrangian reference frame for MPM calculations. Tight coupling can be achieved between the two phases by using a multimaterial CFD formulation, while each phase still enjoys the benefits of computation using its optimal reference frame. When using explicit time integration for both phases, the time step sizes required by the Courant stability condition are often disparate by several orders of magnitude. The time step size imposed by this restriction is relatively severe compared to the time step limitations of most CFD codes. This can result in prohibitive computational solution times for fluid-solid interaction problems involving traditional engineering materials.

To circumvent the time step restrictions imposed by explicit time integration for low rate dynamic and quasi-static problems, the objectives of this work were to develop and implement an implicit time integration strategy with the MPM and to test the implementation against an explicit MPM code and an implicitly integrated FE code. Our implicit time integration strategy exploits similarities between the function of material points in MPM and integration points in FE calculations to adapt implicit time integration for use with MPM. The implementation uses Newton 's method to solve for the incremental grid displacements in the linearized form of the equations of motion and the update of nodal kinematics using the trapezoidal rule. Explicit expressions for the tangent stiffness are derived in terms of the grid displacements used in MPM. Because of the similarities that are identified between MPM and FE methods in this work, further improvements to MPM algorithms and numerical implementations should be able to benefit from the large amount of published research on implicit FE methods. In addition to allowing for much larger time steps, the implicit algorithm has also shown advantage over the explicit algorithm in its ability to obtain a solution for certain types of problems. The accuracy for problems involving large deformation, contact and dynamics is demonstrated through representative numerical simulations. The implicit MPM algorithm described here is intended to give an additional option to analysts for the solution of problems for which MPM is well suited, but hindered by the explicit time step size restriction.

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