My work on MPM
Discover my work on the development of the Material Point Method, in particular the Total Lagrangian version of it.
Our book on MPM
Read more about the book I co-authored on the Material Point Method.
The Material Point Method
What is the Material Point Method (MPM)?
The Material Point Method (MPM) is a particle-based or meshfree method where materials are discretized into collections of points. The material points (or particles) track the physical information (mass, velocities, …) and move over a fixed background grid that serves as a computational pad. This unique combination makes MPM a hybrid Lagrangian/Eulerian method.
The Material Point Method computational step
In the Material Point Method, a computational step consists of 4 sub-steps:
- Particle to Grid (P2G): the information stored at the particle such as its mass, velocity, stresses, etc. is mapped onto the grid.
- Grid Updating: the momentum equations and/or heat equations are solved on the grid nodes.
- Grid to Particles (G2P): the updated information at the grid nodes is mapped back onto the particles.
- Grid Resetting: the grid is reset.
The mathematical equations used in the different sub-steps are listed below.
History of the MPM
The Material Point Method is one of the latest developments in the particle-in-cell (PIC) methods. PIC was first developed in the early 1950s by Harlow at the Los Alamos National Laboratory in New Mexico. It was originally developed for fluid mechanics. However, PIC is affected by large energy dissipation. This was overcome by the Fluid Implicit Particle method introduced in 1986 by Brackbill and Ruppel. These techniques are extremely popular in the computer graphics community for the rending of fluids.
MPM was introduced a little more than 25 years ago as a development of PIC and FLIP for the simulation of solid mechanics by Sulsky and coworkers from the University of New-Mexico in 1994. It is logically built on the main concepts used in PIC that are the Lagrangian material points to carry the physical information and the Eulerian background grid used for the discretization of continuous fields (i.e., displacement field). The main difference though is that in FLIP, the stresses and strains are stored in the background grid cell centres, while they are stored at the particle sites in MPM.
Difference between FEM and MPM
The main difference between the Finite Element Method (FEM) and the Material Point Method (MPM) is that in FEM, solids are discretized in a series of cells connected together that form a mesh, while in MPM they are discretized in a series of particles. These particles have a specific domain attached to them that is explicitly tracked or not depending on the type of MPM used.
However, there are some similarities between FEM and MPM since MPM. MPM is similar to FEM where the material points are moving integration points and where the edges of the mesh do not coincide with that of the solids/fluids. Moreover, the mesh used in MPM remains fixed throughout the simulation and does not deform. Therefore, when solids experience large deformation, in MPM, there is no problem with mesh distortion, unlike in FEM.
Advantages and Disadvantages of MPM
Here are some of the advantages of MPM:
- Absence of mesh distortion problem. Even in the presence of large or extremely large deformation, there won’t be any distortion of the background mesh, and the particles are free to move as the equations of state dictate.
- Exact conservation of mass.
- Exact conservation of linear momentum.
- A no-slip, no-penetration contact algorithm comes at no extra cost with the standard MPM formulation.
- An easy and efficient treatment of frictional contacts is allowed by the presence of the background grid
- Easy to implement, compared to other meshless methods. The MPM algorithm can be easily coded for parallel computing via straight-forward decomposition of the physical domain.
- Highly suitable for problems that are difficult to mesh with FEM.
Some of the disadvantages of MPM are:
- Higher computational cost compared to FEM.
- Large memory footprint due to the background grid having to span the whole domain the fluids/solids are occupying and will occupy.
- Convergence, error and stability analysis of MPM is difficult to perform.
- Difficult enforcement of certain types of boundary conditions.
- Lower accuracy than FEM as the material points move around.
Cell crossing error in MPM
Early versions of MPM where known for being affected by the so-called cell crossing error. This error is characterized by an oscillation in the stresses when a particle (or point) moves from one background grid cell to another.
This error emerges from the abrupt change in the derivative of the shape function at the particle position. It is acute when using linear shape functions. In this case, the shape functions are continuous, but not continuously derivable. Therefore the derivative of these functions are discontinuous. The discontinuity is located at the edge of each background grid cell. In 1D, when a particle moves from one cell to the next, the derivative of the shape function goes from -1 to +1 or vice-versa. This in turns causes a discontinuity in the stresses.
By and large, the use of B-spline shape functions or method such as GIMP mitigate this problem. The use of these higher order shape functions that are continuously derivable limit greatly the stress oscillations.
Example of Material Point Method simulations
All these MPM simulations were performed with the open source code Karamelo.