 Material Point Method

The Material Point Method (MPM), is an extension of the Particleincell (PIC) Method in computational fluid dynamics to computational solid dynamics, and is a Finite element method (FEM)based particle method. It is primarily used for multiphase simulations, because of the ease of detecting contact without interpenetration. It can also be used as an alternative to dynamic FEM methods to simulate large material deformations, because there is no remeshing required by the MPM.
In the MPM, Lagrangian point masses, or material points, are moved through a Eulerian background mesh. At the end of each calculation cycle, a ‘convective’ step occurs, in which the mesh is reset to its original position, while material points remain in their current positions. There are two key differences between the PIC and MPM. The first one is that the MPM is formulated in the weak form similar to that for the FEM so that the FEM and MPM could be combined together for largescale simulations. The second one is that historydependent constitutive models could be formulated on the material points, which results in a robust spatial discretization method for multiphase and multiphysics problems.
Contents
History of PIC/MPM
The PIC was originally conceived to solve problems in fluid dynamics, and developed by Harlow at Los Alamos National Laboratory in 1957.^{[1]} One of the first PIC codes was the FluidImplicit Particle (FLIP) program, which was created by Brackbill in 1986^{[2]} and has been constantly in development ever since. Until the 1990’s, the PIC method was used principally in fluid dynamics.
Motivated by the need for better simulating penetration problems in solid dynamics, Sulsky, Chen and Schreyer started in 1993 to reformulate the PIC and develop the MPM, with funding from Sandia National Laboratories ^{[3]}. The original MPM was then further extended by Bardenhagen et al.. to include frictional contact ^{[4]}, which enabled the simulation of granular flow ^{[5]}, and by Nairn to include explicit cracks^{[6]} and crack propagation (known as CRAMP).
Recently, an MPM implementation based on a micropolar Cosserat continuum ^{[7]} has been used to simulate highshear granular flow, such as silo discharge. MPM's uses were further extended into Geotechnical engineering with the recent development of a quasistatic, implicit MPM solver which provides numerically stable analyses of largedeformation problems in Soil mechanics. ^{[8]}
Annual workshops on the use of MPM are held at various locations in the United States. The Fifth MPM Workshop is scheduled to be held at Oregon State University, in Corvallis, OR, on April 2 and 3, 2009.
Applications of PIC/MPM
The uses of the PIC or MPM method can be divided into two broad categories: firstly, there are many applications involving fluid dynamics, plasma physics, magnetohydrodynamics, and multiphase applications. The second category of applications comprises problems in solid mechanics.
Fluid Dynamics and Multiphase Simulations
The PIC method has been used to simulate a wide range of fluidsolid interactions, including sea ice dynamics ^{[9]}, penetration of biological soft tissues ^{[10]}, fragmentation of gasfilled canisters ^{[11]}, dispersion of atmospheric pollutants ^{[12]}, multiscale simulations coupling molecular dynamics with MPM ^{[13]}^{[14]}, and fluidmembrane interactions ^{[15]}. In addition, the PICbased FLIP code has been applied in magnetohydrodynamics and plasma processing tools, and simulations in astrophysics and freesurface flow ^{[16]}.
Solid Mechanics
MPM has also been used extensively in solid mechanics, to simulate impact, penetration, collision and rebound, as well as crack propagation ^{[17]}^{[18]}. MPM has also become a widelyused method within the field of soil mechanics: it has been used to simulate granular flow, silo discharge, pile driving, bucket filling, and material failure; and to model soil stress distribution, compaction, and hardening. It is now being used in wood mechanics problems such as simulations of transverse compression on the cellular level including cell wall contact ^{[19]} (this work received the George Marra Award for paper of the year from the Society of Wood Science and Technology[1])
Classification of PIC/MPM codes
MPM in the context of numerical methods
One subset of numerical methods are Meshfree methods, which are defined as methods for which “a predefined mesh is not necessary, at least in field variable interpolation”. Ideally, a meshfree method does not make use of a mesh “throughout the process of solving the problem governed by partial differential equations, on a given arbitrary domain, subject to all kinds of boundary conditions,” although existing methods are not ideal and fail in at least one of these respects. Meshless methods, which are also sometimes called particle methods, share a “common feature that the history of state variables is traced at points (particles) which are not connected with any element mesh, the distortion of which is a source of numerical diffculties.” As can be seen by these varying interpretations, some scientists consider MPM to be a meshless method, while others do not. All agree, however, that MPM is a particle method.
The Arbitrary Lagrangian Eulerian (ALE) methods form another subset of numerical methods which includes MPM. Purely Lagrangian methods employ a framework in which a space is discretised into initial subvolumes, whose flowpaths are then charted over time. Purely Eulerian methods, on the other hand, employ a framework in which the motion of material is described relative to a mesh that remains fixed in space throughout the calculation. As the name indicates, ALE methods combine Lagrangian and Eulerian frames of reference.
Subclassification of MPM/PIC
PIC methods may be based on either the strong form collocation or a weak form discretisation of the underlying partial differential equation (PDE). Those based on the strong form are properly referred to as finitevolume PIC methods. Those based on the weak form discretisation of PDEs may be called either PIC or MPM.
MPM solvers can model problems in one, two, or three spatial dimensions, and can also model axisymmetric problems. MPM can be implemented to solve either quasistatic or dynamic equations of motion, depending on the type of problem that is to be modeled.
The timeintegration used for MPM may be either explicit or implicit. The advantage to implicit integration is guaranteed stability, even for large timesteps. On the other hand, explicit integration runs much faster and is easier to implement.
Advantages of MPM
MPM Compared to FEM
Unlike FEM, MPM does not require periodical remeshing steps and remapping of state variables, and is therefore better suited to the modeling of large material deformations. In MPM, particles and not the mesh points store all the information on the state of the calculation. Therefore, no numerical error results from the mesh returning to its original position after each calculation cycle, and no remeshing algorithm is required.
The particle basis of MPM allows it to treat crack propagation and other discontinuities better than FEM, which is known to impose the mesh orientation on crack propagation in a material. Also, particle methods are better at handling historydependent constitutive models.
MPM Compared to Pure Particle Methods
Because in MPM nodes remain fixed on a regular grid, the calculation of gradients is trivial.
In simulations with two or more phases it is rather easy to detect contact between entities, as particles can interact via the grid with other particles in the same body, with other solid bodies, and with fluids.
Disadvantages of MPM
MPM is more expensive in terms of storage than other methods, as MPM makes use of mesh as well as particle data. MPM is more computationally expensive than FEM, as the grid must be reset at the end of each MPM calculation step and reinitialised at the beginning of the following step. Spurious oscillation may occur as particles cross the boundaries of the mesh in MPM, although this effect can be minimized by using generalized interpolation methods (GIMP). In MPM as in FEM, the size and orientation of the mesh can impact the results of a calculation: for example, in MPM, strain localisation is known to be particularly sensitive to mesh refinement.
Notes
 ^ Johnson, N.L. "The legacy and future of computational fluid dynamics at Los Alamos". In Proceedings of the 1996 Canadian CFD Conference, 1996
 ^ Brackbill, J.U. and Ruppel, H.M. "FLIP: A method for adaptively zoned, particleincell calculations in two dimensions". Journal of Computational Physics, 65, 1986
 ^ Sulsky, D., Chen, Z., and Schreyer, H.L. "A particle method for historydependent materials". Computer Methods in Applied Mechanics and Engineering, 118:179196, 1994.
 ^ Bardenhagen, S.G., Brackbill, J.U., and Sulsky, D. "Shear deformation in granular materials". 1998
 ^ Wieckowski, Z. "A particleincell method in analysis of motion of a granular material in a silo". Computational Mechanics: New trends and Applications, 1998
 ^ Nairn, J. A. "Material Point Method Calculations with Explicit Cracks". Computer Modeling in Engineering & Science, 4:649664, 2003.
 ^ Coetzee, C.J. The modelling of granular flow using the particleincell method. PhD thesis, University of Stellenbosch, South Africa, 2004.
 ^ Beuth, L., Coetzee, C.J., Bonnier, P. and van den Berg, P. "Formulation and validation of a quasistatic material point method." In 10th International Symposium on Numerical Methods in Geomechanics, 2007.
 ^ Ligett, K. "A modified particleincell method for sea ice dynamics". Clarkson University Technical Report, 2002.
 ^ Ionescu, I., Guilkey, J., Berzins, M., Kirby, R., and Weiss, J. "Computational simulation of penetrating trauma in biological soft tissues using MPM."
 ^ Banerjee, B. "Material point method simulations of fragmenting cylinders". In 17th ASCE Engineering Mechanics Conference, 2004.
 ^ Patankar,N.A. and Joseph, D.D. "Lagrangian numerical simulation of particulate flows". International Journal of Multiphase Flows, 2001.
 ^ Lu, H., Daphalapurkar, N. P., Wang, B., Roy, S., Komanduri, R. "Multiscale Simulation from atomistic to continuum  coupling molecular dynamics (MD) with material point method (MPM)." Philosophical Magazine. 86(20):29712994, 2006.
 ^ Ma., J. "Multiscale Simulation using the Generalized Interpolation MPM, Discrete Dislocations and Molecular Dynamics." PhD thesis, Oklahoma State University, 2006.
 ^ York, A., Sulsky, D. and Schreyer, H. "Fluidmembrane interaction based on the material point method." International Journal for Numerical Methods in Engineering, 2000.
 ^ Li, S. and Liu, W.K. "Meshfree and particle methods and their applications". American Society of Mechanical Engineers, 2002.
 ^ Karuppiah, V. "Implementation of irregular mesh in MPM for simulation of mixed mode crack opening in tension." Master’s thesis, Oklahoma State University, 2004.
 ^ Daphalapurkar, N. P., Lu, H., Coker, D., Komanduri, R. "Simulation of dynamic crack growth using the generalized interpolation material point (GIMP) method." International Journal of Fracture. 143:79102, 2007.
 ^ Nairn, J. A. "Numerical Simulations of Transverse Compression and Densification in Wood". Wood and Fiber Science. 38:576591, 2006.
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