Material Point Method

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The Material Point Method (MPM), also referred to as the Particle-in-cell (PIC) Method, is a Finite element method (FEM)-based particle method. It is primarily used for multiphase simulations, because of the ease of detecting contact. It can also be used as an alternative to dynamic FEM methods to simulate large material deformations, because there is no remeshing required by MPM.

In MPM, Eulerian point masses, or particles, are moved through a Lagrangian mesh. At the end of each calculation cycle, a ‘convective’ step occurs, in which the mesh is reset to its original position, while particles remain in their current positions.

Contents 1 History of PIC/MPM 2 Applications of PIC/MPM 2.1 Fluid Dynamics and Multiphase Simulations 2.2 Solid Mechanics 3 Classification of PIC/MPM codes 3.1 MPM in the context of numerical methods 3.2 Subclassification of MPM/PIC 4 Advantages of MPM 4.1 MPM Compared to FEM 4.2 MPM Compared to Pure Particle Methods 5 Disadvantages of MPM 6 External links 7 Notes

[edit] History of PIC/MPM

PIC was originally conceived to solve problems in fluid dynamics, and developed by Harlow at Los Alamos Laboratories in 1957 ^[1].

One of the first PIC codes was the Fluid-Implicit 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. Then, in 1995, the FLIP code was reformulated by Sulsky et al to treat solid mechanics applications ^[3]. Sulsky’s model 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 micro-polar Cosserat continuum ^[7] has been used to simulate high-shear granular flow, such as silo discharge. MPM's uses were further extended into Geotechnical engineering with the recent development of a quasi-static, implicit MPM solver which provides numerically stable analyses of large-deformation problems in Soil mechanics. ^[8]

[edit] 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.

[edit] Fluid Dynamics and Multiphase Simulations

The PIC method has been used to simulate a wide range of fluid-solid interactions, including sea ice dynamics ^[9], penetration of biological soft tissues ^[10], fragmentation of gas-filled canisters ^[11], dispersion of atmospheric pollutants ^[12], multiscale simulations coupling molecular dynamics with MPM ^[13][14], and fluid-membrane interactions ^[15]. In addition, the PIC-based FLIP code has been applied in magnetohydrodynamics and plasma processing tools, and simulations in astrophysics and free-surface flow ^[16].

[edit] 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 widely-used 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])

[edit] Classification of PIC/MPM codes

[edit] 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.

[edit] Subclassification of MPM/PIC

PIC methods may be based the 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 finite-volume 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 quasi-static or dynamic equations of motion, depending on the type of problem that is to be modeled.

The time-integration 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.

[edit] Advantages of MPM

[edit] 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 history-dependent constitutive models.

[edit] MPM Compared to Pure Particle Methods

Because in MPM nodes remain fixed on a regular mesh, the calculation of gradients is trivial.

In simulations with two or more phases it is very 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.

[edit] 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.

[edit] External links

• Center for Simulation of Accidental Fires and Explosions - MPM code available
• Open-source MPM code

[edit] Notes