Material Point Method
Encyclopedia
The Material Point Method (MPM), is an extension of the Particle-in-cell
(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 inter-penetration. It can also be used as an alternative to dynamic FEM methods to simulate large material deformations, because there is no re-meshing 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 large-scale simulations. The second one is that history-dependent constitutive models could be formulated on the material points, which results in a robust spatial discretization method for multiphase and multi-physics problems.
at Los Alamos National Laboratory
in 1957. One of the first PIC codes was the Fluid-Implicit Particle (FLIP) program, which was created by Brackbill in 1986 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 . The original MPM was then further extended by Bardenhagen et al.. to include frictional contact , which enabled the simulation of granular flow , and by Nairn to include explicit cracks and crack propagation (known as CRAMP).
Recently, an MPM implementation based on a micro-polar Cosserat continuum 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
.
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.
, 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.
(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.
, 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.
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.
Particle-in-cell
The Particle-in-Cell method refers to a technique used to solve a certain class of partial differential equations. In this method, individual particles in a Lagrangian frame are tracked in continuous phase space, whereas moments of the distribution such as densities and currents are computed...
(PIC) Method in computational fluid dynamics
Computational fluid dynamics
Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...
to computational solid dynamics, and is a Finite element method
Finite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
(FEM)-based particle method. It is primarily used for multiphase simulations, because of the ease of detecting contact without inter-penetration. It can also be used as an alternative to dynamic FEM methods to simulate large material deformations, because there is no re-meshing 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 large-scale simulations. The second one is that history-dependent constitutive models could be formulated on the material points, which results in a robust spatial discretization method for multiphase and multi-physics problems.
History of PIC/MPM
The PIC was originally conceived to solve problems in fluid dynamics, and developed by HarlowFrancis H. Harlow
Francis Harvey Harlow is an American theoretical physicist known for his work in the field of fluid dynamics. He was a researcher at Los Alamos National Laboratory, Los Alamos, New Mexico...
at Los Alamos National Laboratory
Los Alamos National Laboratory
Los Alamos National Laboratory is a United States Department of Energy national laboratory, managed and operated by Los Alamos National Security , located in Los Alamos, New Mexico...
in 1957. One of the first PIC codes was the Fluid-Implicit Particle (FLIP) program, which was created by Brackbill in 1986 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 . The original MPM was then further extended by Bardenhagen et al.. to include frictional contact , which enabled the simulation of granular flow , and by Nairn to include explicit cracks and crack propagation (known as CRAMP).
Recently, an MPM implementation based on a micro-polar Cosserat continuum has been used to simulate high-shear granular flow, such as silo discharge. MPM's uses were further extended into Geotechnical engineering
Geotechnical engineering
Geotechnical engineering is the branch of civil engineering concerned with the engineering behavior of earth materials. Geotechnical engineering is important in civil engineering, but is also used by military, mining, petroleum, or any other engineering concerned with construction on or in the ground...
with the recent development of a quasi-static, implicit MPM solver which provides numerically stable analyses of large-deformation problems in Soil mechanics
Soil mechanics
Soil mechanics is a branch of engineering mechanics that describes the behavior of soils. It differs from fluid mechanics and solid mechanics in the sense that soils consist of a heterogeneous mixture of fluids and particles but soil may also contain organic solids, liquids, and gasses and other...
.
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
Oregon State University
Oregon State University is a coeducational, public research university located in Corvallis, Oregon, United States. The university offers undergraduate, graduate and doctoral degrees and a multitude of research opportunities. There are more than 200 academic degree programs offered through the...
, 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 fluid-solid interactions, including sea ice dynamics , penetration of biological soft tissues Ionescu, I., Guilkey, J., Berzins, M., Kirby, R., and Weiss, J. "Computational simulation of penetrating trauma in biological soft tissues using MPM.", fragmentation of gas-filled canisters , dispersion of atmospheric pollutants , multiscale simulations coupling molecular dynamics with MPM , and fluid-membrane interactions . In addition, the PIC-based FLIP code has been applied in magnetohydrodynamics and plasma processing tools, and simulations in astrophysics and free-surface flow .Solid Mechanics
MPM has also been used extensively in solid mechanics, to simulate impact, penetration, collision and rebound, as well as crack propagation . 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 (this work received the George Marra Award for paper of the year from the Society of Wood Science and Technologyhttp://www.swst.org/marrarecip.html)MPM in the context of numerical methods
One subset of numerical methods are Meshfree methodsMeshfree methods
Meshfree methods are a particular class of numerical simulation algorithms for the simulation of physical phenomena. Traditional simulation algorithms relied on a grid or a mesh, meshfree methods in contrast use the geometry of the simulated object directly for calculations. Meshfree methods exist...
, 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
Lagrangian and Eulerian coordinates
In fluid dynamics and finite-deformation plasticity the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an individual parcel through time gives the...
methods employ a framework in which a space is discretised into initial subvolumes, whose flowpaths are then charted over time. Purely Eulerian
Lagrangian and Eulerian coordinates
In fluid dynamics and finite-deformation plasticity the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an individual parcel through time gives the...
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 equationPartial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
(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
Explicit and implicit methods
Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes....
or implicit
Explicit and implicit methods
Explicit and implicit methods are approaches used in numerical analysis for obtaining numerical solutions of time-dependent ordinary and partial differential equations, as is required in computer simulations of physical processes....
. 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.
MPM Compared to FEM
Unlike FEMFinite element method
The finite element method is a numerical technique for finding approximate solutions of partial differential equations as well as integral equations...
, 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.
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.