The material point method (MPM) is a numerical technique used to simulate the behavior of solids, liquids, gases, and any other continuum material. Especially, it is a robust spatial discretization method for simulating multi-phase (solid-fluid-gas) interactions. In the MPM, a continuum body is described by a number of small Lagrangian elements referred to as 'material points'. These material points are surrounded by a background mesh/grid that is used to calculate terms such as the deformation gradient. Unlike other mesh-based methods like the finite element method, finite volume method or finite difference method, the MPM is not a mesh based method and is instead categorized as a meshless/meshfree or continuum-based particle method, examples of which are smoothed particle hydrodynamics and peridynamics. Despite the presence of a background mesh, the MPM does not encounter the drawbacks of mesh-based methods (high deformation tangling, advection errors etc.) which makes it a promising and powerful tool in computational mechanics.
The MPM was originally proposed, as an extension of a similar method known as FLIP (a further extension of a method called PIC) to computational solid dynamics, in the early 1990 by Professors Deborah L. Sulsky, Zhen Chen and Howard L. Schreyer at University of New Mexico. After this initial development, the MPM has been further developed both in the national labs as well as the University of New Mexico, Oregon State University, University of Utah and more across the US and the world. Recently the number of institutions researching the MPM has been growing with added popularity and awareness coming from various sources such as the MPM's use in the Disney film Frozen .
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 Fluid-Implicit Particle (FLIP) program, which was created by Brackbill in 1986 [2] and has been constantly in development ever since. Until the 1990s, 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 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]
Annual workshops on the use of MPM are held at various locations in the United States. The Fifth MPM Workshop was 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 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]
As a result of a joint effort between UCLA's mathematics department and Walt Disney Animation Studios, MPM was successfully used to simulate snow in the 2013 computer-animated film Frozen . [17] [18] [19]
Solid mechanics
MPM has also been used extensively in solid mechanics, to simulate impact, penetration, collision and rebound, as well as crack propagation. [20] [21] MPM has also become a widely used method within the field of soil mechanics: it has been used to simulate granular flow, quickness test of sensitive clays, [22] landslides, [23] [24] [25] silo discharge, pile driving, fall-cone test, [26] [27] [28] [29] bucket filling, and material failure; and to model soil stress distribution, [30] 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. [31] The work also received the George Marra Award for paper of the year from the Society of Wood Science and Technology. [32]
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 difficulties." 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 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. Several versions of MPM include Generalized Interpolation Material Point Method [33] ;Convected Particle Domain Interpolation Method; [34] Convected Particle Least Squares Interpolation Method. [35]
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.
In plasma physics, the particle-in-cell (PIC) 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 simultaneously on Eulerian (stationary) mesh points.
Smoothed-particle hydrodynamics (SPH) is a computational method used for simulating the mechanics of continuum media, such as solid mechanics and fluid flows. It was developed by Gingold and Monaghan and Lucy in 1977, initially for astrophysical problems. It has been used in many fields of research, including astrophysics, ballistics, volcanology, and oceanography. It is a meshfree Lagrangian method, and the resolution of the method can easily be adjusted with respect to variables such as density.
Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells. Often these cells form a simplicial complex. Usually the cells partition the geometric input domain. Mesh cells are used as discrete local approximations of the larger domain. Meshes are created by computer algorithms, often with human guidance through a GUI, depending on the complexity of the domain and the type of mesh desired. A typical goal is to create a mesh that accurately captures the input domain geometry, with high-quality (well-shaped) cells, and without so many cells as to make subsequent calculations intractable. The mesh should also be fine in areas that are important for the subsequent calculations.
A crash simulation is a virtual recreation of a destructive crash test of a car or a highway guard rail system using a computer simulation in order to examine the level of safety of the car and its occupants. Crash simulations are used by automakers during computer-aided engineering (CAE) analysis for crashworthiness in the computer-aided design (CAD) process of modelling new cars. During a crash simulation, the kinetic energy, or energy of motion, that a vehicle has before the impact is transformed into deformation energy, mostly by plastic deformation (plasticity) of the car body material, at the end of the impact.
In physics and astronomy, an N-body simulation is a simulation of a dynamical system of particles, usually under the influence of physical forces, such as gravity. N-body simulations are widely used tools in astrophysics, from investigating the dynamics of few-body systems like the Earth-Moon-Sun system to understanding the evolution of the large-scale structure of the universe. In physical cosmology, N-body simulations are used to study processes of non-linear structure formation such as galaxy filaments and galaxy halos from the influence of dark matter. Direct N-body simulations are used to study the dynamical evolution of star clusters.
The lattice Boltzmann methods (LBM), originated from the lattice gas automata (LGA) method (Hardy-Pomeau-Pazzis and Frisch-Hasslacher-Pomeau models), is a class of computational fluid dynamics (CFD) methods for fluid simulation. Instead of solving the Navier–Stokes equations directly, a fluid density on a lattice is simulated with streaming and collision (relaxation) processes. The method is versatile as the model fluid can straightforwardly be made to mimic common fluid behaviour like vapour/liquid coexistence, and so fluid systems such as liquid droplets can be simulated. Also, fluids in complex environments such as porous media can be straightforwardly simulated, whereas with complex boundaries other CFD methods can be hard to work with.
Soft-body dynamics is a field of computer graphics that focuses on visually realistic physical simulations of the motion and properties of deformable objects. The applications are mostly in video games and films. Unlike in simulation of rigid bodies, the shape of soft bodies can change, meaning that the relative distance of two points on the object is not fixed. While the relative distances of points are not fixed, the body is expected to retain its shape to some degree. The scope of soft body dynamics is quite broad, including simulation of soft organic materials such as muscle, fat, hair and vegetation, as well as other deformable materials such as clothing and fabric. Generally, these methods only provide visually plausible emulations rather than accurate scientific/engineering simulations, though there is some crossover with scientific methods, particularly in the case of finite element simulations. Several physics engines currently provide software for soft-body simulation.
In the field of numerical analysis, meshfree methods are those that do not require connection between nodes of the simulation domain, i.e. a mesh, but are rather based on interaction of each node with all its neighbors. As a consequence, original extensive properties such as mass or kinetic energy are no longer assigned to mesh elements but rather to the single nodes. Meshfree methods enable the simulation of some otherwise difficult types of problems, at the cost of extra computing time and programming effort. The absence of a mesh allows Lagrangian simulations, in which the nodes can move according to the velocity field.
The fast multipole method (FMM) is a numerical technique that was developed to speed up the calculation of long-ranged forces in the n-body problem. It does this by expanding the system Green's function using a multipole expansion, which allows one to group sources that lie close together and treat them as if they are a single source.
In fracture mechanics, the energy release rate, , is the rate at which energy is transformed as a material undergoes fracture. Mathematically, the energy release rate is expressed as the decrease in total potential energy per increase in fracture surface area, and is thus expressed in terms of energy per unit area. Various energy balances can be constructed relating the energy released during fracture to the energy of the resulting new surface, as well as other dissipative processes such as plasticity and heat generation. The energy release rate is central to the field of fracture mechanics when solving problems and estimating material properties related to fracture and fatigue.
The finite element method (FEM) is a extremely popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
In numerical analysis, transfinite interpolation is a means to construct functions over a planar domain in such a way that they match a given function on the boundary. This method is applied in geometric modelling and in the field of finite element method.
In fluid mechanics, meteorology and oceanography, a trajectory traces the motion of a single point, often called a parcel, in the flow.
Biology Monte Carlo methods (BioMOCA) have been developed at the University of Illinois at Urbana-Champaign to simulate ion transport in an electrolyte environment through ion channels or nano-pores embedded in membranes. It is a 3-D particle-based Monte Carlo simulator for analyzing and studying the ion transport problem in ion channel systems or similar nanopores in wet/biological environments. The system simulated consists of a protein forming an ion channel (or an artificial nanopores like a Carbon Nano Tube, CNT), with a membrane (i.e. lipid bilayer) that separates two ion baths on either side. BioMOCA is based on two methodologies, namely the Boltzmann transport Monte Carlo (BTMC) and particle-particle-particle-mesh (P3M). The first one uses Monte Carlo method to solve the Boltzmann equation, while the later splits the electrostatic forces into short-range and long-range components.
The multiphase particle-in-cell method (MP-PIC) is a numerical method for modeling particle-fluid and particle-particle interactions in a computational fluid dynamics (CFD) calculation. The MP-PIC method achieves greater stability than its particle-in-cell predecessor by simultaneously treating the solid particles as computational particles and as a continuum. In the MP-PIC approach, the particle properties are mapped from the Lagrangian coordinates to an Eulerian grid through the use of interpolation functions. After evaluation of the continuum derivative terms, the particle properties are mapped back to the individual particles. This method has proven to be stable in dense particle flows, computationally efficient, and physically accurate. This has allowed the MP-PIC method to be used as particle-flow solver for the simulation of industrial-scale chemical processes involving particle-fluid flows.
Smoothed finite element methods (S-FEM) are a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining meshfree methods with the finite element method. S-FEM are applicable to solid mechanics as well as fluid dynamics problems, although so far they have mainly been applied to the former.
The Kansa method is a computer method used to solve partial differential equations. Its main advantage is it is very easy to understand and program on a computer. It is much less complicated than the finite element method. Another advantage is it works well on multi variable problems. The finite element method is complicated when working with more than 3 space variables and time.
Equation-free modeling is a method for multiscale computation and computer-aided analysis. It is designed for a class of complicated systems in which one observes evolution at a macroscopic, coarse scale of interest, while accurate models are only given at a finely detailed, microscopic, level of description. The framework empowers one to perform macroscopic computational tasks using only appropriately initialized microscopic simulation on short time and small length scales. The methodology eliminates the derivation of explicit macroscopic evolution equations when these equations conceptually exist but are not available in closed form; hence the term equation-free.
In geology, numerical modeling is a widely applied technique to tackle complex geological problems by computational simulation of geological scenarios.
OceanParcels, “Probably A Really Computationally Efficient Lagrangian Simulator”, is a set of python classes and methods that is used to track particles like water, plankton and plastics. It uses the output of Ocean General Circulation Models (OGCM's). OceanParcels main goal is to process the increasingly large amounts of data that is governed by OGCM's. The flow dynamics are simulated using Lagrangian modelling and the geophysical fluid dynamics are simulated with Eulerian modelling or provided through experimental data. OceanParcels is dependent on two principles, namely the ability to read external data sets from different formats and customizable kernels to define particle dynamics.
