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.
The algorithm An MPM simulation consists of the following stages: (Prior to the time integration phase)
Initialization of grid and material points. A geometry is discretized into a collection of material points, each with its own material properties and initial conditions (velocity, stress, temperature, etc.) The grid, being only used to provide a place for gradient calculations is normally made to cover an area large enough to fill the expected extent of computational domain needed for the simulation. (During the time integration phase - explicit formulation)
Material point quantities are extrapolated to grid nodes. Material point mass ( m m p {\textstyle m_{mp}} ), momenta ( P m p → {\textstyle {\vec {P_{mp}}}} ), stresses ( σ ¯ ¯ m p {\displaystyle {\boldsymbol {\bar {\bar {\sigma }}}}_{mp}} ), and external forces ( b → {\displaystyle {\vec {b}}} ) are extrapolated to the nodes at the corners of the cells within which the material points reside. This is most commonly done using standard linear shape functions ( N n d − m p {\textstyle N_{nd-mp}} ), the same used in FEM. The grid use the material point values to create the masses ( M n o d e {\textstyle M_{\mathrm {node} }} ), velocities ( V → n o d e {\textstyle {\vec {V}}_{\mathrm {node} }} ), internal and external force vectors ( F → n o d e i n t e r n a l {\textstyle {\vec {F}}_{\mathrm {node} }^{\mathsf {internal}}} , F → n o d e e x t e r n a l {\textstyle {\vec {F}}_{\mathrm {node} }^{\mathsf {external}}} ) for the nodes: M n o d e = ∑ m p m m p N m p − n d {\displaystyle M_{\mathrm {node} }=\sum _{mp}m_{mp}~~N_{mp-nd}}
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