Material Point Method


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Solving mechanical problems including large deformation, fracture, and, impact can result in numerical issues. To solve governing equations, a Lagrangian description or an Eulerian description can be used. In Lagrangian methods, the computational mesh deforms with the material. So, in the case of large deformation, fracture, and impact, Lagrangian methods cause mesh distortion leading to mesh entanglement. On the other hand, in Eulerian methods, the governing equations are solved using a fixed grid. Thus, the Eulerian description can handle highly deformed motion. However, tracking material points is difficult to implement, so the Eulerian description has trouble with history dependent constitutive laws. To overcome disadvantages of purely Lagrangian methods and Eulerian methods, a Material Point Method (MPM) is introduced. MPM uses both the Lagrangian description based on a set of material points and the Eulerian description based on the fixed computational grid. In Lagrangian formulation, material points move with the deformation and are used to track the material variables such as position, mass, momentum, stress, and strain. Hence, there is no need to worry about mesh entanglement. In Eulerian formulation, a fixed computational grid is employed to compute a spatial gradient. Furthermore, a convective term in the material time derivative doesn’t have to be considered because a fixed grid performs a role of an Lagrangian frame at every time step. Numerical examples are given to illustrate advantages of MPM.