Computation of multi-material interactions using point method

Calculations of fluid flows are often based on Eulerian description, while calculations of solid deformations are often based on Lagrangian description of the material. When the Eulerian descriptions are used to problems of solid deformations, the state variables, such as stress and damage, need to be advected, causing significant numerical diffusion error. When Lagrangian methods are used to problems involving large solid deformat ions or fluid flows, mesh distortion and entanglement are significant sources of error, and often lead to failure of the calculation. There are significant difficulties for either method when applied to problems involving large deformation of solids. To address these difficulties, particle-in-cell (PIC) method is introduced in the 1960s. In the method Eulerian meshes stay fixed and the Lagrangian particles move through the Eulerian meshes during the material deformation. Since its introduction, many improvements to the method have been made. The work of Sulsky et al. (1995, Comput. Phys. Commun. v. 87, pp. 236) provides a mathematical foundation for an improved version, material point method (MPM) of the PIC method. The unique advantages of the MPM method have led to many attempts of applying the method to problems involving interaction of different materials, such as fluid-structure interactions. These problems are multiphase flow or multimaterial deformation problems. In these problems pressures, material densities and volume fractions are determined by satisfying the continuity constraint. However, due to the difference in the approximations between the material point method and the Eulerian method, erroneous results for pressure will be obtained if the same scheme used in Eulerian methods for multiphase flows is used to calculate the pressure. To resolve this issue, we introduce a numerical scheme that satisfies the continuity requirement to higher order of accuracy in the sense of weak solutions for the continuity equations. Numerical examples are given to demonstrate the new scheme.