Fluid–membrane interaction based on the material point method

Abstract

The material point method (MPM) uses unconnected, Lagrangian, material points to discretize solids, #uids or membranes. All variables in the solution of the continuum equations are associated with these points; so, for example, they carry mass, velocity, stress and strain. A background Eulerian mesh is used to solve the momentum equation. Data mapped from the material points are used to initialize variables on the background mesh. In the case of multiple materials, the stress from each material contributes to forces at nearby mesh points, so the solution of the momentum equation includes all materials. The mesh solution then updates the material point values. This simple algorithm treats all materials in a uniform way, avoids complicated mesh construction and automatically applies a noslip contact algorithm at no additional cost. Several examples are used to demonstrate the method, including simulation of a pressurized membrane and the impact of a probe with a pre-in#ated airbag.

A weighted least squares particle‐in‐cell method for solid mechanics

“…GIMP retains the same generality as MPM, though at additional computational cost. For realistic problems GIMP approaches first-order accuracy [32] but does not enjoy the reproducibility of MLS-based methods.MPM and GIMP have been studied and used by many investigators; a subset of these contributions includes: analysis and improvement of time integration properties by Bardenhagen [33], Sulsky et al [34], Wallstedt and Guilkey [32], and Steffen et al [35]; membranes and fluidstructure interaction by York et al [36,37]; implicit time integration by Guilkey and Weiss [38] and Sulsky and Kaul [39]; conservation properties and plasticity by Love and Sulsky [40,41]; contact by Bardenhagen et al [42]; cracks and fracture by Nairn [43]; tracking of particle extents by Ma et al [44]; and enhanced velocity projection and verification via the method of manufactured solutions by Wallstedt and Guilkey [32,45].In this work a new method is proposed that fits within the GIMP framework: it is based on a cartesian grid and it does not search for nearest neighbors or require a finite element mesh. The new method is based on a Weighted Least Squares approximation of data surrounding each grid node and is referred to in the remainder of this paper as 'WLS'.…”

A new Particle-in-Cell method for modeling magnetized fluids

“…Application of FLIP methodology to solid mechanics produced its most successful child, a Material Point Method (MPM) [16][17] [18]. It is now widely applied in geophysics [19,20], engineering [21] and 3D graphics [22].…”

Second‐order convected particle domain interpolation (CPDI2) with enrichment for weak discontinuities at material interfaces

“…The material point method (MPM) is a hybrid arbitrary Lagrangian/Eulerian method suitable for modeling large deformations of history-dependent solids (see for example [1][2][3][4][5][6][7][8][9][10][11]). The MPM saves all discrete continuum field data (displacement, velocity, stress, temperature, etc.)…”

“…In particular, because an updated Lagrange formulation is used in the numerical algorithm, the material rate of p is zero during a time step. Taking p to be a convected function throughout the duration of a problem would be sufficient to ensure satisfaction of Equation (1). However, it would be also possible to change the p domains at the end of any time step while still having each one associated with a given particle.…”

“…Due to practical considerations described later in the text, they placed no such constraint on the characteristic functions in the deformed configuration. Unlike [15], we will use the phrase 'characteristic function' and the notation p to refer exclusively to admissible characteristic functions satisfying Equation (1). The phrase 'weighting function' and notation p will be used to denote functions that might violate partition of unity but are otherwise similar to characteristic functions in the sense that they have compact support on a domain p containing the particle.…”