The Local Maximum-Entropy Material Point Method

Abstract

En este trabajo se propone el uso de una metodología B-free (o sin componentes) para la resolución de problemas de elastodinámica no lineal en grandes deformaciones mediante el LME-MPM. Esta metodología permite eliminar la brecha existente entre la formulación continua del problema y su versión discretizada. Asimismo, en este trabajo se introduce el uso de una formulación incremental del algoritmo Newmark-β para el LME-MPM. En aras de la exhaustividad, al final de este documento se analiza la evaluación del marco propuesto en el régimen de grandes deformaciones.

The Material Point Method

“…Multi-modal inference Inference of multi-modal densities require particles to separate and form local groups, which is similar to fracture or cracks modelling using MPM: there are multiple internal surfaces across which the displacement field is discontinuous, and multiple velocity fields at nodes whose supports are cut by the cracks. MPM is efficient for modelling problems with moving discontinuities such as fracture evolution [58], and well suited for large motion and shape change problems.…”

“…Advantages of MPM-ParVI (1) Advantages of MPM. MPM provides a particle simulation framework which enjoys the advantages of both Lagrangian and Eulerian descriptions [58]: no element distortion happens and no remeshing is required during simulation. The Lagrangian formulation allows particle motion and material deformation histories to be tracked throughout the simulation process, while the Eulerian grid enables automatic treatment of self-collision and fracture [33].…”

“…Applications Similar to other gradient-based sampling methods such as Hamiltonian Monte Carlo (HMC), Langevin Monte Carlo (LMC), SVGD, and SPH-ParVI 30 Grid-crossing error is one of the well-known shortcomings of MPM, which is induced by (insufficient) smoothness of the interpolation shape function [58]. If a shape function with C 0 degree of smoothness (first-order derivative is discontinuous between cells) is employed, spurious variations (e.g.…”

“…If a shape function with C 0 degree of smoothness (first-order derivative is discontinuous between cells) is employed, spurious variations (e.g. jumps in the stress) can result when particles cross cell boundaries [58]. Remedies have been proposed, e.g.…”

“…While PIC and MPM use both Lagrangian particles and a background Eulerian grid 6 simultaneously, particles in PIC carry only position and mass, MPM allows particles to carry full physical state [13]. Some MPM variants include the generalized interpolation material point method (GIMP [2,51]), B-splines MPM (BSMPM [63]), the convected particle domain interpolation (CPDI [62]), the dual domain MPM (DDMPM [85]), weighted least square MPM [77], moving least square MPM [15], improved MPMs (iMPM [69,79,75]), the total Lagrangian MPM (TLMPM [12]) and discontinuous Galerkin MPM (DGMPM [60,61], isogeometric MPM (IGA-MPM [55]), local maximum-entropy MPM (LME-MPM [58]), affine matrix-based MPM (AM-MPM, [21]), etc. To model fluids and gases, weakly compressible [82] and incompressible [65] MPMs have also been developed.…”