Abstract
Challenging solid mechanics problems exist in areas such as geotechnical and biomedical engineering which require numerical methods that can cope with very large deformations, both stretches and torsion. One candidate for these problems is the Material Point Method (MPM), and to deal with stability issues the standard form of the MPM has been developed into new domain-based techniques which change how information is mapped between the computational mesh and the material points. The latest of these developments are the Convected Particle Domain Interpolation (CPDI) approaches. When these are demonstrated, they are typically tested on problems involving large stretch but little torsion and if these MPMs are to be useful for the challenging problems mentioned above, it is important that their capabilities and shortcomings are clear. Here we present a study of the behaviour of some of these MPMs for modelling problems involving large elasto-plastic deformation including distortion. This is carried out in a unified implicit quasi-static computational framework and finds that domain distortion with the CPDI2 approaches affects some solutions and there is a particular issue with one approach. The older CPDI1 approach and the standard MPM however produce physically realistic results. The primary aim of this paper is to raise awareness of the capabilities or otherwise of these domain-based MPMs.
Highlights
The Material Point Method (MPM) [1,2,3], originally proposed as an extension of a similar method known as the Fluid-Implicit Particle or FLIP method [4], itself an extension of the Particle-in-Cell method [5], is a numerical method combining advantages of Eulerian and Lagrangian approaches to solve solid mechanics problems
We show that the CPDI2 approaches are less accurate than the standard MPM, under certain deformation fields, due to particle domain distortion
The results from the standard MPM (sMPM) and Convected Particle Domain Interpolation (CPDI) approaches are the same as the Finite Element Method (FEM), as this simulation is not affected by mesh distortion
Summary
The Material Point Method (MPM) [1,2,3], originally proposed as an extension of a similar method known as the Fluid-Implicit Particle or FLIP method [4], itself an extension of the Particle-in-Cell method [5], is a numerical method combining advantages of Eulerian and Lagrangian approaches to solve solid mechanics problems. Each of the new developments of the sMPM have been demonstrated in the individual papers referenced above and have been verified on selected problems The purpose of this contribution is to present an investigation of the behaviours of some of the domain-based MPMs on a selection of specific problems that test their predictive abilities when modelling problems involving large stretch, shear and torsion, hopefully providing useful guidance to those wishing to employ the methods on real-world applications of the types mentioned above. In this paper we use a combination of tensor and matrix notation in order to ensure that both the continuum formulation and the steps to numerical implementation are as clear as possible
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