Kirby-37(1) - Copyright c 2008 Tech Science Press CMES,...

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Copyright c ± 2008 Tech Science Press CMES, vol.31, no.2, pp.107-127, 2008 Examination and Analysis of Implementation Choices within the Material Point Method (MPM) M. Steffen 1 ,P .C .Wallstedt 2 ,J .E .Guilkey 2 , 3 ,R .M .Kirby 1 and M. Berzins 1 Abstract: The Material Point Method (MPM) has shown itself to be a powerful tool in the sim- ulation of large deformation problems, especially those involving complex geometries and contact where typical Fnite element type methods fre- quently fail. While these large complex problems lead to some impressive simulations and solu- tions, there has been a lack of basic analysis char- acterizing the errors present in the method, even on the simplest of problems. The large number of choices one has when implementing the method, such as the choice of basis functions and boundary treatments, further complicates this error analysis. In this paper we explore some of the many choices one can make when implementing an MPM al- gorithm and the numerical ramiFcations of these choices. SpeciFcally, we analyze and demonstrate how the smoothing length within the General- ized Interpolation Material Point Method (GIMP) can affect the error and stability properties of the method. We also demonstrate how various choices of basis functions and boundary treat- ments affect the spatial convergence properties of MPM. Keyword: Material Point Method, GIMP, Meshfree Methods, Meshless Methods, Particle Methods, Smoothed Particle Hydrodynamics, Quadrature 1 Introduction The Material Point Method (MPM) [Sulsky, Chen, and Schreyer (1994); Sulsky, Zhou, and 1 School of Computing, University of Utah, Salt Lake City, UT, USA. {msteffen,kirby,[email protected] 2 Department of Mechanical Engineering, Uni- versity of Utah, Salt Lake City, UT, USA. {philip.wallstedt,[email protected] 3 Corresponding Author Schreyer (1995)] is a mixed Lagrangian and Eu- lerian method utilizing Lagrangian particles to carry history-dependent material properties and an Eulerian background mesh to calculate deriva- tives and solve the equations of motion. MPM and its variants have been shown to be extremely successful and robust in simulating a large number of complicated engineering prob- lems (see for example [Bardenhagen, Brydon, and Guilkey (2005); Nairn (2006); Sulsky, Schreyer, Peterson, Kwok, and Coon (2007)]). The most well known of these variants is the General- ized Interpolation Material Point (GIMP) Method [Bardenhagen and Kober (2004)], of which tradi- tional MPM is a special case. GIMP provides im- proved accuracy, stability and robustness to sim- ulations through the introduction of particle char- acteristic functions, which in most cases have the effect of smoothing the grid basis functions. The ability to handle solid mechanics problems in- volving large deformations and/or fragmentation of structures, which are sometimes problematic for Fnite element methods, has led, in part, to the method’s success.
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This note was uploaded on 11/16/2011 for the course MSE 5960 taught by Professor Douglas during the Fall '04 term at University of Florida.

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Kirby-37(1) - Copyright c 2008 Tech Science Press CMES,...

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