Abstract
The material point method (MPM) has been very successful in providing solutions to many challenging problems involving large deformations. Nevertheless there are some important issues that remain to be resolved with regard to its analysis. One key challenge applies to both MPM and particle-in-cell (PIC) methods and arises from the difference between the number of particles and the number of the nodal grid points to which the particles are mapped. This difference between the number of particles and the number of grid points gives rise to a non-trivial null space of the linear operator that maps particle values onto nodal grid point values. In other words, there are non-zero particle values that when mapped to the grid point nodes result in a zero value there. Moreover, when the nodal values at the grid points are mapped back to particles, part of those particle values may be in that same null space. Given positive mapping weights from particles to nodes such null space values are oscillatory in nature. While this problem has been observed almost since the beginning of PIC methods there are still elements of it that are problematical today as well as methods that transcend it. The null space may be viewed as being connected to the ringing instability identified by Brackbill for PIC methods. It will be shown that it is possible to remove these null space values from the solution using a null space filter. This filter improves the accuracy of the MPM methods using an approach that is based upon a local singular value decomposition (SVD) calculation. This local SVD approach is compared against the global SVD approach previously considered by the authors and to a recent MPM method by Zhang and colleagues.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bardenhagen S (2002) Energy conservation error in the material point method for solid mechanics. J Comput Phys 180:383–403
Bardenhagen S, Kober E (2004) The generalized interpolation material point method. Comput Model Eng Sci 5:477–495
Belytschko T, Xiao S (2002) Stability analysis of particle methods with corrected derivatives. Comput Math Appl 43:329–350
Belytschko T, Krongauz Y, Dolbow J, Gerlach C (1998) On the completeness of meshfree particle methods. Int J Numer Methods Eng 43:785–819
Belytschko T, Guo Y, Kam Liu W, Ping Xiao S (2000) A unified stability analysis of meshless particle methods. Int J Numer Methods Eng 48(9):1359–1400
Brackbill J (1988) The ringing instability in particle-in-cell calculations of low-speed flow. J Comput Phys 75:469–492
Brackbill JU (2015) On energy and momentum conservation in particle-in-cell simulation. arXiv:1510.08741
Brackbill J, Lapenta G (1994) A method to supress the finite-grid instability in plasma simulations. J Comput Phys 114:77–84
Brackbill J, Kothe D, Ruppel H (1988) Flip: a low-dissipation, particle-in-cell method for fluid flow. Comput Phys Commun 48:25–38
Chen L, Langdon AB, Birdsall CK (1974) Reduction of the grid-effects in simulation plasma’s. J Comput Phys 14:200–222
Dilts GA (1999) Moving-least-squares-particle hydrodynamics-I. consistency and stability. Int J Numer Methods Eng 44:1115–1155
Golub GH, Loan CFV (1996) Matrix computations, 3rd edn. The John Hopkins University Press, Baltimore
Gritton CE (2014) Ringing Instabilities in particle methods, M.S.Thesis in computational engineering and science, school of computing, University of Utah, Salt Lake City
Gritton CE, Berzins M, Kirby RM (2015) Improving accuracy in particle methods using null spaces and filters. In: Onate E, Bischoff M, Owen DRJ, Wriggers P, Zohdi T (eds) Proceedings of the IV international conference on particle-based methods—fundamentals and applications (CIMNE), Barcelona, Spain, pp 202–213,2015. ISBN 978-84-944244-7-2. http://congress.cimne.com/particles2015/frontal/doc/E-book_PARTICLES_2015,
Harlow FH (1964) The particle-in-cell method for fluid dynamics. Methods Computat Phys 3(3):319–343
Johnson GR, Beissel SR (1996) Normalized smoothing functions for sph impact computations. Int J Numer Methods Eng 39:2725–2741
Krongauz Y, Belytschko T (1997) Consistent pseudo-derivatives in meshless methods. Comput Methods Appl Mech Eng 146:371–386
Langdon A (1970) Effects of the spatial grid in simulation plasmas. J Comput Phys 6:247–267
Lapenta G (2016) Exactly energy conserving implicit moment particle in cell formulation. J Comput Phys. arXiv:1602.06326
Liu WK, Jun S, Li S, Adee J, Belytschko T (1995) Reproducing kernel particle methods for structural dynamics. Int J Numer Methods Eng 38:1655–1679
Love E, Sulsky DL (2006) An energy-consistent material-point method for dynamic finite deformation plasticity. Int J Numer Methods Eng 65:1608–1638
Love E, Sulsky DL (2006) An unconditionally stable, energymomentum consistent implementation of the material-point method. Comput Methods Appl Mech Eng 195:3903–3925
Mast CM, Mackenzie-Helnwein P, Arduino P, Miller GR, Shin W (2012) Mitigating kinematic locking in the material point method. J Comput Phys 231(16):5351–5373
Monaghan JJ (2000) SPH without a tensile instability. J Comput Phys 159(2):290–311
Ortiz M (1986) A note on energy conservation and stability of nonlinear time-stepping algorithms. Comput Struct 24(1):167–168
Randles P, Libersky L (1996) Smoothed particle hydrodynamics: Some recent improvements and applications. Comput Methods Appl Mech Eng 139:375–408
Sadeghirad A, Brannon RM, Guilkey JE (2013) Second-order convected particle domain interpolation (CPDI2) with enrichment for weak discontinuities at material interfaces. Int J Numer Methods Eng 95(11):928–952
Steffen M, Wallstedt PC, Guilkey JE, Kirby RM, Berzins M (2008) Examination and analysis of implementation choices within the material point method (MPM). Comput Model Eng Sci 31(2):107–127
Steffen M, Kirby RM, Berzins M (2008) Analysis and reduction of quadrature errors in the material point method (mpm). Int J Numer Methods Eng 76:922–948
Stomakhin A, Schroeder C, Chai L, Teran J, Selle A (2013) A material point method for snow simulation. ACM Trans Graph 32(4):102:1–102:10
Sulsky D, Chen Z, Schreyer H (1994) A particle method for history-dependent materials. Comput Methods Appl Mech Eng 118:179–196
Trefethen LN, Bau I D (1997) Numerical linear algebra. SIAM, Philadelphia
Wallstedt PC, Guilkey JE (2008) An evaluation of explicit time integration schemes for use with the generalized interpolation material point method. J Comput Phys 227:9628–9642. doi:10.1016/j.jcp.2008.07.019
Zhang DZ, Ma X, Giguere PT (2011) Material point method enhanced by modified gradient of shape function. J Comput Phys 230(16):6379–6398
Acknowledgments
We would like to thank the referees for their thoughtful and helpful comments that have helped to greatly improve this paper. This research was primarily sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-12-2-0023. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The authors would like to thank ARL for their support and their colleague Mike Kirby for suggesting the use of the global SVD approach.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gritton, C., Berzins, M. Improving accuracy in the MPM method using a null space filter. Comp. Part. Mech. 4, 131–142 (2017). https://doi.org/10.1007/s40571-016-0134-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40571-016-0134-3