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A Smoothed Particle Galerkin Formulation for Extreme Material Flow Analysis in Bulk Forming Applications

C. T. Wu

Livermore Software Technology Corporation, 7374 Las Positas Road, Livermore, CA 94551, USA

E-mail Address: ctwu@lstc.com

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W. Hu

Livermore Software Technology Corporation, 7374 Las Positas Road, Livermore, CA 94551, USA

E-mail Address: whu@lstc.com

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M. Koishi

Koishi Laboratory, The Yokohama Rubber Co. Ltd., 2-1 Oiwake Hiratsuka, Japan

E-mail Address: koishi@hpt.yrc.co.jp

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https://doi.org/10.1142/S0219876216500195Cited by:2 (Source: Crossref)

Abstract

This paper presents a new particle formulation for extreme material flow analyses in the bulk forming applications. The new formulation is first established by an introduction of a smoothed displacement field to the standard Galerkin formulation to eliminate zero-energy modes in conventional particle methods. The discretized system of linear equations is consistently derived and integrated using a direct nodal integration scheme. The linear formulation is next extended to the large deformation quasi-static analysis of inelastic materials. As quasi-static Lagrangian simulation proceeds in the severe deformation range, the analysis method is switched to explicit dynamics formulation and an adaptive Lagrangian kernel approach is preformed to reset the reference configuration and maintain the injective deformation mapping at the particles. Both nonconvex and convex meshfree approximations are investigated in this study. Several numerical benchmarks are provided to demonstrate the effectiveness and accuracy of the proposed method.

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References

Arroyo, M. and Ortiz, M. [2006] “ Local maximum-entropy approximation schemes: A seamless bridge between finite elements and meshfree methods,” Int. J. Numer. Methods Eng. 65, 2167–2202. Crossref, Web of Science, Google Scholar
Belytschko, T., Lu, Y. Y. and Gu, L. [1994] “ Element-free Galerkin methods,” Int. J. Numer. Methods Eng. 37, 229–256. Crossref, Web of Science, Google Scholar
Belytschko, T., Guo, Y., Liu, W. K. and Xiao, S. P. [2000] “ A unified stability analysis of meshless particle methods,” Int. J. Numer. Methods Eng. 48, 1359–1400. Crossref, Web of Science, Google Scholar
Beissel, S. and Belytschko, T. [1996] “ Nodal integration of the element-free Galerkin method,” Comput. Methods Appl. Mech. Eng. 139, 49–74. Crossref, Web of Science, Google Scholar
Bonet, J. and Kilasegaram, S. [2000] “ Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations,” Int. J. Numer. Methods Eng. 47, 1189–1214. Crossref, Web of Science, Google Scholar
Brenner, S. C. and Scott, L. R. [2008] The Mathematical Theory of Finite Element Methods (Springer-Verlag, New York). Crossref, Google Scholar
Chen, J. S., Pan, C., Wu, C. T. and Liu, W. K. [1996] “ Reproducing kernel particle methods for large deformation analysis of non-linear structures,” Comput. Methods Appl. Mech. Eng. 139, 195–227. Crossref, Web of Science, Google Scholar
Chen, J. S., Wu, C. T., Yoon, S. and You, Y. [2001] “ A stabilized conforming nodal integration for Galerkin meshfree methods,” Int. J. Numer. Methods Eng. 50, 435–466. Crossref, Web of Science, Google Scholar
Chen, J. S., Yoon, S. and Wu, C. T. [2002] “ Nonlinear version of stabilized conforming nodal integration for Galerkin meshfree methods,” Int. J. Numer. Methods Eng. 53, 2587–2615. Crossref, Web of Science, Google Scholar
Chen, J. S., Hillman, M. and Ruter, M. [2013] “ An arbitrary order variationally consistent integration method for Galerkin meshfree methods,” Int. J. Numer. Methods Eng. 95, 387–418. Crossref, Web of Science, Google Scholar
Dyka, C. T. and Ingel, R. P. [1995] “ An approach for tension instability in smoothed particle hydrodynamics,” Comput. Struct. 57, 573–580. Crossref, Web of Science, Google Scholar
Guan, P. C., Chi, S. W., Chen, J. S., Slawson, T. R. and Roth, M. J. [2011] “ Semi-Lagrangian reproducing kernel particle method for fragment-impact problems,” Int. J. Impact Eng. 38, 1033–1047. Crossref, Web of Science, Google Scholar
Hallquist, J. O. [2013] LS-DYNA User’s Manual (Livermore Software Technology Corporation, Livermore, CA). Google Scholar
Krongazu, Y. and Belytschko, T. [1997] “ Consistent pseudo derivatives in meshless methods,” Comput. Methods Appl. Mech. Eng. 146, 371–386. Crossref, Web of Science, Google Scholar
Li, S. and Liu, W. K. [2004] Meshfree Particle Method (Springer, Berlin). Google Scholar
Libersky, L. D., Petscheck, A. G., Carney, T. C., Hipp, J. R. and Allahadai, F. A. [1993] “ High strength Lagrangian hydrodynamics,” J. Comput. Phys. 109, 67–75. Crossref, Web of Science, Google Scholar
Liu, G. R. and Liu, M. B. [2003] Smoothed Particle Hydrodynamics (World Scientific Publishing, Singapore). Link, Google Scholar
Liu, G. R., Dai, K. Y. and Nguyen-Thoi, T. [2007] “ A smoothed finite element method for mechanics problems,” Comput. Mech. 39, 859–877. Crossref, Web of Science, Google Scholar
Liu, G. R. [2008a] “ A generalized gradient smoothing technique and the smoothed bilinear form for Galerkin formulation of a wide class of computational methods,” Int. J. Comput. Methods 5, 199–236. Link, Web of Science, Google Scholar
Liu, G. R. and Zhang, G. Y. [2008b] “ Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LS-PIM),” Int. J. Numer. Methods Eng. 74, 1128–1161. Crossref, Web of Science, Google Scholar
Liu, G. R., Nguyen-Thoi, T., Nguyen-Xuan, H. and Lam, K. Y. [2009] “ A node-based smoothed finite element method (ns-fem) for upper bound solutions to solid mechanics problems,” Comput. Struct. 87, 14–26. Crossref, Web of Science, Google Scholar
Liu, G. R. [2010] Meshfree Methods: Moving Beyond the Finite Element Method (CRC Press, Florida). Google Scholar
Liu, W. K., Jun, S. and Zhang, Y. F. [1995] “ Reproducing kernel particle methods,” Int. J. Numer. Methods Fluids 20, 1081–1106. Crossref, Web of Science, Google Scholar
Monaghan, J. J. [1982] “ Why particle methods work,” SIAM J. Sci. Stat. Comput. 3, 422–433. Crossref, Web of Science, Google Scholar
Monaghan, J. J. [2000] “ SPH without a tensile instability,” J. Comput. Phys. 159, 290–311. Crossref, Web of Science, Google Scholar
Nguyen-Thoi, T., Liu, G. R., Lam, K. Y. and Zhang, G. Y. [2008] “ A face-based smoothed finite element method (fs-fem) for 3D linear and geometrically non-linear solid mechanics problems using 4-noded tetrahedral elements,” Int. J. Numer. Methods Eng. 78, 324–353. Crossref, Web of Science, Google Scholar
Park, C. K., Wu, C. T. and Kan, C. D. [2011] “ On the analysis of dispersion property and stable time step in meshfree method using generalized meshfree approximation,” Finite Elem. Anal. Des. 47, 683–697. Crossref, Web of Science, Google Scholar
Parshikov, A. N. and Medin, S. A. [2002] “ Smoothed particle hydrodynamics using interparticle contact algorihm,” J. Comput. Phys. 180, 358–382. Crossref, Web of Science, Google Scholar
Peco, C., Rosolen, A. and Arroyo, M. [2013] “ An adaptive meshfree method for phase-field models of biomembranes. Part II: A Lagrangian approach for membranes in viscous fluids,” J. Comput. Phys. 249, 320–336. Crossref, Web of Science, Google Scholar
Puso, M. A., Chen, J. S., Zywicz, E. and Elmer, W. [2008] “ Meshfree and finite element nodal integration methods,” Int. J. Numer. Methods Eng. 74, 416–446. Crossref, Web of Science, Google Scholar
Rabczuk, T., Belytschko, T. and Xiao, S. P. [2004] “ Stable particle methods based on Lagrangian kernels,” Comput. Methods Appl. Mech. Eng. 193, 1035–1063. Crossref, Web of Science, Google Scholar
Randles, P. W. and Libersky, L. D. [2000] “ Recent improvements in SPH modeling of hypervelocity impact,” Int. J. Impact Eng. 48, 1445–1462. Google Scholar
Sukumar, N. [2004] “ Construction of polygonal interpolants: A maximum entropy approach,” Int. J. Numer. Methods Eng. 61, 2159–2181. Crossref, Web of Science, Google Scholar
Swegle, J. W., Hicks, D. L. and Attaway, S. W. [1995] “ Smoothed particle hydrodynamics stabiliy analysis,” J. Comput. Phys. 116, 123–134. Crossref, Web of Science, Google Scholar
Timoshenko, S. P. and Goodier, J. N. [1970] Theory of Elasticity (McGraw-Hill, New York). Google Scholar
Vidal, Y., Bonet, J. and Huerta, A. [2006] “ Stabilized updated Langrangian corrected SPH for explicit dynamic problems,” Int. J. Numer. Methods Eng. 69, 2687–2710. Crossref, Web of Science, Google Scholar
Vignjevic, R., Campbell, J. and Libersky, L. D. [2000] “ A treatment of zero-energy modes in the smoothed particle hydrodynamics method,” Comput. Methods Appl. Mech. Eng. 184, 67–85. Crossref, Web of Science, Google Scholar
Violeau, D. [2012] Fluid Mechanics and the SPH Method: Theory and Applications (Oxford University Press, Oxford). Crossref, Google Scholar
Vishal, M., Sijoy, C. D., Vinayak, M. and Shashank, C. [2012] “ Tensile instability and artificial stresses in impact problems in SPH,” J. Phys. Conf. Ser. 377, 012012, doi: 10.1088/1742-6596/377/1/012102. Crossref, Google Scholar
Wang, D. and Chen, J. S. [2004] “ Locking-free stabilized conforming nodal integration for meshfree Mindlin–Reissner plate formulation,” Comput. Methods Appl. Mech. Eng. 193, 1065–1083. Crossref, Web of Science, Google Scholar
Wang, H. P., Wu, C. T., Guo, Y. and Botkin, M. [2009] “ A coupled meshfree/finite element method for automotive crashworthiness simulations,” Int. J. Impact Eng. 36, 1210–1222. Crossref, Web of Science, Google Scholar
Wu, C. T., Park, C. K. and Chen, J. S. [2011a] “ A generalized approximation for the meshfree analysis of solids,” Int. J. Numer. Methods Eng. 85, 693–722. Crossref, Web of Science, Google Scholar
Wu, C. T. and Hu, W. [2011b] “ Meshfree-enriched simplex elements with strain smoothing for the finite element analysis of compressible and nearly incompressible solids,” Comput. Methods Appl. Mech. Eng. 200, 2991–3010. Crossref, Web of Science, Google Scholar
Wu, C. T., Hu, W. and Chen, J. S. [2012a] “ A meshfree-enriched finite element method for compressible and nearly incompressible elasticity,” Int. J. Numer. Methods Eng. 90, 882–914. Crossref, Web of Science, Google Scholar
Wu, C. T. and Koishi, M. [2012b] “ Three-dimensional meshfree-enriched finite element formulation for micromechanical hyperelastic modeling of particulate rubber composites,” Int. J. Numer. Methods Eng. 91, 1137–1157. Crossref, Web of Science, Google Scholar
Wu, C. T. and Hu, W. [2013a] “ Multi-scale finite element analysis of acoustic waves using global residual-free meshfree enrichments,” Interact. Multiscale Mech. 6, 83–105. Crossref, Google Scholar
Wu, C. T., Guo, Y. and Askari, E. [2013b] “ Numerical modeling of composite solids using an immersed meshfree Galerkin method,” Compos. B Eng. 45, 1397–1413. Crossref, Web of Science, Google Scholar
Wu, C. T. and Wang, H. P. [2013c] “ An enhanced cell-based smoothed finite element method for the analysis of Reissner–Mindlin plate bending problems involving distorted mesh,” Int. J. Numer. Methods Eng. 95, 288–312. Crossref, Web of Science, Google Scholar
Wu, C. T., Hu, W. and Liu, G. R. [2014] “ Bubble-enhanced smoothed finite element method: A variational multi-scale approach for volume-constrained problems in two-dimensional linear elasticity,” Int. J. Numer. Methods Eng. 100, 374–398. Crossref, Web of Science, Google Scholar