A Lagrangian Hamiltonian Computational Method for Hyper-Elastic Structural Dynamics

Authors: Hosein Falahaty, Hitoshi Gotoh, Abbas Khayyer

Abstract:

Performance of a Hamiltonian based particle method in simulation of nonlinear structural dynamics is subjected to investigation in terms of stability and accuracy. The governing equation of motion is derived based on Hamilton's principle of least action, while the deformation gradient is obtained according to Weighted Least Square method. The hyper-elasticity models of Saint Venant-Kirchhoff and a compressible version similar to Mooney- Rivlin are engaged for the calculation of second Piola-Kirchhoff stress tensor, respectively. Stability along with accuracy of numerical model is verified by reproducing critical stress fields in static and dynamic responses. As the results, although performance of Hamiltonian based model is evaluated as being acceptable in dealing with intense extensional stress fields, however kinds of instabilities reveal in the case of violent collision which can be most likely attributed to zero energy singular modes.

Keywords: Hamilton's principle of least action, particle based method, hyper-elasticity, analysis of stability.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1126930

References:

[1] L.B. Lucy, “A numerical approach to the testing of fission hypothesis”, Astronom. J., vol. 82, 1977, pp. 1013–1024.
[2] R.A. Gingold, J.J. Monaghan, “Smoothed particle hydrodynamics: theory and applications to non-spherical stars”, Mon. Not.R. Astr. Soc., vol. 181, 1977, pp. 375–389.
[3] S. Koshizuka and Y. Oka, “Moving particle semi-implicit method for fragmentation of incompressible fluid”, Nuclear Science and Engineering, vol. 123, 1996, pp. 421-434.
[4] T. Belytschko, Y.Y. Lu, L. Gu, “Element-free Galerkin methods”, Int. J. Numer. Methods Engrg., vol. 37, 1994, pp. 229–256.
[5] L.D. Libersky, A.G. Petschek, “Smooth particle hydrodynamics with strength of materials, Advances in the Free Lagrange Method”, Lecture Notes in Physics, vol. 395, 1990.
[6] T. Belytschko, Y. Guo, W.K. Liu, S.P. Xiao, “A unified stability analysis of meshless particle methods”, Int. J. Numer. Methods Engrg., vol. 48, 2000, pp. 1359–1400.
[7] S.P. Xiao, T. Belytschko, “Material stability analysis of particle methods”, Adv. Comput. Math., vol. 23, 2005, pp. 171–190.
[8] G.R. Johnson, S.R. Beissel, “Normalized smoothing functions for SPH impact computations”, Int. J. Numer. Methods Engrg., vol. 39, 1996, pp. 2725–2741.
[9] P.W. Randles, L.D. Libersky, “Recent improvements in SPH modeling of hypervelocity impact”, Int. J. Impact Engrg., vol. 20, 1997, pp. 525–532.
[10] Y. Krongauz, T. Belytschko, “Consistent pseudo derivatives in meshless methods”, Comput. Methods Appl. Mech. Engrg., vol. 146, 1997, pp. 371–386.
[11] T. Belytschko, Y. Krongauz, D. Organ, P. Krysl, “Meshless methods: An overview and recent developments”, Comput. Methods Appl. Mech. Engrg., vol. 139, 1996, pp. 3–47.
[12] G.A. Dilts, “Moving least squares particle hydrodynamics II: Conservation and boundaries”, Int. J. Numer. Methods Engrg., vol. 48, 2000, pp. 1503–1524.
[13] G.A. Dilts, “Moving-least-squares-particle hydrodynamics I: Consistency and stability”, Int. J. Numer. Methods Engrg., vol. 44, 1999, pp. 1115–1155.
[14] G.A. Dilts, “Some recent developments for moving-least-squares particle methods”, First M.I.T. Conference on Computational Fluid and Soil Mechanics, Preprint, Massachusetts Institute of Technology, Cambridge, MA 02139, USA, June 12–14, 2001.
[15] J.W. Swegle, S.W. Attaway, M.W. Heinstein, F.J. Mello, Hicks D.L., “An Analysis of Smoothed Particle Hydrodynamics”, Sandia Report SAND93-2513, 1994, SNL, Albuquerque, NM 87185.
[16] T. Belytschko, S.P. Xiao, “Stability analysis of particle methods with corrected derivatives”, Comput. Math. Appl., vol. 43, 2000, pp. 329–350.
[17] M.S. Song, S. Koshizuka, Y. Oka, “A particle method for dynamic simulation of elastic solids”, in Proc. 6th World Congress on Computational Mechanics (WCCM VI), Beijing, 5–10 September 2004.
[18] M. Kondo, Y. Suzuki, S. Koshizuka, “Application of symplectic scheme to three-dimensional elastic analysis using MPS method (in Japanese)”, Transactions of the Japan Society of Mechanical Engineers, vol. 72, 2006, pp. 65–71.
[19] Y. Suzuki, S. Koshizuka, “A Hamiltonian particle method for non-linear elastodynamics”, Int. J. Numer. Methods Engrg., 74, 2008, 1344–1373.
[20] J.E. Marsden, T.J.R. Hughes, “Mathematical Foundations of Elasticity”, Prentice Hall: Englewood Cliffs, NJ, 1983.
[21] M. Kondo, Y. Suzuki and S. Koshizuka, “Suppressing local particle oscillations in the Hamiltonian particle method for elasticity”, Int. J. Numer. Meth. Engng., vol. 81, 2010, pp. 1514–1528.
[22] A. Khayyer, H. Gotoh, “Enhancement of stability and accuracy of the moving particle semi-implicit method”, J. Comput. Phys., vol. 230, 2011, pp. pp. 3093–3118.
[23] J.W. Swegle, “Conservation of momentum and tensile instability in particle methods”, Sandia Report SAND 2000-1223, 2000.
[24] J. Bonet, T.S. Lok, “Variational and momentum preservation aspects of smooth particle hydrodynamic formulation”, Comput. Methods Appl. Mech. Engrg., vol. 180, 1999, pp. 97–115.
[25] S.C. Hwang, A. Khayyer, H. Gotoh and J.C. Park, “Development of a fully Lagrangian MPS-based coupled method for simulation of fluid-structure interaction problems”, Journal of Fluids and Structures, vol. 50, 2014, pp. 497-511.
[26] G. A. Holzapfel, “Nonlinear solid mechanics: a continuum approach for engineering”, Meccanica, vol. 37, 2000, pp. 489-490.
[27] J.P. Gray, J.J. Monaghan, R.P. Swift, “SPH elastic dynamics”, Comput. Meth. Appl. Mech. Engrg., vol. 190, 2001, pp. 6641–6662.
[28] T. Rabczuk, T. Belytschko, S.P. Xiao, “Stable particle methods based on Lagrangian kernels”, Comput. Methods Appl. Mech. Engrg., vol. 193, 2004, pp. 1035–1063.
[29] L.D. Landau, E.M. Lifshitz, “Theory of Elasticity; Course of Theoretical Physics”, Pergamon Press, Oxford, vol. 7, 1970.
[30] C. Antoci, M. Gallati, S. Sibilla, “Numerical simulation of fluid–structure interaction by SPH”, Comput. Struct., vol. 85, 2007, pp. 879–890.
[31] A. Rafiee, P. K. Thiagarajan, “An SPH projection method for simulating fluid-hypoelastic structure interaction”, Comput. Methods Appl. Mech. Engrg., vol. 198, 2009, pp. 2785–2795.