×

zbMATH — the first resource for mathematics

High order curvilinear finite elements for elastic-plastic Lagrangian dynamics. (English) Zbl 1351.76057
Summary: This paper presents a high-order finite element method for calculating elastic-plastic flow on moving curvilinear meshes and is an extension of our general high-order curvilinear finite element approach for solving the Euler equations of gas dynamics in a Lagrangian frame [1,2]. In order to handle transition to plastic flow, we formulate the stress-strain relation in rate (or incremental) form and augment our semi-discrete equations for Lagrangian hydrodynamics with an additional evolution equation for the deviatoric stress which is valid for arbitrary order spatial discretizations of the kinematic and thermodynamic variables. The semi-discrete equation for the deviatoric stress rate is developed for 2D planar, 2D axisymmetric and full 3D geometries. For each case, the strain rate is approximated via a collocation method at zone quadrature points while the deviatoric stress is approximated using an \(L2\) projection onto the thermodynamic basis. We apply high order, energy conserving, explicit time stepping methods to the semi-discrete equations to develop the fully discrete method. We conclude with numerical results from an extensive series of verification tests that demonstrate several practical advantages of using high-order finite elements for elastic-plastic flow.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
74S05 Finite element methods applied to problems in solid mechanics
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
76A10 Viscoelastic fluids
Software:
TENSOR; MFEM
PDF BibTeX Cite
Full Text: DOI
References:
[1] Dobrev, V. A.; Kolev, Tz. V.; Rieben, R. N., High order curvilinear finite element methods for Lagrangian hydrodynamics, SIAM J. Sci. Comp., 34, 5, B606-B641, (2012), (Also available as LLNL technical report LLNL-JRNL-516394) · Zbl 1255.76076
[2] V.A. Dobrev, T.E. Ellis, Tz. V. Kolev, R.N. Rieben, High order curvilinear finite elements for axisymmetric Lagrangian hydrodynamics. Comput. Fluids, in press. <http://dx.doi.org/10.1016/j.compfluid.2012.06.004>.
[3] Wilkins, M. L., Calculation of elastic-plastic flow, Methods in Computational Physics, vol. 3, (1964), Academic Press
[4] J.T. Cherry, S. Sack, G. Maenchen, V. Kransky, Two-dimensional stress-induced adiabatic flow, Technical Report UCRL-50987, Lawrence Livermore National Laboratory, 1970.
[5] Dobrev, V. A.; Ellis, T. E.; Kolev, Tz. V.; Rieben, R. N., Curvilinear finite elements for Lagrangian hydrodynamics, Int. J. Numer. Methods Fluids, 65, 11-12, 1295-1310, (2010) · Zbl 1255.76075
[6] Ockendon, H.; Ockendon, J., Waves and compressible flow, Texts in Applied Mathematics, vol. 47, (2004), Springer-Verlag · Zbl 1041.76001
[7] Benson, D. J., Computational methods in Lagrangian and Eulerian hydrocodes, Comput. Methods Appl. Mech. Eng., 99, 235-394, (1992) · Zbl 0763.73052
[8] Spencer, A. J.M., Continuum mechanics, (1980), Dover Publications Inc. Mineola, New York · Zbl 0427.73001
[9] Simo, J. C.; Hughes, T. J.R., Computational Inelasticity, (1998), Springer · Zbl 0934.74003
[10] Maire, P.-H.; Abgrall, R.; Breil, J.; Loubère, R.; Rebourcet, B., A nominally second-order cell-centered Lagrangian scheme for simulating elastic-plastic flows on two-dimensional unstructured grids, J. Comput. Phys., (2012)
[11] Maenchen, G.; Sack, S., The TENSOR code, Methods in Computational Physics, vol. 3, (1964), Academic Press
[12] Sambasivan, S. K.; Shashkov, M. J.; Burton, D. E., Exploration of new limiter schemes for stress tensors in Lagrangian and ALE hydrocodes, Comput. Fluids, (2012) · Zbl 1290.76107
[13] Kelmanson, M. A.; Maunder, S. B., Modeling high-velocity impact phenomena using unstructured dynamically adaptive Eulerian meshes, Comput. Methods Appl. Mech. Eng., 142, 269-301, (1997)
[14] Udaykumar, H. S.; Tran, L.; Belk, D. M.; Vanden, K. J., An Eulerian method for computation of multimaterial impact with ENO shock-capturing and sharp interfaces, J. Comput. Phys., 186, 1, 136-177, (2003) · Zbl 1047.76558
[15] Howell, B. P.; Ball, G. J., A free-Lagrange augmented Godunov method for the simulation of elastic-plastic solids, J. Comput. Phys., 175, 128-167, (2002) · Zbl 1043.74048
[16] Camacho, G. T.; Ortiz, M., Adaptive Lagrangian modeling of ballistic penetration of metallic targets, Comput. Methods Appl. Mech. Eng., 142, 269-301, (1997) · Zbl 0892.73056
[17] Kluth, G.; Despres, B., Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme, J. Comput. Phys., 229, 24, 9092-9118, (2010) · Zbl 1427.74029
[18] Landau, L. D.; Lifshitz, E. M., Theory of Elasticity, (1986), Pergamon Press
[19] M.L. Wilkins. Calculation of elastic-plastic flow, Technical Report UCRL-7322, Rev. 1, Lawrence Livermore National Laboratory, 1969.
[20] R. Tipton. CALE Lagrange step, Technical report, Lawrence Livermore National Laboratory, October 1990. Unpublished.
[21] Dienes, J. K., On the analysis of rotation and stress rate in deforming bodies, Acta Mech., 32, 217-232, (1978) · Zbl 0414.73005
[22] Caramana, E. J.; Burton, D. E.; Shashkov, M. J.; Whalen, P. P., The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J. Comput. Phys., 146, 227-262, (1998) · Zbl 0931.76080
[23] D.J. Steinberg, Equation of state and strength properties of selected materials, Technical Report UCRL-MA-106439, Lawrence Livermore National Laboratory, 1996.
[24] BLAST: High-order finite element shock hydrocode. <http://www.llnl.gov/CASC/blast>
[25] MFEM: Modular parallel finite element methods library. <http://mfem.googlecode.com>
[26] M. Kennamond, T. McAbee, Personal Communication, 2011, LLNL.
[27] G. Bazan, Personal Communication, 2010, LLNL.
[28] Verney, D., Evaluation de la limite elastique du cuivre et de l‘uranium par des experiences d‘implosion ’lente’, (Behavior of Dense Media Under High Dynamic Pressures, Symposium, H.D.P., IUTAM, (1968), Gordon and Breach Paris, New York), 293-303
[29] J.R. Kamm, J.S. Brock, S.T. Brandon, D.L. Cotrell, B.M. Johnson, P. Knupp, T.G. Trucano, W.J. Rider, V.G. Weirs, Enhanced verification test suite for physics simulations codes, Technical Report SAND2008-7813, Sandia National Laboratory, 2009.
[30] D.E. Burton, T.C. Carney, N.R. Morgan, S.K. Sambasivan, M.J. Shashkov, A cell-centered lagrangian godunov-like method for solid dynamics. Comput. Fluids, in press. http://dx.doi.org/10.1016/j.compfluid.2012.09.008. · Zbl 1290.76095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.