zbMATH — the first resource for mathematics

Exploration of new limiter schemes for stress tensors in Lagrangian and ALE hydrocodes. (English) Zbl 1290.76107
Summary: Limiter schemes are chiefly responsible for making high-resolution computations realizable in Lagrangian, Eulerian and ALE hydrocodes. Robust limiter schemes that are frame invariant and preserve symmetry have been established for scalars and to an extent, for vectors. However such limiter schemes have not been formulated and reported in the literature for tensor variables. In this work, new designs for limiter schemes for stress tensors are explored. Novel design principles are introduced and several limiter schemes are constructed based on these guiding principles. A scaling technique based on invariants and two new designs for slope limiter are proposed. In contrast to conventional slope limiters, the scaling technique designed by constraining the second invariant of stress tensor ensures monotonicity compliance by scaling the eigenvalues of the reconstructed stress tensors. Scalar slope limiter constructed based on constraining the second invariant is a formulation to extract a slope limiter from the scaling procedure. The tensor slope limiter scheme proposed for limiting velocity gradients is also extended for constraining stress tensor and the results from the same are considered as the basis for comparing and establishing the various limiter schemes formulated in this work. These limiter formulations are compared and contrasted in a cell-centered Lagrangian framework augmented with hypo-elastic model, by virtue of several one- and two-dimensional example problems.

MSC:
 76M20 Finite difference methods applied to problems in fluid mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
CAVEAT; ReALE
Full Text:
References:
 [1] Addessio FL, Baumgardner JR, Dukowicz JK, Johnson NL, Kashiwa BA, Rauenzahn RM, et al. Caveat: a computer code for fluid dynamics problems with large distortion and internal slip. Technical report, Los Alamos National Laboratory; 1986. [2] Despres, B.; Mazeran, C., Symmetrization of Lagrangian gas dynamic in dimension two and multidimensional solvers, CR Meca, 331, 7, 475-480, (2003) · Zbl 1293.76089 [3] Timothy J Barth. Von Karman lecture series 1994-05, chapter Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes equations. Von Karman Institute for Fluid Dynamics, Rhode-Saint-Genese, Belgium; 1994. [4] Barth, Timothy J., An introduction to recent developments in theory and numerics for conservation laws, proceedings of the international school on theory and numerics for conservation laws, Lecture notes in computational science and engineering, (1997), Springer-Verlag Berlin, [chapter Numerical Methods for Gasdynamic Systems on Unstructured Meshes, pp. 195-284] [5] Barth Timothy J, Fredrickson Paul O. Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction. In: 28th AIAA aerospace sciences meeting, Reno, NV; January 8-11 1990. [6] Burton DE. Multidimensional discretization of conservation laws for unstructured polyhedral grids. Technical report, Lawrence Livermore National Laboratory, CA, Aug 22 1994. Presented at the SAMGOP-94: 2nd international workshop on analytical methods and process optimization in fluid and gas mechanics, Arzamas (Russian Federation), 10-16 September 1994. [7] 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 [8] Carney, T. C.; Burton, D. E.; Morgan, N. R.; Runnels, S. R.; Shashkov, M. J., Cell-centered Lagrangian hydro applied to flow of real materials, (2011), Los Alamos National Laboratory Los Alamos, NM [9] Michel Delanaye, Yen Liu. Quadratic reconstruction finite volume schemes on 3d arbitrary unstructured polyhedral grids. In: 14th Computational fluid dynamics conference, Norfolk, VA; June 28-July 1 1999. [10] Godunov, S. K.; Romenskii, Evgenii I., Elements of continuum mechanics and conservation laws, (2003), Springer · Zbl 1031.74004 [11] Gurtin, Morton E.; Fried, Eliot; Anand, Lalit, The mechanics and thermodynamics of continua, (2009), Cambridge University Press [12] Khan, Akhtar S.; Huang, Sujian, Continuum theory of plasticity, (1995), Wiley-Interscience · Zbl 0856.73002 [13] Kluth, G.; Després, B., Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme, J Comput Phys, 229, 9092-9118, (2010) · Zbl 1427.74029 [14] Liu, X. D.; Osher, S., Convex ENO high order multi-dimensional schemes without field by field decomposition or staggered grids, J Comput Phys, 142, 304-330, (1998) · Zbl 0941.65082 [15] Raphaël Loubère, Pierre-Henri Maire, Mikhail Shashkov. ReALE: a reconnection arbitrary-Lagrangian? Eulerien method in cylindrical geometry. Comput Fluids 2011;46 (1):59-69. [10th ICFD conference series on numerical methods for fluid dynamics (ICFD 2010)]. [16] Loubère, Raphaël; Maire, Pierre-Henri; Shashkov, Mikhail; Breil, Jérôme; Galera, Stéphane, Reale: a reconnection-based arbitrary-Lagrangian-eulerien method, J Comput Phys, 229, 12, 4724-4761, (2010) · Zbl 1305.76067 [17] Loubère, Raphaël; Shashkov, Mikhail J., A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-eulerien methods, J Comput Phys, 209, 1, 105-138, (2005) · Zbl 1329.76236 [18] Luttwak, Gabi; Falcovitz, Joseph, Slope limiting for vectors: a novel vector limiting algorithm, Int J Numer Methods Fluids, 65, 11-12, (2011) · Zbl 1453.76100 [19] Maire, Pierre-Henri, A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry, J Comput Phys, 228, 6882-6915, (2009) · Zbl 1261.76021 [20] Maire, Pierre-Henri, A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, J Comput Phys, 228, 7, 2391-2425, (2009) · Zbl 1156.76434 [21] Maire, Pierre-Henri, A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, Comput Fluids, 46, 1, 341-347, (2011) · Zbl 1433.76137 [22] Maire, Pierre-Henri; Loubère, Raphaël; Váchal, Pavel, Staggered Lagrangian discretization based on cell-centered Riemann solver and associated hydrodynamics scheme, Commun Comput Phys, 10, 4, 940-978, (2011) · Zbl 1373.76138 [23] Margolin, L. G.; Shashkov, Mikhail, Second-order sign-preserving conservative interpolation (remapping) on general grids, J Comput Phys, 184, 266-298, (2003) · Zbl 1016.65004 [24] Margolin LG, Flower FC. Numerical simulation of plasticity at high strain rate. Technical Report LA-UR-91-1292, Los Alamos National Laboratory, December 1-6 1991. [submitted to High Temperature Constitutive Modelling (1991 ASME Winter Annual Mtg.), Atlanta, GA]. [25] O’Rourke, Joseph, Computational geometry in C, (1998), Cambridge University Press · Zbl 0912.68201 [26] Ponthot, Jean-Philippe, Unified stress update algorithms for the numerical simulation of large deformation elasto-plastic and elasto-viscoplastic processes, Int J Plast, 18, 1, 91-126, (2002) · Zbl 1035.74012 [27] Press, William H.; Flannery, Brian P.; Teukolsky, Saul A.; Vetterling, William T., Numerical recipes in FORTRAN 77: the art of scientific computing, vol. 25, (1992), Cambridge University Press · Zbl 0778.65002 [28] Shiv Kumar Sambasivan. Sharp interface cartesian grid hydrocode. PhD thesis, The University of Iowa; 2010. [29] Shiv Kumar Sambasivan, Shashkov Mikhail J. A Lagrangian cell centered mimetic formulation for computing elasto-plastic deformation of solids, 2011. Presented at the international conference on numerical methods for multi-material fluid flows (multimat); September 5-9, Arcachon, France. [30] Sambasivan, Shiv Kumar; Udaykumar, H. S., A sharp interface method for high-speed multi-material flows: strong shocks and arbitrary material pairs, Int J Comput Fluid Dyn, 25, 3, 139-162, (2011) · Zbl 1271.76255 [31] Shashkov, Mikhail; Wendroff, Burton, The repair paradigm and application to conservation laws, J Comput Phys, 198, 1, 265-277, (2004) · Zbl 1107.65341 [32] Shu, C. W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes II, J Comput Phys, 83, 32-78, (1989) · Zbl 0674.65061 [33] Sokolov, I. V.; Powell, K. G.; Gombosi, T. I.; Roussev, I. I., Short note: A tvd principle and conservative tvd schemes for adaptive Cartesian grids, J Comput Phys, 220, 1-5, (2006) · Zbl 1106.65075 [34] Tyndall, M. B., Numerical modelling of shocks in solids with elastic-plastic conditions, Shock Waves, 3, 55-66, (1993) · Zbl 0782.73023 [35] Leer, Bram van, Towards the ultimate conservative difference scheme v. a second-order sequel to godunov’s method, J Comput Phys, 135, 229-248, (1997) · Zbl 0939.76063 [36] Venkatakrishnan, V., Convergence to steady state solutions of the Euler equations on unstructured grids with limiters, J Comput Phys, 118, 1, 120-130, (1995) · Zbl 0858.76058 [37] Wilkins, Mark L., Computer simulation of dynamic phenomena, Springer Series in Scientific Computing, (1999), Springer · Zbl 0926.76001
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.