Abstract
We analyze several new and existing approaches for limiting tensor quantities in the context of deviatoric stress remapping in an ALE numerical simulation of elastic flow. Remapping and limiting of the tensor component-by-component are shown to violate radial symmetry of derived variables such as elastic energy or force. Therefore, we have extended the symmetry-preserving Vector Image Polygon algorithm, originally designed for limiting vector variables. This limiter constrains the vector (in our case a vector of independent tensor components) within the convex hull formed by the vectors from surrounding cells—an equivalent of the discrete maximum principle in scalar variables. We compare this method with a limiter designed specifically for deviatoric stress limiting which aims to constrain the \(J_2\) invariant that is proportional to the specific elastic energy and scale the tensor accordingly. We also propose a method which involves remapping and limiting the \(J_2\) invariant independently using known scalar techniques. The deviatoric stress tensor is then scaled to match this remapped invariant, which guarantees conservation in terms of elastic energy.
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References
T.J. Barth, Numerical methods for gasdynamic systems on unstructured meshes, in An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, Proceedings of the International School on Theory and Numerics for Conservation Laws, ed. by C. Rohde, D. Kroner, M. Ohlberger. Lecture Notes in Computational Science and Engineering (Springer, Berlin, 1997). ISBN 3-540-65081-4
E.J. Caramana, D.E. Burton, M.J. Shashkov, P.P. Whalen, The construction of compatible hydrodynamics algorithms utilizing conservation of total energy. J. Comput. Phys. 146(1), 227–262 (1998)
C.W. Hirt, A.A. Amsden, J.L. Cook, An arbitrary Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 14(3), 227–253 (1974)
M. Kucharik, Arbitrary Lagrangian-Eulerian (ALE) Methods in Plasma Physics. PhD thesis, Czech Technical University in Prague (2006)
G. Luttwak, J. Falcovitz, Slope limiting for vectors: a novel vector limiting algorithm. Int. J. Numer. Methods Fluids 65(11–12), 1365–1375 (2011)
L.G. Margolin, Introduction to “An arbitrary Lagrangian-Eulerian computing method for all flow speeds”. J. Comput. Phys. 135(2), 198–202 (1997)
D.J. Mavriplis, Revisiting the least-squares procedure for gradient reconstruction on unstructured meshes, in AIAA 2003–3986 2003, 16th AIAA Computational Fluid Dynamics Conference, June 23–26, Orlando, Florida (2003)
R. Menikoff, Equations of state and fluid dynamics. Technical Report LA-UR-07-3989, Los Alamos National Laboratory (2007)
L.G. Margolin, M. Shashkov, Second-order sign-preserving remapping on general grids. Technical Report LA-UR-02-525, Los Alamos National Laboratory (2002)
P.-H. Maire, R. Abgrall, J. Breil, R. Loubere, B. Rebourcet, A nominally second-order cell-centered Lagrangian scheme for simulating elasticplastic flows on two-dimensional unstructured grids. J. Comput. Phys. 235, 626–665 (2013)
S.K. Sambasivan, M. Shashkov, D.E. Burton, Exploration of new limiter schemes for stress tensors in Lagrangian and ALE hydrocodes. Comput. Fluids 83, 98–114 (2013)
J. Velechovsky, M. Kucharik, R. Liska, M. Shashkov, Symmetry-preserving momentum remap for ALE hydrodynamics. J. Phys.: Conf. Ser. 454, 012003 (2013). IOP Publishing
M.L. Wilkins, Calculation of elastic-plastic flow. Technical Report UCRL-7322, California. University Livermore. Lawrence Radiation Laboratory (1963)
Acknowledgements
This work was performed under the auspices of the National Nuclear Security Administration of the US Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396 and supported by the DOE Advanced Simulation and Computing (ASC) program. The authors acknowledge the partial support of the DOE Office of Science ASCR Program. This work was partially supported by the Czech Technical University grant SGS16/247/OHK4/3T/14, the Czech Science Foundation project 14-21318S and by the Czech Ministry of Education project RVO 68407700.
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Klima, M., Kucharik, M., Shashkov, M., Velechovsky, J. (2018). Bound-Preserving Reconstruction of Tensor Quantities for Remap in ALE Fluid Dynamics. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems II. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 237. Springer, Cham. https://doi.org/10.1007/978-3-319-91548-7_11
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