Geodesic
CABARET scheme for computational modelling of linear elastic deformation problems
Matematičeskoe modelirovanie, Tome 29 (2017) no. 11, pp. 53-70.

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A generalisation of the CABARET scheme for linear elasticity equations with accounting for plastic deformations is suggested in a Lagrangian framework. In accordance with the conservative characteristic decomposition CABARET method, conservation variables are defined in cell centres and 'active' flux variables are defined in cell faces. Linear elasticity equations, which correspond to the hyperbolic part of the problem, are solved in a strong conservation form to update the cell-centre variables in time at the predictor-corrector stages. Cell-face variables are updated in time using the characteristic decomposition along each of the characteristic directions. For plastic deformation, the classical Prandtl–Reuss model is used to restrict the deformation stress components in accordance with the elastic limit at each step of the scheme. The Lagrangian step includes update of the coordinates of vertices of each control volume, which are slowly varying in time. Validation examples of the new method are provided for several test problems including the hard sphere denting into an elastic medium, shell deformation under the blast wave loading, and a spherical seismic wave propagation from a point source. The solutions of the new method are compared with the reference solutions available in the literature based on the artificial viscosity approaches and also on the Discontinuous Galerkin approach. Scalability results of the new algorithm for massively parallel computations are provided.
Keywords: elastic-plastic solid, deformation, modeling, parallel computation
Mots-clés : CABARET scheme. 
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M. A. Zaitsev; S. A. Karabasov. CABARET scheme for computational modelling of linear elastic deformation problems. Matematičeskoe modelirovanie, Tome 29 (2017) no. 11, pp. 53-70. http://geodesic.mathdoc.fr/item/MM_2017_29_11_a3/

[1] E.N. Vedeniapin, V.N. Kukudzhanov, “Metod chislennogo integrirovania nestatsionarnykh zadach dinamiki uprugoi sredy”, Zh. vychisl. matem. I matem. fiz., 21:5 (1981), 1233–1248 | Zbl

[2] I.B. Petrov, A.S. Kholodov, “Chislennoe issledovanie nekotorych dinamicheskikh zadach mekhaniki deformiruemogo tverdogo tela setochno-kharacteristicheskim metodom”, Zh. vychisl. matem. I matem. fiz., 24:5 (1984), 722–739 | Zbl

[3] V.D. Ivanov, V.I. Kondaurov, I.B. Petrov, A.S. Kholodov, “Raschet dinamicheskogo deformirovaniia i razrusheniia uprugoplasticheskikh tel setochnokharacteristicheskimi metodami”, Matem. Model., 2:11 (1990), 10–29 | Zbl

[4] A.V. Vasyukov, A.S. Ermakov, A.P. Potapov, I.B. Petrov, A.V. Favorskaya, A. V. Shevtsov, “Combined Grid-Characteristic Method for the Numerical Solution of Three-Dimensional Dynamical Elastoplastic Problems”, Comp. Math. Math. Phys., 54:7 (2014), 1176–1189 | DOI | MR | Zbl

[5] V.A. Miryaha, A.V. Sannikov, I. B. Petrov, “Discontinuous Galerkin Method for Numerical Simulation of Dynamic Processes in Solids”, Math Models. Comp. Simul., 7:5 (2015), 446–455 | DOI | MR

[6] M. Dumbser, I. Peshkov, E. Romenski, O. Zanotti, “High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conductiong fluids and elastic solids”, J. of Computational Physics, 314 (2016), 864–862 | DOI | MR

[7] V.M. Goloviznin, M.A. Zaitsev, S.A. Karabasov, I.A. Korotkin, Novye algoritmy vychislitelnoi gidrodinamiki dlia mnogoprotsessornykh vychislitelnykh kompleksov, Izd-vo MGU, M., 2013

[8] V. Goloviznin, “Balanced characteristic method for systems of hyperbolic conservation laws”, Doklady Math., 72 (2005), 619–623

[9] A. Iserles, “Generalized Leapfrog methods”, IMA J. Numer. Anal., 1986, no. 6, 381–392 | DOI | MR | Zbl

[10] P.L. Roe, “Linear bicharacteristic schemes without dissipation”, SIAM J. Sci. Comput., 19 (1998), 1405–1427 | DOI | MR | Zbl

[11] V.M. Goloviznin, A.A. Samarskii, “Difference approximation of convective transport with spatial splitting of time derivative”, Math. Model., 1998, no. 10, 86–100 | MR | Zbl

[12] V.M. Goloviznin, A.A. Samarskii, “Some properties of the CABARET scheme”, Math. Model., 1998, no. 10, 101–116 | MR | Zbl

[13] S. Karabasov, V. Goloviznin, “Compact Accurately Boundary-Adjusting high-Resolution Technique for fluid dynamics”, Journal of Computational Physics, 228:19 (2009), 7426–7451 | DOI | MR | Zbl

[14] V.V. Ostapenko, “On the strong monotonicity of the CABARET scheme”, Computational Mathematics and Mathematical Physics, 52:3 (2012), 387–399 | DOI | MR | Zbl

[15] N.A. Zyuzina, V.V. Ostapenko, “Modification of the CABARET scheme ensuring its strong monotonicity and high accuracy on local extrema”, Doklady Mathematics, 90:1 (2014), 453–457 | DOI | MR | Zbl

[16] V.M. Goloviznin, V.N. Semenov, I.A. Korotkin, S. A. Karabasov, “A novel computational method for modelling stochastic advection in heterogeneous media”, Transport in Porous Media, 66:3 (2007), 439–456 | DOI | MR

[17] S.A. Karabasov, V.M. Goloviznin, “New efficient high-resolution method for nonlinear problems in aeroacoustics”, AIAA J., 45 (2007), 2861–2871 | DOI

[18] V.M. Goloviznin, M.A. Zaitsev, S. A. Karabasov, “A highly scalable hybrid mesh CABARET MILES method for MATIS-H problem”, Proc. of the CFD4NRS-4 WORKSHOP (Daejon, South Korea, 2012), 104 | Zbl

[19] V. Semiletov, S. Karabasov, “CABARET scheme with conservation-flux asynchronous time-stepping for nonlinear aeroacoustics problems”, J. of Computational Physics, 253 (2013), 157–165 | DOI | MR | Zbl

[20] G.A. Faranosov, V.M. Goloviznin, S.A. Karabasov, V.G. Kondakov, V.F. Kopiev, M. A. Zaitsev, “CABARET method on unstructured hexahedral grids for jet noise computation”, Computers and Fluids, 88 (2013), 165–179 | DOI

[21] A.P. Markesteijn, V. Semiletov, S. A. Karabasov, “GPU CABARET Solutions for the SILOET Jet Noise Experiment: Flow and Noise Modelling”, 22nd AIAA/CEAS Aeroacoustics Conference, Aeroacoustics Conferences, AIAA 2016-2967

[22] M.L. Wilkins, Computer Simulation of Dynamic Phenomena, Springer-Verlag, Berlin–Heidelberg–New York, 1999 | MR | Zbl

[23] N.I. Drobyshevskii, M.A. Zaitsev, A.S. Fillipov, Konechno-elementnoe modelirovanie teplovogo I mekhanicheskogo vozdeistviia na obiekty i konstruktsii atomnoi tekhniki, Preprint IBRAE, 1993

[24] M.B. Bakirov, M.A. Zaitsev, I.V. Frolov, “Matematicheskoe modelirovanie protsessov identirovaniia sfery v uprugoplasticheskoe poluprostranstvo”, Zavodskaia laboratoriia, 67:1 (2001), 37–47

[25] M.A. Zaitsev, S.M. Goldberg, “Matematicheskoe modelirovanie vzaimodeistviia detoniruiushchikh sred s uprugimi obolochrami”, Matem. Modelirovanie, 5:6 (1993), 56–68 | Zbl

[26] H. Lamb, “On the propagation of tremors over thes urface of an elastic solid”, Phil. Trans. R. Soc. Lond., Ser. A, 203 (1904), 1–42 | DOI

[27] M. Kaser, M. Dumbser, “An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two-dimensional isotropic case with external source terms”, Geophys. J. Int., 166 (2006), 855–877 | DOI