zbMATH — the first resource for mathematics

Extension of centered hydrodynamical schemes to unstructured deforming conical meshes: the case of circles. (English. French summary) Zbl 1330.76082
Summary: In a prior work [B. Boutin et al., ESAIM, Proc. 32, 31–55 (2011; Zbl 1235.76079)], a curvilinear bi-dimensional finite volume extension of Lagrangian centered schemes GLACE [G. Carré et al., J. Comput. Phys. 228, No. 14, 5160–5183 (2009; Zbl 1168.76029)] on unstructured cells, whose edges are parameterized by rational quadratic Bézier curves was proposed and we showed numerical results for this scheme. Now, we extend the EUCCLHYD scheme [P.-H. Maire et al., SIAM J. Sci. Comput. 29, No. 4, 1781–1824 (2007; Zbl 1251.76028)] to these cells. To simulate flows with evolving large deformations, we write a formalism allowing the time evolution of the conic parameter. As an example, this allows an edge changing from an ellipse segment to a hyperbolic one. In this framework, we consider the case of a mesh whose edges are circle segments with non fixed centers. We show that this formalism extends also the previous work [A. Claisse et al., J. Comput. Phys. 231, No. 11, 4324–4354 (2012; Zbl 1426.76350)] (which is equivalent to [Boutin, loc. cit.] when conic edges are all circles). This is a necessary first step toward general conical deformation.
76M12 Finite volume methods applied to problems in fluid mechanics
76Bxx Incompressible inviscid fluids
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] G. Duvaut, M\'{}ecanique des milieux continus, Coll. Math. appl. pour la ma\hat{}ıtrise, MASSON 1990. · Zbl 0281.73005
[2] Wang Guojin, T.W. Sederberg, Computing areas bounded by rational B\'{}ezier curves, CADDM, Vol 4, No. 2, pp. 18-27, September 1994.
[3] Wang Guojin, Computing integral values involving nurbs curves, Jour. of Software, Vol 7, No. 9, pp. 542-546, September 1996.
[4] Ming Li, Xiao-Shan Gao, Shang-Ching Chou, Quadratic approximation to plane parametric curves and its application in approximate implicitization, Visual Comput., Vol 22, pp. 906-917, 2006.
[5] B. Despr\'{}es, C. Mazeran., Lagrangian gas dynamics in two dimensions and lagrangian systems. Arch. Rational Mech. Anal., Vol 178, pp 327-372, 2005. · Zbl 1096.76046
[6] B.Despr\'{}es,E.Labourasse,StabilizationofthemeshforLagrangiancomputations, http://multimat2011.celia.u-bordeaux1.fr/Multimat2011/Thursday AM/Despres.pdf,Multi-MatConference,Arcachon September 2011.
[7] G. Carre, S. Del Pino, B. Despr\'{}es, E. Labourasse, A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension, Jour. Comp. Physic., Vol 228,pp 5160-5183, 2009. · Zbl 1168.76029
[8] P.H. Maire, R. Abgrall, J. Breil, J. Ovadia, A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J.Sci.Comput., Vol 29,pp 1781-1824, 2007. · Zbl 1251.76028
[9] A. Chorin, J. Marsden, A Mathematical Introduction to Fluid Mechanics., Springer Verlag, 1992. · Zbl 0417.76002
[10] A. Claisse, B. Despr\'{}es, E. Labourasse, F. Ledoux, A new exceptional point method with application to cell-centered Lagrangian schemes and curved meshessubmitted to J.Comp.Phys 2011. · Zbl 1426.76350
[11] S. Del Pino, A curvilinear finite-volume method to solve compressible gas dynamics in semi-Lagrangian coordinates, CRAS, Vol 348, num 17-18, pp. 1027-1032, 2010. · Zbl 1426.76652
[12] V. Dobrev, T. Ellis, T. Kolev, R. Rieben, Energy conserving finite element discretizations of Lagrangian hydrodynamics. Part 1: Theoretical framework, Downloadable presentation of Multimat’09 conference.
[13] B. Despr\'{}es, Weak consistency of the cell centered Lagrangian GLACE scheme on general meshes in any dimension, Comp. Meth. Appl. Mech. Engr, 199, pp. 2669-2679, 2010. · Zbl 1231.76177
[14] J. Cheng, C.W. Shu, A third order Conservative Lagrangian type scheme on curvilinear meshes for the compressible Euler equations, comm. on comput. phys., vol 4, no5, pp 1008-1024, 2008. · Zbl 1364.76111
[15] P.-H. Maire, Contribution to the numerical modeling of Inertial Confinement Fusion. Habilitation ‘a Diriger des Recherches. Bordeaux University 2011. optics type, M2AN, 36(5):883-905, 2002.
[16] L.G. Margolin, M. Shashkov, Using a curvilinear grid to construct symmetry-preserving discretizations for Lagrangian gas dynamics, J. Comput. Phys., vol 149, number 2, pp 389-417, 1999. · Zbl 0936.76057
[17] S. K. Godunov, A. V. Zabrodin, M. Ya. Ivanov, et al., R\'{}esolution num\'{}erique des probl‘emes multidimensionnels de la dynamiques des gaz(Editions Mir, Moscou, 1979).
[18] B. Larrouturou, How to preserve the mass fractions positivity when computing compressible multi-component flows, J. Comput. Phys. 95 (1991), 59-84. · Zbl 0725.76090
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.