Evolution of CFD numerical methods and physical models towards a full discrete approach
Comptes Rendus. Mécanique, Online first (2022), pp. 1-10.

The physical models and numerical methodologies of Computational Fluid Dynamics (CFD) are historically linked to the concept of continuous medium and to analysis where continuity, derivation and integration are defined as limits at a point. The first consequence is the need to extend these notions in a multidimensional space by establishing a global inertial frame of reference in order to project the variables there. In recent decades, the emergence of methodologies based on differential geometry or exterior calculus has changed the point of view by starting with the creation of entangled polygonal and polyhedral structures where the variables are located. Mimetic methods and Discrete Exterior Calculus, notably, have intrinsic conservation properties which make them very efficient for solving fluid dynamics equations. The natural extension of this discrete vision relates to the derivation of the equations of mechanics by abandoning the notion of continuous medium. The Galilean frame of reference is replaced by a local frame of reference composed of an oriented segment where the acceleration of the material medium or of a particle is defined. The extension to a higher dimensional space is carried from cause to effect, from one local structure to another. The conservation of acceleration over a segment and the Helmholtz–Hodge decomposition are two essential principles adopted for the derivation of a discrete law of motion. As the fields covered by CFD are increasingly broad, it is natural to return to the deeper meaning of physical phenomena to try a new research or new path which would preserve the properties of current formulations.

Revised:
Accepted:
Online First:
DOI: 10.5802/crmeca.131
Keywords: Conservation of acceleration, Navier–Stokes equations, Mimetic methods, Discrete exterior calculus, Discrete mechanics, Helmholtz–Hodge decomposition

Jean-Paul Caltagirone 1

1 Bordeaux INP, University of Bordeaux, CNRS UMR-5295, Arts et Métiers Institute of Technology, INRAE, I2M Bordeaux, 33400 Talence, France
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Jean-Paul Caltagirone. Evolution of CFD numerical methods and physical models towards a full discrete approach. Comptes Rendus. Mécanique, Online first (2022), pp. 1-10. doi : 10.5802/crmeca.131.

[1] L. Landau; E. Lifchitz Fluid Mechanics, Pergamon Press, London, 1959

[2] C. Truesdell; W. Noll The Non-Linear Field Theories of Mechanics, Encyclopedia of Physics, III/3, Springer-Verlag, New York, 1992 | DOI

[3] P. Germain; P. Muller Introduction à la Mécanique des Milieux Continus, Masson, Paris, 1995

[4] J. Salencon Mécanique des milieux continus, Editions de l’Ecole Polytechnique, Palaiseau, 2002

[5] A. Chorin The numerical solution of the Navier–Stokes equations for an incompressible fluid, Bull. Am. Math. Soc., Volume 73 (1967), pp. 928-931 | MR | Zbl

[6] R. Temam Navier–Stokes Equations, Theory and Numerical Analysis, North-Holland, Amsterdam, New-York, 1984

[7] F. Harlow; J. Welch Numerical calculation of time-dependent viscous incompressible flow of fluid with a free surface, Phys. Fluids, Volume 8 (1965), pp. 2182-2189 | DOI | MR | Zbl

[8] A. Chorin A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., Volume 135 (1967), pp. 118-125 | DOI | MR | Zbl

[9] J. Guermond; P. Minev; J. Shen An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., Volume 195 (2006), pp. 6011-6045 | DOI | MR | Zbl

[10] M. Fortin; R. Glowinski Méthodes de lagrangien augmenté, Application à la résolution numérique de problèmes aux limites, Dunod, Paris, 1982

[11] R. Nicolaides Flow discretization by complementary volume techniques, 9th AIAA Computational Fluid Dynamics Conference, Volume 1 (1989), pp. 464-470 | DOI

[12] B. Perot Conservation properties of unstructured staggered mesh schemes, J. Comput. Phys., Volume 159 (2000), pp. 58-89 | DOI | MR | Zbl

[13] B. Perot; V. Subramanian Discrete calculus methods for diffusion, J. Comput. Phys., Volume 224 (2007), pp. 59-81 | DOI | MR | Zbl

[14] B. Perot Discrete conservation properties of unstructured mesh schemes, Annu. Rev. Fluid Mech., Volume 43 (2011), pp. 299-318 | DOI | MR | Zbl

[15] G. R. McNamara; G. Zanetti Use of the Boltzmann equation to simulate lattice-gas automata, Phys. Rev. Lett., Volume 61 (1988), pp. 2332-2335 | DOI

[16] S. Succi The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Clarendon Press, Oxford, 2001

[17] M. Shashkov Conservative Finite-Difference Methods on General Grids, CRC Press, Boca Raton, 1996 | DOI

[18] J. M. Hyman; M. Shashkov Natural discretizations for the divergence, gradient ans curl on logically rectangular grids, SIAM J. Numer. Anal., Volume 36 (1999), pp. 788-818 | DOI

[19] J. M. Hyman; M. Shashkov The orthogonal decomposition theorems for mimetic finite difference methods, SIAM J. Numer. Anal., Volume 36 (1999), pp. 788-818 | DOI | MR | Zbl

[20] J. M. Hyman; M. Shashkov Mimetic discretizations for Maxwell’s equations, J. Comput. Phys., Volume 151 (1999), pp. 881-909 | DOI | MR | Zbl

[21] J. M. Hyman; M. Shashkov Mimetic discretizations for Maxwell’s equations and the equations of magnetic diffusion, Prog. Electromagn. Res., Volume 32 (2001), pp. 89-121 | DOI

[22] K. Lipnikov; G. Manzini; M. Shashkov Mimetic finite difference method, J. Comput. Phys., Volume 257 (2014), pp. 1163-1227 | DOI | MR | Zbl

[23] A. Palha; M. Gerritsma A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier–Stokes equations, J. Comput. Phys., Volume 328 (2017), pp. 200-220 | DOI | MR | Zbl

[24] G. de Diegoa; A. Palha; M. Gerritsma Inclusion of no-slip boundary conditions in the MEEVC scheme, J. Comput. Phys., Volume 378 (2019), pp. 615-633 | DOI | MR | Zbl

[25] H. Ranocha; K. Ostaszewski; P. Heinisch Discrete vector calculus and Helmholtz Hodge decomposition for classical finite difference summation by parts operators, Commun. Appl. Math. Comput., Volume 2 (2020), pp. 581-611 | DOI | MR | Zbl

[26] J. Bonelle; A. Ern Analysis of compatible discrete operator schemes for the stokes equations on polyhedral meshes, IMA J. Numer. Anal., Volume 35 (2015), pp. 1672-1697 | DOI | MR | Zbl

[27] J. Bonelle; D. Di Pietro; A. Ern Low-order reconstruction operators on polyhedral meshes: application to compatible discrete operator schemes, Comput. Aided Geom. Des., Volume 35 (2015), pp. 27-41 | DOI | MR | Zbl

[28] M. Mohamed; A. Hirani; R. Samtaney Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes, J. Comput. Phys., Volume 312 (2016), pp. 175-191 | DOI | MR | Zbl

[29] J. P. Caltagirone Discrete Mechanics, Concepts and Applications, ISTE, John Wiley & Sons, London, 2019 | DOI

[30] J. P. Caltagirone On a reformulation of Navier–Stokes equations based on Helmholtz–Hodge decomposition, Phys. Fluids, Volume 33 (2021), 063605 | DOI

[31] J. P. Caltagirone; S. Vincent On primitive formulation in fluid mechanics and fluid–structure interaction with constant piecewise properties in velocity-potentials of acceleration, Acta Mech., Volume 231 (2020), pp. 2155-2171 | DOI | MR | Zbl

[32] J. P. Caltagirone Non-Fourier heat transfer at small scales of time and space, Int. J. Heat Mass Transfer, Volume 160 (2020), 120145 | DOI

[33] J. P. Caltagirone On Helmholtz–Hodge decomposition of inertia on a discrete local frame of reference, Phys. Fluids, Volume 32 (2020), 083604 | DOI

[34] J. P. Caltagirone Application of discrete mechanics model to jump conditions in two-phase flows, J. Comput. Phys., Volume 432 (2021), 110151 | DOI | MR

[35] P. Angot; J. P. Caltagirone; P. Fabrie Fast discrete Helmholtz–Hodge decomposition in bounded domains, Appl. Math. Lett., Volume 26 (2013), pp. 445-451 | DOI | MR | Zbl

[36] H. Bhatia; G. Norgard; V. Pascucci; P. Bremer The Helmholtz–Hodge decomposition—a survey, IEEE Trans. Visual. Comput. Graph., Volume 19 (2013), pp. 1386-1404 | DOI

[37] H. Ranocha; P. Öffner; T. Sonar Summation-by-parts operators for correction procedure via reconstruction, J. Comput. Phys., Volume 311 (2016), pp. 299-328 | DOI | MR | Zbl

[38] J. Maxwell A dynamical theory of the electromagnetic field, Phil. Trans. R. Soc. Lond., Volume 155 (1865), pp. 459-512

[39] A. Liénard Champ électrique et magnétique produit par une charge électrique concentrée en un point et animée d’un mouvement quelconque, L’Éclairage électrique, Volume XVI (1898), pp. 5-112

[40] Y. Kosmann-Schwarzbach Noether Theorems. Invariance and Conservations Laws in the Twentieth Century, Springer-Verlag, New York, 2011 | DOI

[41] G. Sardanashvily Noether’s Theorems. Applications in Mechanics and Field Theory, Lomonosov Moscow State University, Moscow, 2016 | DOI

[42] L. Rosenhead Introduction. The second coefficient of viscosity: a brief review of fundamentals, Proc. R. Soc. Lond. A: Math. Phys. Sci., Volume 226 (1954), pp. 1-6 | MR

[43] M. Gad-El-Hak Stokes hypothesis for a newtonian, isotropic fluid, J. Fluids Eng., Volume 117 (1995), pp. 3-5 | DOI

[44] K. Rajagopal A new development and interpretation of the Navier–Stokes fluid which reveals why the “Stokes assumption” is inapt, Int. J. Non-Linear Mech., Volume 50 (2013), pp. 141-151 | DOI

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