# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2021062

## A new Hodge operator in discrete exterior calculus. Application to fluid mechanics

 Laboratoire des Sciences de l'Ingénieur pour l'Environnement, UMR 7356 La Rochelle Université – CNRS, Avenue Michel Crépeau, 17042 La Rochelle Cedex 1, France

* Corresponding author

1 Current address: École Nationale Supérieure de Mécanique et des Microtechniques, 26, rue de l'Épitaphe - 25030 Besan¸con Cedex, France

Received  August 2020 Revised  February 2021 Published  April 2021

Fund Project: The first author is partially supported by the Nouvelle-Aquitaine region and the European Union through CPER Bâtiment Durable, Axe 3 "Qualité des Environnements Intérieurs (QEI)", convention number P-2017-BAFE-102

This article introduces a new and general construction of discrete Hodge operator in the context of Discrete Exterior Calculus (DEC). This discrete Hodge operator permits to circumvent the well-centeredness limitation on the mesh with the popular diagonal Hodge. It allows a dual mesh based on any interior point, such as the incenter or the barycenter. It opens the way towards mesh-optimized discrete Hodge operators. In the particular case of a well-centered triangulation, it reduces to the diagonal Hodge if the dual mesh is circumcentric. Based on an analytical development, this discrete Hodge does not make use of Whitney forms, and is exact on piecewise constant forms, whichever interior point is chosen for the construction of the dual mesh. Numerical tests oriented to the resolution of fluid mechanics problems and thermal transfer are carried out. Convergence on various types of mesh is investigated. Flat and non-flat domains are considered.

Citation: Rama Ayoub, Aziz Hamdouni, Dina Razafindralandy. A new Hodge operator in discrete exterior calculus. Application to fluid mechanics. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021062
##### References:
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Hiptmair, Discrete Hodge operators, Numerische Mathematik, 90 (2001), 265-289.  doi: 10.1007/s002110100295.  Google Scholar [31] A. Hirani, Discrete Exterior Calculus, Phd thesis, California Institute of Technology, Pasadena, CA, USA, 2003.  Google Scholar [32] A. Hirani, K. Kalyanaraman and E. VanderZee, Delaunay Hodge star, Computer-Aided Design, 45 (2013), 540-544.  doi: 10.1016/j.cad.2012.10.038.  Google Scholar [33] A. Hirani, K. Nakshatrala and J. Chaudhry, Numerical method for Darcy flow derived using discrete exterior calculus, International Journal for Computational Methods in Engineering Science and Mechanics, 16 (2015), 151-169.  doi: 10.1080/15502287.2014.977500.  Google Scholar [34] J. Lee, Introduction to Smooth Manifolds, 2nd edition, Graduate Texts in Mathematics, Springer, 2012.  Google Scholar [35] J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, 2nd edition, Springer Verlag, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar [36] M. Mohamed, A. Hirani and R. Samtaney, Comparison of discrete Hodge star operators for surfaces, Computer-Aided Design, 78 (2016), 118-125.   Google Scholar [37] M. Mohamed, A. Hirani and R. Samtaney, Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes, J. Comput. Phys., 312 (2016), 175-191.  doi: 10.1016/j.jcp.2016.02.028.  Google Scholar [38] M. Mohamed, A. Hirani and R. Samtaney, Numerical convergence of discrete exterior calculus on arbitrary surface meshes, Int. J. Comput. Method. Eng. Sci. Mech., 19 (2018), 194-206.  doi: 10.1080/15502287.2018.1446196.  Google Scholar [39] S. Morita, Geometry of Differential Forms, American Mathematical Society, 2001. doi: 10.1090/mmono/201.  Google Scholar [40] P. Mullen, P. Memari, F. Goes and M. Desbrun, HOT: Hodge-optimized triangulations, ACM Trans. Graph., 30 (2011). Google Scholar [41] I. Nitschke, S. Reuther and A. Voigt, Discrete Exterior Calculus (DEC) for the Surface Navier-Stokes Equation, Springer International Publishing, 2017.  Google Scholar [42] A. R, M. JE and R. R., Manifolds, Tensor Analysis and Applications, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1029-0.  Google Scholar [43] V. Rajan, Optimality of the Delaunay triangulation in $\mathbb{R}^d$, Discrete Comput. Geom., 12 (1994), 189-202.  doi: 10.1007/BF02574375.  Google Scholar [44] D. Razafindralandy, A. Hamdouni and M. Chhay, A review of some geometric integrators, Adv. Model. Simul. Eng. Sci., 5 (2018), 16. Google Scholar [45] D. Razafindralandy, V. Salnikov, A. Hamdouni and A. Deeb, Some robust integrators for large time dynamics, Adv. Model. Simul. Eng. Sci., 6 (2019), 5. Google Scholar [46] V. Salnikov and A. Hamdouni, From modelling of systems with constraints to generalized geometry and back to numerics, J. Appl. Math. Mech., 99 (2019), e201800218. doi: 10.1002/zamm.201800218.  Google Scholar [47] M. Spivak, A Comprehensive Introduction to Differential Geometry, 3rd edition, Publish or Perish, 1999.  Google Scholar [48] T. Tarhasaari, L. Kettunen and A. Bossavit, Some realizations of a discrete Hodge operator: a reinterpretation of finite element techniques, IEEE Trans. Magnet., 35 (1999), 1494-1497.   Google Scholar [49] E. VanderZee, A. Hirani, D. Guoy and E. Ramos, Well-centered triangulation, SIAM J. Sci. Comput., 31. doi: 10.1137/090748214.  Google Scholar [50] H. Whitney, Geometric Integration Theory, Princeton University Press, 1957.   Google Scholar [51] L. Yuan, Acute triangulations of trapezoids, Discrete Appl. Math., 158 (2010), 1121-1125.  doi: 10.1016/j.dam.2010.02.008.  Google Scholar [52] C. T. Zamfirescu, Survey of two-dimensional acute triangulations, Discrete Math., 313 (2013), 35-49.  doi: 10.1016/j.disc.2012.09.016.  Google Scholar

show all references

##### References:
 [1] P. Alotto and I. Perugia, Matrix properties of a vector potential cell method for magnetostatics, IEEE Trans. Magnet., 40 (2004), 1045-1048.   Google Scholar [2] D. Arnold, Finite Element Exterior Calculus, SIAM-Society for Industrial and Applied Mathematics, 2018. doi: 10.1137/1.9781611975543.ch1.  Google Scholar [3] D. Arnold, R. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15 (2006), 1-155.  doi: 10.1017/S0962492906210018.  Google Scholar [4] D. Arnold, R. Falk and R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bullet. Amer. Math. Soc., 47 (2010), 281-354.  doi: 10.1090/S0273-0979-10-01278-4.  Google Scholar [5] B. Auchmann and S. Kurz, A geometrically defined discrete Hodge operator on simplicial cells, IEEE Trans. Magnet., 42 (2006), 643-646.   Google Scholar [6] C. Blanes and S. Fernando, A Concise Introduction to Geometric Numerical Integration, CRC Press, 2016.   Google Scholar [7] P. Bochev and J. Hyman, Compatible Spatial Discretizations, Springer, 2006. doi: 10.1007/0-387-38034-5_5.  Google Scholar [8] A. Bossavit, Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism, Science, Measurement and Technology, IEE Proceedings A, 135 (1988), 493-500.   Google Scholar [9] A. Bossavit, Computational electromagnetism. Variational formulations, complementarity, edge elements, Academic Press, 1998.   Google Scholar [10] A. Bossavit, On the geometry of electromagnetism: (1) Affine space, J. Jpn. Soc. Appl. Electrom. Mech., 6 (1998), 17-28.   Google Scholar [11] A. Bossavit, On the geometry of electromagnetism: (2) Geometrical objects, J. Jpn. Soc. Appl. Electrom. Mech., 6 (1998), 114-123.   Google Scholar [12] A. Bossavit, On the geometry of electromagnetism: (3) Integration, Stokes, Faraday's law, J. Jpn. Soc. Appl. Electrom. Mech., 6 (1998), 233-240.   Google Scholar [13] A. Bossavit, On the geometry of electromagnetism: (4) Maxwell's house, J. Jpn. Soc. Appl. Electrom. Mech., 6 (1998), 318-326.   Google Scholar [14] A. Bossavit, Computational electromagnetism and geometry : (3) Convergence, J. Jpn. Soc. Appl. Electrom. Mech., 7 (1999), 401-408.   Google Scholar [15] A. Bossavit, Computational electromagnetism and geometry: (1) Network equations, J. Jpn. Soc. Appl. Electrom. Mech., 7 (1999), 150-159.   Google Scholar [16] A. Bossavit, Computational electromagnetism and geometry: (2) Network constitutive laws, J. Jpn. Soc. Appl. Electrom. Mech., 7 (1999), 204-301.   Google Scholar [17] A. Bossavit, Computational electromagnetism and geometry : (4): From degrees of freedom to fields, J. Jpn. Soc. Appl. Electrom. Mech., 8 (2000), 102-109.   Google Scholar [18] A. Bossavit, Computational electromagnetism and geometry : (5): The Galerkin Hodge, J. Jpn. Soc. Appl. Electrom. Mech., 8 (2000), 203-209.   Google Scholar [19] A. Bossavit, Discretization of electromagnetic problems: The "generalized finite differences" approach, in Numer. Method. Electrom., Elsevier, 2005, 105-197.  Google Scholar [20] C. Cassidy and G. Lord, A Square Acutely Triangulated, Baywood Publishing Co.. Inc., 1980.  Google Scholar [21] M. Cinalli, F. Edelvik, R. Schuhmann and T. Weiland, Consistent material operators for tetrahedral grids based on geometrical principles, Int. J. Numer. Model., 17 (2004), 487-507.   Google Scholar [22] K. Crane, F. de Goes, M. Desbrun and P. Schr$\ddot{o}$der, Digital geometry processing with discrete exterior calculus, in ACM SIGGRAPH 2013 courses, SIGGRAPH'13, ACM, 2013. Google Scholar [23] M. Desbrun, A. Hirani, M. Leok and J. Marsden, Discrete exterior calculus, arXivmath/0508341. Google Scholar [24] S. Elcott, Y. Tong, E. Kanso, P. Schr$\ddot{o}$der and M. Desbrun, Stable, circulation-preserving, simplicial fluids, ACM Trans. Graph., 26 (2015), 377-388.   Google Scholar [25] M. Fecko, Differential Geometry and Lie Groups for Physicists, Cambridge University Press, 2006.  doi: 10.1017/CBO9780511755590.  Google Scholar [26] C. Geuzaine and J.-F. Remacle, Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Int. J. Numer. Method. Eng., 79 (2009), 1309-1331.  doi: 10.1002/nme.2579.  Google Scholar [27] A. Gillette, Notes on Discrete Exterior Calculus, Technical report, University of Texas at Austin, 2009. Google Scholar [28] E. Grispun, P. Schr$\ddot{o}$der and M. Desbrun, Discrete differential geometry: An applied introduction, in ACM SIGGRAPH 2005 course notes, SIGGRAPH'05, ACM, 2005. Google Scholar [29] W. Hairer, G. Wanner and C. Lubich, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd edition, Springer Series in Computational Mathematics, Springer, 2006.  Google Scholar [30] R. Hiptmair, Discrete Hodge operators, Numerische Mathematik, 90 (2001), 265-289.  doi: 10.1007/s002110100295.  Google Scholar [31] A. Hirani, Discrete Exterior Calculus, Phd thesis, California Institute of Technology, Pasadena, CA, USA, 2003.  Google Scholar [32] A. Hirani, K. Kalyanaraman and E. VanderZee, Delaunay Hodge star, Computer-Aided Design, 45 (2013), 540-544.  doi: 10.1016/j.cad.2012.10.038.  Google Scholar [33] A. Hirani, K. Nakshatrala and J. Chaudhry, Numerical method for Darcy flow derived using discrete exterior calculus, International Journal for Computational Methods in Engineering Science and Mechanics, 16 (2015), 151-169.  doi: 10.1080/15502287.2014.977500.  Google Scholar [34] J. Lee, Introduction to Smooth Manifolds, 2nd edition, Graduate Texts in Mathematics, Springer, 2012.  Google Scholar [35] J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, 2nd edition, Springer Verlag, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar [36] M. Mohamed, A. Hirani and R. Samtaney, Comparison of discrete Hodge star operators for surfaces, Computer-Aided Design, 78 (2016), 118-125.   Google Scholar [37] M. Mohamed, A. Hirani and R. Samtaney, Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes, J. Comput. Phys., 312 (2016), 175-191.  doi: 10.1016/j.jcp.2016.02.028.  Google Scholar [38] M. Mohamed, A. Hirani and R. Samtaney, Numerical convergence of discrete exterior calculus on arbitrary surface meshes, Int. J. Comput. Method. Eng. Sci. Mech., 19 (2018), 194-206.  doi: 10.1080/15502287.2018.1446196.  Google Scholar [39] S. Morita, Geometry of Differential Forms, American Mathematical Society, 2001. doi: 10.1090/mmono/201.  Google Scholar [40] P. Mullen, P. Memari, F. Goes and M. Desbrun, HOT: Hodge-optimized triangulations, ACM Trans. Graph., 30 (2011). Google Scholar [41] I. Nitschke, S. Reuther and A. Voigt, Discrete Exterior Calculus (DEC) for the Surface Navier-Stokes Equation, Springer International Publishing, 2017.  Google Scholar [42] A. R, M. JE and R. R., Manifolds, Tensor Analysis and Applications, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1029-0.  Google Scholar [43] V. Rajan, Optimality of the Delaunay triangulation in $\mathbb{R}^d$, Discrete Comput. Geom., 12 (1994), 189-202.  doi: 10.1007/BF02574375.  Google Scholar [44] D. Razafindralandy, A. Hamdouni and M. Chhay, A review of some geometric integrators, Adv. Model. Simul. Eng. Sci., 5 (2018), 16. Google Scholar [45] D. Razafindralandy, V. Salnikov, A. Hamdouni and A. Deeb, Some robust integrators for large time dynamics, Adv. Model. Simul. Eng. Sci., 6 (2019), 5. Google Scholar [46] V. Salnikov and A. Hamdouni, From modelling of systems with constraints to generalized geometry and back to numerics, J. Appl. Math. Mech., 99 (2019), e201800218. doi: 10.1002/zamm.201800218.  Google Scholar [47] M. Spivak, A Comprehensive Introduction to Differential Geometry, 3rd edition, Publish or Perish, 1999.  Google Scholar [48] T. Tarhasaari, L. Kettunen and A. Bossavit, Some realizations of a discrete Hodge operator: a reinterpretation of finite element techniques, IEEE Trans. Magnet., 35 (1999), 1494-1497.   Google Scholar [49] E. VanderZee, A. Hirani, D. Guoy and E. Ramos, Well-centered triangulation, SIAM J. Sci. Comput., 31. doi: 10.1137/090748214.  Google Scholar [50] H. Whitney, Geometric Integration Theory, Princeton University Press, 1957.   Google Scholar [51] L. Yuan, Acute triangulations of trapezoids, Discrete Appl. Math., 158 (2010), 1121-1125.  doi: 10.1016/j.dam.2010.02.008.  Google Scholar [52] C. T. Zamfirescu, Survey of two-dimensional acute triangulations, Discrete Math., 313 (2013), 35-49.  doi: 10.1016/j.disc.2012.09.016.  Google Scholar
Example of 2D simplicial complex embedded in $\mathbb{R} ^3$
Example of a consistently oriented mesh. Arrows represent the orientation of edges and faces
Primal simplices (in blue) of a triangle $f$ (left), an edge $e$ (middle) and a vertex $v$ (right) and their duals $f^*$, $e^*$, $v^*$ (in red) in a 2D mesh
Sample 2D mesh (in blue) on a square and its circumcentric dual (in red)
Primal simplices and their circumcentric dual cells in a 2D mesh composed of a single triangle. $c_i$ is the circumcenter of the primal edge $e_i$, for $i = 1, 2, 3$, and $c_t$ is the circumcenter of the triangle. $e_i^*$ is perpendicular to $e_i$
Primal simplices and their arbitrary-centered dual cells in a 2D mesh composed of a single triangle. The $c_i$'s and $c_t$ are respectively arbitrary interior points of the edges $e_i$'s and of the triangle. The triangle is oriented counterclockwise. Arrows indicate the orientations of the primal edges and the induced orientations of dual edges. The angles $\theta _i$ defined in (3.1) are drawn in red
Right triangle with side lengths $m$ and $n$. Left: with a circumcentric dual mesh ($\text{e}_3^*$ has a zero length). Right: with a barycentric dual mesh
Typical acute and right triangulations of the unit square
Poisson equation with with $u_{exact} = x^2+y^2$: error evolution
Poisson equation with $u_{exact} = \sin (πx)\, \sinh (πx)$: error evolution
Poiseuille flow: error on $\psi$
Poiseuille flow: error on $u$
Taylor-Green vortex: error on $\psi$
Taylor-Green vortex: error on $u$
Traveling wave. Profile of the relative error on the stream function along $y = 0$ and at $t = T$
Traveling wave. Profile of the relative error on the horizontal velocity along $y = 0$ and at $t = T$
Traveling wave. Profile of the relative error on the temperature along $y = 0$ and at $t = T$
Unstructured meshes with respectively 36.36%, 40.69%, 39.41%, 43.46% and 44.20% of non-Delaunay triangles
Unstructured meshes. Convergence of the stream function and the temperature
Relative error on mesh (e) of Table 18
Set of meshes with 15% of non-Delaunay triangles
Set of meshes with 25% of non-Delaunay triangles
Set of meshes with 50% of non-Delaunay triangles
), the second set (Figure 22) and the third set (Figure 23) of meshes">Figure 24.  Convergence of the stream function and of the temperature. From top to bottom: with the first set (Figure 21), the second set (Figure 22) and the third set (Figure 23) of meshes
Evolution of the convergence rate with the ratio of non-Delaunay triangles
Discretized ellipsoid. Non-well centered triangles are darkened
Streamlines for $t$ from 0 to 11
$\ell^2$-norm of the error of the discrete Hodge operators on the unit right triangle and on a right-triangularized square with 20 points in each direction
 ${\text{Single triangle}}$ ${\text{Right mesh}}$ $\text{Differential form}$ $\text{Barycentric}$ $\text{Incentric}$ $\text{Barycentric}$ $\text{Incentric}$ $(x-y)(\text{d}x-\text{d}y)$ $0.2946$ $0.3232$ $1.5243·10^{-2}$ $1.5715·10^{-2}$ $(x+y)(\text{d}x+\text{d}y)$ $0.0589$ $0.0303$ $6.6882·10^{-4}$ $3.4424·10^{-4}$
 ${\text{Single triangle}}$ ${\text{Right mesh}}$ $\text{Differential form}$ $\text{Barycentric}$ $\text{Incentric}$ $\text{Barycentric}$ $\text{Incentric}$ $(x-y)(\text{d}x-\text{d}y)$ $0.2946$ $0.3232$ $1.5243·10^{-2}$ $1.5715·10^{-2}$ $(x+y)(\text{d}x+\text{d}y)$ $0.0589$ $0.0303$ $6.6882·10^{-4}$ $3.4424·10^{-4}$
Poisson equation with with $u_{exact} = x^2+y^2$: convergence rate
 Acute mesh Right mesh Circumcentric dual 1.995 – Barycentric dual 1.985 1.923 Incentric dual 1.992 1.921
 Acute mesh Right mesh Circumcentric dual 1.995 – Barycentric dual 1.985 1.923 Incentric dual 1.992 1.921
Poisson equation with $u_{exact} = \sin (πx)\, \sinh (πx)$: convergence rate
 Acute mesh Right mesh Circumcentric dual 1.975 – Barycentric dual 1.979 1.809 Incentric dual 1.982 1.840
 Acute mesh Right mesh Circumcentric dual 1.975 – Barycentric dual 1.979 1.809 Incentric dual 1.982 1.840
Poiseuille flow: convergence rates
 Acute mesh Right mesh Stream function Velocity Stream function Velocity Circumcentric 2.006 1.421 – – Barycentric 2.184 1.106 1.989 1.553 Incentric 2.185 1.096 1.989 1.539
 Acute mesh Right mesh Stream function Velocity Stream function Velocity Circumcentric 2.006 1.421 – – Barycentric 2.184 1.106 1.989 1.553 Incentric 2.185 1.096 1.989 1.539
Taylor-Green vortex: convergence rates
 Acute mesh Right mesh Stream function Velocity Stream function Velocity Circumcentric 1.996 1.187 – – Barycentric 2.065 1.130 2.018 1.735 Incentric 2.088 1.118 2.067 1.736
 Acute mesh Right mesh Stream function Velocity Stream function Velocity Circumcentric 1.996 1.187 – – Barycentric 2.065 1.130 2.018 1.735 Incentric 2.088 1.118 2.067 1.736
Traveling wave. Mean relative error
 Dual mesh Stream function Velocity Temperature Barycentric $2.651\cdot 10^{-5}$ $7.270\cdot10^{-5}$ $5.529\cdot10^{-3}$ Incentric $8.875\cdot 10^{-5}$ $2.132\cdot10^{-4}$ $5.589\cdot10^{-3}$
 Dual mesh Stream function Velocity Temperature Barycentric $2.651\cdot 10^{-5}$ $7.270\cdot10^{-5}$ $5.529\cdot10^{-3}$ Incentric $8.875\cdot 10^{-5}$ $2.132\cdot10^{-4}$ $5.589\cdot10^{-3}$
Unstructured meshes. Convergence rates of the stream function and the temperature
 Stream function Temperature Convergence rate 1.3690 1.1430
 Stream function Temperature Convergence rate 1.3690 1.1430
Relative errors on mesh (e) of Table 18
 Stream function Temperature Velocity Relative error $3.581\cdot 10^{-3}$ $3.682\cdot10^{-3}$ $2.316\cdot10^{-2}$
 Stream function Temperature Velocity Relative error $3.581\cdot 10^{-3}$ $3.682\cdot10^{-3}$ $2.316\cdot10^{-2}$
Convergence rate
 Stream function Temperature 15% non-Delaunay meshes 1.9005 1.5159 25% non-Delaunay meshes 1.6729 1.2154 50% non-Delaunay meshes 1.6591 0.8660
 Stream function Temperature 15% non-Delaunay meshes 1.9005 1.5159 25% non-Delaunay meshes 1.6729 1.2154 50% non-Delaunay meshes 1.6591 0.8660
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