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

Rama Ayoub; Aziz Hamdouni; Dina Razafindralandy

doi:10.3934/cpaa.2021062

Article Contents

2021, Volume 20, Issue 6: 2155-2185. Doi: 10.3934/cpaa.2021062

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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
^* 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

Early access: April 2021

Published: June 2021

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 53A70, 58F17; Secondary: 53Z30.

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Figure 1. Example of 2D simplicial complex embedded in $ \mathbb{R} ^3 $

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Figure 2. Example of a consistently oriented mesh. Arrows represent the orientation of edges and faces

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Figure 3. 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

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Figure 4. Sample 2D mesh (in blue) on a square and its circumcentric dual (in red)

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Figure 5. 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 $

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Figure 6. 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

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Figure 7. 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

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Figure 8. Typical acute and right triangulations of the unit square

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Figure 9. Poisson equation with with $ u_{exact} = x^2+y^2 $: error evolution

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Figure 10. Poisson equation with $ u_{exact} = \sin (πx)\, \sinh (πx) $: error evolution

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Figure 11. Poiseuille flow: error on $ \psi $

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Figure 12. Poiseuille flow: error on $ u $

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Figure 13. Taylor-Green vortex: error on $ \psi $

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Figure 14. Taylor-Green vortex: error on $ u $

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Figure 15. Traveling wave. Profile of the relative error on the stream function along $ y = 0 $ and at $ t = T $

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Figure 16. Traveling wave. Profile of the relative error on the horizontal velocity along $ y = 0 $ and at $ t = T $

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Figure 17. Traveling wave. Profile of the relative error on the temperature along $ y = 0 $ and at $ t = T $

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Figure 18. Unstructured meshes with respectively 36.36%, 40.69%, 39.41%, 43.46% and 44.20% of non-Delaunay triangles

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Figure 19. Unstructured meshes. Convergence of the stream function and the temperature

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Figure 20. Relative error on mesh (e) of Table 18

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Figure 21. Set of meshes with 15% of non-Delaunay triangles

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Figure 22. Set of meshes with 25% of non-Delaunay triangles

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Figure 23. Set of meshes with 50% of non-Delaunay triangles

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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

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Figure 25. Evolution of the convergence rate with the ratio of non-Delaunay triangles

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Figure 26. Discretized ellipsoid. Non-well centered triangles are darkened

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Figure 27. Streamlines for $ t $ from 0 to 11

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Table 1. $ \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} $

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Table 2. 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

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Table 3. 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

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Table 4. 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

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Table 5. 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

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Table 6. 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} $

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Table 7. Unstructured meshes. Convergence rates of the stream function and the temperature

	Stream function	Temperature
Convergence rate	1.3690	1.1430

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Table 8. 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} $

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Table 9. 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

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