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

  • * 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

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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  • 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.

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

    Citation:

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

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

    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

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

    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 $

    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

    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

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

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

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

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

    Figure 12.  Poiseuille flow: error on $ u $

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

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

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

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

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

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

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

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

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

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

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

    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

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

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

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

    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} $
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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} $
     | Show Table
    DownLoad: CSV

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

    Stream function Temperature
    Convergence rate 1.3690 1.1430
     | Show Table
    DownLoad: CSV

    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} $
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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