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Fluid Flows with Complex Free Surfaces 191 ρ vn+1 − v n+1/2 δt n − 2div ( µD(v n+1 ) ) + ∇p n+1 = f(t n+1 ) in Ω n+1 , (7) div v n+1 =0 inΩ n+1 , (8) where v n+1/2 is the prediction of the velocity obtained with (6) after the advection step. The boundary conditions on the free surface are given by (4). The weak formulation corresponding to (7), (8) and (4), therefore, consists in finding v n+1 and p n+1 such that v n+1 is vanishing on ∂Λ and v ρ ∫Ω n+1 − v n+1/2 ∫ δt n · w dx +2 µD(v n+1 ):D(w) dx n+1 Ω ∫ ∫ n+1 ∫ − p n+1 div w dx − f · w dx − q div v n+1 dx =0, (9) Ω n+1 Ω n+1 Ω n+1 for all test functions (w,q) such that w vanishes on the boundary of the cavity where essential boundary conditions are enforced. 2.3 A Two-Grids Method for Space Discretization Advection and diffusion phenomena being now decoupled, the equations (5) are first solved using the method of characteristics on a structured mesh of small cells in order to reduce numerical diffusion of the interface Γ t between the liquid and the gas, and have an accurate approximation of the liquid region, see Figure 2 (left). The bounding box of the cavity Λ is meshed into a structured grid made out of small cubic cells of size h, each cell being labeled by indices (ijk). Let ϕ n ijk and vn ijk be the approximate values of ϕ and v at the center of cell number (ijk) at time t n . The unknown ϕ n ijk is the volume fraction of liquid in the cell ijk and is the numerical approximation of the characteristic function ϕ at Fig. 2. Two-grids method. The advection step is solved on a structured mesh of small cubic cells composed of blocks whose union covers the physical domain Ω h (left), while the diffusion step is solved on a finite element unstructured mesh of tetrahedra (right).
192 A. Bonito et al. 3 1 16 4 1 1 16 4 0 0 1 1 4 9 1 16 4 3 1 16 4 0 ϕ 1 = 4 1 Fig. 3. Effect of the SLIC algorithm on numerical diffusion. An example of two dimensional advection and projection when the volume fraction of liquid in the cell is ϕ n ij = 1 . Left: without SLIC, the volume fraction of liquid is advected and 4 projected on four cells, with contributions (from the top left cell to the bottom right cell) 3 1 , 1 1 , 9 1 , 3 1 . Right: with SLIC, the volume fraction of liquid is 16 4 16 4 16 4 16 4 first pushed at one corner, then it is advected and projected on one cell only, with contribution 1 . 4 time t n , which is piecewise constant on each cell of the structured grid. The advection step for the cell number (ijk) consists in advecting ϕ n ijk and vn ijk by δt n v n ijk and then projecting the values on the structured grid, to obtain ϕn+1 ijk and a prediction of the velocity v n+ 1 2 ijk . A simple implementation of the SLIC (Simple Linear Interface Calculation) algorithm, described in [MPR03] and inspired by [NW76], allows to reduce the numerical diffusion of the domain occupied by the liquid by pushing the fluid along the faces of the cell before advecting it. The choice of how to push the fluid depends on the volume fraction of liquid of the neighboring cells. The cell advection and projection with SLIC algorithm are presented in Figure 3, in two space dimensions for the sake of simplicity. We refer to [AMS04] for a recent improvement of the SLIC algorithm. Remark 1. A post-processing technique allows to avoid the compression effects and guarantees the conservation of the mass of liquid. Related to global repair algorithms [SW04], this technique produces final values ϕ n+1 ijk which are between zero and one, even when the advection of ϕ n gives values strictly larger than one. The technique consists in moving the fraction of liquid in excess in the cells that are over-filled to receiver cells in a global manner by sorting the cells according to ϕ n+1 . Details can be found in [MPR99, MPR03]. Once values ϕ n+1 ijk and vn+1/2 ijk have been computed on the cells, values of the fraction of liquid ϕ n+1 P and of the velocity field v n+ 1 2 P are computed at the nodes P of the finite element mesh with approximated projection methods. We take advantage of the difference of refinement between a coarse finite element
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Partial Differential Equations
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Partial Differential Equations Mode
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Dedicated to Olivier Pironneau
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VIII Preface computers has been at
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Contents List of Contributors .....
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List of Contributors Yves Achdou UF
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List of Contributors XV Claude Le B
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Discontinuous Galerkin Methods Vive
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Discontinuous Galerkin Methods 5 2
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Discontinuous Galerkin Methods 7 g
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Discontinuous Galerkin Methods 9 (
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Discontinuous Galerkin Methods 15 W
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Let a h and b h denote the bilinear
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Discontinuous Galerkin Methods 19 t
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Discontinuous Galerkin Methods 21 l
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Table 1. Primal DG for transport Di
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Discontinuous Galerkin Methods 25 [
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Mixed Finite Element Methods on Pol
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Mixed FE Methods on Polyhedral Mesh
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4 Hybridization and Condensation Mi
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is symmetric and positive definite,
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with some coefficient α ∈ R wher
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u n+1 h Optimal Higher Order Time D
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