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Fluid Flows with Complex Free Surfaces 195 Fig. 5. S-shaped channel: 3D results when the cavity is initially filled with vacuum. Time equals 8.0 ms,26.0 ms,44.0 ms and 53.9 ms. constant velocity 8.7 m/s. Density and viscosity are taken to be respectively ρ = 1000 kg/m 3 and µ =0.01 kg/(ms). Slip boundary conditions are enforced to avoid boundary layers and a turbulent viscosity is added, the coefficient α T being equal to 4h 2 , as proposed in [CPR05]. Since the ratio between Capillary number and Reynolds number is very small, surface tension effects are neglected. The final time is T =0.0054 s and the time step is τ =0.0001 s. The mesh is made out of 96030 elements. In Figure 5, 3D computations are presented when a valve is placed at the end of the cavity, thus allowing the gas to exit. The CPU time for the simulations in three space dimensions is approximately 319 minutes for 540 time steps. Most of the CPU time is spent to solve the Stokes problem. A comparison with experimental results shows that the bubbles of gas trapped by the liquid vanish too rapidly. In order to obtain more realistic results, the effect of the gas compressibility onto the liquid must be considered. This is the scope of the next section. 3 Extension to the Modeling of an Incompressible Liquid Surrounded by a Compressible Gas 3.1 Extension of the Model In Section 2, the zero force condition (4) was applied on the liquid-gas interface. Going back to the simulation of Figure 5, this corresponds to filling with liquid a cavity under vacuum. When considering industrial mold filling processes, the mold is not initially under vacuum, but contains some compressible air that interacts with the liquid. Therefore, the model has to be extended. The velocity in the gas is disregarded here, since it is CPU time consuming to solve the Euler compressible equations in the gas domain. The model presented in Section 2 is extended by adding the normal forces due to the gas pressure on the free surface Γ t , still neglecting tangential and capillary forces. The relationship (4) is replaced by
196 A. Bonito et al. −pn +2µD(v)n = −P n on Γ t , t ∈ (0,T), (12) where P is the pressure in the gas. For instance, consider the experiment of Figure 5 where the cavity is being filled with liquid. The gas present in the cavity at initial time can either escape if a valve is placed at the end of the cavity (in which case the gas does exert very little resistance on the liquid) or be trapped in the cavity. When a bubble of gas is trapped by the liquid, the gas pressure prevents the bubble to vanish rapidly, as it is the case for vacuum. The pressure in the gas is assumed to be constant in space in each bubble of gas, that is to say in each connected component of the gas domain. Let k(t) be the number of bubbles of gas at time t and let B i (t) denote the domain occupied by the bubble number i (the i-th connected component). Let P i (t) denote the pressure in B i (t). At initial time, P i (0) is constant in each bubble i. The gas is assumed to be an ideal gas. If V i (t) is defined as the volume of B i (t), the pressure in each bubble at time t is thus computed by using the law of ideal gases at constant temperature: P i (t)V i (t) =constant i =1,...,k(t). (13) The above relationship is an expression of the conservation of the number of molecules of trapped gas (gas that cannot escape through a valve) between time t and t + δt. However, this simplified model requires the tracking of the position of the bubbles of gas between two time steps. When δt is small enough, three situations appear between two time steps: first, a single bubble remains a single bubble; or a bubble splits into two bubbles, or two bubbles merge into one. Combinations of these three situations may appear. For instance, in the case of a single bubble, if the pressure P (t) inthe bubble at time t and the volumes V (t) andV (t + δt) are known, the gas pressure at time t+δt is easily computed from the relation P (t+δt)V (t+δt) = P (t)V (t). The other cases are described at the discrete level in the following. Details can be found in [CPR05]. The additional unknowns in our model are the bubbles of gas B i (t) and the constant pressure P = P i (t) in the bubble of gas number i. The equations (1)–(3) are to be solved together with (12), (13). 3.2 Modification of the Numerical Method The tracking of the bubbles of gas and the computation of their internal pressure introduce an additional step in our time splitting scheme. This procedure is inserted between the advection step (6) and the diffusion step (7), (8), in order to compute an approximation of the pressure to plug into (12). Let us denote by k n , P n i , Bn i , i =1, 2,...,kn , the approximations of k, P i , B i , i =1, 2,...,k, respectively at time t n .Letξ(t) beabubble numbering
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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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44 E.J. Dean and R. Glowinski so fa
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46 E.J. Dean and R. Glowinski 2 A L
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48 E.J. Dean and R. Glowinski S:T=
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50 E.J. Dean and R. Glowinski minim
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52 E.J. Dean and R. Glowinski 6 On
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54 E.J. Dean and R. Glowinski Fig.
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56 E.J. Dean and R. Glowinski and
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58 E.J. Dean and R. Glowinski 7 Num
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60 E.J. Dean and R. Glowinski Fig.
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62 E.J. Dean and R. Glowinski Assum
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Higher Order Time Stepping for Seco
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u n+1 h Optimal Higher Order Time D
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Optimal Higher Order Time Discretiz
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96 I. Sazonov et al. To provide a p
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98 I. Sazonov et al. In the first s
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100 I. Sazonov et al. Fig. 1. An ex
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102 I. Sazonov et al. H z 1 exact F
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104 I. Sazonov et al. (a) (b) Fig.
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106 I. Sazonov et al. Scattering Wi
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108 I. Sazonov et al. 6.4 Scatterin
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110 I. Sazonov et al. (a) (b) Fig.
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112 I. Sazonov et al. [MHP96] K. Mo
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114 R. Sanders and A.M. Tesdall I R
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116 R. Sanders and A.M. Tesdall imp
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118 R. Sanders and A.M. Tesdall alo
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120 R. Sanders and A.M. Tesdall (a)
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122 R. Sanders and A.M. Tesdall D C
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124 R. Sanders and A.M. Tesdall 8.6
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126 R. Sanders and A.M. Tesdall 0.3
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128 R. Sanders and A.M. Tesdall [TR
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132 S. Lapin et al. Ω R γ Ω 2 Γ
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134 S. Lapin et al. ∫ ∂ 2 ∫
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136 S. Lapin et al. Ω R γ Fig. 3.
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138 S. Lapin et al. 4 Energy Inequa
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Reduced-Order Modelling of Dispersi
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Calibration of Lévy Processes with
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We have proved Calibration of Lévy
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280 S. Ikonen and J. Toivanen the p
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