Partial Differential Equations - Modelling and ... - ResearchGate

More documents

Recommendations

Info

Fluid Flows with Complex Free Surfaces 197 function, defined as negative in the liquid region Ω t and equal to i in bubble B i (t). The approximations k n+1 , P n+1 i , B n+1 i , i =1, 2,...,k n+1 and ξ n+1 are computed as follows. Numbering of the Bubbles of gas Given the new liquid domain Ω n+1 , the key point is to find the number of bubbles k n+1 (that is to say the number of connected components) and the bubbles B n+1 i , i =1,...,k n+1 .GivenapointP in the gas domain Λ \ Ω n+1 , we search for a function u such that −∆u = δ P in Λ \ Ω n+1 , with u =0on Ω n+1 and u continuous. Since the solution u to this problem is strictly positive in the connected component containing point P and vanishes outside, the first bubble is found. The procedure is repeated iteratively until all the bubbles are recognized. The algorithm is written as follows: Set k n+1 =0,ξ n+1 =0inΛ \ Ω n+1 and ξ n+1 = −1 inΩ n+1 ,andΘ n+1 = {x ∈ Λ : ξ n+1 (x) =0}. While Θ n+1 ≠ ∅, do: 1. Choose a point P in Θ n+1 ; 2. Solve the following problem: Find u : Λ → R which satisfies: ⎧ ⎪⎨ −∆u = δ P , in Θ n+1 , u =0, in Λ \ Θ n+1 , (14) ⎪⎩ [u] =0, on ∂Θ n+1 , where δ P is Dirac delta function at point P ,[u] is the jump of u through ∂Θ n+1 ; 3. Increase the number of bubbles k n+1 at time t n+1 : k n+1 = k n+1 +1; 4. Define the bubble of gas number k n+1 : B n+1 k = {x ∈ Θ n+1 : u(x) ≠0}; n+1 5. Update the bubble numbering function ξ n+1 (x) =k n+1 , for all x ∈ B n+1 k ; n+1 6. Update Θ n+1 for the next iteration: Θ n+1 = {x ∈ Λ : ξ n+1 (x) =0}. The cost of this original numbering algorithm is bounded by the cost of solving k n+1 times a Poisson problem in the gas domain. The corresponding CPU time used to solve the Poisson problems is usually less than 10 percent of the total CPU time. This numbering algorithm is implemented on the finite element mesh. The Poisson problems (14) are solved on T h , using standard continuous, piecewise linear finite elements. Computation of the Pressure in the Gas Once the connected components of gas are numbered, an approximation P n+1 i of the constant pressure in bubble i at time t n+1 has to be computed with (13). In the case of a single bubble in the liquid, (13) yields P1 n+1 V1 n+1 = P1 n V1 n . In the case when two bubbles merge, this relation becomes P1 n+1 V1 n+1 = P1 n V1 n + P2 n V2 n . When a bubble B1 n splits onto two,
198 A. Bonito et al. each of its parts at time t n contributes to bubbles B1 n+1 and B2 n+1 .The volume fraction of bubble B1 n which contributes to bubble B n+1 j is noted V n+1/2 1,j , j =1, 2. The pressure in the bubble B n+1 j is computed by taking into account the compression/decompression of the two fractions of bubbles P n+1 j = P1 n V n+1/2 1,j /V n+1 j , j =1, 2. Details of the implementation require to take into account several situations, when two bubbles at time t n and t n+1 do or do not intersect between two time steps, and are detailed in [CPR05]. The value of the pressure can be inserted as a boundary term in (9) for the resolution of the generalized Stokes problem (7), (8). Remark 3. By using the divergence theorem in the variational formulation (9) and the fact that P n+1 is piecewise constant, the integral on the free surface Γ n+1 h is transformed into an integral on Ω n+1 h and, therefore, an approximation of the normal vector n is not explicitly needed. 3.3 Numerical Results Numerical results are presented here for mold filling simulations in order to show the influence of the gas pressure and to compare with results in Section 2.4. The same S-shaped channel is initially filled with gas at atmospheric pressure P = 101300 Pa. A valve is located at the upper extremity of the channel allowing gas to escape. Numerical results (cf. Figure 6) show the persistence of the bubbles. The CPU time for the simulations is approximately 344 min with the bubbles computations (to compare with 319 min in Section 2). Fig. 6. S-shaped channel: 3D results when the cavity is initially filled with compressible gas at atmospheric pressure. Time equals 8.0 ms,26.0 ms,44.0 ms and 53.9 ms.
Page 1 and 2:
Partial Differential Equations
Page 3 and 4:
Partial Differential Equations Mode
Page 5 and 6:
Dedicated to Olivier Pironneau
Page 7 and 8:
VIII Preface computers has been at
Page 9 and 10:
Contents List of Contributors .....
Page 11 and 12:
List of Contributors Yves Achdou UF
Page 13 and 14:
List of Contributors XV Claude Le B
Page 15 and 16:
Discontinuous Galerkin Methods Vive
Page 17 and 18:
Discontinuous Galerkin Methods 5 2
Page 19 and 20:
Discontinuous Galerkin Methods 7 g
Page 21 and 22:
Discontinuous Galerkin Methods 9 (
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Discontinuous Galerkin Methods 15 W
Page 29 and 30:
Let a h and b h denote the bilinear
Page 31 and 32:
Discontinuous Galerkin Methods 19 t
Page 33 and 34:
Discontinuous Galerkin Methods 21 l
Page 35 and 36:
Table 1. Primal DG for transport Di
Page 37 and 38:
Discontinuous Galerkin Methods 25 [
Page 39 and 40:
Mixed Finite Element Methods on Pol
Page 41 and 42:
Mixed FE Methods on Polyhedral Mesh
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
4 Hybridization and Condensation Mi
Page 49 and 50:
Page 51 and 52:
is symmetric and positive definite,
Page 53 and 54:
with some coefficient α ∈ R wher
Page 55 and 56:
44 E.J. Dean and R. Glowinski so fa
Page 57 and 58:
46 E.J. Dean and R. Glowinski 2 A L
Page 59 and 60:
48 E.J. Dean and R. Glowinski S:T=
Page 61 and 62:
50 E.J. Dean and R. Glowinski minim
Page 63 and 64:
52 E.J. Dean and R. Glowinski 6 On
Page 65 and 66:
54 E.J. Dean and R. Glowinski Fig.
Page 67 and 68:
56 E.J. Dean and R. Glowinski and
Page 69 and 70:
58 E.J. Dean and R. Glowinski 7 Num
Page 71 and 72:
60 E.J. Dean and R. Glowinski Fig.
Page 73 and 74:
62 E.J. Dean and R. Glowinski Assum
Page 75 and 76:
Higher Order Time Stepping for Seco
Page 77 and 78:
u n+1 h Optimal Higher Order Time D
Page 79 and 80:
Optimal Higher Order Time Discretiz
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
96 I. Sazonov et al. To provide a p
Page 105 and 106:
98 I. Sazonov et al. In the first s
Page 107 and 108:
100 I. Sazonov et al. Fig. 1. An ex
Page 109 and 110:
102 I. Sazonov et al. H z 1 exact F
Page 111 and 112:
104 I. Sazonov et al. (a) (b) Fig.
Page 113 and 114:
106 I. Sazonov et al. Scattering Wi
Page 115 and 116:
108 I. Sazonov et al. 6.4 Scatterin
Page 117 and 118:
110 I. Sazonov et al. (a) (b) Fig.
Page 119 and 120:
112 I. Sazonov et al. [MHP96] K. Mo
Page 121 and 122:
114 R. Sanders and A.M. Tesdall I R
Page 123 and 124:
116 R. Sanders and A.M. Tesdall imp
Page 125 and 126:
118 R. Sanders and A.M. Tesdall alo
Page 127 and 128:
120 R. Sanders and A.M. Tesdall (a)
Page 129 and 130:
122 R. Sanders and A.M. Tesdall D C
Page 131 and 132:
124 R. Sanders and A.M. Tesdall 8.6
Page 133 and 134:
126 R. Sanders and A.M. Tesdall 0.3
Page 135 and 136:
128 R. Sanders and A.M. Tesdall [TR
Page 137 and 138:
132 S. Lapin et al. Ω R γ Ω 2 Γ
Page 139 and 140:
134 S. Lapin et al. ∫ ∂ 2 ∫
Page 141 and 142:
136 S. Lapin et al. Ω R γ Fig. 3.
Page 143 and 144:
138 S. Lapin et al. 4 Energy Inequa
Page 145 and 146:
140 S. Lapin et al. 5 Numerical Exp
Page 147 and 148: 142 S. Lapin et al. Fig. 6. Contour
Page 149 and 150: 144 S. Lapin et al. Fig. 9. Obstacl
Page 151 and 152: Domain Decomposition and Electronic
Page 153 and 154: Domain Decomposition Approach for C
Page 169 and 170: Numerical Analysis of a Finite Elem
Page 181 and 182: so that |u| 2 1,ω h ≤ 2 Numerica
Page 189 and 190: 188 A. Bonito et al. of the model a
Page 191 and 192: 190 A. Bonito et al. Fig. 1. The sp
Page 193 and 194: 192 A. Bonito et al. 3 1 16 4 1 1 1
Page 195 and 196: 194 A. Bonito et al. ∫ v n+1 h
Page 197: 196 A. Bonito et al. −pn +2µD(v)
Page 201 and 202: 200 A. Bonito et al. The normal vec
Page 203 and 204: 202 A. Bonito et al. with initial c
Page 205 and 206: 204 A. Bonito et al. Fig. 8. Jet bu
Page 207 and 208: 206 A. Bonito et al. References [AM
Page 209 and 210: 208 A. Bonito et al. [Set96] J. A.
Page 211 and 212: 210 J. Hao et al. due to shear flow
Page 213 and 214: 212 J. Hao et al. The backward reac
Page 215 and 216: 214 J. Hao et al. where u and p den
Page 217 and 218: 216 J. Hao et al. * * * * * * * * *
Page 219 and 220: 218 J. Hao et al. and solve for V n
Page 221 and 222: 220 J. Hao et al. Table 2. The calc
Page 223 and 224: 222 J. Hao et al. References [ASS80
Page 225 and 226: Computing the Eigenvalues of the La
Page 227 and 228: Eigenvalues of the Laplace-Beltrami
Page 233 and 234: A Fixed Domain Approach in Shape Op
Page 235 and 236: Shape Optimization Problems with Ne
Page 243 and 244: Reduced-Order Modelling of Dispersi
Page 249 and 250:
Reduced-Order Modelling of Dispersi
Page 251 and 252:
Page 253 and 254:
Page 255 and 256:
Calibration of Lévy Processes with
Page 257 and 258:
Page 259 and 260:
Page 261 and 262:
Page 263 and 264:
We have proved Calibration of Lévy
Page 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Note that p ∗ satisfies Calibrati
Page 273 and 274:
Page 275 and 276:
280 S. Ikonen and J. Toivanen the p
Page 277 and 278:
282 S. Ikonen and J. Toivanen Merto
Page 279 and 280:
284 S. Ikonen and J. Toivanen For H
Page 281 and 282:
286 S. Ikonen and J. Toivanen and {
Page 283 and 284:
288 S. Ikonen and J. Toivanen 1.6 1
Page 285 and 286:
290 S. Ikonen and J. Toivanen 8 Con
Page 287:
292 S. Ikonen and J. Toivanen [Mer7
show all

Partial Differential Equations - Modelling and ... - ResearchGate

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?