Imagery for 3D geometry design: application to fluid flows.

Transcription

1 Imagery for 3D geometry design: application to fluid flows. C. Galusinski, C. Nguyen IMATH, Université du Sud Toulon Var, Supported by ANR Carpeinter May 14, 2010

2 Toolbox Ginzburg-Landau. Skeleton 3D extension Boundary condition Super-Skeleton Geometry from Skeleton Model examples

3 Toolbox Ginzburg-Landau. An imagery soft C++ with wxwidgets, OpenGL (Mesa) VTK. Tools: Contrast. Ginzburg-Landau. Connected component (Scanning, Front propagation). Skeleton.

4 Toolbox Ginzburg-Landau. Ginzburg-Landau. t u L 2 u = u(u 1)(u θ). with Neuman boundary conditions. forcing black (u = 0) and white (u = 1). thickening {u = 0}: choose θ > 1 2. until {u = 0} is connected sliming {u = 0}:(choose θ < 1 2 ).

5 Example Toolbox Ginzburg-Landau.

6 Example Toolbox Ginzburg-Landau.

7 Contrast Toolbox Ginzburg-Landau.

8 Ginzburg-Landau Toolbox Ginzburg-Landau.

9 Ginzburg-Landau Thikening Toolbox Ginzburg-Landau.

10 Ginzburg-Landau Sliming Toolbox Ginzburg-Landau.

11 Image processing result Toolbox Ginzburg-Landau.

12 Defintion Skeleton 3D extension Let Ω d R d, the skelton of Ω d is the smallest set S such that Ω d = x S B d (x, r(x)), with maximal radius r(x). B d (x, r(x)) is the bigger ball centered in x included in Ω d. Ω d = x Ωd B d (x, r(x)), with r(x) = dist(x, Ω d ). If x S, then B d (x, r(x)) is tangent to Ω d in two points at least

13 Example Skeleton 3D extension

14 2D-3D distance function Skeleton 3D extension The following function ψ is associated to the skeleton: y R d, ψ(y) = inf { x y r(x)}, (1) x S The function ψ: the signed distance Level Set function to Ω d. d = 2: ψ is known the Skeleton S is computed (set points where ψ is singular and ψ < 0) d = 3: ψ is evaluated.

15 Example continued Skeleton 3D extension

16 Skeleton 3D extension Computation of skeleton Imagery processing black (0) and white (1) image u 0 = 0 in black region, u 0 = 1 in white region Solve Eikonal equation with Fast Marching Method: ψ = 1 where u 0 = 1 ψ = 0 where u 0 = 0. Compute discrete gradients: ++ ψ, + ψ, + ψ, ψ Normalize discrete gradients and compute minimal scalar product PS. if PS < 0.8 the point belongs to Skeketon!

17 Example Skeleton 3D extension

18 Flow rate or pressure Boundary condition Super-Skeleton Geometry from Skeleton Dirichlet boundary conditions: - the user precises the flow rate near inlet or outlet - the flow rate is associated to points on skeleton - the velocity field is given with parabolic profile depending on ψ ( Poiseuille flow ): The direction of the skeketon and ψ give the velocity field direction Pressure can be imposed instead of velocities.

19 Skeleton direction Boundary condition Super-Skeleton Geometry from Skeleton The direction of the skeleton, for x S: ds(x) = 1ψ 1 ψ + 2ψ 2 ψ or ds(x) = ( 1ψ) + ( 2 ψ) ψ is singular, 1, 2 are such that 1ψ 2 ψ 1 ψ 2 ψ is minimal.

20 Super-Skeleton Boundary condition Super-Skeleton Geometry from Skeleton position: x R 3 radius: r(x) R + flow rate: d(x) R direction: ds(x) R 3 transverse direction: dst(x) R 3 choice of L p norm: p(x) 1 The L p norm is the chosen distance in the transverse direction: p = 2 for circular section p = 1 or p = for square section... (design of the unit ball in R 2 ).

21 Geometry from Skeleton Boundary condition Super-Skeleton Geometry from Skeleton Rotation on transverse direction of Super-Skeleton with p = 1:

22 Geometry from Skeleton Boundary condition Super-Skeleton Geometry from Skeleton Rotation on transverse direction of Super-Skeleton with p = 1:

23 Model Model examples ( ) u ρ t + u. u.(2ηd u) + p = F (t, x) R + Ω, (2) With the incompressibility condition :. u = 0 (t, x) R + Ω, (3) where the field u = (u, v, w) is the velocity, p the pressure, ρ the density, η the viscosity, F any body force detailed hereafter and D u = ( u + T u)/2.

24 Bifluid Model Model examples Incompressible two-phase flows (Sussman, Smereka and Osher (94)) for two-phase flows: Phases are located by the sign of a Level Set function φ. φ t + u. φ = 0 (t, x) R+ Ω. (4) Forces are gravity and surface tension: F σ = ρ g + σκδ(φ) n (5)

25 Model examples Complete model ( ) u ρ(φ) t + u. u.(2ηd u) + 1 H( ψ) u + p ε. u = 0 u = 0 if ψ 0, = ρ g + σκ H(φ), (t, x) R + B (t, x) R + B (t, x) R + B u = u b if ψ < 0, (t, x) R + B, (6) where u b is defined thanks to direction of Skeleton.

26 Discretization Model examples Time Discretization ρ(ϕ n ) un+1 u n t div u n+1 = 0, κ n = div ϕn ϕ n, ϕ n+1 ϕ n + u n+1 ϕ n = 0. t div(ν(ϕ n )( u n+1 + ( u n+1 ) t ) + ρ(ϕ n ) u n u n + p n+1 = σκ n (H(ϕ n )), Augmented Lagrangian for incompressibility.

27 Model examples Space Discretization Cartesian uniform grid on a box containing the domain. Discretization on uniform (MAC) staggered grid for fluid solver. WENO5 (G-S. JIANG, D. PENG) scheme for transport of smooth function (signed distance Level Set function) on grids 3 times thiner.

28 Numerical stability Model examples Proposition (C.G., P. Vigneaux 08) For low Reynolds, the above numerical scheme is stable under the condition: t min ( t c, t σ ), avec t c = c 0 u 1 L (Ω) x et t σ = 1 ( ) η c 2 2 σ x + η ρ (c 2 σ x)2 + 4c 1 σ x 3 where t is the time step, x the space step and c 0 c 1, c 2 do not depend on physical and numerical parameter.

29 Numerical stability Model examples Known time step: Brackbill (BKZ) capillary time step and Capillary time step for Stokes t BKZ = c 1 ρ σ x 3 = t σ (ρ, 0), t STK = c 2 η σ x = t σ(0, η). Remark: t σ max( t STK, t BKZ ). (7)

30 Numerical instability example Model examples

31 Stable flow Model examples

32 Examples Model examples

33 Examples Model examples

34 Examples Model examples

35 Examples Model examples

36 Examples Model examples

37 Examples Model examples

38 Examples Model examples

39 Examples Model examples Asymetric flow in Y

40 Examples Model examples Symetric flow in Y

41 Examples Model examples Symetric flow in Y

42 Conclusion Skeleton and Level Set for Geometries: A simple tool for complex geometries, no mesh to fit the geometries Level Set for fluid interfaces: simple and efficient thiner grid for Level Set than for flow (very low Reynolds) A user friendly interactive imagery software for geometry generation and flow informations

43 Improvement AMR An adaptative mesh refinement to reduce computation cost A new fluid solver adapted to AMR (being developped with DDFV scheme) DDFV schemes (F. Hubert et al., F. Boyer...): - allow nonconforming meshes - verify a discrete variationnal formulation - verify exact discrete Green formula ( u, p) L 2 = (u, p) L 2 - MAC scheme generalization -verify flux continuity (good for viscosity discontinuity) Modified boundary conditions... ψ u ψ + u = 0 on walls...

44 Improvement Vessel reconstruction Extend Vessel reconstruction to really 3D geometries use normal cuts of vessel on medical images connect the 3D Skeleton between normal cuts Skeleton defined by a Monge-Kantorovich problem Developpement of an imagery software (user friendly) Non Newtonian flows...