Mesh and Fluid Dynamics eonore Gauci + El eric Alauzet + and - - PowerPoint PPT Presentation

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Mesh and Fluid Dynamics eonore Gauci + El eric Alauzet + and - - PowerPoint PPT Presentation

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Mesh and Fluid Dynamics eonore Gauci + El eric Alauzet + and Alain Dervieux Supervisors : Fr ed ( + ) INRIA, Team Gamma3, Saclay, France ( ) INRIA, Team Ecuador, Sophia-Antipolis, France Ph.D Seminar November, 21 st 2016,

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Mesh and Fluid Dynamics

´ El´ eonore Gauci∗+

Supervisors : Fr´ ed´ eric Alauzet+ and Alain Dervieux∗ (+) INRIA, Team Gamma3, Saclay, France (∗) INRIA, Team Ecuador, Sophia-Antipolis, France

Ph.D Seminar November, 21st 2016, Sophia-Antipolis

1 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Motivations

Aim : Numerical Simulations for Fluid Dynamics.

2 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Contents

1/ Introduction to Numerical Simulation 2/ Mesh : Definition 3/ Mesh : Creation 4/ Mesh : Adaptation

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Numerical Simulation

Principle of the Numerical Simulation :

4 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Numerical Simulation

Principle of the Numerical Simulation :

CREATE THE MESH SOVE THE NUMERICAL SOLUTION COMPARE/ VALIDATE THE RESULTS

5 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Mesh : Definition

6 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Simplicial Recovering and Triangulation

Let S be a finite collection of points. A Simplicial Recovering from S is a set Tr of elements K (triangles or tetrahedra), such that : Definition (Simplical Recovery)

1 The collection of vertices of Tr is S. 2 ∀K ∈ Tr, ˚

K = ∅.

3 ∪K∈Tr K ⊂ Conv(S)

A such recovering is conform. This is a Triangulation if in addition Definition (Triangulation) 1 + 2 + 3+ ∀K1, K2 ∈ Tr, K1 ∩ K2 = either ∅ or the joint vertex, edge, face.

7 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Mesh : Definition

Let Ω be a domain we want to mesh. Ω is a given constraint. A conform Mesh for Ω is a set M of elements K (triangles or tetrahedra) which satisfies : Definition (Mesh)

1 ∀K ∈ M, ˚

K = ∅.

2 ∀K1, K2 ∈ M, K1 ∩ K2 = either ∅ or the joint vertex, edge,

face.

3 Ω ⊂ M.

Remark : In practise, M is a geometric approximation of Ω.

8 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Folkloric Meshes

9 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Mesh Difficulties

Mesh of a complex surface Mesh of the fluid around the body

10 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Mesh : Conform/Structured

non conform conform and structured conform and unstructured

11 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Mesh = Synthetic image

Famous character 3D-Modeling Button details

12 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Mesh : Existence in 2D

Convex polygon Concave polygon

13 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Mesh : Existence in 3D

Sch¨

  • nhardt polyhedron

14 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Non-convex polyhedra combinatorially equiv. to regular polyhedra [Rambau 2005]

Top : Octahedron/Sch¨

  • nhardt polyhedron

Bottom : Icosahedron/Jessen polyhedron

15 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Mesh : Creation

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Mesh : Creation

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Mesh : Quality

18 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Mesh : Quality

19 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Mesh : Creation

Advancing Front Mesh Generation Method

20 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Mesh : Creation

Delaunay Mesh Generation Method Definition (Delaunay triangulation) A Delaunay triangulation for a set S of points in a plane is a triangulation Tr such that no point in S is inside the circumcircle

  • f any triangle in Tr.

21 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Mesh : Creation

22 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Anisotropic Mesh

Isotropic Mesh Anisotropic Mesh

23 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Idea of adapted mesh

24 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Motivations

Mesh need to capture the details of one solution ⇒ need to adapt the mesh to the solution .

25 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Mesh Adaptation

Main idea : introduce the use of metrics field, and notion of unit mesh.

[George, Hecht and Vallet., Adv Eng. Software 1991]

Riemannian metric space: M : d × d symmetric definite positive matrix u, vM =t uMv ⇒ ℓM(a, b) = 1

  • tab M(a + tab) ab dt

|K|M =

  • K

√ det M d|K|

continuous Metric Field→ discrete Mesh.

26 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Metric Field

Metric Field Corresponding Mesh

27 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Feature-based Mesh Adaptation

Deriving the best mesh to compute the characteristics of a given solution w in space

[Tam et al.,CMAME 2000], [Picasso, SIAMJSC 2003], [Formaggia et al, ANM 2004 ], [Frey and Alauzet CMAME 2005], [Gruau and Coupez, CMAME 2005 ], [Huang, JCP 2005 ], [Compere et al., 2007 ]

Discrete mesh adaptation problem : Find Hopt

Lp having N vertices such that

Hopt

Lp = ArgminH||u − Πhu||H,Lp(Ω)

Well-posed Continuous mesh adaptation problem : Find Mopt

Lp of complexity N such that

ELp(Mopt

Lp ) = minM

  • Ω

Trace(M(x)− 1

2 |Hu(x)|M(x)− 1 2 )pdx

  • 1

p

⇒ Solved by variational calculus

28 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics

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Unsteady Feature-based mesh adaptation in : Two F117s crossing flight paths

Two planes moved at Mach 0.4 inside an inert air. The planes are translated and rotating 50 sub intervals and 3 adaptation loops Total space time complexity: 36, 000, 000 vertices, average mesh size: 732, 000 vertices, 80, 000 timesteps

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Unsteady Feature-based mesh adaptation in : Shuttle

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Unsteady Feature-based mesh adaptation in : Vortex shedding

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Thank you for your attention

32 ´ El´ eonore Gauci∗+ Mesh and Fluid Dynamics