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

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, Sophia-Antipolis eonore Gauci ∗ + ´ 1 El´ Mesh and Fluid Dynamics

Motivations Aim : Numerical Simulations for Fluid Dynamics. eonore Gauci ∗ + ´ 2 El´ Mesh and Fluid Dynamics

Contents 1/ Introduction to Numerical Simulation 2/ Mesh : Definition 3/ Mesh : Creation 4/ Mesh : Adaptation eonore Gauci ∗ + ´ 3 El´ Mesh and Fluid Dynamics

Numerical Simulation Principle of the Numerical Simulation : eonore Gauci ∗ + ´ 4 El´ Mesh and Fluid Dynamics

Numerical Simulation Principle of the Numerical Simulation : SOVE THE COMPARE/ CREATE THE NUMERICAL VALIDATE MESH SOLUTION THE RESULTS eonore Gauci ∗ + ´ 5 El´ Mesh and Fluid Dynamics

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

Simplicial Recovering and Triangulation Let S be a finite collection of points. A Simplicial Recovering from S is a set T r of elements K (triangles or tetrahedra), such that : Definition (Simplical Recovery) 1 The collection of vertices of T r is S . 2 ∀ K ∈ T r , ˚ K � = ∅ . 3 ∪ K ∈T r K ⊂ Conv ( S ) A such recovering is conform. This is a Triangulation if in addition Definition (Triangulation) 1 + 2 + 3+ ∀ K 1 , K 2 ∈ T r , K 1 ∩ K 2 = either ∅ or the joint vertex, edge, face. eonore Gauci ∗ + ´ 7 El´ Mesh and Fluid Dynamics

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 ∀ K 1 , K 2 ∈ M , K 1 ∩ K 2 = either ∅ or the joint vertex, edge, face. 3 Ω ⊂ M . Remark : In practise, M is a geometric approximation of Ω. eonore Gauci ∗ + ´ 8 El´ Mesh and Fluid Dynamics

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

Mesh Difficulties Mesh of a complex surface Mesh of the fluid around the body eonore Gauci ∗ + ´ 10 El´ Mesh and Fluid Dynamics

Mesh : Conform/Structured non conform conform and structured conform and unstructured eonore Gauci ∗ + ´ 11 El´ Mesh and Fluid Dynamics

Mesh � = Synthetic image Button details Famous character 3D-Modeling eonore Gauci ∗ + ´ 12 El´ Mesh and Fluid Dynamics

Mesh : Existence in 2D Convex polygon Concave polygon eonore Gauci ∗ + ´ 13 El´ Mesh and Fluid Dynamics

Mesh : Existence in 3D Sch¨ onhardt polyhedron eonore Gauci ∗ + ´ 14 El´ Mesh and Fluid Dynamics

Non-convex polyhedra combinatorially equiv. to regular polyhedra [Rambau 2005] Top : Octahedron/Sch¨ onhardt polyhedron Bottom : Icosahedron/Jessen polyhedron eonore Gauci ∗ + ´ 15 El´ Mesh and Fluid Dynamics

Mesh : Creation eonore Gauci ∗ + ´ 16 El´ Mesh and Fluid Dynamics

Mesh : Creation eonore Gauci ∗ + ´ 17 El´ Mesh and Fluid Dynamics

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

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

Mesh : Creation Advancing Front Mesh Generation Method eonore Gauci ∗ + ´ 20 El´ Mesh and Fluid Dynamics

Mesh : Creation Delaunay Mesh Generation Method Definition (Delaunay triangulation) A Delaunay triangulation for a set S of points in a plane is a triangulation T r such that no point in S is inside the circumcircle of any triangle in T r . eonore Gauci ∗ + ´ 21 El´ Mesh and Fluid Dynamics

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

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

Idea of adapted mesh eonore Gauci ∗ + ´ 24 El´ Mesh and Fluid Dynamics

Motivations Mesh need to capture the details of one solution ⇒ need to adapt the mesh to the solution . eonore Gauci ∗ + ´ 25 El´ Mesh and Fluid Dynamics

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 � 1 � u , v � M = t u M v ⇒ ℓ M ( a , b ) = � t ab M ( a + tab ) ab d t 0 √ � | K | M = det M d | K | K continuous Metric Field → discrete Mesh . eonore Gauci ∗ + ´ 26 El´ Mesh and Fluid Dynamics

Metric Field Metric Field Corresponding Mesh eonore Gauci ∗ + ´ 27 El´ Mesh and Fluid Dynamics

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 H opt L p having N vertices such that H opt L p = Argmin H || u − Π h u || H , L p (Ω) Well-posed Continuous mesh adaptation problem : Find M opt L p of complexity N such that �� � Trace ( M ( x ) − 1 2 | H u ( x ) |M ( x ) − 1 1 E L p ( M opt 2 ) p d x L p ) = min M p Ω ⇒ Solved by variational calculus eonore Gauci ∗ + ´ 28 El´ Mesh and Fluid Dynamics

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 eonore Gauci ∗ + ´ 29 El´ Mesh and Fluid Dynamics

Unsteady Feature-based mesh adaptation in : Shuttle eonore Gauci ∗ + ´ 30 El´ Mesh and Fluid Dynamics

Unsteady Feature-based mesh adaptation in : Vortex shedding eonore Gauci ∗ + ´ 31 El´ Mesh and Fluid Dynamics

Thank you for your attention eonore Gauci ∗ + ´ 32 El´ Mesh and Fluid Dynamics