ECCOMAS 2012

A second-order cell-centered finite volume scheme for anisotropic diffusion on three-dimensional unstructured meshes

´ ECCOMAS 2012 | JACQ Pascal, MAIRE Pierre-Henri, ABGRALL Remi, CLAUDEL Jean 10 SEPTEMBER 2012

Overview

1

Introduction

2

Finite Volume scheme for diffusion on 3D meshes

3

Numerical results

4

Conclusions and perspectives

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 0/23

Context and objectives Aerothermal coupling during atmospheric reentry Hypersonic flow around the solid : Navier Stokes equations Removal of material from the surface of the solid : simplified ablation model Thermal transfer in the solid : anisotropic heat equation

F IGURE: Intermediate eXperimental Vehicle

F IGURE: Mars descent vehicule

Development of a 3D anisotropic diffusion scheme 2nd order Finite Volume scheme Unstructured meshes to handle complex geometries Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 1/23

Introduction : 3D diffusion scheme Needs Handle complex geometries : unstructured meshes Handle multi-materials : cell-centered (material interfaces defined by the mesh)

The origins CCLAD scheme : Cell-Centered LAgrangian Diffusion Developped to solve diffusion equations in the context of Lagrangian hydrodynamics

2D isotropic diffusion J. Breil and P.-H. Maire. A cell-centered diffusion scheme on two-dimensional unstructured meshes. J. Comp. Phys., 224(2) :785-823

Anisotropic diffusion - Cylindrical geometry J. Breil and P.-H. Maire. A nominally second-order accurate finite volume cell-centered scheme for anisotropic diffusion on two-dimensional unstructured grids. J. Comp. Phys., 231(5) : 2259-2299 Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 2/23

Introduction : 3D diffusion scheme

Finite volume scheme for anisotropic diffusion Extension of the CCLAD scheme to 3D geometries for meshes made of tetrahedra hexahedra pyramids prisms

Parallel (MPI) version of the algorithm Numerical validation with numerous test cases

Work in progress : Development of an ALE method (Arbitrary Lagrangian-Eulerian) Finite Volume scheme on a moving mesh to take into account the ablation phenomenon Boundary motion comes from the ablation model Mesh motion obtained by solving the linear elastodynamic equation

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 3/23

Developpement of a 3D diffusion scheme Diffusion equation in the domain D ⊂ R3 ∂T +∇·q =0 ∂t T (x, t) = T 0 (x),

(x, t) ∈ D × [0, T],

ρC

x ∈ D,

T (x, t) = T ? (x, t),

x ∈ ∂DD , Dirichlet,

q(x, t) · n = qN? (x, t),

x ∈ ∂DN , Neumann,

αT (x, t) + βq(x, t) · n = qR? (x, t),

x ∈ ∂DR , Robin

Fourier’s law for the heat flux q = −K∇T , where K (conductivity tensor) is positive definite : Kφ · φ ≥ 0 for all φ ∈ D.

Continuity of the normal flux at the interfaces q 1 · n12 = q 2 · n12 . n12 is the unit normal at the interface between the materials 1 and 2. Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 4/23

3D diffusion scheme

Finite volume method ∂T + ∇ · q = 0 over a cell ωc ∂t Z d We obtain mc Cvc Tc + q·n =0 dt ∂ωc

p

We integrate ρCv

n~3

ωpc

n~1

n~2

2 ∂ωpc

1 ∂ωpc

Notations Cell’s temperature : Tc (t) =

1 |ωc |

R

1 |ωc |

R

ωc

T (x, t) dv

ωc

Cell’s heat capacity : Cvc = ωc Cv (x) dv R 1 Cell’s density : ρc = |ω | ωc ρ(x) dv c

Cell’s mass : mc = ρc | ωc |

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 5/23

3D diffusion scheme : Sub-cell discretization Notations around a point p

Flux discretization

p

It remains to express the flux Z q·n

n~3

ωpc

n~1

n~2

∂ωc 2 ∂ωpc

1 ∂ωpc

To do so we will introduce a sub-cell discretization [ ωc = ωpc

ωc

p∈P(c)

where P(c) is the set of nodes in cell c where We introduce the Sub-Face Normal Flux Z 1 f qpc = f q · n ds f Spc ∂ωpc

f ∈ F (p, c) the set of sub-faces in cell c inpinging on node p f = |∂ω f | Spc pc

The sub-faces are triangulated to f obtain n and Spc Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 6/23

3D diffusion scheme

With these notations we have : Z q·n = ∂ωc

X

X

f f Spc qpc

p∈P(c) f ∈F (p,c)

where P(c) is the list of nodes of the cell c F (p, c) is the list of sub-faces of the cell c connected to the node p

Semi-discretization The scheme writes : m c Cc

X d Tc + dt

X

f f Spc qpc =0

p∈P(c) f ∈F (p,c)

f We will now express the normal fluxes qpc

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 7/23

3D diffusion scheme : the normal fluxes Notations p ωpc 1 Tpc

Auxiliary unknowns 2 Tpc

S1

We introduce the sub-faces Temperatures Z 1 f Tpc = f T (x, t) ds. f Spc ∂ωpc

S2

Tc ωc

Sub-faces Normal Flux Approximation We would like to express the flux under the form F

f f 1 qpc = hpc (Tpc − Tc , ..., Tpcpc − Tc )

We will use a local Variationnal Formulation over the sub-cell ωpc the continuity conditions to eliminate the sub-faces temperatures Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 8/23

3D diffusion scheme : Variational formulation From the heat flux definition :

K −1 q + ∇T = 0

We multiply by a test function φ and integrate by part over ωpc

Sub-cell variational formulation Z

φ · K −1 qdv = Tc

Z

Z φ · nds − ∂ωpc ∩∂ωc

ωpc

T φ · nds ∂ωpc ∩∂ωc

Leads to : Z ωpc

Z

φ · K −1 qdv = wpc φpc · K c−1 q pc

Z φ · nds −

Tc ∂ωpc ∩∂ωc

T φ · nds = − ∂ωpc ∩∂ωc

X

f f Spc (Tpc − Tc )φfpc

f ∈F (p,c)

Where wpc is a volume weight wpc φpc · K −1 c q pc = −

X

f f Spc (Tpc − Tc )φfpc

f ∈F (p,c) Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 9/23

(1) (2)

3D diffusion scheme : Variational formulation Normal component decomposition Let φ ∈ R3 a vector and φpc its piecewise approximation over the sub-cell ωpc  1  We define φfpc as φpc · nfpc = φfpc φpc h i   Fpc −t  ..  1 Let’s note J pc = npc , . . . , npc so φpc = J pc  .  F φpcpc Makes sense only when Fpc = 3, it works with tetrahedra, hexahedra, prisms but not in pyramids or more generic meshes. The sub-cell variational formulation becomes  1   1   1 1 − T ) φ1  φpc Spc (Tpc c  −1 q pc pc t 2 2 2 2 q pc  · φpc  = − Spc (Tpc − Tc ) · φ2pc  wpc J pc K c J pc 3 (T 3 − T ) φ3pc Spc φ3pc q 3pc c pc Finally we get :

With αpc =

1 wpc

  1 1 − T ) q 1pc Spc (Tpc c 2 (T 2 − T ) q 2pc  = −αpc K pc Spc c pc 3 (T 3 − T ) Spc q 3pc c pc 

and K pc = J tpc K c J pc , K pc is positive definite iff K c is positive definite Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 10/23

3D diffusion scheme : sub-face temperature elimination Continuity conditions Normal flux is continuous on every sub-face Temperature is continuous on every sub-face If c and d are the cells surrounding a face f , the continuity conditions of the normal flux writes : f f f f qpd =0 Spc qpc + Spd

The continuity of the temperature writes : f f Tpc = Tpd = TF

where F is the position of the sub-face f in the list of sub-faces surrounding the node p We can write a system of equations under the form MT = ST with T = (T1 , . . . , TCp )t the vector of cell-centered temperatures around node p T = (T 1 , . . . , T Fp )t the vector of sub-face temperatures around node p M is a Fp × Fp matrix, Fp beeing the number of sub-faces surrounding node p S is a Fp × Cp matrix, Cp beeing the number of sub-cells surrounding node p Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 11/23

3D diffusion scheme : Details on local matrices Definition of matrix S S has 2 terms on each row (2 cells around a face) : SFc = αpc

3 X

fg

g

f Spc K pc Spc

SFd = αpd

g=1

3 X

fg

g

f Spd K pd Spd

g=1

e Definition of matrix S P f qf We are interested by the quantity Qpc = 3f =1 Spc pc   It rewrites 3 3 X X fg g ¯ F f  Qpc = − Spc αpc K pc Spc (T − Tc ) g=1

f =1

=−

3 X g=1

=−

3 X

αpc

3 X

! g

fg

f (Spc K pc Spc )

¯ F − Tc ) (T

f =1

e t (T ¯F S Fc

− Tc )

g=1

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 12/23

3D diffusion scheme Scheme final expression mc Cc

X d Tc + dt

X

p

Γc,d (Td − Tc ) = 0

p∈P(c) d∈C(p)

With

e t M −1 S Γp = S

The global system writes : MCv

dT + AT = 0 dt

Global diffusion matrix A The matrix assembly is made by adding the contributions obtained at each node The global matrix is sparse, positive semi-definite (and symetric iff K symetric) The semi-discrete scheme is stable for L2 norm

Time discretization Implicit scheme, first order in time Global system is solved using iterative methods (PETSc library) Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 13/23

3D diffusion scheme : dealing with meshes where Fpc > 3 The pyramidal problem p n~4

n~3

ωpc

n~1

n~2

2 ∂ωpc

1 ∂ωpc

The top node has 4 faces surrounding it : the normal component decomposition leads to an overdeterminated system We don’t want to change all the scheme

ωc

By dividing every problematic sub-cell in tetrahedra we can express the normal component decomposition

sub-cell sub-discretization For each sub-face we introduce a fictious sub-cell p

The sub-cell has 3 faces : The sub-face, and 2 fictious faces

n~1 T1

T˜ 1

1 ∂ωpc

ωc

For each fictious sub-cell we introduce a fictious cell temperature, and for each fictious sub-face we introduce a fictious face temperature We build the local matrices with theses fictious cells and faces : the normal component decomposition now exists We add only the contributions of the real sub-faces to the global system, the contributions of the fictious-faces are eliminated Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 14/23

Numerical results : Stationnary test case

Test case 2 materials In this test we have 2 isotropic materials defined as : K = K1 = 1 for x ≤

1 2

K = K2 = 10 for x >

1 2

The boundary conditions are : Dirichlet at x = 0 T = 0, at x = 1 T = 1 Neumann at y = 0 and y = 1 φ = 0 The solution is : T (x, y) =

2K2 x K1 +K2

T (x, y) =

K2 −K1 K1 +K2

for x ≤

+

1 2

2K1 x K1 +K2

for x >

1 2

This test show the ability of our scheme to preserve piecewise linear solutions

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 15/23

Numerical results : Stationnary test case

Results on tetrahedral and hexahedral meshes

Scheme order on tetrahedral and hexahedral meshes Round-off error obtained

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 16/23

Numerical results on hybrid meshes : Hexahedra - Tetrahedra - Pyramids Test case 2 materials

Scheme order on an hybrid mesh h 1.00e − 01 5.00e − 02 2.50e − 02

E∞ 3.75e − 04 1.88e − 04 9.44e − 05

q∞ 0.99 1.00 −

EL2 4.83e − 05 1.71e − 05 6.04e − 06

qL2 1.50 1.50 −

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 17/23

Numerical results : Stationnary test case Test case homogeneous boundary conditions Isotropic material with a conductivity K = 1 On the boundary q · n = 0 We add a source term : r (x, y) =

cos(1) − 1 cos(x) + sin(x) + T (x, y) sin(1)

The solution is : T (x, y) = −x +

cos(1) − 1 cos(x) + sin(x) sin(1)

Scheme order on kershaw meshes h 1.00e − 01 5.00e − 02 2.50e − 02

E∞ 4.48e − 03 1.14e − 03 4.09e − 04

q∞ 1.97 1.48 −

EL2 1.43e − 03 3.70e − 04 1.06e − 04

qL2 1.95 1.80 −

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 18/23

Numerical results : Sphere Definition of the test The domain is a truncated sphere with an internal radius Ri = 0.1 and an external radius Re = 1. The temperatures are imposed on the boundaries : T (Ri ) = Ti = 0 and T (Re ) = Te = 1. T −T Analytic solution : T (r ) = Ti + Re Ri ( R1 − 1r ) Re −Ri . i

Contours of T

e

i

T versus rc =

p

xc2 + yc2 + zc2

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 19/23

Numerical results : Sphere Convergence analysis h 2.96e − 03 2.38e − 03 1.76e − 03 1.16e − 03

E∞ 3.64e − 03 2.33e − 03 1.34e − 03 5.90e − 04

q∞ 2.02 1.84 1.97 −

EL2 2.49e − 04 1.57e − 04 8.50e − 05 3.69e − 05

qL2 2.10 2.03 2.01 −

Series of anisotropic meshes made of N tetrahedra N = 81862; 156045; 379666; 1296987. Only

1 8

of the sphere was meshed

We applied homogeneous boundary condition on the symmetry planes The characteristic size of the mesh is computed using the following formula :  h=

Vtruncated sphere

1

3

N Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 20/23

Parallelization MPI Every processor owns different part of the mesh, they exchange data at the interfaces Mesh partitionning (Scotch https ://gforge.inria.fr/projects/scotch/ ) + overlaps Matrix construction : we just compute the local rows Parallel Matrix-Vector multiplication PetSC library

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 21/23

Parallelization

Efficiency of the parallelization in 3D

Partitionning computed with the SCOTCH library (INRIA)

Datas Tests where launched on PLAFRIM (IMB/LABRI/INRIA)

Mesh made of 1.3 106 tetrahedra Tests used 16 nodes made of 8 cores Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 22/23

Conclusions and perspectives Conclusions Developpement of a 3D anisotropic diffusion scheme Finite Volume scheme of order 2, the implementation is parallelized Validated on a set of tests cases

Work in progress Developpement of the ALE version of the scheme Developpement of a flow model based on a modified Newton law a Developpement of a simplified ablation model (sublimation only) b a. Hypersonic and High-Temperature Gas Dynamics, Second Edition, J. Anderson Jr. ´ ´ b. A. Velghe, Modelisation de l’interation entre un ecoulement turbulent et une paroi ablatable, PhD Thesis (2007)

Future work Developpement of an elastodynamic equation to propagate the boundary speed to the mesh for the ALE scheme Developpement of a finite volume scheme for the Navier stokes equations Coupling all the physics together : fluid dynamics, ablation and heat transfer Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 23/23

ALE formulation Finite Volume scheme on a moving mesh Due to the ablation phoenomenon, the simulation domain is evolving with time. We use the Reynolds transport formula : Z Z Z ∂φ dx d φ(x, t) dv = dv + φ · n ds dt ω(t) dt ω(t) ∂t ∂ω(t) Where dx = V is the mesh motion dt The scheme rewrites : Z Z Z d ρCT dv + q · n ds − ρCT V · n ds = 0 dt ω(t) ∂ω(t) ∂ω(t) | {z } ALE modification

The ALE part is discretised with an explicit scheme (small mesh motion)

Discrete GCL The ALE scheme has to satisfy the discrete Geometric Conservation Law : Z Z d dv − V · n ds = 0 dt ω(t) ∂ω(t) Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 23/23

A simplifed model for atmospheric reentry Modified Newton Law This law gives the pressure coefficient CP along the body a : CP =

p − p∞ 1 ρ V2 2 ∞ ∞

= Cpmax sin2 θ

We use the perfect gas approximation Cpmax is the pressure coefficient at the stagnation point obtained by the Rayleigh Pitot tube formula : ! γ   γ−1 2 2 (γ + 1)2 M∞ p 1 − γ + 2γM∞ =1+ − 1 sin2 θ 2 − 2(γ − 1) p∞ 4γM∞ γ+1

(3)

(4)

We can also obtain the density at the stagnation point : ρs = ρ∞



2 (γ + 1)2 M∞ 2 − 2(γ − 1) 4γM∞

1  γ−1

2 (γ + 1)M∞ 2 2 + (γ − 1)M∞

To obtain the repartition of the density along the body we consider that the boundary is a streamline so : ρpγ = Cte = ρpγs s

The temperature is then obtained with the EOS : T =

p ρr

a. Hypersonic and High-Temperature Gas Dynamics, Second Edition, J. Anderson Jr.

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 23/23

(5)

A simplifed model for atmospheric reentry

Ablation law for a Carbon heat shield We consider that the sublimation of the C3 species is the preponderant reaction : 3C −→ C3 a s MC3 ˙ sub = α3 ¯C3 − pC3 ) (6) m (p 2πRT where α3 is the accommodation coefficient of the C3 species MC3 is the molar mass of C3 R = 8.3143J/(K.mol) is the perfect gas constant pC3 and T are given by the Newton model   ¯C3 = 2.821x105 A3 T n3 exp − ET3 is the saturated vapour pressure of C3 p where A3 = 4.3x1015 , n3 = −1.5, E3 = 97597.0K ´ ´ a. A. Velghe, Modelisation de l’interation entre un ecoulement turbulent et une paroi ablatable, PhD Thesis (2007)

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 23/23

A simplifed model for atmospheric reentry

Stardust Return Sample Capsule Preliminary results

The Stardust mission was launch by NASA in february 1999. Rep It’s goal was to return particle samples from Comet Wild 2 of the temperature in the capsule and from interstellar dust. a a. http ://stardust.jpl.nasa.gov/mission/capsule.html

Pascal JACQ | 10 SEPTEMBER 2012 | PAGE 23/23