A Second Order Finite Volume Method for a Multi-material Heat Equation on Cartesian Grids: Application to Stefan problems.

ECCOMAS 2012 | Manuel LATIGE, Gerard GALLICE, Thierry COLIN September 10-14 2012

Scientific Background

General problem

We need to ... handle several interfaces. Impose the jump conditions across the interfaces

Goals ... Develop a spatial second-order Finite Volume method with a compact stencil for the heat equation in 2D. The solid part can be composed by several materials with different properties One of the solids undergoes a phase change (fusion) CEA, INRIA | September 10-14 2012 | PAGE 1/18

Heat equation on multi-material domain Heat equation on multi-material domain ρ1,2 C1,2 ∂t T − ∇. (K1,2 ∇T ) = 0 on Ω1,2 /Γ.

(1)

[T ]Γ = T |Ω1 − T |Ω2 = 0, ∂T ∂T2 = K1 − K = 0, 2 ∂n Ω1 ∂n Ω2

(2)

 K

∂T ∂n

 Γ

(3)

where T the temperature field, K1 , ρ1 and C1 (respectively K2 , ρ2 and C2 ) the scalar thermal conductivity, the mass density and the heat capacity in Ω1 (respectively in Ω2 ).

Elliptic equation with variable coefficients for the spatial discretization.

 K

∂T ∂n

 Γ

−∇. (K1,2 ∇T ) = 0 on Ω1,2 /Γ,

(4)

[T ]Γ = T |Ω1 − T |Ω2 = h, ∂T ∂T2 = K1 − K = g. 2 ∂n Ω1 ∂n Ω2

(5)

where the functions h and g represent the general jump conditions.

(6) CEA, INRIA | September 10-14 2012 | PAGE 2/18

Stefan problem A non linear model : the unknowns are the temperature field and the location of the interface.

Assumptions : Density remains constant and the thermophysical properties (conductivities and heat capacities) are different for each phase. No convection in the liquid phase. The interface thickness is null.

Stefan problem ρsol,liq Csol,liq ∂t T − ∇. (Ksol,liq ∇T ) = 0 on Ωsol,liq /Γ T = Tfusion on Γ, 

K ∂T ∂n

 Γ

~ .~n, = ρLV

⇐ ⇐

(7)

Dirichlet boundary condition decoupling the two subdomains (liquid and solid), Stefan condition (energy balance) governing the evolution of the interface,

~ the interface velocity. where Tfusion the fusion temperature, L the latent heat and V CEA, INRIA | September 10-14 2012 | PAGE 3/18

Outline

1 Elliptic equation with variable coefficients

Discrete form for a control volume Mixed Finite Volume – Finite Element Method Numerical results 2 Stefan problem

Stefan problem algorithm Dirichlet boundary condition Numerical results 3 Conclusion

CEA, INRIA | September 10-14 2012 | PAGE 4/18

Discrete form for a control volume

Equations : −∇. (K1,2 ∇T ) = 0 on ωi,j ⊂ (Ω1,2 /Γ), [T ]Γ = T |Ω1 T ωi,j − T |Ω2 T ωi,j = h,   ∂T ∂T2 ∂T − K = g. = K1 K 2 ∂n Γ ∂n Ω1 T ωi,j ∂n Ω2 T ωi,j

Discrete form for the control volume ωi,j −

X XZ m=I ,IV s=1,2

Z k1/2 ∇T .n dl = esM

Z f dS +

g dl,

(8)

[T ]Γ = T |Ω1 T ωi,j − T |Ω2 T ωi,j = h,

(9)

ωi,j

Γ∩ωi,j

where esM , s = 1, 2, the two boundary edges of ωi,j lying inside the dual cell M. CEA, INRIA | September 10-14 2012 | PAGE 5/18

Mixed Finite Volume – Finite Element Method

Finite Element Approach Polynomial functions T h are used in each dual cell to approximate the temperature field. Those polynomials depend of the four corner values of the dual cell and the jump conditions across the interface.

⇓ Finite Volume Approach The flux is analytically calculated on each edges with the polynomial functions.

⇓  Evaluation of



R

 esM

k1/2 ∇T .~n dl, s = 1, 2.



CEA, INRIA | September 10-14 2012 | PAGE 6/18

Without interface case : regular dual cell Here, the temperature field is approximated by a Q1 polynomial T h : T (x, y ) ≈ T h (x, y ) = α0 + α1 x + α2 y + α3 xy in $M

Determination of the four unknowns coefficients αγ , γ = 1, 4, of the bilinear representation. T h (xcr , ycr ) = Tcr r = 1, 4,     Tc1 α0 −1  ..   .  =⇒  ..  = A  . . Tc4 α3

Tcr , r = 1, 4, are the corner values defined at (xcr , ycr ). $M the dual cell domain.

(10) (11)

Laplacian discretization in the case ∆x = ∆y : R ωi,j

∇.(k ∇T )dS = k

[

1 T 4 i−1,j+1 1 T 4 i−1,j 1 T 4 i−1,j−1

+ − +

1 T 2 i,j+1

3Ti,j 1 T 2 i,j−1

+ + +

1 T 4 i+1,j+1 1 T 4 i+1,j 1 T 4 i+1,j−1

+ + . (12) ]

E. S¨ uli, Convergence of finite volume schemes for poisson’s equation on nonuniforme meshes, SIAM Journal on Numerical Analysis 28 (5) (1991) 1419-1430

CEA, INRIA | September 10-14 2012 | PAGE 7/18

With interface case : two geometric configurations

Type I interface :

We want ...

Type II interface :

the same processing for the 2 geometric configurations. to avoid singularities in the definition of the polynomial coefficients. a continuous processing of the interface during its movement. Here, we choose T h in P2 polynomial space : T (x, y ) ≈ T h (x, y ) = T1h (x, y ) $M T Ω + T2h (x, y ) $M T Ω where 1

2

Tςh (x, y ) = β0ς + β1ς x + β2ς y + β3ς xy + β4ς x 2 + β5ς y 2 , ς = 1, 2. =⇒ 12 unknowns coefficients βγς , ς = 1, 2, γ = 0, ..., 5. M. Oevermann, R. Klein. A cartesian grid finite volume method for elliptic equation with variable coefficients and embedded interfaces. Journal of Computationnal Physics, 219 :749-769, 2006. CEA, INRIA | September 10-14 2012 | PAGE 8/18

With interface case : parametrization of the interface 4 relations thanks to the corner values : T h (xcr , ycr ) = Tcr r = 1, 4.

5 relations thanks to the interface parametrization and the jump conditions : ~ = s~τ , ∀M ∈ Γ. For ς = 1, 2 : We define s such as OM  ς t β1 Tςh (s) = β0ς + .~τ s β2ς
+ β3ς τx τy + β4ς τx2 + β5ς τy2 s 2 . We obtain for the jump conditions ∀s :    h ∂T h ∂T h ∂T h k T Γ = T1h (s) − T2h (s) = k1 1 − k2 2 ∂n Γ ∂n ∂n = a + b s + c s 2, = d + e s, where {a, b, c, d, e} are given by the functions h and g (here e is chosen null).

3 closure relations to ensure a continuous processing and avoid singularities : find

n

β01 , . . . , β51 , β02 , . . . , β52

where $ςM = $M

T

Ως , ς = 1, 2.

o

β4ς = β5ς , ς = 1, 2, which minimizes $1M β51 + $2M β52 ,

(13) (14)

CEA, INRIA | September 10-14 2012 | PAGE 9/18

Numerical results

We have developed a Finite Volume method with a compact 9-point stencil for 2D Elliptic equations with variable coefficients.

Grid 64 × 64 128 × 128 256 × 256 512 × 512 1024 × 1024 64 × 192 128 × 384 256 × 768 512 × 1536

L2 1.0772e-03 3.1098e-04 8.6661e-05 2.5887e-05 7.9134e-06 8.5133e-04 2.4627e-04 6.4761e-05 1.7885e-05

Order 1.79 1,84 1,74 1,70 1,79 1,92 1,85

L∞ 4.6612e-04 1.1993e-04 2.7800e-05 7.4453e-06 2.3704e-06 5.0004e-04 1.2730e-04 3.1171e-05 8.5367e-06

Order 1,96 2,11 1,90 1,65 1,97 2,03 1,87

Table: Convergence results for the solution u in the L2 and L∞ -norm on two different sets of grids with K1 = 10 and K2 = 1.

CEA, INRIA | September 10-14 2012 | PAGE 10/18

Numerical results

Grid 64 × 64 128 × 128 256 × 256 512 × 512 1024 × 1024

K1 /K2 = 10−1 7.4567e-04 1.7658e-04 4.1495e-05 1.0900e-05 2.7083e-06

Order

K1 /K2 = 1000

Order

2.07 2.09 1,93 2.01

2.9247e-03 9.5600e-04 2.6487e-04 3.1096e-05 1.1598e-05

1,61 1.85 3.09 1,42

Table: Convergence results for the solution u in the L2 -norm with two different ratios K1 /K2 , K1 = 1

Results... A compact 9-point method Robustness Second order accuracy

CEA, INRIA | September 10-14 2012 | PAGE 11/18

Outline

1 Elliptic equation with variable coefficients

Discrete form for a control volume Mixed Finite Volume – Finite Element Method Numerical results 2 Stefan problem

Stefan problem algorithm Dirichlet boundary condition Numerical results 3 Conclusion

CEA, INRIA | September 10-14 2012 | PAGE 12/18

Stefan problem algorithm

Initialisation of a time step

%

Temperature field and interface position.

&

New temperature fields

Interface velocity field

On Ωsol,liq /Γ   ρsol,liq Csol,liq ∂t T − ∇. (Ksol,liq ∇T ) = 0,  T =T fusion on Γ.

Both domains, solid and liquid ones, are solved with a Dirichlet boundary condition on Γ.

The interface velocity is define by the Stefan condition :  K

∂T ∂n



~ .~n. = ρLV Γ

.

Evolving of the interface Γ The Level Set method is used to evolve the interface.

CEA, INRIA | September 10-14 2012 | PAGE 13/18

Dirichlet boundary condition Problem to solve : .

 ∇. (k1 ∇T ) = f1 ,     T = TΓ on Γ,     T = w on ∂Ω1 /Γ.

Asymptotic equivalent problem :

− − −− −→

 ∇. (k1,2 ∇T ) = f1,2 ,        [T ]Γ = TΓ on Γ,        K ∂T ∂n Γ = 0 on Γ,        T = w on ∂Ω1 /Γ,      T = 0 on ∂Ω2 /Γ.

Here f2 = 0and k2 = +∞ on Ω2 . 



− − −− −→

Implementation :  ∇. (k1,2 ∇T ) = f1,2 ,        [T ]Γ = TΓ on Γ,        K ∂T ∂n Γ = 0 on Γ,        T = w on ∂Ω1 /Γ,      T = 0 on ∂Ω2 /Γ.

Here f2 = 0 and 



k1 k2 = on Ω2 . min (∆x 2 , ∆y 2 )

CEA, INRIA | September 10-14 2012 | PAGE 14/18

Numerical results : Infinite corner problem Datas : Tinitial = Tfusion Tfusion = 0 ◦ C Tw = 10◦ C ρ = 1000 kg /m3 L = 3350000 J/kg kwater = 0.6 W /m/K kice = 2.18 W /m/K Cwater = 4186 J/kg /K Cice = 2260 J/kg /K

Figure: The interface location at different time steps with 35 × 35 grid-points in the (x, y ).

Similarity variables : x , ξx = √ 4αt y kwater ξy = √ with α = C water ρ 4αt

Figure: The interface location at different time steps with 35 × 35 grid-points in the (ξx , ξy ). CEA, INRIA | September 10-14 2012 | PAGE 15/18

Outline

1 Elliptic equation with variable coefficients

Discrete form for a control volume Mixed Finite Volume – Finite Element Method Numerical results 2 Stefan problem

Stefan problem algorithm Dirichlet boundary condition Numerical results 3 Conclusion

CEA, INRIA | September 10-14 2012 | PAGE 16/18

Conclusion

What have been done ... A compact 9-point method, A second order accuracy method, An easy way to impose Dirichlet and Neumann boundary conditions, Use the method to solve Stefan problems .

What remains to do ... =⇒ realise a coupling between a multi-material solid phase and a liquid phase with convection.

CEA, INRIA | September 10-14 2012 | PAGE 17/18

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