Numerical Simulation of Heat Transfer in Materials with Anisotropic Thermal Conductivity: A Finite Volume Scheme to Handle Complex Geometries

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1 Numerical Simulation of Heat Transfer in Materials with Anisotropic Thermal Conductivity: A Finite Volume Scheme to Handle Complex Geometries Olaf Klein, Jürgen Geiser, Peter Philip 2 Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin, Germany 2 University of Minnesota Institute for Mathematics and its Applications (IMA) Minneapolis, USA IMA Workshop: New Paradigms in Computation Minneapolis, March 28 30, 2005

2 Complex Sample Domain from Crystal Growth Blind hole (for cooling and measurements). Ω 3 Ω Ω Ω 2 Ω 2 Ω 4 Ω 5 Ω : Insulation (often anisotropic). Ω 2 : Graphite crucible. Ω 3 : SiC crystal seed. Ω 4 : Gas enclosure. Ω 5 : SiC powder source. Ω 6 : Quartz. Ω = 6 S m= Ω m. Ω 6 Figure : Axisymmetric domain representing a growth apparatus used in silicon carbide single crystal growth by physical vapor transport (PVT). The geometry is a modified version of K. Semmelroth et al., J. Phys.-Condes. Matter 6 (2004).

3 Model for Stationary Anisotropic Heat Conduction div(k m (θ) θ) = f m in Ω m (m M), θ: absolute temperature, K m : symmetric and positive definite tensor of thermal conductivity, f m : heat sources, Ω m : domain of material m. Assumed form of K m : K m (θ) = ą κ m i,j (θ)ć, where κ m i,j (θ) = 8 < α m i κ m iso (θ) for i = j, : 0 for i j. Interface Conditions on Ω m Ω m2 : ą Km (θ) θ ć Ωm n m = ą K m2 (θ) θ ć Ωm2 n m. : restriction, n m : outer unit normal vector to material m. Boundary Conditions: Dirichlet, Robin θ = θ Dir on Γ Dir, ą K m (θ) θ ć n m = ξ m (θ θ ext ) on Γ Rob Ω m, m M,

4 Finite Volume Discretization Σ m = (σ m,i ) i Im conforming triangulation of Ω m satisfying the constrained Delaunay property: If γ is an interior edge of Σ m, α and β the angles opposite to γ, then α + β π. If γ Ω m is a boundary edge of Σ m, α the angle opposite γ, then α π/2. V (σ m,i ) = ľ v m i,j : j {, 2, 3}ł : Set of vertices. V := S m M, i I m V (σ m,i ). ω v := ľ x Ω : x v 2 < x z 2 for each z V \ {v} ł, ω m,v := ω v Ω m, V m := {z V : ω m,z }. A m = (a m i,j ), am i,j := 8 < α m i for i = j, : 0 for i j. w ω,w σ = conv{v, w, u }, σ 2 = conv{v, w, u 2 } Ω = σ σ 2 u ω,u σ ω,u σ 2 γ,v,u σ γ,v,u σ 2 ω,v γ,v,w γ,w,u2 γ,v,u2 ω,u2 u 2 v Figure 2: Illustration of the space discretization.

5 Approximation of Anisotropic Terms φ σ,v : σ [0, ]: Affine coordinates on triangle σ w.r.t. v V (σ). For each edge [v, w] of some σ Σ m : Letting Σ m,v,w := ľ σ Σ m : {v, w} V (σ) ł. Σ γm,v,w := ľ σ Σ m,v,w : λ (H v,w,σ γ m,v,w ) 0 ł, decompose γ m,v,w : γ m,v,w = [ σ Σ γ m,v,w σ γ m,v,w. Approximation: (A m θ) σ n ωv γm,v,w X θ(ṽ) (A m φ σ,ṽ ) ṽ V (σ) w v w v 2. Finite Volume Scheme Find (θ v ) v V satisfying: θ v = θ Dir (v) for each v V Dir, 0 = X m M X m M X m M ξ m ą θv θ ext (v) ć λ ( ω m,v Γ Rob ) X σ Σ γ m,v,w X ṽ V (σ) 2 ą κ m iso (θ v ) + κ m iso (θ w) ć θṽ (A m φ σ,ṽ ) w v w v 2 λ (H v,w,σ γ m,v,w ) f m,v λ 2 (ω m,v ) for each v V Dir = V \ V Dir.

6 Comparison with Closed-Form Solution Axisymmetric Single-Material Domain Ω = {(r, z) : 0 < r < 0.2, 0.2 < z < 0.2}: ţ ű θ r α r ţ ű θ α z = 0 r r r z z in Ω, (a) θ Dir (r, z) := 2 Solution: θ(r, z) = 2 Numerical solution θ 0 num r 2 z 2 on Ω. (b) α r α z r 2 z 2 on Ω. α r α z Exact solution θ Exact Stationary Solution z = 20 T_min=-0.04 T_max=0.002 Levels: : 0.0 2: : : : z = -20 r = 0 r = 20 Figure 3: Solution of (): Numerical θnum, 0 37 triangles (left); exact θ (right). Isolevel difference: Discrete L -error: ɛ l L := X vol(ω v ) θnum l (v) θ(v), v V l v V l : vertices, vol(ω v ): r-weighted area of Voronoï cell. Numerical convergence rate: ρ l L := (ln(ɛ l L ) ln(ɛ l L ))/(ln(h l ) ln(h l )), h l : upper bound for triangle area of level l.

7 Axisymmetric Multi-Material Domain r r 00 Ω = {(r, z) : 0 < r < r 0, 0 < z < z 0 }, Ω 2 = {(r, z) : r 0 < r < r max, 0 < z < z 0 }, Ω 3 = {(r, z) : 0 < r < r 0, z 0 < z < z max }, Ω 4 = {(r, z) : r 0 < r < r max, z 0 < z < z max }, ţ θ r α m,r α m,r 0 0 α m,z 00 α m,r 0 0 α m,z Solution: ű z A θ Ωm ţ ű θ α m,z z A n m A θ Ω m = f m in Ω m, (2a) A n m on Ω m Ω m, (2b) θ Dir,m (r, z) := a m r 2 + b m z 2 + c m on Ω Ω m. (2c) θ(r, z) := a m r 2 + b m z 2 + c m on Ω m, θ Dir,m (r, z) := a m r 2 + b m z 2 + c m on Ω Ω m, where r 0 = z 0 = 0., r max = z max = 0.2, α,r = 2, α 2,r =, α 3,r = 4, α 4,r = 2, α,z =, α 2,z = 2, α 3,z = 3, α 4,z = 6, a =, a 2 = 2, a 3 =, a 4 = 2, b =, b 2 =, b 3 = /3, b 4 = /3, c = 0, c 2 = /00, c 3 = 2/300, c 4 = /300, f = 0, f 2 = 2.0, f 3 = 8.0, f 4 = 20.0

8 0 Numerical solution θnum Exact solution θ Levels: : Ω3 Ω Ω 2: : : : Ω2 0 Figure 4: Solution of (2): Numerical θnum, 37 triangles (left); exact θ (right). Isolevel difference: level number l of triangles max area hl L -error ²lL numerical convergence rate ρll Table : L -error and numerical convergence rate for the numerical solution of () with anisotropy (αr, αz ) = (0, ). level number l of triangles max area hl L -error ²lL numerical convergence rate ρll Table 2: L -error and numerical convergence rate for the numerical solution of (2).

9 Stationary heat field Stationary temperature field Heat source 580 K αr αz Maximal Temperature [K] 220 K 820 K Figure 5: Left: Domain of heat sources highlighted. Right: T -field for isotropic insulation, i.e. αr = αz =. Stationary temperature field Stationary temperature field 660 K Stationary temperature field 580 K 580 K 740 K 740 K 700 K 900 K 900 K 580 K Figure 6: T -field for anisotropic insulation with αr = 000 (left), with αr = 000 for sides, αz = 000 for top and bottom (middle), αz = 000 (right).

10 Publications P. PHILIP: Transient Numerical Simulation of Sublimation Growth of SiC Bulk Single Crystals. Modeling, Finite Volume Method, Results, Thesis, Department of Mathematics, Humboldt University of Berlin, Germany, 2003 Report No. 22, Weierstrass Institute for Applied Analysis and Stochastics, Berlin. J. GEISER, O. KLEIN, P. PHILIP: Numerical simulation of heat transfer in materials with anisotropic thermal conductivity: A finite volume scheme to handle complex geometries. In preparation. J. GEISER, O. KLEIN, P. PHILIP: Influence of anisotropic thermal conductivity in the apparatus insulation for sublimation growth of SiC: Numerical investigation of heat transfer. In preparation. Funding: Supported by the DFG Research Center Matheon: Mathematics for Key Technologies in Berlin, by the IMA in Minneapolis, and by the WIAS in Berlin.