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Test case C1.3: Flow over Naca0012 airfoil
Aravind Balan, Michael Woopen, Jochen Schutz and Georg May
AICES Graduate School, RWTH Aachen University, Germany
2nd International Workshop on High-Order CFD Methods
May 27, 2013
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 1 / 14
Outline
1 Solver Description
2 Discontinuous Galerkin
3 Hybridized Discontinous Galerkin
4 Numerical Results
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 2 / 14
Solver Description
Hybridized Discontinuous Galerkin method for convection-diffusion equations
∇ · (fc(u)− fv(u,∇u)) = S
Hybridization to reduce the globally coupled degrees of freedom
λ ≈ u|ΓLocal solvers to solve for u from λ
Adjoint-based error estimator
Netgen mesh generator, Ngsolve FE library
Nonlinear equations solved by Damped Newton procedure
GMRES with ILU(n), using PETSc
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 3 / 14
Discontinous GalerkinGeneral convection-diffusion equation
σ = ∇u∇ · (fc(u)− fv(u, σ)) = s(u, σ)
The solution spaces : uh ∈ Vh, σh ∈ Hh
Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm(Ωk)
Hh := τ ∈ L2(Ω)× L2(Ω) : τ |Ωk ∈ Pm(Ωk)× Pm(Ωk)
Discontinuous Galerkin method
∑k
∫Ωk
σh · τ +
∫Ωk
(∇ · τ )uh −∫∂Ωk
(τ · n)uh = 0
∑k
∫Ωk
−(fc − fv) · ∇ϕ+
∫∂Ωk
ϕ(fc − fv)−∫
Ωk
(∇ · σh)ϕ =∑k
∫Ωk
sϕ
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 4 / 14
Discontinous GalerkinGeneral convection-diffusion equation
σ = ∇u∇ · (fc(u)− fv(u, σ)) = s(u, σ)
The solution spaces : uh ∈ Vh, σh ∈ Hh
Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm(Ωk)
Hh := τ ∈ L2(Ω)× L2(Ω) : τ |Ωk ∈ Pm(Ωk)× Pm(Ωk)
Discontinuous Galerkin method
∑k
∫Ωk
σh · τ +
∫Ωk
(∇ · τ )uh −∫∂Ωk
(τ · n)uh = 0
∑k
∫Ωk
−(fc − fv) · ∇ϕ+
∫∂Ωk
ϕ(fc − fv)−∫
Ωk
(∇ · σh)ϕ =∑k
∫Ωk
sϕ
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 4 / 14
Hybridizing...
The solution spaces : uh ∈ Vh, σh ∈ Hh, λh ∈Mh
Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm(Ωk)
Hh := τ ∈ L2(Ω)× L2(Ω) : τ |Ωk ∈ Pm(Ωk)× Pm(Ωk)
Mh := µ ∈ L2(Γ) : µ|Γk ∈ Pm(Γk)
Hybridized Discontinuous Galerkin method
∑k
∫Ωk
σh · τ +
∫Ωk
(∇ · τ )uh −∫∂Ωk
(τ · n)λh = 0
∑k
∫Ωk
−(fc − fv) · ∇ϕ+
∫∂Ωk
ϕ(fc − fv)−∫
Ωk
(∇ · σh)ϕ =∑k
∫Ωk
sϕ
∑k
∫∂Ωk
(fc − fv)µ = 0
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 5 / 14
Hybridizing...
The solution spaces : uh ∈ Vh, σh ∈ Hh, λh ∈Mh
Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm(Ωk)
Hh := τ ∈ L2(Ω)× L2(Ω) : τ |Ωk ∈ Pm(Ωk)× Pm(Ωk)
Mh := µ ∈ L2(Γ) : µ|Γk ∈ Pm(Γk)
Hybridized Discontinuous Galerkin method
∑k
∫Ωk
σh · τ +
∫Ωk
(∇ · τ )uh −∫∂Ωk
(τ · n)λh = 0
∑k
∫Ωk
−(fc − fv) · ∇ϕ+
∫∂Ωk
ϕ(fc − fv)−∫
Ωk
(∇ · σh)ϕ =∑k
∫Ωk
sϕ
∑k
∫∂Ωk
(fc − fv)µ = 0
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 5 / 14
Naca0012 Subsonic: M=0.5, α = 2
Figure: Mach number
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 6 / 14
Naca0012 Subsonic: M=0.5, α = 2
10−3 10−2 10−1
10−10
10−8
10−6
10−4
10−2
1/√
ndof
cder
ror
p = 1
p = 2
p = 3
p = 4
(a) Drag coefficient
10−3 10−2 10−110−5
10−4
10−3
10−2
1/√
ndofcl
erro
r
p = 1
p = 2
p = 3
p = 4
(b) Lift coefficient
Figure: Error Vs Dofs
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 7 / 14
Naca0012 Subsonic: M=0.5, α = 2
10−3 10−2 10−1
10−10
10−8
10−6
10−4
10−2
1/√
ndof
cder
ror
UniformResidualAdjoint
Error estimateCorrected
Figure: Error Vs Dofs, p = 2
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 8 / 14
Naca0012 Subsonic : Work units
10−2 10−1 100 101 102 103
10−11
10−9
10−7
10−5
10−3
work units
cder
ror
p = 1
p = 2
p = 3
p = 4
Figure: Error Vs Work units
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 9 / 14
Naca0012 Transonic: M=0.8, α = 1.25
10−3 10−2 10−110−6
10−5
10−4
10−3
10−2
1/√
ndof
cder
ror
p = 1
p = 2
p = 3
p = 4
(a) Drag coefficient
10−3 10−2 10−110−4
10−3
10−2
10−1
1/√
ndofcl
erro
r
p = 1
p = 2
p2newp = 3
(b) Lift coefficient
Figure: Error Vs Dofs
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 10 / 14
Naca0012 Transonic: M=0.8, α = 1.25
(a) Initial mesh (b) Adjoint adapted mesh on Drag, p2
Figure: Mesh
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 11 / 14
Naca0012 Transonic: M=0.8, α = 1.25
10−3 10−2 10−110−5
10−4
10−3
10−2
1/√
ndof
cder
ror
ResidualAdjoint
Corrected
Figure: Error Vs Dofs, p = 2
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 12 / 14
Naca0012 Laminar: M=0.5, α = 1, Re=5000
10−3 10−2 10−110−6
10−5
10−4
10−3
10−2
1/√
ndof
cder
ror
p = 1
p = 2
p = 3
p = 4
(a) Drag coefficient
10−3 10−2 10−110−6
10−5
10−4
10−3
10−2
10−1
1/√
ndofcl
erro
r
p = 1
p = 2
p = 3
p = 4
(b) Lift coefficient
Figure: Error Vs Dofs
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 13 / 14
Acknowledgement
Financial support from the Deutsche Forschungsgemeinschaft (GermanResearch Association) through grant GSC 111 is gratefully acknowledged
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 14 / 14
Adjoint Equations
Error in targeteh = J(w)− J(wh)
Adjoint EquationN ′[wh](dw; zh) = J ′[wh](dw)
Global error estimateeh ≈ Nh(wh, zh)
Aravind Balan (AICES, RWTH Aachen) HDG Method May 27, Cologne 15 / 14