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Hp-Adaptivity on Anisotropic Meshes for HybridizedDiscontinuous Galerkin Scheme
Aravind Balan, Michael Woopen and Georg May
AICES Graduate School, RWTH Aachen University,Germany
22nd AIAA Computational Fluid Dynamics Conference,Dallas, Texas
June 23rd, 2015
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 1 / 15
Introduction
Background and Motivation
Theme of Present work:
Adjoint-based anisotropic (h and hp) adaptation
Context:
Discontinuous Galerkin method for convection dominated flows
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 2 / 15
Introduction
Scope of Present Work
Anisotropic adaptation based on a continuous mesh model for triangulargrids
Use two error estimators:
Density of continuous mesh is determined by adjoint-based estimate
For triangle of given area, anisotropy follows from analyticminimization of the interpolation error in the Lq-norm
The latter is based on recently proposed error model and estimates1
For hp adaptation, choose the configuration - polynomial degree and theanisotropy - with smallest error estimate.
1V. Dolejsi., Applied Numerical Mathematics. 82, 2014.
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 3 / 15
Proposed Adaptation Method
Outline
1 Proposed Adaptation Method
2 Numerical Results
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 4 / 15
Proposed Adaptation Method
Mesh-Metric Duality
The discrete setting:
Metric tensors encode information about mesh elements
h2
h1 θ
e1 e2
The triangular element is equilateral w.r.t the norm induced by the metric.
||ek ||2M = eTk Mek = C , k = 1, 2, 3,
The element is unit with respect to M
Continuous Mesh:
Reconstruct continuous metric
Unit Mesh generated using length constraint in Riemannian geometry.
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 5 / 15
Proposed Adaptation Method
Defining Size and Anisotropy
Eigenvalue decomposition of the metric gives the size and shape of thetriangle.
M =
(cos θ − sin θsin θ cos θ
)T (λ1 00 λ2
)(cos θ − sin θsin θ cos θ
).
λ1 =1
h21
λ2 =1
h22
,
The area of the triangle is given for C = 3,
I =3√
3
4h1h2
We define the aspect ratio of the mesh element as
β = h2/h1,
The triangle is described by I , β, θ
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 6 / 15
Proposed Adaptation Method
Algorithm for h-Adaptation
1 Find the implied metric of the current mesh.
2 Determine new area, I of mesh elements using adjoint-based error estimate.
If the elemental error > the cut off error, then refine I = Ic/rf
If the elemental error < the cut off error, then coarsen I = Ic × cf
3 Determine anisotropy β, θ by minimization of error model in the Lq norm.
4 I , β, θ is used to evaluate the desired metric on the current mesh and isfed to a metric-conforming mesh generator.
5 Do computation on new mesh
6 Done? Otherwise go to step 1
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 7 / 15
Proposed Adaptation Method
Error Model and Error bound
Define the error function E , of degree p, as
Ex,p(x) =1
(p + 1)!
p+1∑l=0
(p + 1
l
)∂p+1u(x)
∂x l∂yp+1−l (x − x)l(y − y)p+1−l
The error model in radial coordinates (x , y) = (|x| cosφ, |x| sinφ)
Ex,p(x) =1
(p + 1)!u(p+1,φ)|x− x|p+1 (1)
where the directional derivative of order k is given by
u(k,φ) =k∑
l=0
(k
l
)∂ku
∂x l∂yk−l (cosφ)l (sinφ)k−l
Error model for interpolation of u(x) is that of Taylor series πx,pu(x)
u(x)− πx,pu(x) ≈ Ex,p(x)
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 8 / 15
Proposed Adaptation Method
Error Model and Error bound
Define the error function E , of degree p, as
Ex,p(x) =1
(p + 1)!
p+1∑l=0
(p + 1
l
)∂p+1u(x)
∂x l∂yp+1−l (x − x)l(y − y)p+1−l
The error model in radial coordinates (x , y) = (|x| cosφ, |x| sinφ)
Ex,p(x) =1
(p + 1)!u(p+1,φ)|x− x|p+1 (1)
where the directional derivative of order k is given by
u(k,φ) =k∑
l=0
(k
l
)∂ku
∂x l∂yk−l (cosφ)l (sinφ)k−l
Error model for interpolation of u(x) is that of Taylor series πx,pu(x)
u(x)− πx,pu(x) ≈ Ex,p(x)
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 8 / 15
Proposed Adaptation Method
Interpolation error function and bound
A bound for the interpolation error function Ex,p(x) can be written, in terms of 3parameters Ap, ρp, φp as
|Ex,p(x)| ≤ Ap
((x− x)T QφpDρpQ
Tφp
(x− x)) p+1
2 ∀x ∈ Ω
where,
Qφp =
[cosφp − sinφpsinφp cosφp
], Dρp =
[1 0
0 ρ− 2
p+1p
]
and
Ap =1
(p + 1)!|u(p+1,φp)| ρp =
u(p+1,φp)
u(p+1,φp−π2 )
φp = argmaxφ|u(p+1,φ)| (2)
Note: Ap, ρp modified from Ap, ρp , such that bound is safe (and sharp)
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 9 / 15
Proposed Adaptation Method
Interpolation error function and bound
A bound for the interpolation error function Ex,p(x) can be written, in terms of 3
parameters Ap, ρp, φp as
|Ex,p(x)| . Ap
((x− x)T QφpDρpQ
Tφp
(x− x)) p+1
2 ∀x ∈ Ω
where,
Qφp =
[cosφp − sinφpsinφp cosφp
], Dρp =
[1 0
0 ρp− 2
p+1
]
and
Ap =1
(p + 1)!|u(p+1,φp)| ρp =
u(p+1,φp)
u(p+1,φp−π2 )
φp = argmaxφ|u(p+1,φ)| (2)
Note: Ap, ρp modified from Ap, ρp , such that bound is safe (and sharp)
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 9 / 15
Proposed Adaptation Method
Interpolation error function and bound
A bound for the interpolation error function Ex,p(x) can be written, in terms of 3
parameters Ap, ρp, φp as
|Ex,p(x)| . Ap
((x− x)T QφpDρpQ
Tφp
(x− x)) p+1
2 ∀x ∈ Ω
where,
Qφp =
[cosφp − sinφpsinφp cosφp
], Dρp =
[1 0
0 ρp− 2
p+1
]and
Ap =1
(p + 1)!|u(p+1,φp)| ρp =
u(p+1,φp)
u(p+1,φp−π2 )
φp = argmaxφ|u(p+1,φ)| (2)
Note: Ap, ρp modified from Ap, ρp , such that bound is safe (and sharp)
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 9 / 15
Proposed Adaptation Method
Triangle with minimum interpolation error
For given density, we can analytically find the optimal anisotropy.
done by minimizing the bound in Lq-norm taken over metric ellipse.
The optimal anisotropy of the triangle is
β = ρ1
p+1p , θ = φp − π/2,
The minimum bound is given as
‖Ex,p‖Lq(K) ≤ ω
whereω = cp,qApρ
−0.5p I (
p+12 + 1
q )
and cp,q depends only on p and q.
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 10 / 15
Proposed Adaptation Method
Hp-Adaptation
Algorithm for Hp-Adaptation
1 Set the area I , of the triangle, κ using the adjoint error estimate.
2 do a ”dry run”: Compute optimal triangle p = pκ − 1, pκ, pκ + 1
3 Evaluate the error bound ω using Ap, ρp, φp for the above three cases
4 Select, for each element κ, the p and the corresponding metric I , β, θ,which gives the smallest error bound ω.
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 11 / 15
Numerical Results
Outline
1 Proposed Adaptation Method
2 Numerical Results
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 12 / 15
Numerical Results
Solver details
HDG scheme, with adjoint based estimator
Damped Newton time integration
GMRES, ILU(3) from PETSc as linear solver
Netgen2 as (isotropic) mesh generator, Ngsolve as FE library
BAMG3 as anisotropic mesh generator
2Schoberl, Comput. Visual Sci., 1, 19973F. Hecht, Technical report, Inria, France, 2006
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 13 / 15
Numerical Results
Verification of the bound
The bound for the interpolation error is given as,
|Ex,p(x)| ≤ Ap
((x− x)T QφpDρpQ
Tφp
(x− x)) p+1
2 ∀x ∈ Ω
Scalar boundary layer
w(x , y) =
(x +
ex/ε − 1
1− e1/ε
)·(y +
ey/ε − 1
1− e1/ε
)
Figure: ε = 0.1
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 14 / 15
Numerical Results
Verification of the bound
Interpolation Error and Numerical Bound with Ap, ρpUniform Isotropic Refinement, ε = 0.1
102 103 104
10−3
10−2
Number of elements
L2
nor
mp = 1 error
p = 1 bound
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 15 / 15
Numerical Results
Verification of the bound
Interpolation Error and Numerical Bound with Ap, ρpUniform Isotropic Refinement, ε = 0.1
102 103 104
10−5
10−4
10−3
Number of elements
L2
nor
mp = 2 error
p = 2 bound
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 15 / 15
Numerical Results
Verification of the bound
Interpolation Error and Numerical Bound with Ap, ρpUniform Isotropic Refinement, ε = 0.1
102 103 104
10−7
10−6
10−5
10−4
Number of elements
L2
nor
mp = 3 error
p = 3 bound
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 15 / 15
Numerical Results
Verification of the bound
Interpolation Error and Numerical Bound with Ap, ρpUniform Isotropic Refinement, ε = 0.1
102 103 104
10−8
10−7
10−6
10−5
10−4
Number of elements
L2
nor
mp = 4 error
p = 4 bound
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 15 / 15
Numerical Results
Verification of the bound
Interpolation Error and Numerical Bound with Ap, ρpUniform Isotropic Refinement, ε = 0.1
102 103 104 105 10610−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
Number of elements
L2
nor
mp = 1 error
p = 1 bound
p = 2 error
p = 2 bound
p = 3 error
p = 3 bound
p = 4 error
p = 4 bound
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 16 / 15
Numerical Results
Verification of the bound
Interpolation Error, Numerical Bound with Ap, ρp and Modified Bound withAp, ρpAdaptive Anisotropic Refinement, ε = 0.1
103 104
10−5
10−4
Number of elements
L2
nor
mp = 2 error
p = 2 bound
p = 2 modified bound
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 17 / 15
Numerical Results
Verification of the bound
Interpolation Error, Numerical Bound with Ap, ρp and Modified Bound withAp, ρpAdaptive Anisotropic Refinement, ε = 0.1
103 104
10−7
10−6
10−5
Number of elements
L2
nor
mp = 3 error
p = 3 bound
p = 3 modified bound
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 17 / 15
Numerical Results
Verification of the bound
Interpolation Error, Numerical Bound with Ap, ρp and Modified Bound withAp, ρpAdaptive Anisotropic Refinement, ε = 0.1
103 104
10−8
10−7
10−6
Number of elements
L2
nor
mp = 4 error
p = 4 bound
p = 4 modified bound
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 17 / 15
Numerical Results
Verification of the bound
Interpolation Error, Numerical Bound with Ap, ρp and Modified NumericalBound with Ap, ρpAdaptive Anisotropic Refinement, ε = 0.1
103 104
10−8
10−7
10−6
10−5
10−4
10−3
Number of elements
L2
nor
m
p = 1 error
p = 1 bound
p = 1 modified bound
p = 2 error
p = 2 bound
p = 2 modified bound
p = 3 error
p = 3 bound
p = 3 modified bound
p = 4 error
p = 4 bound
p = 4 modified bound
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 18 / 15
Numerical Results
Verification of the bound
Element-wise distribution of interpolation error and numerical bound (sorted)Adapted Anisotropic Mesh, 1581 elements, ε = 0.1, p = 3
0 500 1,000 1,50010−11
10−10
10−9
10−8
10−7
Element number
L2
nor
m
Error Function Ex,p
Bound
Error vs. Numerical Bound withAp, ρp
0 500 1,000 1,50010−11
10−10
10−9
10−8
10−7
Element number
L2
nor
m
Error Function Ex,p
Modified Bound
Error vs. Modified numerical boundwith Ap, ρp
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 19 / 15
Numerical Results
Verification of the bound
Element-wise distribution of interpolation error and numerical bound (sorted)Adapted Anisotropic Mesh, 1450 elements, ε = 0.01, p = 3
400 600 800 1,000 1,200 1,400
10−11
10−10
10−9
10−8
10−7
10−6
Element number
L2
nor
m
Error Function Ex,p
Bound
Error vs. Numerical Bound withAp, ρp
400 600 800 1,000 1,200 1,400
10−11
10−10
10−9
10−8
10−7
10−6
Element number
L2
nor
m
Error Function Ex,p
Modified Bound
Error vs. Modified numerical boundwith Ap, ρp
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 20 / 15
Numerical Results
Verification of the bound
Element-wise distribution of interpolation error and numerical bound (sorted)Adapted Anisotropic Mesh, 1402 elements, ε = 0.005, p = 3
400 600 800 1,000 1,200 1,400
10−12
10−11
10−10
10−9
10−8
10−7
10−6
Element number
L2
nor
m
Error Function Ex,p
Bound
Error vs. Numerical Bound withAp, ρp
400 600 800 1,000 1,200 1,400
10−12
10−11
10−10
10−9
10−8
10−7
10−6
Element number
L2
nor
m
Error Function Ex,p
Modified Bound
Error vs. Modified numerical boundwith Ap, ρp
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 21 / 15
Numerical Results
Scalar convection-diffusion
2D viscous Burger’s equation with source
∇ · (w ,w)− ε∆w = s (x , y) ∈ Ω = [0, 1]2
w(x , y) = 0 (x , y) ∈ ∂Ω
We take the solution as
w(x , y) =
(x +
ex/ε − 1
1− e1/ε
)·(y +
ey/ε − 1
1− e1/ε
),
The target functional of interest is the weighted total boundary flux i.e.
J =
∫∂Ω
ψ(w − εn · q)dσ
where the weighting function, ψ = cos(2πx) cos(2πy).
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 22 / 15
Numerical Results
Scalar convection-diffusion
h-Adaptation, p = 2, ε = 0.005Initial and adapted mesh
Initial mesh - 512 elements h-adapted mesh, 2nd adaptation step - 1417elements
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 23 / 15
Numerical Results
Scalar convection-diffusion
h-Adaptation, p = 2, ε = 0.005Solution on initial and h-adapted mesh
Initial mesh - 512 elements Adapted mesh - 1417 elements
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 24 / 15
Numerical Results
Scalar convection-diffusion
Comparison of isotropic and anisotropic adaptations.
103 104
10−9
10−8
10−7
10−6
10−5
ndof
|J(w
)−
J(w
h)|
p = 2 iso
p = 2
Error Vs. Degrees of freedom
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 25 / 15
Numerical Results
Scalar convection-diffusion
Comparison of isotropic and anisotropic adaptations.
103 104
10−9
10−8
10−7
10−6
10−5
ndof
|J(w
)−
J(w
h)|
hp iso
hp
Error Vs. Degrees of freedom
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 25 / 15
Numerical Results
Scalar convection-diffusion
Comparison of h- and hp-Adaptation, ε = 0.005Solution at same error level of ≈ 10−9
H-adapted mesh - 1417 elements, p = 2 Hp-adapted mesh - 554 elements,p = 2, .., 6
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 26 / 15
Numerical Results
Flow over NACA0012 airfoil
Target functional - Drag coefficient
Flow cases :
Subsonic viscous flow, M∞ = 0.5, α = 1.0,Re = 5000
Subsonic inviscid flow, M∞ = 0.5, α = 2
Transonic inviscid flow, M∞ = 0.8, α = 1.25
Supersonic inviscid flow, M∞ = 1.5, α = 0
Initial Mesh - 2155 elements, Far field at a radius of 1000 chords.
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 27 / 15
Numerical Results
Subsonic viscous flow over NACA0012
h-Adaptation, p = 2Adapted mesh and Mach number
Adapted mesh - 11945 elements Mach number
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 28 / 15
Numerical Results
Subsonic viscous flow over NACA0012
h-Adaptation, p = 2Capture of flow recirculation
Adapted mesh Streamline
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 29 / 15
Numerical Results
Subsonic viscous flow over NACA0012
Comparison of error convergence for isotropic and anisotropic adaptations.For hp-adaptation, p = 1, .., 6
104.2 104.4 104.6 104.8 105
10−6
10−5
10−4
10−3
10−2
ndof
|∆Cd|
p = 2 iso
p = 2
Error in drag coefficient Vs Degrees of freedom
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 30 / 15
Numerical Results
Subsonic viscous flow over NACA0012
hp-AdaptationCapture of flow recirculation
Adapted mesh - 5361 elements Streamline
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 31 / 15
Numerical Results
Subsonic viscous flow over NACA0012
Comparison of error convergence for isotropic and anisotropic adaptations.For hp-adaptation, p = 1, .., 6
104 10510−8
10−7
10−6
10−5
10−4
10−3
10−2
ndof
|∆Cd|
hp iso
hp
Error in drag coefficient Vs Degrees of freedom
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 32 / 15
Numerical Results
Subsonic viscous flow over NACA0012
Comparison of error convergence for isotropic and anisotropic adaptations.For hp-adaptation, p = 1, .., 6
104 10510−8
10−7
10−6
10−5
10−4
10−3
10−2
ndof
|∆Cd|
p = 1 iso
p = 2 iso
p = 3 iso
hp iso
p = 1
p = 2
p = 3
hp
Error in drag coefficient Vs Degrees of freedom
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 33 / 15
Numerical Results
Inviscid subsonic flow over NACA0012
Hp-Adaptation, p = 1, ..., 6Adapted mesh - 1674 elements
Mach number Polynomial map
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 34 / 15
Numerical Results
Inviscid subsonic flow over NACA0012
Comparison of Anisotropic and Isotropic Hp-adaptationPolynomial map
Anisotropic adaptation, using Interpolationerror
Isotropic adaptation, using Jump indicator
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 35 / 15
Numerical Results
Inviscid subsonic flow over NACA0012
Comparison of error convergence for isotropic and anisotropic adaptations.For hp-adaptation, p = 1, .., 6
104 105
10−7
10−6
10−5
10−4
10−3
ndof
|∆Cd|
p = 1 iso
p = 2 iso
p = 3 iso
hp iso
p = 1
p = 2
p = 3
hp
Error in drag coefficient Vs Degrees of freedom
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 36 / 15
Numerical Results
Inviscid transonic flow over NACA0012
H-Adaptation, p = 2Adapted mesh and Mach number
Adapted mesh - 4394 elements Mach number
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 37 / 15
Numerical Results
Inviscid transonic flow over NACA0012
Comparison of error convergence for isotropic and anisotropic adaptations.
104.2 104.4 104.6 104.8 10510−5
10−4
10−3
ndof
|∆Cd|
p = 2 iso
p = 2
Error in drag coefficient Vs Degrees of freedom
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 38 / 15
Numerical Results
Inviscid transonic flow over NACA0012
Hp-Adaptation, p = 1, 2, ..., 6Adapted mesh, 4258 elementsMach number and Polynomial map
Mach number Polynomial map
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 39 / 15
Numerical Results
Inviscid transonic flow over NACA0012
Comparison of error convergence for isotropic and anisotropic adaptations.For hp-adaptation, p = 1, .., 6
104 105
10−4
10−3
ndof
|∆Cd|
hp iso
hp
Error in drag coefficient Vs Degrees of freedom
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 40 / 15
Numerical Results
Inviscid transonic flow over NACA0012
Comparison of error convergence for isotropic and anisotropic adaptations.For hp-adaptation, p = 1, .., 6
104 10510−5
10−4
10−3
ndof
|∆Cd|
p = 1 iso
p = 2 iso
p = 3 iso
hp iso
p = 1
p = 2
p = 3
hp
Error in drag coefficient Vs Degrees of freedom
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 41 / 15
Numerical Results
Inviscid supersonic flow over NACA0012
H-Adaptation, p = 2Adapted mesh and Mach number
Adapted mesh - 7380 elements Mach number
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 42 / 15
Numerical Results
Inviscid supersonic flow over NACA0012
H-Adaptation, p = 2Adapted mesh near bow-shock
Adapted mesh Adapted mesh
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 43 / 15
Numerical Results
Inviscid supersonic flow over NACA0012
Hp-Adaptation, p = 1, ..., 6Mach number and Polynomial map (8267 elements)
Mach number Polynomial map
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 44 / 15
Numerical Results
Inviscid supersonic flow over NACA0012
Comparison of error convergence for isotropic and anisotropic adaptations.For hp-adaptation, p = 1, .., 6
104 10510−7
10−6
10−5
10−4
10−3
10−2
ndof
|∆Cd|
p = 1 iso
p = 2 iso
p = 3 iso
hp iso
p = 1
p = 2
p = 3
hp
Error in drag coefficient Vs Degrees of freedom
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 45 / 15
Numerical Results
Conclusions
Interpolation bound and verification
Anisotropic refinement - superior to isotropic refinement
Hp-refinement - superior to pure h-refinement
Future work :
Extension to 3D
Global optimization
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 46 / 15
Numerical Results
Acknowledgement
Financial support from the Deutsche Forschungsgemeinschaft(German Research Association) through grant GSC 111 is gratefully
acknowledged
Balan, Woopen, May (AICES) Hp-Adaptivity on Anisotropic Meshes June 23rd, 2015 47 / 15