hp-adaptivity on anisotropic meshes for hybridized ... · hp-adaptivity on anisotropic meshes for...

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


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.


Proposed Adaptation Method

Outline

1 Proposed Adaptation Method

2 Numerical Results



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.



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



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



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)



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)




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


Tφp

(x− x)) p+1

2 ∀x ∈ Ω

where,

Qφp =


], 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 )






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


Tφp

(x− x)) p+1

2 ∀x ∈ Ω

where,

Qφp =


], 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 )





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.



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.


Numerical Results

Outline

1 Proposed Adaptation Method

2 Numerical Results


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


Numerical Results

Verification of the bound

The bound for the interpolation error is given as,

|Ex,p(x)| ≤ Ap


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


Numerical Results


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


Numerical Results



102 103 104

10−5

10−4

10−3

Number of elements

L2

nor

mp = 2 error

p = 2 bound


Numerical Results



102 103 104

10−7

10−6

10−5

10−4

Number of elements

L2

nor

mp = 3 error

p = 3 bound


Numerical Results



102 103 104

10−8

10−7

10−6

10−5

10−4

Number of elements

L2

nor

mp = 4 error

p = 4 bound


Numerical Results



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


Numerical Results


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


Numerical Results



103 104

10−7

10−6

10−5

Number of elements

L2

nor

mp = 3 error

p = 3 bound



Numerical Results



103 104

10−8

10−7

10−6

Number of elements

L2

nor

mp = 4 error

p = 4 bound



Numerical Results


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 = 2 error

p = 2 bound


p = 3 error

p = 3 bound


p = 4 error

p = 4 bound



Numerical Results


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


Numerical Results



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


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



Numerical Results



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


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



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).


Numerical Results


h-Adaptation, p = 2, ε = 0.005Initial and adapted mesh

Initial mesh - 512 elements h-adapted mesh, 2nd adaptation step - 1417elements


Numerical Results


h-Adaptation, p = 2, ε = 0.005Solution on initial and h-adapted mesh

Initial mesh - 512 elements Adapted mesh - 1417 elements


Numerical Results


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


Numerical Results


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


Numerical Results


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


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.


Numerical Results

Subsonic viscous flow over NACA0012

h-Adaptation, p = 2Adapted mesh and Mach number

Adapted mesh - 11945 elements Mach number


Numerical Results


h-Adaptation, p = 2Capture of flow recirculation

Adapted mesh Streamline


Numerical Results


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


Numerical Results


hp-AdaptationCapture of flow recirculation

Adapted mesh - 5361 elements Streamline


Numerical Results



104 10510−8

10−7

10−6

10−5

10−4

10−3

10−2

ndof

|∆Cd|

hp iso

hp



Numerical Results



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



Numerical Results

Inviscid subsonic flow over NACA0012

Hp-Adaptation, p = 1, ..., 6Adapted mesh - 1674 elements

Mach number Polynomial map


Numerical Results


Comparison of Anisotropic and Isotropic Hp-adaptationPolynomial map

Anisotropic adaptation, using Interpolationerror

Isotropic adaptation, using Jump indicator


Numerical Results



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



Numerical Results

Inviscid transonic flow over NACA0012

H-Adaptation, p = 2Adapted mesh and Mach number



Numerical Results


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



Numerical Results


Hp-Adaptation, p = 1, 2, ..., 6Adapted mesh, 4258 elementsMach number and Polynomial map



Numerical Results



104 105

10−4

10−3

ndof

|∆Cd|

hp iso

hp



Numerical Results



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



Numerical Results

Inviscid supersonic flow over NACA0012

H-Adaptation, p = 2Adapted mesh and Mach number



Numerical Results


H-Adaptation, p = 2Adapted mesh near bow-shock

Adapted mesh Adapted mesh


Numerical Results


Hp-Adaptation, p = 1, ..., 6Mach number and Polynomial map (8267 elements)



Numerical Results



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



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


Numerical Results

Acknowledgement

Financial support from the Deutsche Forschungsgemeinschaft(German Research Association) through grant GSC 111 is gratefully

acknowledged


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