Hp-Adaptivity on Anisotropic Meshes for Hybridized Discontinuous Galerkin Scheme

56 Pages·2015·4.14 MB·English

Hp-Adaptivity on Anisotropic Meshes for Hybridized
Discontinuous Galerkin Scheme
Aravind Balan, Michael Woopen and Georg May
AICESGraduateSchool,RWTHAachenUniversity,
Germany
22ndAIAAComputationalFluidDynamicsConference,
Dallas,Texas
June 23rd, 2015
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Introduction
Background and Motivation
Theme of Present work:
Adjoint-based anisotropic (h and hp) adaptation
Context:
Discontinuous Galerkin method for convection dominated flows
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Introduction
Scope of Present Work
Anisotropic adaptation based on a continuous mesh model for triangular
grids
Use two error estimators:
Density of continuous mesh is determined by adjoint-based estimate
For triangle of given area, anisotropy follows from analytic
minimization 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 the
anisotropy - with smallest error estimate.
1V.Dolejsi.,AppliedNumericalMathematics. 82,2014.
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ProposedAdaptationMethod
Outline
1 Proposed Adaptation Method
2 Numerical Results
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ProposedAdaptationMethod
Mesh-Metric Duality
The discrete setting:
Metric tensors encode information about mesh elements

e1 e2

h1 h 2θ
The triangular element is equilateral w.r.t the norm induced by the metric.
||e ||2 =eTMe =C, k =1,2,3,
k M k k
The element is unit with respect to M
Continuous Mesh:
Reconstruct continuous metric
Unit Mesh generated using length constraint in Riemannian geometry.
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ProposedAdaptationMethod
Defining Size and Anisotropy
Eigenvalue decomposition of the metric gives the size and shape of the
triangle.
(cid:18) (cid:19)T (cid:18) (cid:19)(cid:18) (cid:19)
cosθ −sinθ λ 0 cosθ −sinθ
M= 1 .
sinθ cosθ 0 λ sinθ cosθ
2
1 1
λ = λ = ,
1 h2 2 h2
1 2
The area of the triangle is given for C =3,
√
3 3
I = h h
4 1 2
We define the aspect ratio of the mesh element as
β =h /h ,
2 1
The triangle is described by {I,β,θ}
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ProposedAdaptationMethod
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 =I /r
c f
If the elemental error < the cut off error, then coarsen I =I ×c
c f
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 is
fed to a metric-conforming mesh generator.
5 Do computation on new mesh
6 Done? Otherwise go to step 1
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The error model in radial coordinates (x,y)=(|x|cosφ,|x|sinφ)
1
E (x)= u(p+1,φ)|x−¯x|p+1 (1)
¯x,p (p+1)!
where the directional derivative of order k is given by
u(k,φ) =(cid:88)k (cid:18)k(cid:19) ∂ku (cosφ)l(sinφ)k−l
l ∂xl∂yk−l
l=0
Error model for interpolation of u(x) is that of Taylor series π u(x)
¯x,p
u(x)−π u(x)≈E (x)
¯x,p ¯x,p
ProposedAdaptationMethod
Error Model and Error bound
Define the error function E, of degree p, as
1 (cid:88)p+1(cid:18)p+1(cid:19) ∂p+1u(¯x)
E (x)= (x −x¯)l(y −y¯)p+1−l
¯x,p (p+1)! l ∂xl∂yp+1−l
l=0
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ProposedAdaptationMethod
Error Model and Error bound
Define the error function E, of degree p, as
1 (cid:88)p+1(cid:18)p+1(cid:19) ∂p+1u(¯x)
E (x)= (x −x¯)l(y −y¯)p+1−l
¯x,p (p+1)! l ∂xl∂yp+1−l
l=0
The error model in radial coordinates (x,y)=(|x|cosφ,|x|sinφ)
1
E (x)= u(p+1,φ)|x−¯x|p+1 (1)
¯x,p (p+1)!
where the directional derivative of order k is given by
u(k,φ) =(cid:88)k (cid:18)k(cid:19) ∂ku (cosφ)l(sinφ)k−l
l ∂xl∂yk−l
l=0
Error model for interpolation of u(x) is that of Taylor series π u(x)
¯x,p
u(x)−π u(x)≈E (x)
¯x,p ¯x,p
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and
1 u(p+1,φp)
A(cid:101)p = (p+1)!|u(p+1,φp)| ρ(cid:101)p = u(p+1,φp−π2)
φ =argmax |u(p+1,φ)| (2)
p φ
Note: {Ap,ρp} modified from {A(cid:101)p,ρ(cid:101)p} , such that bound is safe (and sharp)
ProposedAdaptationMethod
Interpolation error function and bound
A bound for the interpolation error function E (x) can be written, in terms of 3
¯x,p
parameters {A ,ρ ,φ } as
p p p
(cid:16) (cid:17)p+1
|E (x)|≤ A (x−¯x)T Q D QT (x−¯x) 2 ∀x∈Ω
¯x,p p φp ρp φp
where,
(cid:20) cosφ −sinφ (cid:21) (cid:34) 1 0 (cid:35)
Qφp = sinφpp cosφpp , Dρp = 0 ρp−p+21
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