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Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Anisotropic Goal-Oriented Mesh Adaptation inFiredrake
Joe Wallwork1 Nicolas Barral2 David Ham1
Matthew Piggott1
1Imperial College London, UK2University of Bordeaux, France
Firedrake ’19, Durham
1 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Mesh Adaptation
Riemannian Metric Field
Metric-Based Mesh Adaptation.
Riemannian metric fields M = M(x)x∈Ω are SPD ∀x ∈ Rn.∴ Orthogonal eigendecomposition M(x) = VΛV T .
Steiner ellipses [Barral, 2015] Resulting mesh [Barral, 2015]
(1,0)
(0,1)M
12
M−12
(λ1, 0)(0, λ2)
E2 = (R2, I2) (R2,M)
node insertion
edgeswap
node deletion
node movement
Rokos and Gorman [2013] 2 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Mesh Adaptation
Hessian-Based Adaptation
The Hessian.
Consider interpolating u ≈ Ihu ∈ P1.
It is shown in [Frey and Alauzet, 2005] that
‖u − Πhu‖L∞(K) ≤ γmaxx∈K
maxe∈∂K
eT |H(x)|e
where γ > 0 is a constant related to the spatial dimension.
A metric tensor M = M(x)x∈Ω may be defined as
M(x) =γ
ε|H(x)|,
where ε > 0 is the tolerated error level.
3 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Model Validation
Test Case
Point Discharge with Diffusion
Analytical solution.
Finite element forward solution.
(Presented on a 1,024,000 element uniform mesh.)
Test case taken fromTELEMAC-2D Valida-tion Document 7.0[Riadh et al., 2014].
u · ∇φ−∇ · (ν∇φ) = f
νn · ∇φ|walls = 0φ|inflow = 0
f = δ(x − 2, y − 5)
4 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Model Validation
Test Case
Point Discharge with Diffusion: Adjoint Problem
Adjoint solution for J1.
Adjoint solution for J2.
(Presented on a 1,024,000 element uniform mesh.)
Ji (φ) =
∫Ri
φ dx
R1 = B 12((20, 5))
R2 = B 12((20, 7.5))
5 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Model Validation
Test Case
Point Discharge with Diffusion: Convergence
Elements J1(φ) J1(φh) J2(φ) J2(φh)
4,000 0.20757 0.20547 0.08882 0.0890116,000 0.16904 0.16873 0.07206 0.0720564,000 0.16263 0.62590 0.06924 0.06922
256,000 0.16344 0.16343 0.06959 0.069581,024,000 0.16344 0.16345 0.06959 0.06958
Ji (φ) : analytical solutionsJi (φh) : P1 finite element solutions
6 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Theory
Dual Weighted Residual (DWR).
Given a PDE Ψ(u) = 0 and its adjoint written in Galerkin forms
ρ(uh, v) :=L(v)− a(uh, v) = 0, ∀v ∈ Vh
ρ∗(u∗h, v) :=J(v)− a(v , u∗h) = 0, ∀v ∈ Vh
=⇒ a posteriori error results [Becker and Rannacher, 2001]
J(u)− J(uh) = ρ(uh, u∗ − u∗h) + R(2)
J(u)− J(uh) =1
2ρ(uh, u
∗ − u∗h) +1
2ρ∗(u∗h, u − uh) + R(3)
[Remainders R(2) and R(3) depend on errors u − uh and u∗ − u∗h.]
7 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Theory
DWR Integration by Parts
J(u)− J(uh) = ρ(uh, u∗ − u∗h) + R(2)
Applying integration by parts (again) elementwise:
|J(u)− J(uh)|∣∣∣K≈ |〈Ψ(uh), u∗ − u∗h〉K + 〈ψ(uh), u∗ − u∗h〉∂K |.
Ψ(uh) is the strong residual on K ;
ψ(uh) embodies flux terms over elemental boundaries.
8 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Isotropic Goal-Oriented Mesh Adaptation
Isotropic Metric
|J(u)− J(uh)|∣∣∣K≈ η := |〈Ψ(uh), u∗ − u∗h〉K + 〈ψ(uh), u∗ − u∗h〉∂K |.
Isotropic case:
M =
[ΠP1η 0
0 ΠP1η
].
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Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Isotropic Goal-Oriented Mesh Adaptation
Isotropic Meshes
Centred receiver (12,246 elements).
Offset receiver (19,399 elements).
10 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Anisotropic Goal-Oriented Mesh Adaptation
A posteriori Approach
Motivated by the approach of [Power et al., 2006], consider theinterpolation error:
|J(u)− J(uh)| ≈ |〈Ψ(uh), u∗ − u∗h︸ ︷︷ ︸u∗−Πhu∗
〉K + 〈ψ(uh), u∗ − u∗h︸ ︷︷ ︸u∗−Πhu∗
〉∂K |
This suggests the node-wise metric,
M = |Ψ(uh)||H(u∗)|,
and correspondingly for the adjoint,
M = |Ψ∗(u∗h)||H(u)|.
11 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Anisotropic Goal-Oriented Mesh Adaptation
A posteriori Anisotropic Meshes
Centred receiver (16,407 elements).
Offset receiver (9,868 elements).
12 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Anisotropic Goal-Oriented Mesh Adaptation
A priori Approach
Alternative a priori error estimate [Loseille et al., 2010]:
J(u)− J(uh) = 〈(Ψh −Ψ)(u), u∗〉+ R.
Assume we have the conservative form Ψ(u) = ∇ · F(u), so
J(u)− J(uh) ≈ 〈(F − Fh)(u),∇u∗〉Ω − 〈n · (F − Fh(u), u∗)〉∂Ω.
This gives Riemannian metric fields
Mvolume =n∑
i=1
|H(Fi (u))|∣∣∣∣∂u∗∂xi
∣∣∣∣ ,Msurface =|u∗|
∣∣∣∣∣H(
n∑i=1
F i (u) · ni
)∣∣∣∣∣13 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Anisotropic Goal-Oriented Mesh Adaptation
A priori Anisotropic Meshes
Centred receiver (44,894 elements).
Offset receiver (29,143 elements).
14 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Anisotropic Goal-Oriented Mesh Adaptation
Meshes from Combined Metrics (Offset Receiver)
Averaged isotropic (13,980 elements). Superposed isotropic (19,588 elements).
Averaged a posteriori (9,289 elements). Superposed a posteriori (14,470 elems).
Averaged a priori (25,204 elements). Superposed a priori (49,793 elements).
15 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Results
Convergence Analysis: Centred Receiver.
16 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Results
Convergence Analysis: Offset Receiver.
17 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Results
Three Dimensions.
Anisotropic mesh resulting fromaveraging a posteriori metrics.
Uniform mesh (1,920,000 elements).
Adapted mesh (1,766,396 elements).
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Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Results
Outlook.
To appear in proceedings of the 28th International MeshingRoundtable:JW, N Barral, D Ham, M Piggott, “AnisotropicGoal-Oriented Mesh Adaptation in Firedrake” (2019).
Future work:
Time dependent (tidal) problems.
Other finite element spaces, e.g. DG.
Boundary and flux terms for anisotropic methods.
Realistic desalination application.
19 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Goal-Oriented Adaptation
Results
References
Nicolas Barral. Time-accurate anisotropic mesh adaptation for three-dimensional moving mesh problems. PhDthesis, Universite Pierre et Marie Curie, 2015.
Roland Becker and Rolf Rannacher. An optimal control approach to a posteriori error estimation in finiteelement methods. Acta Numerica 2001, 10:1–102, May 2001. ISSN 0962-4929.
Pascal-Jean Frey and Frederic Alauzet. Anisotropic mesh adaptation for cfd computations. Computer methodsin applied mechanics and engineering, 194(48-49):5068–5082, 2005.
Adrien Loseille, Alain Dervieux, and Frederic Alauzet. Fully anisotropic goal-oriented mesh adaptation for 3dsteady euler equations. Journal of computational physics, 229(8):2866–2897, 2010.
PW Power, Christopher C Pain, MD Piggott, Fangxin Fang, Gerard J Gorman, AP Umpleby, Anthony JHGoddard, and IM Navon. Adjoint a posteriori error measures for anisotropic mesh optimisation. Computers &Mathematics with Applications, 52(8):1213–1242, 2006.
A Riadh, G Cedric, and MH Jean. TELEMAC modeling system: 2D hydrodynamics TELEMAC-2D softwarerelease 7.0 user manual. Paris: R&D, Electricite de France, page 134, 2014.
Georgios Rokos and Gerard Gorman. Pragmatic–parallel anisotropic adaptive mesh toolkit. In Facing theMulticore-Challenge III, pages 143–144. Springer, 2013.
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Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Appendix
Combining Metrics
Metric Combination.
Consider metrics M1 and M2. How to combine these in ameaningful way?
Metric average:M := 1
2 (M1 +M2).
Metric superposition:intersection of Steiner ellipses.
Metric superposition [Barral, 2015]
21 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Appendix
Hessian recovery
Double L2 projection
We recover H = ∇T∇u by solving the auxiliary problemH = ∇Tgg = ∇u
as∫Ωτ : Hh dx +
∫Ωdiv(τ) : gh dx −
n∑i=1
n∑j=1
∫∂Ω
(gh)iτijnj ds = 0, ∀τ∫Ωψ · gh dx =
∫∂Ω
uhψ · n ds −∫
Ωdiv(ψ)uh dx , ∀ψ.
22 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake
Appendix
A priori Metric
Accounting for Source Term
Forward equation:
∇ · F(φ) = f , F(φ) = uφ− ν∇φ.
M = |H(F1(φ))|∣∣∣∣∂φ∗∂x
∣∣∣∣+ |H(F2(φ))|∣∣∣∣∂φ∗∂y
∣∣∣∣+ |H(f )| |φ∗|.
Adjoint equation:
∇ · G(φ∗) = g , G(φ∗) = −uφ∗ − ν∇φ∗,
M = |H(G1(φ∗))|∣∣∣∣∂φ∂x
∣∣∣∣+ |H(G2(φ∗))|∣∣∣∣∂φ∂y
∣∣∣∣+ |H(g)| |φ|,
23 / 20