Anisotropic Goal-Oriented Mesh Adaptation in Firedrake Anisotropic Goal-Oriented Mesh Adaptation in Firedrake Joe Wallwork 1 Nicolas Barral 2 David Ham 1 Matthew Piggott 1 1 Imperial College London, UK 2 University 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 ∈ R n . ∴ Orthogonal eigendecomposition M ( x ) = V Λ V T . Steiner ellipses [Barral, 2015] Resulting mesh [Barral, 2015] (0,1) 1 node node ( λ 1 , 0) (0 , λ 2 ) insertion deletion M 2 (1,0) node edge swap movement M − 1 E 2 = ( R 2 , I 2 ) ( R 2 , M ) 2 Rokos and Gorman [2013] 2 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake Mesh Adaptation Hessian-Based Adaptation The Hessian. Consider interpolating u ≈ I h u ∈ P 1. It is shown in [Frey and Alauzet, 2005] that e ∈ ∂ K e T | H ( x ) | e � u − Π h u � L ∞ ( K ) ≤ γ max x ∈ K max 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 ǫ | H ( x ) | , where ǫ > 0 is the tolerated error level. 3 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake Model Validation Test Case Point Discharge with Diffusion Test case taken from TELEMAC-2D Valida- tion Document 7.0 Analytical solution. [Riadh et al., 2014]. u · ∇ φ − ∇ · ( ν ∇ φ ) = f ν � n · ∇ φ | walls = 0 φ | inflow = 0 Finite element forward solution. f = δ ( x − 2 , y − 5) (Presented on a 1,024,000 element uniform mesh.) 4 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake Model Validation Test Case Point Discharge with Diffusion: Adjoint Problem � Adjoint solution for J 1 . J i ( φ ) = φ d x R i R 1 = B 1 2 ((20 , 5)) Adjoint solution for J 2 . R 2 = B 1 2 ((20 , 7 . 5)) (Presented on a 1,024,000 element uniform mesh.) 5 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake Model Validation Test Case Point Discharge with Diffusion: Convergence Elements J 1 ( φ ) J 1 ( φ h ) J 2 ( φ ) J 2 ( φ h ) 4,000 0.20757 0.20547 0.08882 0.08901 16,000 0.16904 0.16873 0.07206 0.07205 64,000 0.16263 0.62590 0.06924 0.06922 256,000 0.16344 0.16343 0.06959 0.06958 1,024,000 0.16344 0.16345 0.06959 0.06958 J i ( φ ) : analytical solutions J i ( φ h ) : P 1 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 ( u h , v ) := L ( v ) − a ( u h , v ) = 0 , ∀ v ∈ V h ρ ∗ ( u ∗ h , v ) := J ( v ) − a ( v , u ∗ h ) = 0 , ∀ v ∈ V h = ⇒ a posteriori error results [Becker and Rannacher, 2001] J ( u ) − J ( u h ) = ρ ( u h , u ∗ − u ∗ h ) + R (2) J ( u ) − J ( u h ) = 1 h ) + 1 2 ρ ( u h , u ∗ − u ∗ h , u − u h ) + R (3) 2 ρ ∗ ( u ∗ [Remainders R (2) and R (3) depend on errors u − u h and u ∗ − u ∗ h .] 7 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake Goal-Oriented Adaptation Theory DWR Integration by Parts J ( u ) − J ( u h ) = ρ ( u h , u ∗ − u ∗ h ) + R (2) Applying integration by parts (again) elementwise : � � K ≈ |� Ψ( u h ) , u ∗ − u ∗ h � K + � ψ ( u h ) , u ∗ − u ∗ | J ( u ) − J ( u h ) | � h � ∂ K | . Ψ( u h ) is the strong residual on K ; ψ ( u h ) 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 � � K ≈ η := |� Ψ( u h ) , u ∗ − u ∗ h � K + � ψ ( u h ) , u ∗ − u ∗ | J ( u ) − J ( u h ) | � h � ∂ K | . Isotropic case: � Π P 1 η � 0 M = . 0 Π P 1 η 9 / 20
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 the interpolation error: | J ( u ) − J ( u h ) | ≈ |� Ψ( u h ) , u ∗ − u ∗ � K + � ψ ( u h ) , u ∗ − u ∗ � ∂ K | h h � �� � � �� � u ∗ − Π h u ∗ u ∗ − Π h u ∗ This suggests the node-wise metric, M = | Ψ( u h ) || 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 ( u h ) = � (Ψ h − Ψ)( u ) , u ∗ � + � R . Assume we have the conservative form Ψ( u ) = ∇ · F ( u ), so J ( u ) − J ( u h ) ≈ � ( F − F h )( u ) , ∇ u ∗ � Ω − � � n · ( F − F h ( u ) , u ∗ ) � ∂ Ω . This gives Riemannian metric fields � � � n � � ∂ u ∗ M volume = � � | H ( F i ( u )) | � , � ∂ x i i =1 � � n �� � � � � � M surface = | u ∗ | � H F i ( u ) · n i � � � i =1 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. Uniform mesh (1,920,000 elements). Anisotropic mesh resulting from averaging a posteriori metrics. Adapted mesh (1,766,396 elements). 18 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake Goal-Oriented Adaptation Results Outlook. To appear in proceedings of the 28 th International Meshing Roundtable: JW, N Barral, D Ham, M Piggott, “Anisotropic Goal-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 . PhD thesis, Universit´ e Pierre et Marie Curie, 2015. Roland Becker and Rolf Rannacher. An optimal control approach to a posteriori error estimation in finite element methods. Acta Numerica 2001 , 10:1–102, May 2001. ISSN 0962-4929. Pascal-Jean Frey and Fr´ ed´ eric Alauzet. Anisotropic mesh adaptation for cfd computations. Computer methods in applied mechanics and engineering , 194(48-49):5068–5082, 2005. Adrien Loseille, Alain Dervieux, and Fr´ ed´ eric Alauzet. Fully anisotropic goal-oriented mesh adaptation for 3d steady 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 JH Goddard, 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 software release 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 the Multicore-Challenge III , pages 143–144. Springer, 2013. 20 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake Appendix Combining Metrics Metric Combination. Consider metrics M 1 and M 2 . How to combine these in a meaningful way? Metric average: M := 1 2 ( M 1 + M 2 ) . Metric superposition: intersection of Steiner ellipses. Metric superposition [Barral, 2015] 21 / 20
Anisotropic Goal-Oriented Mesh Adaptation in Firedrake Appendix Hessian recovery Double L 2 projection We recover H = ∇ T ∇ u by solving the auxiliary problem � H = ∇ T g g = ∇ u as � � � � n � n τ : H h d x + div ( τ ) : g h d x − ( g h ) i τ ij n j d s = 0 , ∀ τ Ω Ω ∂ Ω i =1 j =1 � � � ψ · g h d x = u h ψ · � n d s − div ( ψ ) u h d x , ∀ ψ. Ω ∂ Ω Ω 22 / 20
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