Adaptive mesh redistribution on the sphere for global atmospheric modelling Phil Browne & Hilary Weller (Reading), Colin Cotter & Jemma Shipton (Imperial), Chris Budd & Andrew McRae (Bath) Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Why r-adaptivity Does not create load balancing problems on parallel computers, Does not require mapping solutions between different meshes, Does not lead to sudden changes in resolution, Can be retro-fitted into existing models Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Objectives of the NERC project Solve optimal transport equations on the sphere to efficiently redistribute a mesh Assess mesh quality for the equations of the atmosphere Develop mimetic finite element/volume methods on moving meshes Compare with established test cases Establish suitable refinement criteria for the atmosphere Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
r-adaptive mesh redistribution Original computational mesh T c Adapted physical mesh T p F ( T c ) = T p ; ∀ ξ ∈ T c ∃ x ∈ T p s.t. x = F ( ξ ) (1) Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Monitor function equidistribution Given m ( x ) > 0 , find F : Ω c → Ω p such that m ( x ) | J ( ξ ) | = c. (2) Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Optimally transported meshes Seek F ∗ such that � F ∗ = arg min | ξ − F ( ξ ) | 2 d ξ. || F − I || = (3) F Ω c Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Optimally transported meshes Seek F ∗ such that � F ∗ = arg min | ξ − F ( ξ ) | 2 d ξ. || F − I || = (3) F Ω c Theorem (Brenier (1991) [in cuboid domains]) There exists a unique optimally transported map F ( ξ ) which minimises (3) , and the Jacobian of which satisfies the equidistribution equation (2). Furthermore, F ( ξ ) can be written as the gradient (with respect to ξ ) of a convex scalar (mesh) potential φ ( ξ ) , so that x ( ξ ) = ∇ ξ φ ( ξ ) , H ξ ( φ ( ξ )) ≻ 0 . (4) Brenier, Y. (1991). Polar Factorization and Monotone Rearrangement of Vector-Valued Functions. Communications on Pure and Applied Mathematics , XLIV:375–417 Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
A Monge-Amp` ere equation As x = ∇ φ and m ( x ) | J ( ξ ) | = c , c m ( ∇ φ ) | H ( φ ) | = c ⇐ ⇒ | H ( φ ) | = (5) m ( ∇ φ ) m ( ∇ φ ) V i ( x ) V i ( x ) c V i ( ξ ) = c ⇐ ⇒ V i ( ξ ) = (6) m ( ∇ φ ) Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Optimal transport on S n 1 Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Optimal transport on S n 1 Theorem (McCann (2001)) Let M be a connected, complete smooth Riemannian manifold, equipped with its standard volume measure dx . Let µ, ν be two probability measures on M with compact support, and let the objective function c ( ξ, x ) be equal to d ( ξ, x ) 2 , where d is the geodesic distance on M . Further, assume that µ is absolutely continuous with respect to the volume measure on M . Then, there is a unique optimal transport map F where F pushes forward the measure µ onto ν . Then, (using classical optimal transport notation): F # ( µ ) = ν i.e. x = F ( ξ ) = exp ξ [ ∇ φ ( ξ )] (7) for some d 2 / 2 -convex potential φ . McCann, R. (2001). Polar factorization of maps on Riemannian manifolds. Geometric & Functional Analysis GAFA , 11(3):589–608 Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Optimal transport on S n 2 Corollary (Weller, B., Budd, Cullen (2015)) There exists a unique, optimally transported mesh on the sphere that satisfies the equidistribution principle. Moreover, that mesh is defined by a c -convex scalar potential function that satisfies the Monge-Amp` ere type equation m (exp ξ [ ∇ φ ( ξ )]) | J ( ξ ) | = c. (8) Corollary (Weller, B., Budd, Cullen (2015)) The optimally transported mesh on the sphere satisfying the equidistribution principle does not exhibit tangling. Weller, H., Browne, P., Budd, C., and Cullen, M. (2015). Mesh adaptation on the sphere using optimal transport and the numerical solution of a Monge-Amp` ere type equation. J Comp Phys , (In Press) Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Solution techniques for the Monge-Amp` ere equation Parabolic Relaxation, Budd & Williams (2009) 1 ( I − γ ∇ 2 ) φ n +1 = ( I − γ ∇ 2 ) φ n + δt [ m ( x n ) | I + H ( φ n ) | ] d . (9) Linearisation about 0 | I + H ( φ n +1 ) | = 1 + ∇ 2 φ n +1 + N ( φ n +1 ) (10) Linearisation about φ n | I + H ( φ n +1 ) | = | I + H ( φ n ) | + ε ∇ · A n ∇ ψ + N ( εψ ) (11) Budd, C. and Williams, J. (2009). Moving mesh generation using the parabolic Monge-Amp` ere Equation. SIAM Journal on Scientific Computing , 31(5):3438–3465 Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Solution techniques for the Monge-Amp` ere equation Parabolic Relaxation, Budd & Williams (2009) 1 ( I − γ ∇ 2 ) φ n +1 = ( I − γ ∇ 2 ) φ n + δt [ m ( x n ) | I + H ( φ n ) | ] d . (9) Fixed point iterations, Weller, B., Budd, Cullen (2015) c γ ∇ 2 φ n +1 = γ ∇ 2 φ n − | I + H ( φ n ) | + m ( x n ) , ∀ n ∈ N . (10) Adaptive linearisation fixed point iterations c A n ∇ φ n +1 � � = ∇· ( A n ∇ φ n ) −| I + H ( φ n ) | + ∇· m ( x n ) , ∀ n ∈ N . (11) Budd, C. and Williams, J. (2009). Moving mesh generation using the parabolic Monge-Amp` ere Equation. SIAM Journal on Scientific Computing , 31(5):3438–3465 Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Convergence 1 1 10 PMA ( γ =0 . 45 ,δt =0 . 1 ) PMA ( γ =0 . 7 ,δt =0 . 1 ) 10 0 PMA ( γ =0 . 45 ,δt =0 . 15 ) PMA ( γ =0 . 7 ,δt =0 . 15 ) PMA ( γ =0 . 5 ,δt =0 . 1 ) PMA ( γ =0 . 7 ,δt =0 . 2 ) -1 10 PMA ( γ =0 . 5 ,δt =0 . 15 ) PMA ( γ =0 . 75 ,δt =0 . 1 ) Equidistribution (CV of m | I + H | ) -2 10 PMA ( γ =0 . 55 ,δt =0 . 1 ) PMA ( γ =0 . 75 ,δt =0 . 15 ) PMA ( γ =0 . 55 ,δt =0 . 15 ) PMA ( γ =0 . 75 ,δt =0 . 2 ) -3 10 PMA ( γ =0 . 6 ,δt =0 . 1 ) PMA ( γ =0 . 75 ,δt =0 . 25 ) -4 10 PMA ( γ =0 . 6 ,δt =0 . 15 ) PMA ( γ =0 . 8 ,δt =0 . 1 ) PMA ( γ =0 . 6 ,δt =0 . 2 ) PMA ( γ =0 . 8 ,δt =0 . 15 ) -5 10 PMA ( γ =0 . 65 ,δt =0 . 1 ) PMA ( γ =0 . 8 ,δt =0 . 2 ) PMA ( γ =0 . 65 ,δt =0 . 15 ) PMA ( γ =0 . 8 ,δt =0 . 25 ) 10 -6 PMA ( γ =0 . 65 ,δt =0 . 2 ) AL -7 10 -8 10 -9 10 0 200 400 600 800 1000 Iteration number Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Convergence 2 1 10 FP ( γ =2 . 6 ) FP ( γ =2 . 95 ) 10 0 FP ( γ =2 . 65 ) FP ( γ =3 . 0 ) FP ( γ =2 . 7 ) FP ( γ =3 . 05 ) -1 10 FP ( γ =2 . 75 ) FP ( γ =3 . 1 ) Equidistribution (CV of m | I + H | ) -2 10 FP ( γ =2 . 8 ) FP ( γ =3 . 15 ) FP ( γ =2 . 85 ) AL -3 10 FP ( γ =2 . 9 ) -4 10 -5 10 10 -6 -7 10 -8 10 -9 10 0 100 200 300 400 500 Iteration number Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Convergence 3 1 10 PMA ( γ =0 . 45 ,δt =0 . 15 ) PMA ( γ =0 . 65 ,δt =0 . 25 ) 10 0 PMA ( γ =0 . 45 ,δt =0 . 2 ) PMA ( γ =0 . 7 ,δt =0 . 15 ) PMA ( γ =0 . 5 ,δt =0 . 15 ) PMA ( γ =0 . 7 ,δt =0 . 2 ) -1 10 PMA ( γ =0 . 5 ,δt =0 . 2 ) PMA ( γ =0 . 7 ,δt =0 . 25 ) Equidistribution (CV of m | I + H | ) -2 10 PMA ( γ =0 . 55 ,δt =0 . 15 ) PMA ( γ =0 . 7 ,δt =0 . 3 ) PMA ( γ =0 . 55 ,δt =0 . 2 ) PMA ( γ =0 . 75 ,δt =0 . 15 ) -3 10 PMA ( γ =0 . 6 ,δt =0 . 15 ) PMA ( γ =0 . 75 ,δt =0 . 2 ) -4 10 PMA ( γ =0 . 6 ,δt =0 . 2 ) PMA ( γ =0 . 75 ,δt =0 . 25 ) PMA ( γ =0 . 6 ,δt =0 . 25 ) PMA ( γ =0 . 75 ,δt =0 . 3 ) -5 10 PMA ( γ =0 . 65 ,δt =0 . 15 ) AL PMA ( γ =0 . 65 ,δt =0 . 2 ) 10 -6 -7 10 -8 10 -9 10 0 50 100 150 200 250 300 Iteration number Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Convergence 4 1 10 FP ( γ =0 . 8 ) FP ( γ =1 . 05 ) 10 0 FP ( γ =0 . 85 ) FP ( γ =1 . 1 ) FP ( γ =0 . 9 ) FP ( γ =1 . 15 ) -1 10 FP ( γ =0 . 95 ) AL Equidistribution (CV of m | I + H | ) -2 10 FP ( γ =1 . 0 ) -3 10 -4 10 -5 10 10 -6 -7 10 -8 10 -9 10 0 20 40 60 80 100 120 140 160 Iteration number Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
Mesh redistribution on the sphere V i ( x ) c on S 2 x = exp ξ ( ∇ φ ) V i ( ξ ) = where (12) m ( x ) Finite volume method (OpenFOAM) Fixed point iterations Geometric version of the Hessian Linearisation about 0 on a tangent plane Exponential mappings of the points Monitor function derived from reanalysis precipitation data Hexagonal isocohedral T c Phil Browne Adaptive algorithms for computational PDEs Adaptive mesh redistribution on the sphere
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