High-order finite volume method for curved boundaries and non-matching domain-mesh problems

Transcription

1 High-order finite volume method for curved boundaries and non-matching domain-mesh problems May 3-7, 0, São Félix, Portugal Ricardo Costa,, Stéphane Clain,, Gaspar J. Machado, Raphaël Loubère Institut de Mathématiques de Toulouse, Université de Toulouse, France Centro de Matemática, Universidade do Minho, Guimarães, Portugal

2 Outline Motivation and Background Problem Formulation 3 Polynomial Reconstruction Machinery ARCH Method 5 Finite Volume Scheme Numerical Benchmark 7 Conclusions and Final Remarks /8

3 Motivation and Background Replacing curved boundaries by polygonal edges associated to the mesh provides at most nd-order accuracy O(h ) Ω M Isoparametric elements are widely applied for FEM and DG Very few methods have been developed in the context of HO-FVM Motivation and Background 3/8

4 State-of-art in HO-FVM Ghost cells approach: add extra cells between the geometric boundary and the computational domain C.F. Ollivier-Gooch and M. Van Altena, A high-order accurate unstructured mesh finite-volume scheme for the advection-diffusion equation, Journal of Computational Physics, (00). Motivation and Background /8

5 State-of-art in HO-FVM Ghost cells approach: add extra cells between the geometric boundary and the computational domain Ollivier-Gooch approach : enforce the BC by constraining the LSM associated to the PR can be very time consuming if the LSM matrix has to be updated (moving boundaries/interfaces, tracking interfaces/discontinuities problems) C.F. Ollivier-Gooch and M. Van Altena, A high-order accurate unstructured mesh finite-volume scheme for the advection-diffusion equation, Journal of Computational Physics, (00). Motivation and Background /8

6 State-of-art in HO-FVM Ghost cells approach: add extra cells between the geometric boundary and the computational domain Ollivier-Gooch approach : enforce the BC by constraining the LSM associated to the PR can be very time consuming if the LSM matrix has to be updated (moving boundaries/interfaces, tracking interfaces/discontinuities problems) New approach: detach the BC conservation from the LSM matrix Easier, flexible, more efficient, more elegant C.F. Ollivier-Gooch and M. Van Altena, A high-order accurate unstructured mesh finite-volume scheme for the advection-diffusion equation, Journal of Computational Physics, (00). Motivation and Background /8

7 Problem Formulation Poisson s equation and BCs: φ = f, φ = φ D, φ n = g N, αφ + β φ n = g R, in Ω on Γ D on Γ N on Γ R Ω, Ω = {Γ D Γ N Γ R } real domain and its boundary n unit normal vector to Ω φ D Dirichlet BC g N Neumann BC g R Robin BC with coefficients α and β Problem Formulation 5/8

8 Generic FV Scheme Γ D eid c i eij n id nij m i q id,r q ij,r c j m j c i, e ij cell, edge q i, q ij quadrature points n ij normal vector ( φ) n i ds = f dx [ R ] e ij ζ r F ij,r c i c i Physical fluxes: j ν(i) r= f i c i = O(h R i ) Source term: F ij,r = φ(q ij,r ) n ij, S f i = ζ s f (q i,s ) s= Problem Formulation /8

9 Polynomial Reconstruction Machinery Non-conservative PR for inner edges e ij ϕ ij (x) = R α ij (x m ij ) α 0 α d α = (α, α ), α = α + α, x α = x α x α R ij = (R α ij ) 0 α d coefficients (to be determined) Polynomial Reconstruction Machinery 7/8

10 Polynomial Reconstruction Machinery Non-conservative PR for inner edges e ij ϕ ij (x) = R α ij (x m ij ) α 0 α d α = (α, α ), α = α + α, x α = x α x α R ij = (R α ij ) 0 α d coefficients (to be determined) E ij (R ij ) = q S ij ñ ô ω ij,q ϕ c q ij (x) dx φ q c q ϕ ij (x) defined with R ij = arg min R ij [E ij (R ij )] Polynomial Reconstruction Machinery 7/8

11 Polynomial Reconstruction Machinery Conservative PR for Dirichlet boundary edges e id ϕ id (x) = φ id + R α id [(x m id ) α MiD] α α d α = (α, α ), α = α + α, x α = x α x α R id = (R α id ) α d coefficients (to be determined) Polynomial Reconstruction Machinery 8/8

12 Polynomial Reconstruction Machinery Conservative PR for Dirichlet boundary edges e id ϕ id (x) = φ id + R α id [(x m id ) α MiD] α α d α = (α, α ), α = α + α, x α = x α x α R id = (R α id ) α d coefficients (to be determined) E id (R id ) = q S id ω id,q ñ ô ϕ c q id (x) dx φ q c q ϕ id (x) defined with R id = arg min R id [E id (R id )] Polynomial Reconstruction Machinery 8/8

13 Polynomial Reconstruction Machinery Naive mean-value conservation nd-order! φ id = φ D (x) ds, ϕ e id e id e id id (x) ds = φ id e id MiD α = (x m id ) α dx e id e id Polynomial Reconstruction Machinery 9/8

14 Polynomial Reconstruction Machinery Naive mean-value conservation nd-order! φ id = φ D (x) ds, ϕ e id e id e id id (x) ds = φ id e id MiD α = (x m id ) α dx e id e id Wise mean-value conservation Quadrature points on Ω! φ id = φ D (x) ds, ϕ Ûe id Ûe id Ûe id id (x) ds = φ id Ûe id MiD α = (x m id ) α dx Ûe id Ûe id Polynomial Reconstruction Machinery 9/8

15 Polynomial Reconstruction Machinery Naive mean-value conservation nd-order! φ id = φ D (x) ds, ϕ e id e id e id id (x) ds = φ id e id MiD α = (x m id ) α dx e id e id Wise mean-value conservation Quadrature points on Ω! φ id = φ D (x) ds, ϕ Ûe id Ûe id Ûe id id (x) ds = φ id Ûe id MiD α = (x m id ) α dx Ûe id Ûe id Wise point-value conservation: φ id = φ D (p id ), p id Ûe id, ϕ id (p id ) = φ id M α id = (p id m id ) α Polynomial Reconstruction Machinery 9/8

16 Polynomial Reconstruction Machinery Conservation of Dirichlet BC only Coefficients MiD α are boundary dependent LSM matrix has to be updated for moving boundaries/interfaces or dynamic BC in time-dependent and unsteady problems... optimization problems... tracking interfaces/discontinuities problems... etc. Ollivier-Gooch method consists in an augmented LSM matrix by constraints rows (equivalent to the wise conservation) Polynomial Reconstruction Machinery 0/8

17 ARCH Method ARCH Adaptive Reconstruction for Conservation of High-order The aim of ARCH is to improve the boundary treatment The main ideas are conserve the HO accuracy... detach the boundary from the LSM matrix... easy handling of moving boundaries... generic treatment of Dirichlet, Neumann and Robin BCs ARCH Method /8

18 ARCH Method ARCH for boundary edges e ib ϕ ib (x; ψ ib ) = ψ ib + ϕ ib (x) ϕ ib (x) non-conservative/naive conservative/wise conservative PR ψ ib free parameter (to be determined) E ib (R ib ; ψ ib ) = q S ib ω ib,q ñ ψ ib + c q c q ϕ ib (x; ψ ib ) dx φ q ô ϕ ib (x; ψ ib ) defined with R ib = arg min R ib [E ib (R ib ; ψ ib )] ψ ib is a RHS the LSM matrix remains the same :D ARCH Method /8

19 ARCH Method Generic BC on Ω to satisfy: g(x; α, β) = α(x)φ(x) + β(x) φ n Dirichlet BC: α =, β = 0 Neumann BC: α = 0, β 0 Robin BC: α 0, β 0 ARCH Method 3/8

20 ARCH Method Generic BC on Ω to satisfy: g(x; α, β) = α(x)φ(x) + β(x) φ n Dirichlet BC: α =, β = 0 Neumann BC: α = 0, β 0 Robin BC: α 0, β 0 For a given p ib Ω, the parameter ψ ib is prescribed such that α(p ib )ϕ ib (p ib ; ψ ib ) + β(p ib ) ϕ ib (p ib ; ψ ib ) n = g(p ib ; α, β) ψ ib = ψib 0 B(p ib ; α, β, ψib 0 )(ψ0 ib ψ ib ) B(p ib ; α, β, ψib 0 ) B(p ib; α, β, ψib 0 + ϕ ib ) ψ 0 ib ψ ib ARCH Method 3/8

21 Finite Volume Scheme Numerical fluxes: inner edge e ij: F ij,r = ϕ ij (q ij,r ) n ij boundary edge e ib : F ib,r = ϕ ib (q id,r ; ψ ib ) n ib PR and fluxes are linear identities Linear residual operator Φ G i (Φ) for vector Φ R I G i (Φ) = j ν(i) [ R ] e ij ζ r F ij,r f i c i, r= Free matrix method G(Φ) = (G i (Φ)) i=,...,i Compute vector Φ R I solution of G(Φ) = 0 Finite Volume Scheme /8

22 Numerical Benchmark Numerical Benchmark 5/8

23 Numerical Benchmark Manufactured solution: φ(r) = a + b ln(r) a, b such that φ [0, ]. 0. x x E0 E- E- E- E-8 E-0 E- Naive PR E3 E E5 E, P E, P E, P3 E, P5 Numerical Benchmark /8

24 Numerical Benchmark Manufactured solution: φ(r) = a + b ln(r) a, b such that φ [0, ]. 0. x x E0 E- E- E- Naive PR E0 E- E- E- E, P E, P E, P3 E, P5 Wise PR E0 E- E- E- E, P E, P E, P3 E, P5 ARCH E, P E, P E, P3 E, P5 E-8 E-8 E-8 E-0 E- E-0 E- E3 E E5 E-0 E- E3 E E5 E3 E E5 Numerical Benchmark /8

25 Numerical Benchmark. 0. x 0 E0 E- E- E- Wise PR E0 E- E- E- E, P E, P E, P3 E, P5 ARCH E, P E, P E, P3 E, P x E-8 E-0 E- E-8 E-0 E- E3 E E5 E3 E E5 Numerical Benchmark 7/8

26 Numerical Benchmark. 0. x 0 E0 E- E- E- Wise PR E0 E- E- E- E, P E, P E, P3 E, P5 ARCH E, P E, P E, P3 E, P x E-8 E-0 E- E-8 E-0 E- E3 E E5 E3 E E5. 0. x 0 E0 E- E- E- Wise PR E0 E- E- E- E, P E, P E, P3 E, P5 ARCH E, P E, P E, P3 E, P x E-8 E-0 E- E-8 E-0 E- E3 E E5 E3 E E5 Numerical Benchmark 7/8

27 Numerical Benchmark = 9730 P P 3 P 5 E E E E E E Naive PR.9E E 03.9E E 03.9E E 03 Wise PR.0E 05.E 05.5E E 08.98E.0E 0 ARCH.0E 05.37E E E E.5E 09 = 9730 P P 3 P 5 E E E E E E Naive PR.87E E 03.87E E 03.87E E 03 Wise PR.07E 05.3E 0 5.5E 09.E E.E 0 ARCH.03E 05.03E 03 7.E 0.0E 08.93E.00E 0 Numerical Benchmark 8/8

28 Numerical Benchmark. 0. x 0 E0 E- E- E- Wise PR E0 E- E- E- E, P E, P E, P3 E, P5 ARCH E, P E, P E, P3 E, P x E-8 E-0 E- E-8 E-0 E- E3 E E5 E3 E E5 Numerical Benchmark 9/8

29 Numerical Benchmark. 0. x 0 E0 E- E- E- Wise PR E0 E- E- E- E, P E, P E, P3 E, P5 ARCH E, P E, P E, P3 E, P x E-8 E-0 E- E-8 E-0 E- E3 E E5 E3 E E5. 0. x 0 E0 E- E- E- Wise PR E0 E- E- E- E, P E, P E, P3 E, P5 bfbfarch E, P E, P E, P3 E, P x E-8 E-0 E- E-8 E-0 E- E3 E E5 E3 E E5 Numerical Benchmark 9/8

30 Numerical Benchmark = 9730 P P 3 P 5 E E E E E E Naive PR 5.88E 03.88E E 03.88E E 03.88E 0 Wise PR.7E 05.77E 0 3.8E 08.37E 07.80E 0.E 09 ARCH.8E 05.80E 0.E 08.7E 07.E 0.E 08 = 305 P P 3 P 5 E E E E E E Naive PR.3E 0 7.0E 0.E E 0.E E 0 Wise PR 9.7E 0 9.7E E 0.0E 0 3.3E 08.30E 0 ARCH 9.E 0.35E E E 05.8E E 07 Numerical Benchmark 0/8

31 Numerical Benchmark. 0. x 0 E E0 E- E- E- Wise PR E E0 E- E- E- E, P E, P E, P3 E, P5 ARCH E, P E, P E, P3 E, P x E-8 E-0 E- E-8 E-0 E- E3 E E5 E3 E E5 = 305 P P 3 P 5 E E E E E E Naive PR.3E 0 7.0E 0.E E 0.E E 0 Wise PR 9.7E 0 9.7E E 0.0E 0 3.3E 08.30E 0 ARCH 9.E 0.35E E E 05.8E E 07 Numerical Benchmark /8

32 Numerical Benchmark Γ I, Γ E interior and exterior boundaries Γ I : x = R I (θ; α I ) cos(θ), Γ E : x = R E (θ; α E ) cos(θ) sin(θ) sin(θ) x R I (θ; α I ) = r I Å + ã 0 sin(α Iθ), R E (θ; α E ) = r E Å + ã 0 sin(α Eθ) Manufactured solution: x φ(r, θ) = a(θ) + b(θ) ln(r) a(θ) and b(θ) are chosen such that φ [0, ] in Ω Numerical Benchmark /8

33 Numerical Benchmark. 0. x x Numerical Benchmark 3/8

34 Numerical Benchmark. 0. x x Wise PR E- E- E- E- E- E-8 E-0 E- E, E, E, E, E, E, E- ARCH E0 P P P3 P3 P5 P5 E- E-8 E-0 E3 Numerical Benchmark E E5 E- E, E, E, E, E, E, E- E0 Naive PR E0 E- P P P3 P3 P5 P5 E, E, E, E, E, E, P P P3 P3 P5 P5 E-8 E-0 E3 E E5 E- E3 E E5 3/8

35 Numerical Benchmark. 0. x x Wise PR E- E- E- E- E- E-8 E-0 E- E, E, E, E, E, E, E- ARCH E0 P P P3 P3 P5 P5 E- E-8 E-0 E3 Numerical Benchmark E E5 E- E, E, E, E, E, E, E- E0 Naive PR E0 E- P P P3 P3 P5 P5 E, E, E, E, E, E, P P P3 P3 P5 P5 E-8 E-0 E3 E E5 E- E3 E E5 /8

36 Numerical Benchmark Mesh Pick-up. 0. x x Numerical Benchmark 5/8

37 Numerical Benchmark Mesh Pick-up. E0 E- E- E- Wise PR E, P E, P E, P3 E, P5 0. E-8 x 0 E-0 E- E3 E E x E0 E- E- E- E-8 E-0 E- ARCH E3 E E5 E, P E, P E, P3 E, P5 Numerical Benchmark 5/8

38 Numerical Benchmark Mesh Pick-up. E0 E- E- E- Wise PR E, P E, P E, P3 E, P5 0. E-8 x 0 E-0 E- E3 E E x 0.. E0 E- E- E- E-8 E-0 E- ARCH E3 E E5 E, P E, P E, P3 E, P5 Numerical Benchmark /8

39 Numerical Benchmark Mesh Pick-up. E0 E- E- E- Wise PR E, P E, P E, P3 E, P5 0. E-8 x 0 E-0 E- E3 E E x E0 E- E- E- E-8 E-0 E- ARCH E3 E E5 E, P E, P E, P3 E, P5 Numerical Benchmark 7/8

40 Conclusions and Final Remarks HO is fully restored with the ARCH method Wise PR and ARCH methods have comparable accuracy Extrapolation situations are less robust Conclusions and Final Remarks 8/8

41 Conclusions and Final Remarks HO is fully restored with the ARCH method Wise PR and ARCH methods have comparable accuracy Extrapolation situations are less robust Benchmarks for Neumann and Robin BCs are in progress (promising results) Improved mesh pick-up algorithms are in progress Conclusions and Final Remarks 8/8

42 Conclusions and Final Remarks HO is fully restored with the ARCH method Wise PR and ARCH methods have comparable accuracy Extrapolation situations are less robust Benchmarks for Neumann and Robin BCs are in progress (promising results) Improved mesh pick-up algorithms are in progress Obrigado Conclusions and Final Remarks 8/8