Hybrid Discontinuous Galerkin methods for incompressible flow problems

Transcription

1 Hybrid Discontinuous Galerkin methods incompressible flow problems Christoph Lehrenfeld, Joachim Schöberl MathCCES Computational Mathematics in Engineering Workshop Linz, May 31 - June 1, 2010

2 Contents The methods: Cockburn, Lazarov, Gopalakrishnan, 2008 Egger, Schöberl, 2009 Scalar convection diffusion (IP) (U) (IP) (U) Cockburn, Kanschat, Schötzau, 2005 incompressible Navier Stokes Equations (IP) (IP) (U) (U) Schöberl, Zaglmayr, 2005 Reduced Reduced 'high order 'high order divfree' divfree' basis basis Software, Examples, Conclusion

3 Contents The methods: Cockburn, Lazarov, Gopalakrishnan, 2008 Egger, Schöberl, 2009 Scalar convection diffusion (IP) (U) (IP) (U) Cockburn, Kanschat, Schötzau, 2005 incompressible Navier Stokes Equations (IP) (IP) (U) (U) Schöberl, Zaglmayr, 2005 Reduced Reduced 'high order 'high order divfree' divfree' basis basis Software, Examples, Conclusion

4 Hybrid scalar convection diffusion equations u + div(bu) = f

5 Discretization space Trial functions are: discontinuous piecewise polynomials (on each facet and element) with appropriate mulations we get more unknowns but less matrix entries implementation fits into standard element-based assembling structure allows condensation of element unknowns

6 mulation u Symmetric Interior Penalty Formulation (τ h h ): p 2 B (u, v) = u v dx T T + { { u } { v } } [[v]] ds [[u]] ds + τ h [[u]][[v]] ds E E n E n E Hybrid Symmetric Interior Penalty Formulation (τ h h ): p 2 B ( (u,u F ),(v,v F ) ) = { u v dx T T u T n (v v v F ) ds T n (u u F ) ds } + τ h (u u F )(v v F ) ds T This and other hybridizations of CG, mixed and methods were discussed in [Cockburn+Gopalakrishnan+Lazarov, 08]

7 mulation u Symmetric Interior Penalty Formulation (τ h h ): p 2 B (u, v) = u v dx T T + { { u } { v } } [[v]] ds [[u]] ds + τ h [[u]][[v]] ds E E n E n E Hybrid Symmetric Interior Penalty Formulation (τ h h ): p 2 B ( (u,u F ),(v,v F ) ) = { u v dx T T u T n (v v v F ) ds T n (u u F ) ds } + τ h (u u F )(v v F ) ds T This and other hybridizations of CG, mixed and methods were discussed in [Cockburn+Gopalakrishnan+Lazarov, 08]

8 mulation div(bu) = f (div(b) = 0) After partial integration on each element we get div(bu) v dx = { bu v dx + Ω T T Upwind Hybridized Upwind : T } b n u? v ds u U = { unb if b n 0 u if b n > 0 = u HU = { uf if b n 0 u if b n > 0

9 mulation div(bu) = f (div(b) = 0) After partial integration on each element we get div(bu) v dx = { bu v dx + Ω T T Upwind Hybridized Upwind : T } b n u? v ds u U = { unb if b n 0 u if b n > 0 = u HU = { uf if b n 0 u if b n > 0 + T T out b n (u F u)v F ds This gives us an appropriate bilinearm C ( (u, u F ), (v, v F ) ) the convective term. Already used in [Egger+Schöberl, 09]

10 mulation div(bu) = f (div(b) = 0) After partial integration on each element we get div(bu) v dx = { bu v dx + Ω T T Upwind Hybridized Upwind : T } b n u? v ds u U = { unb if b n 0 u if b n > 0 = u HU = { uf if b n 0 u if b n > 0 + T T out b n (u F u)v F ds This gives us an appropriate bilinearm C ( (u, u F ), (v, v F ) ) the convective term. Already used in [Egger+Schöberl, 09]

11 mulation u + div(bu) = f (div(b) = 0) Now we can add the introduced bilinearms together and easily get a mulation the convection diffusion problem: A ( (u, u F ), (v, v F ) ) := B ( (u, u F ), (v, v F ) ) + C ( (u, u F ), (v, v F ) ) = (f, v)

12 Contents The methods: Cockburn, Lazarov, Gopalakrishnan, 2008 Egger, Schöberl, 2009 Scalar convection diffusion (IP) (U) (IP) (U) Cockburn, Kanschat, Schötzau, 2005 incompressible Navier Stokes Equations (IP) (IP) (U) (U) Schöberl, Zaglmayr, 2005 Reduced Reduced 'high order 'high order divfree' divfree' basis basis Software, Examples, Conclusion

13 Hybrid incompressible Navier Stokes Equations div( ν u + u u + pi ) = f div u = 0 + b.c.

14 (steady) incompressible Navier Stokes Equations variational mulation: B(u, v) + C(u; u, v) + D(p, v) + D(q, u) = f (v) where B is a suitable bilinearm the viscous term, D is the bilinearm the incompressibility-constraint and the pressure term and C is the convective Trilinearm. tasks: Find an appropriate discretization space Find appropriate bilinearms Find an iterative procedure to solve the nonlinearity ( Oseen its.)

15 Contents The methods: Cockburn, Lazarov, Gopalakrishnan, 2008 Egger, Schöberl, 2009 Scalar convection diffusion (IP) (U) (IP) (U) Cockburn, Kanschat, Schötzau, 2005 incompressible Navier Stokes Equations (IP) (IP) (U) (U) Schöberl, Zaglmayr, 2005 Reduced Reduced 'high order 'high order divfree' divfree' basis basis Software, Examples, Conclusion

16 H(div)-conming elements Navier Stokes [Cockburn, Kanschat, Schötzau, 2005]: + inc. Navier-Stokes + local conservation + energy-stability = need exactly divergence-free convective velocity C(u; u, v)

17 Contents The methods: Cockburn, Lazarov, Gopalakrishnan, 2008 Egger, Schöberl, 2009 Scalar convection diffusion (IP) (U) (IP) (U) Cockburn, Kanschat, Schötzau, 2005 incompressible Navier Stokes Equations (IP) (IP) (U) (U) Schöberl, Zaglmayr, 2005 Reduced Reduced 'high order 'high order divfree' divfree' basis basis Software, Examples, Conclusion

18 H(div)-conming elements Navier Stokes Trial and test functions: normal-cont., tangential-discont. velocity element functions, piecewise polynomial (degree k) u, v facet velocity functions the tangential component only, piecewise polynomial (degree k) u t F, v t F discont. element pressure functions, piecewise polynomial (degree k 1) p, q

19 Hybrid Navier Stokes bilinearms Viscosity: B ( (u, u F ), (v, v F ) ) = Convection: T T C ( w; (u, u F ), (v, v F ) ) = T pressure / incompressibility constraint: D ( (u, u F ), q ) = T { ν u : v dx ν u n (v t vf t ) ds T T } ν v n (u t uf t ) ds + ντ h (u t uf t ) (v t vf t ) ds T { } u w : v dx+ w nu up v ds+ w n(uf t u t )vf t ds T T T out T div(u)q dx weak incompressibility (+ H(div)-conmity) exactly divergence-free solutions

20 Contents The methods: Cockburn, Lazarov, Gopalakrishnan, 2008 Egger, Schöberl, 2009 Scalar convection diffusion (IP) (U) (IP) (U) Cockburn, Kanschat, Schötzau, 2005 incompressible Navier Stokes Equations (IP) (IP) (U) (U) Schöberl, Zaglmayr, 2005 Reduced Reduced 'high order 'high order divfree' divfree' basis basis Software, Examples, Conclusion

21 Reduced basis H(div)-conming Finite Elements We use the construction of high order finite elements of [Schöberl+Zaglmayr, 05, Thesis Zaglmayr 06]: It is founded on the de Rham complex H 1 H(curl) curl H(div) div L 2 W h curl div Vh S h Q h a a b b c c + 1 Leading to a natural separation of the space ( higher order) W hp = W L1 + span{ϕ W h.o.} V hp = V N0 + span{ ϕ W h.o.} + span{ϕ V h.o.} S hp = S RT 0 + span{curl ϕ V h.o.} + span{ϕ S h.o.} Q hp = Q P0 + span{div ϕ S h.o.}

22 Reduced basis H(div)-conming Finite Elements We use the construction of high order finite elements of [Schöberl+Zaglmayr, 05, Thesis Zaglmayr 06]: It is founded on the de Rham complex H 1 H(curl) curl H(div) div L 2 W h curl div Vh S h Q h a a b b c c + 1 Leading to a natural separation of the space ( higher order) W hp = W L1 + span{ϕ W h.o.} V hp = V N0 + span{ ϕ W h.o.} + span{ϕ V h.o.} S hp = S RT 0 + span{curl ϕ V h.o.} + span{ϕ S h.o.} Q hp = Q P0 + span{div ϕ S h.o.}

23 Reduced basis H(div)-conming Finite Elements We use the construction of high order finite elements of [Schöberl+Zaglmayr, 05, Thesis Zaglmayr 06]: It is founded on the de Rham complex H 1 H(curl) curl H(div) div L 2 W h curl div Vh S h Q h a a b b c c + 1 Leading to a natural separation of the space ( higher order) W hp = W L1 + span{ϕ W h.o.} V hp = V N0 + span{ ϕ W h.o.} + span{ϕ V h.o.} S hp = S RT 0 + span{curl ϕ V h.o.} + span{ϕ S h.o.} Q hp = Q P0 + span{div ϕ S h.o.}

24 Reduced basis H(div)-conming Finite Elements We use the construction of high order finite elements of [Schöberl+Zaglmayr, 05, Thesis Zaglmayr 06]: It is founded on the de Rham complex H 1 H(curl) curl H(div) div L 2 W h curl div Vh S h Q h a a b b c c + 1 Leading to a natural separation of the space ( higher order) W hp = W L1 + span{ϕ W h.o.} V hp = V N0 + span{ ϕ W h.o.} + span{ϕ V h.o.} S hp = S RT 0 + span{curl ϕ V h.o.} Q hp = Q P0 + span{div ϕ S h.o.}

25 space separation in 2D div(σ k h ) = T RT 0 DOF higher order edge DOF higher order div.-free DOF higher order DOF with nonz. div RT 0 shape fncs. curl of H 1 -edge fncs. curl of H 1 -el. fncs. remainder P 0 (T ) {0} {0} T [P k 1 P 0 ](T ) 1 #DOF = 3 + 3k + 2 k(k 1) k(k + 1) 1 discrete functions have only piecewise constant divergence only piecewise constant pressure necessary exact incompressibility

26 Summary of ingredients and properties So, we have presented a new Finite Element Method Navier Stokes, with H(div)-conming Finite Elements Hybrid Discontinuous Galerkin Method viscous terms Upwind flux (in -sence) the convection term leading to solutions, which are locally conservative energy-stable ( d dt u 2 L 2 C ν f 2 L 2 ) exactly incompressible static condensation standard finite element assembly is possible less matrix entries than std. approaches reduced basis possible

27 Software, Examples and Conclusion

28 Software Netgen Downloads: Help & Instructions: NGSolve (including the presented scalar methods) Downloads: Help & Instructions: NGSflow (including the presented methods) Downloads: Help & Instructions: NGSflow is the flow solver add-on to NGSolve. It includes: The presented Navier Stokes Solver A package solving heat (density) driven flow

29 Examples 2D Driven Cavity (u top = 0.25, ν = 10 3 )

30 Examples 2D laminar flow around a disk (Re=100): 3D laminar flow around a cylinder (Re=100):

31 Heat driven flow changes in density are small: incompressibility model is still acceptable changes in density just cause some buoyancy ces Boussinesq-Approximation: Incompressible Navier Stokes Equations are just modified at the ce term: f = g (1 β(t T 0 ))g β : heat expansion coefficient Convection-Diffusion-Equation the temperature T t + div( λ T + u T ) = q Discretization of Convection Diffusion Equation with method Weak coupling of unsteady Navier Stokes Equation and unsteady Convection Diffusion Equation Higher Order (p 1,.., 5) in time with IMEX schemes

32 Examples Benard-Rayleigh example: Top temperature: constant 20 C Bottom temperature: constant 20.5 C Initial mesh and initial condition (p = 5):

33 Conclusion and ongoing work Conclusion: efficient methods scalar convection diffusion equations Navier Stokes Equations using Hybrid H(div)-conming Finite Elements reduced basis related/future work: heat driven flow (Boussinesq Equations) time integration (IMEX schemes,...) 3D Solvers (Preconditioners,...)

34 Thank you your attention!