DG Methods for Aerodynamic Flows: Higher Order, Error Estimation and Adaptive Mesh Refinement

Transcription

1 16th International Conference on Finite Elements in Flow Problems DG Methods for Aerodynamic Flows: Higher Order, Error Estimation and Adaptive Mesh Refinement Institute of Aerodynamics and Flow Technology DLR Braunschweig 25. March / 42

2 Research group working on discontinuous Galerkin methods for aerodynamic flows at DLR The current group members are: Dr. Tobias Leicht (PhD student) Stefan Schoenawa (PhD student) Marcel Wallraff (PhD student) former group member were: Dr. Joachim Held Florian Prill (PhD student) Numerical results are based on: The DLR-PADGE code which is based on a modified version of deal.ii. 2 / 42

3 Overview Higher-order discontinuous Galerkin methods Error estimation and adaptive mesh refinement for force coefficients Residual-based mesh refinement Numerical results for aerodynamic test cases considered in the EU-project ADIGMA laminar flow around a delta wing turbulent flow around the 3-element L1T2 high-lift configuration turbulent flow around the DLR-F6 wing-body configuration considered in the EU-project IDIHOM subsonic turbulent flow around the VFE-2 delta wing configuration transonic turbulent flow around the VFE-2 delta wing configuration 3 / 42

4 DG discretization of the RANS-kω equations RANS and Wilcox k-ω turbulence model equations: (F c (u) F v (u, u) ) = S(u, u) Discontinuous Galerkin discretization of order p + 1: Find u h V p h such that R(u h, v h ) R(u h ) v h dx + r(u h ) v + h + ρ(u h) : v + h ds κ T h with the element residual, Ω κ\γ + r Γ (u h ) v + h + ρ (u Γ h) : v + h ds = 0 v h V p h, Γ R(u h ) = S(u h, u h ) F c (u h ) + F v (u h, u h ), and face and boundary residuals r(u h ), ρ(u h ) and r Γ (u h ), ρ Γ (u h ). 4 / 42

5 Error estimation with respect to target quantities Target quantities J(u) of interest are the drag, lift and moment coefficients pressure induced and viscous stress induced parts of the force coefficients 5 / 42

6 Error estimation with respect to target quantities Target quantities J(u) of interest are the drag, lift and moment coefficients pressure induced and viscous stress induced parts of the force coefficients We want to quantity the error of the discrete function u h in terms of a target quantity J( ), i.e. we want to quantity the error Here, J(u) J(u h ) J(u h ) is the computed force coefficient, and J(u) is the exact (but unknown) value of the force coefficient 5 / 42

7 Error estimation for single target quantities Given a discretization: find u h V h,p such that N (u h, v h ) = 0 v h V h,p. and a target quantity J. Using a duality argument we obtain an error representation wrt. J( ): J(u) J(u h ) = R(u h, z) := N (u h, z) R(u h, z h ) = κ η κ. where z h is the solution to the discrete adjoint problem: find z h V h,p such that N [u h ](w h, z h ) = J [u h ](w h ) w h V h,p, and η κ are adjoint-based indicators which are particularly suited for the accurate and efficient approximation of the target quantity J(u). 6 / 42

8 Error estimation for multiple target quantities For N target quantities J i (u), i = 1,..., N: Instead of computing N adjoint solutions N [u h ](w h, z i,h ) = J i [u h ](w h ) w h V h,p, i = 1,..., N, to obtain error estimates for the N target quantities J i (u) J i (u h ) = R(u h, z i ) R(u h, z i,h ), i = 1,..., N, we now solve one discrete error equation: find ē h V h,p such that N [u h ](ē h, w h ) = R(u h, w h ) w h V h,p, to obtain error estimates for the N target quantities J i (u) J i (u h ) J i [u h ](e) J i [u h ](ē h ), i = 1,..., N. 7 / 42

9 Adaptive mesh refinement for multiple target functionals Goal-oriented mesh refinement tailored to reducing e.g. a) the sum of relative errors or b) the (weighted) sum of absolute errors a) N J i (u) J i (u h ) / J i (u), b) N α i J i (u) J i (u h ). i=1 Define the combined target functional: N a) J c (v) = s i J i (v)/ J i (u h ), b) J c (v) = i=1 with s i = sign(j i (u) J i (u h )). We now solve the adjoint problem: find z c,h V h,p such that i=1 N α i s i J i (v), N [u h ](w h, z c,h ) = J c[u h ](w h ) w h V h,p, i = 1,..., N, and obtain the error estimate i=1 J c (u) J c (u h ) = R(u h, z c ) R(u h, z c,h ) = κ T h η κ. 8 / 42

10 Residual-based mesh refinement The DG discretization: Find u h V p h such that R(u h, v h ) R(u h ) v h dx + r(u h ) v + h + ρ(u h) : v + h ds κ T h Error representation: Residual-based indicators: Ω κ\γ + r Γ (u h ) v + h + ρ (u Γ h) : v + h ds = 0 v h V p h, Γ J(u) J(u h ) = R(u h, z) ( ) 1/2 J(u) J(u h ) (ηκ res ) 2 κ T h ηκ res = h κ R(u h ) κ + hκ 1/2 r(u h ) κ\γ + hκ 1/2 ρ(u h ) κ\γ + hκ 1/2 r Γ (u h ) κ Γ + hκ 1/2 ρ Γ (u h ) κ Γ 9 / 42

11 Example: ADIGMA BTC3 test case Laminar flow at M = 0.3, Re = 4000 and α = 12.5 around a delta wing Reference values by fine grid computations: Cl ref = , Cd ref = , and Cm ref = ADIGMA industrial accuracy requirements: TOL Cl = 10 2, TOL Cd = TOL Cm = 10 3 Performance of residual-based refinement adjoint-based refinement (single-target and multi-target) error estimation (single-target and multi-target) 10 / 42

12 Example: ADIGMA BTC3 test case Laminar flow at M = 0.3, Re = 4000, α = 12.5 around a delta wing Multi-target adjoint-based mesh refinement for the sum of relative errors of C l, C d and C m Error in C d : Error in C l : error in Cd 0.01 global residual-based adjoint-based(multi) adjoint-based(cd) adj(multi)+est adj(cd)+est TOL_Cd error in Cl 0.01 global residual-based adjoint-based(multi) adjoint-based(cl) adj(multi)+est adj(cl)+est TOL_Cl e+06 1e+07 1e+08 number of dofs e+06 1e+07 1e+08 number of dofs 11 / 42

13 Example: ADIGMA BTC3 test case Laminar flow at M = 0.3, Re = 4000, α = 12.5 around a delta wing Multi-target adjoint-based mesh refinement for the sum of relative errors of C l, C d and C m Error in C d : Error in C d : error in Cd 0.01 global residual-based adjoint-based(multi) adjoint-based(cd) adj(multi)+est adj(cd)+est TOL_Cd error in Cd 0.01 global residual-based adjoint-based(multi) adjoint-based(cd) adj(multi)+est adj(cd)+est TOL_Cd e+06 1e+07 1e+08 number of dofs 1e-05 1e time (fraction of global refinement to meet TOL_Cd) 12 / 42

14 Example: ADIGMA BTC3 test case Laminar flow at M = 0.3, Re = 4000, α = 12.5 around a delta wing Multi-target adjoint-based mesh refinement for the sum of relative errors of C l, C d and C m After 5 residual-based ref. steps: 14.7 mio. DoFs After 4 adjoint-based ref. steps: 6.6 mio. DoFs Sum of relative errors: 5% Sum of relative errors: 1.6% Rel. computing time: 0.06 (no error est.) Rel. computing time: (incl. error est.) 13 / 42

15 Example: ADIGMA BTC3 test case Laminar flow at M = 0.3, Re = 4000, α = 12.5 around a delta wing After 5 residual-based mesh refinement steps: 14.7 mio. DoFs 14 / 42

16 The L1T2 high lift configuration full mesh zoom of mesh Coarse mesh of 4740 elements. Grid lines are given by polynomials of degree / 42

17 Turbulent flow around the L1T2 high lift configuration Freestream conditions: M = 0.197, α = and Re = Mach number and streamlines turbulent intensity 16 / 42

18 Turbulent flow around the L1T2 high lift configuration Freestream conditions: M = 0.197, α = and Re = / 42

19 Turbulent flow around the L1T2 high lift configuration Freestream conditions: M = 0.197, α = and Re = / 42

20 Turbulent flow around the L1T2 high lift configuration Freestream conditions: M = 0.197, α = and Re = / 42

21 Turbulent flow around the L1T2 high lift configuration Turbulent flow at M = 0.197, α = and Re = Adjoint-based refinement targeted at C l : Error in C l : Cl global adjoint-based(cl) adj(cl) + est. Cl_ref-TOL_Cl Cl_ref Cl_ref+TOL_Cl e+06 degrees of freedom 20 / 42

22 Turbulent flow around the L1T2 high lift configuration Turbulent flow at M = 0.197, α = and Re = Adjoint-based refinement targeted at C d : 21 / 42

23 Anisotropic mesh refinement on 3d turbulent flows In the following test cases the meshes are refined anisotropically the anisotropic element subdivision is based on jump indicators Refer to the talk of Tobias Leicht, this room, 10:00, on : hp-adaptive meshes with anisotropic element subdivision 22 / 42

24 The DLR-F6 wing-body configuration without fairing The original mesh of elements has been agglomerated twice. The elements of the coarse mesh of elements are curved based on additional points taken from the original mesh geometry curved mesh with lines given by polynomials of degree 4 23 / 42

25 Subsonic turbulent flow around the DLR-F6 wing-body Modification of the DPW III test case: M = 0.5 (instead of M = 0.75) α = (instead of target lift C l = 0.5) Re = coarse mesh 2 nd order solution DG solutions on coarse mesh of curved elements. 3 rd order solution 4 th order solution 24 / 42

26 Subsonic turbulent flow around the DLR-F6 wing-body TAU on original grid 3 rd order solution after one refinement of coarse mesh 25 / 42

27 Subsonic turbulent flow around the DLR-F6 wing-body TAU on original grid 3 rd order solution after one refinement of coarse mesh 26 / 42

28 Subsonic turbulent flow around the DLR-F6 wing-body Adjoint-based refinement for C d : Mesh after 2 adjoint-based refinement steps Density adjoint 27 / 42

29 Subsonic turbulent flow around the DLR-F6 wing-body Adjoint-based refinement for C d : Convergence of C d (global mesh refinement): p=1, global p=2, global p=3, global TAU 0.04 Mesh after 2 adjoint-based refinement steps Density adjoint e+06 1e+07 degrees of freedom 27 / 42

30 Subsonic turbulent flow around the DLR-F6 wing-body Adjoint-based refinement for C d : Mesh after 2 adjoint-based refinement steps Convergence of C d (global & local mesh refinement): p=1, global p=2, global p=3, global residual-based adjoint-based(cd) adj(cd) + est. TAU Density adjoint e+06 1e+07 degrees of freedom 27 / 42

31 The VFE-2 delta wing with medium rounded leading edge The original mesh of elements has been agglomerated twice. The elements of the coarse mesh of elements are curved based on additional points taken from the original mesh geometry original mesh curved coarse mesh with lines with straight lines given by polynomials of degree 4 28 / 42

32 Fully turbulent flow around the VFE-2 delta wing configuration Underlying flow case U.1 in the EU-project IDIHOM The VFE-2 delta wing with medium rounded leading edge at two different flow conditions: U.1b: RANS-kω, subsonic flow at M = 0.4, α = 13.3 and Re = U.1c: RANS-kω, transonic flow at M = 0.8, α = 20.5 and Re = / 42

33 Subsonic flow around the VFE-2 delta wing U.1b: Fully turbulent flow at M = 0.4, α = 13.3 and Re = coarse mesh with curved elements c p distribution 5 th order solution vs. experiment (PSP) 30 / 42

34 Subsonic flow around the VFE-2 delta wing U.1b: Fully turbulent flow at M = 0.4, α = 13.3 and Re = residual-based refined mesh with curved elements c p distribution 4 th order solution vs. experiment (PSP) 31 / 42

35 Subsonic flow around the VFE-2 delta wing U.1b: Fully turbulent flow at M = 0.4, α = 13.3 and Re = th -order solution on residual-based refined mesh with curved elements 32 / 42

36 Subsonic flow around the VFE-2 delta wing U.1b: Fully turbulent flow at M = 0.4, α = 13.3 and Re = th -order solution on residual-based refined mesh with curved elements 2 nd -order solution on residual-based refined mesh with curved elements 32 / 42

37 Fully turbulent flow around the VFE-2 delta wing configuration Underlying flow case U.1 in the EU-project IDIHOM The VFE-2 delta wing with medium rounded leading edge at two different flow conditions: U.1b: RANS-kω, subsonic flow at M = 0.4, α = 13.3 and Re = U.1c: RANS-kω, transonic flow at M = 0.8, α = 20.5 and Re = requires shock capturing 33 / 42

38 Shock-capturing based on artificial viscosity (1) N sc (u h, v) ε(u h ) u h : vdx ε klm (u h ) xl uh m xk v m dx, κ κ κ κ For the compressible Navier-Stokes equations (2nd order DG discretization), 1 ε klm (u h ) = C ε δ kl h 2 β k R m (u h ), k, l = 1,..., d, m = 1,..., n, R m (u h ) = n R q (u h ), m = 1,..., n, q=1 where R(u h ) = (R q (u h ), q = 1,..., n) is the residual of the PDE given by R(u h ) = (F c (u h ) F v (u h, u h )). 1 R. Hartmann. Adaptive discontinuous Galerkin methods with shock-capturing for the compressible Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 51(9 10): , / 42

39 Shock-capturing based on artificial viscosity (2) N sc (u h, v) ε(u h ) u h : vdx ε klm (u h ) xl uh m xk v m dx, κ κ κ κ For the RANS-kω equations (2nd and higher order discretization), 2 ε klm (u h ) = C ε b k b l hκ 2 f p(u h ) Rp(u h) + s p(u + h, u h ) p, b = p p + ε R p(u h ) = Xd+2 m=1 p Xd+2 R m(u h ), s p(u + h, u h ) = u m m=1 p u m s m(u + h, u h ), with R(u h ) = F c (u h ), Z Z s m(u + h, u + h ) v h dx = `H(u h, u h, n ) F c (u + h ) n + m v h ds κ κ 2 F. Bassi et. al. Very high-order accurate Discontinuous Galerkin Computation of transonic turbulent flows on Aeronautical configurations, ADIGMA, NNFMMD 113, / 42

40 Shock-capturing based on artificial viscosity (combines 1 and 2) N sc (u h, v) κ κ ε(u h ) u h : vdx κ κ ε klm (u h ) xl u m h xk v m dx, For the RANS-kω equations (2nd and higher order discretization) ε klm (u h ) = C ε δ kl h2 k f p(u h ) Rp(u h), k, l = 1,..., d, m = 1,..., d + 2, p R p(u h ) = Xd+2 m=1 p u m R m(u h ), R(u h ) = S(u h, u h ) F c (u h ) + F v (u h, u h ), h i = h i /(degree + 1), where h i is the dimension of the element κ in the x i -coordinate direction 36 / 42

41 Transonic flow around the VFE-2 delta wing U.1c: Fully turbulent flow at M = 0.8, α = 20.5 and Re = refined mesh with curved elements c p distribution 4 th order solution vs. experiment (PSP) 37 / 42

42 Transonic flow around the VFE-2 delta wing U.1c: Fully turbulent flow at M = 0.8, α = 20.5 and Re = th -order solution on residual-based refined mesh with curved elements 38 / 42

43 Transonic flow around the VFE-2 delta wing U.1c: Fully turbulent flow at M = 0.8, α = 20.5 and Re = th -order solution on residual-based refined mesh with curved elements 38 / 42

44 Transonic flow around the VFE-2 delta wing U.1c: Fully turbulent flow at M = 0.8, α = 20.5 and Re = th -order solution on residual-based refined mesh with curved elements 38 / 42

45 Transonic flow around the VFE-2 delta wing U.1c: Fully turbulent flow at M = 0.8, α = 20.5 and Re = th -order solution on residual-based refined mesh with curved elements 38 / 42

46 Transonic flow around the VFE-2 delta wing U.1c: Fully turbulent flow at M = 0.8, α = 20.5 and Re = c p 39 / 42

47 Transonic flow around the VFE-2 delta wing U.1c: Fully turbulent flow at M = 0.8, α = 20.5 and Re = c p and M = 1 isosurface 40 / 42

48 Transonic flow around the VFE-2 delta wing U.1c: Fully turbulent flow at M = 0.8, α = 20.5 and Re = c p and M = 1 isosurface zoom of M = 1 isosurface 41 / 42

49 Summary Higher-order discontinuous Galerkin methods Error estimation and adaptive mesh refinement for force coefficients Residual-based mesh refinement Numerical results for aerodynamic flows around a laminar delta wing the 3-element L1T2 high-lift configuration the DLR-F6 wing-body configuration the VFE-2 delta wing configuration (subsonic and transonic) Computations have been performed with the DLR-PADGE code Thank you 42 / 42