Leicht, Tobias

Wallraff, Marcel

English

Institute of Transport Research:Publications

www.DLR.de • Folie 1 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013Higher Order Multigrid Algorithmsfor a 2D and 3D RANS-kω DG-SolverMarcel Wallraff, Tobias LeichtDLR Braunschweig (AS - C2A2S2E)www.DLR.de • Folie 2 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013Multigrid algorithmswww.DLR.de • Folie 3 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013DG discretizationBasis functionsnon-parametric ortho-normal basis functionsdirectly formulated in physical spacealso referred to as Taylor-DGneed to be evaluated for each mesh elementRANS-kω equationskω turbulence modelsecond scheme of Bassi and Rebay (BR2) for the viscoustermsRoe flux as a convective flux, based on aneigen-decomposition of the full jacobianwww.DLR.de • Folie 4 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013Non-linear multigrid methodnested hierarchy of linear spacesVlmin ⊂ Vlmin+1 ⊂ · · · ⊂ Vlmax−1 ⊂ VlmaxRnlmin Rnlmin+1 · · · Rnlmax−1 Rnlmaxintergrid transfer operators:prolongation: natural injection Ill−1 : Rnl−1 → Rnlcanonical restriction operator Il−1l :=(Ill−1)>non-linear multigrid algorithm also requires:restricted nonlinear state vector:orthogonal L2-projection Iˆl−1l on the space Vl−1www.DLR.de • Folie 5 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013Non-linear multigrid methodLet the non-linear problem to be solved on thefine level lmax be given byLlmax(ulmax) = flmax .restrict solution approximation ul−1 := Iˆl−1l ulcompute forcing function for the coarse level:fl−1 ← fl−1 + Il−1l (fl − Ll(ul))−(fl−1 − Ll−1(u0l−1))Galerkin-transfer for the Jacobian: Rl−1 = Il−1l Rl Ill−1www.DLR.de • Folie 6 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013Non-linear smoother / solversmoother / solverlinearized Backward-EulerSolve[(αi∆t)−1M + Rl](ul,i − ul,i−1) =[fl − Ll(ul,i−1)],where Rl is the fully implicit Jacobian matrix and M is themass matrix. In addition to that ul,j is a state vector, withul,j ∈ Vl ∀ j ∈ N.local pseudo-time steps, adaptive CFL numberwww.DLR.de • Folie 7 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013Linear smoother / solverKrylov method as linear solver (GMRES method)line-Jacobi as preconditioner / smootherlet Ll,k(ul,k) = fl,k the underlying linear problem on line k,solve δul,k,i := ul,k,i − ul,k,i−1 = R−1l,k (fl,k − Ll,kul,k,i−1)set ul,k,i := ul,k,i−1 + δul,k,i,where R−1l,k is the inverse of the Jacobian matrix computedone line k in the meshwww.DLR.de • Folie 8 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013Relaxation schemeJacobian / system matrix structure element diagonalline neighboroff-line off-diagonalmatrix blockswww.DLR.de • Folie 9 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013Numerical algorithmspossible solver choicessingle grid Backward-Eulerstart up streategy in mesh or order sequencingfor improved initial conditionslinear MG as preconditionernon-linear MG to accelerate process in pseudo-timenon-linear MG with linear MG on each levelwww.DLR.de • Folie 10 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013Numerical parameters for the non-linear problemsnon-linear multigridonly V-cycles will be presentedone pre- and post-smoothing iteration on each levelone smoothing iteration on the lowest levela linearized Backward-Euler scheme as smootherusing an SER time steping scheme for the Backward-EulerGalerkin-transfer to obtain the Jacobian on the lower levelswww.DLR.de • Folie 11 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013Numerical parameters for the linear problemsparameters for solving the resulting linear problems on every levelfrom the Backward-Euler linearizationGMRES method with a fixed number of max steps on everylevellinear multigrid as a preconditioner for the GMRES methodone V-cycle is done for preconditioningno pre-smoothing iterationsfour post-smoothing iterationsfour smoothing iterations on the lowest levelline-Jacobi scheme as smootherGalerkin-transfer to obtain the lower level matriceswww.DLR.de • Folie 12 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013L1T2 high-lift configurationflow conditions:Mach: 0.197Reynolds number: 3,520,000α = 20.18◦testcase from EC funded ADIGMA projectcomputations will be shown on an unstructured mesh with23824 mixed elementsp = 2 solution is desired for all computationswww.DLR.de • Folie 13 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013L1T2 high-lift configurationmixed-element mesh(CENAERO via GMSH)third order142 944 DoFthree levelswww.DLR.de • Folie 14 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013L1T2 high-lift configurationrun time comparison: h-MG0 0.5 1 1.5 2 2.5 3 3.510−1210−910−610−3100normalized CPU timedensityresidualSGh-sequenceh-NMGh-LMGh-NMG + LMGwww.DLR.de • Folie 15 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013L1T2 high-lift configurationrun time comparison: p-MG0 0.5 1 1.5 2 2.5 3 3.510−1210−910−610−3100normalized CPU timedensityresidualSGp-sequencep-NMGp-LMGp-NMG + LMGwww.DLR.de • Folie 16 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013VFE-2 Delta-Wing with rounded leading edgeflow conditions:Mach: 0.4Reynolds number: 3,000,000α = 13.3◦, β = 0◦testcase from EC funded IDIHOM projectcomputations will be shown on a structured mesh with13816 elementsp = 2 solution is desired for all computationswww.DLR.de • Folie 17 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013VFE-2 Delta-Wing with rounded leading edgeoriginal FV mesh with 884 224elements was two timesagglomeratedresulting in a higher order meshwith 13 816 curved elements geometryoriginal mesh curved mesh with faceswith straight faces represented by polynomials of order 4www.DLR.de • Folie 18 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013VFE-2 Delta-Wing with rounded leading edge0 20 40 60 80 10010−1110−910−710−510−310−1top level non-linear iterationsdensityresidualh-sequenceh-LMGh-NMGh-NMG + LMGp-sequencep-LMGp-NMGp-NMG + LMGwww.DLR.de • Folie 19 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013VFE-2 Delta-Wing with rounded leading edge0 10 20 30 40 50 60 7010−1110−910−710−510−310−1top level non-linear iterationsdensityresidualh-NMG + LMGh-NMG + LMGp-NMG + LMGp-NMG + LMGwww.DLR.de • Folie 20 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013VFE-2 Delta-Wing with rounded leading edgeThe blue computation on the mesh with 13816 elements and thered computation on an adjoint refined mesh with 23877elemtents.0 20 40 60 80 100 120 140 16010−1210−910−610−3100V-cycledensityresidualp-NMG + LMGp-NMG + LMGwww.DLR.de • Folie 21 >HONOM 2013>Marcel Wallraff, Tobias Leicht• 21. 03. 2013Thanks for your attention!

Higher Order Multigrid Algorithms for a 2D and 3D RANS-kw DG-Solver

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