A DG-based incompressible Navier-Stokes solver

Transcription

1 A DG-based incompressible Navier-Stokes solver Higer-order DG metods and finite element software for modern arcitectures - Bat 2016 Marian Piatkowski 1 Peter Bastian 1 1 Interdisciplinary Center for Scientific Computing, Heidelberg University Tursday, June 2, 2016 Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

2 Navier-Stokes DG-discretization Instationary incompressible Navier-Stokes equations ρb t v µ v ` ρpv qv ` p f v 0 in Ω ˆ p0, T s in Ω ˆ p0, T s vpx, t 0q v 0 pxq # + # + v g for Γ D BΩ v g on Γ D tbω, Hu pdx 0 or v n p n 0 on Γ N BΩzΓ D Ω Stationary Stokes equations as a minimization problem under constraints 1 min v 2 p v, vq 0,Ω pf, vq 0,Ω s.t. v 0 Lagrangian Lpv, pq 1 2 p v, vq 0,Ω pf, vq 0,Ω pp, vq 0,Ω First order optimality conditions lead to weak formulation Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

3 Navier-Stokes DG-discretization DG discretization I SIPG / NIPG for diffusion apu, vq ÿ ÿ u : vdx E t uu n e rvs ɛt vu n e rusds e EPE epγ int u int n e v int ds ` ɛ v int n e pu int gqds Γ D Γ D J 0 pu, vq ÿ σ e ÿ σ e rus rvsds ` e e epγ int epγ e e puint gq v int ds D $ & minp E peq, E `peq q e, e P Γ int e, E peq X E `peq e % e P BΩ, E peq X BΩ e E peq e Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

4 Navier-Stokes DG-discretization DG discretization II Velocity-pressure coupling + convective term b m pv, pq ÿ ÿ p vdx ` E tpurvs n eds ` p int v int nds e Γ D EPE epγ int bpv, pq b m pv, pq p int g nds Γ D cpw, v, φq conservative form + upwind discretization of pw qv. Variational formulation: Find pv, p q P X ˆ M s.t. ρpb t v, φ q ` µ pa ` J 0 qpv, φ q ` ρcpv ; v, φ q ` b m pφ, p q pf, φ q P X bpv, q q P M X, M piecewise polynomial spaces Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

5 Operator Splitting / Projection Metods Wy Discontinuous Galerkin and wy projection metods? Local mass conservation per element Block vector and block matrix structure, contiguous memory access Wit te projection metods: Easier solution of te transient Navier-Stokes equations at ig Reynolds numbers All metods presented in tis talk ave been implemented in te DUNE software framework Dune Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

6 Operator Splitting / Projection Metods Pressure-correction scemes Pressure-correction scemes: Fractional-step tecnique General approac: Split between incompressibility and dynamics 1 One time step wit explicit pressure p,k`1 in momentum equation, straigtforward in DG-discretization. Gives tentative velocity ṽ k`1. 2 Projection of ṽ k`1 onto solenoidal space, requires investigation for DG. Can be reformulated as a Poisson problem for te pressure correction ψ k`1. 3 L 2 -projection wit ψ k`1 Variants: obtain pressure update p k`1 and optional divergence correction of ṽ k`1 Implicit Euler, p,k`1 0, no divergence correction Corin s projection metod p,k`1 p k, no divergence correction Incremental Pressure Correction Sceme (IPCS) p,k`1 p k, wit divergence correction Rotational Incremental Pressure Correction Sceme (RIPCS) Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18 to

7 Operator Splitting / Projection Metods Solution approac Solution approaces in te subproblems Momentum equation Impl. Euler, BDF2, DIRK-metods suc as Alexander2, Alexander3 for time stepping Navier-Stokes case: Newton s metod wit analytical evaluation of jacobians (no numerical differentiation etc.) Pressure Poisson equation Zero mean condition pif Γ D BΩq is imposed in a post-processing step Hybrid AMG for DG wit correction in conforming Q 1 subspace E AMG4DG pi W 1 DG A DGq ν 2 pi R T B AMG RA DG qpi W 1 DG A DGq ν 1 R T prolongation from CG-subspace to DG W DG smooter on DG level Compute A CG RA DG R T directly pa CG q ij ÿ EPE E ϕcg j ϕ CG i dx ` ÿ epγ N e σ e ϕ CG j e ϕ CG i ϕ CG j nϕ CG i B AMG one V-cycle of AMG preconditioner applied to A CG ϕ CG i nϕ CG j ds Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

8 Towards a fast DG-implementation Fast DG assembler Assembler tailored to te data structure in DG Relevant properties of DG Contiguous memory access Block vector and block matrix structure Taking tese properties into account No data copying for coefficients vector, output result Treat in block-based fasion, blocks interpreted as one DOF Can andle te scalar case, and systems of te form X ˆ... ˆ X Furter exploitation of tese ideas No data copying in evaluation of explicit parameter functions (e.g. pressure in momentum equation, velocity in pressure Poisson equation) Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

9 Towards a fast DG-implementation Sum factorization Fast evaluation of residuals and jacobians: Exploiting tensor product structure Tensor product basis: ϕ j pxq ϕ j1,...,j d px 1,..., x d q θ jd px d q... θ j1 px 1 q Tensor product quadrature: Ξ i pξ i1,..., ξ id q T j pj 1,..., j d q P J d t1,.., nu d, i pi 1,..., i d q P I d t1,.., mu d Evaluate finite element function on reference element at quadrature points: u pξ i q ÿ x j ϕ j pξ i q jpj d ÿ... ÿ θ jd pξ id q... θ j1 pξ i1 q x j1,...,j d j d PJ j 1 PJ ÿ... ÿ j d PJ j 1 PJ A pdq i d,j d... A p1q i 1,j 1 X p0q j 1,...,j d : Y i1,...,i d All basis functions at all quadrature points as matrix-matrix product. Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

10 Towards a fast DG-implementation Sum factorization Fast evaluation of residuals and jacobians II: Sum factorization Y i1,...,i d ÿ... ÿ j d PJ ÿ j d PJ j 1 PJ... ÿ A pdq i d,j d... ÿ j d PJ A pdq i d,j d... A p1q i 1,j 1 X p0q j 1,...,j d j 2 PJ A p2q i 2,j 2 ÿ A pdq i d,j d X pd 1q j d,i 1...i d 1 A p1q i 1,j 1 X p0q j 1,...,j d j 1 PJ loooooooooomoooooooooon X p1q j 2,i 1...,j d Dim by dim approac as matrix-matrix product A pkq X pk 1q X pkq. B k u pξ i q and residual computation similar Face integrals similar, reduction to pd 1q dimensions Same for evaluation of explicit parameter functions Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

11 Towards a fast DG-implementation Sum factorization Fast evaluation of residuals and jacobians III Vectorization over solution and its gradient: pu pξ i q, u pξ i qq T ÿ jpj d x j pϕ j pξ i q, ϕ j pξ i qq T, i P I d Use vector class library by Agner Fog, ttp:// Summary: Assembly of volume and face terms 1 Compute solution and its gradient at all quadrature points 2 Gauss-quadrature loop, compute part witout test functions 3 Compute residuals, compute jacobian If jacobian is set up explicitly compute trial functions and and teir gradients on-te-fly at quadrature points for columnwise accumulation Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

12 Towards a fast DG-implementation Performance comparison Performance comparison: Velocity mom residual evaluation Compute µpa ` J 0 qpv, φ q ` b m pφ, p q pf, φ q P X for given rational v, p and f. Compare against (I) Fast implementation: Sum factorization and fast DG assembler (II) Default implementation: Naive version and standard assembler Polynomial degrees: Q 2 {Q 1 in 2D and Q 3 {Q 2 in 3D Intel Xeon E v3 16 core processor (30.4 GFLOPS peak / core) Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

13 Towards a fast DG-implementation Performance comparison Velocity mom., residual evaluation, 2D, Q 2 {Q 1 -DG space - Fast implementation - Default implementation Part Time GFLOPS alpa boundary alpa skeleton alpa volume lambda boundary lambda skeleton lambda volume local operator total Part Time GFLOPS alpa boundary alpa skeleton alpa volume lambda boundary lambda skeleton lambda volume local operator total absolute overead: s relative overead: 0.58 absolute overead: s relative overead: 0.34 Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

14 Towards a fast DG-implementation Performance comparison Residual evaluation: Detailed measurement Part Time GFLOPS total alpa boundary alpa skeleton skeleton coefficients skeleton gauss skeleton residual alpa volume volume coefficients volume gauss volume residual Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

15 Towards a fast DG-implementation Performance comparison Velocity mom., residual evaluation, 3D, Q 3 {Q 2 -DG space - Fast implementation - Default implementation Part Time GFLOPS alpa boundary alpa skeleton alpa volume lambda boundary lambda skeleton lambda volume local operator total Part Time GFLOPS alpa boundary alpa skeleton alpa volume lambda boundary lambda skeleton lambda volume local operator total absolute overead: s relative overead: 0.31 absolute overead: s relative overead: 0.04 Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

16 Towards a fast DG-implementation Simulation results 2D Driven Cavity flow at Re = Q 2 {Q 1 discretization, anisotropic mes wit aspect ratios up to factor 10 Plot of te vorticity ω ˆ v at two different time snapsots. Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

17 Outlook Outlook Extend matrix-based / matrix-free approac in subproblems of pressure-correction scemes Velocity momentum equation: Matrix-free Newton + GMRes-Ricardson, NIPG discretization Pressure Poisson equation: Hybrid AMG for DG, matrix-based or matrix-free approac on te DG level Matrix-free L 2 -projection of tentative velocity onto solenoidal subspace 3D DNS of atmospere-subsurface coupling in te ABL Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

18 Outlook Outlook Extend matrix-based / matrix-free approac in subproblems of pressure-correction scemes Velocity momentum equation: Matrix-free Newton + GMRes-Ricardson, NIPG discretization Pressure Poisson equation: Hybrid AMG for DG, matrix-based or matrix-free approac on te DG level Matrix-free L 2 -projection of tentative velocity onto solenoidal subspace 3D DNS of atmospere-subsurface coupling in te ABL Tank you for your attention! Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18

19 Outlook Residual evaluation 3D Q 3 {Q 2 -DG space: Detailed measurement Part Time GFLOPS total alpa boundary alpa skeleton skeleton coefficients skeleton gauss skeleton residual alpa volume volume coefficients volume gauss volume residual Marian Piatkowski (IWR) Navier-Stokes-DG Projection Tursday, June 2, / 18