Efficient BDF time discretization of the Navier Stokes equations with VMS LES modeling in a High Performance Computing framework

Transcription

1 Introduction VS LES/BDF Implementation Results Conclusions Swiss Numerical Analysis Day 2015, Geneva, Switzerland, 17 April Efficient BDF time discretization of the Navier Stokes equations with VS LES modeling in a High Performance Computing framework L. Dede, D. Forti, S. Deparis, and A. Quarteroni CCS Chair of odelling and Scientific Computing ATHICSE athematics Institute of Computational Science and Engineering EPFL E cole Polytechnique Fe de rale de Lausanne Switzerland

2 Introduction VS LES/BDF Implementation Results Conclusions Introduction Applications Turbulent or transitional flows occur in many physical contexts: External flows (civil Engineering, Hydrodynamics, or Aeronautical applications); Internal flows (e.g. Haemodynamics). DNS vs. LES The Direct Numerical Simulation (DNS) is generally computationally challenging or not affordable, for which Large Eddy Simulations (LES) models can instead be used. Numerical Approximation (LES) Even with LES modeling, efficient and flexible solvers for the incompressible Navier Stokes equations, e.g. based on the Finite Elements method, are required to make feasible the numerical simulation, also in a High Performance Computing HPC setting.

3 Introduction VS LES/BDF Implementation Results Conclusions otivations Goals Develop an efficient solver for LES modeling of the incompressible Navier Stokes equations. Use the Finite Elements method for the spatial approximation. Use efficient time discretization schemes. Use a scalable solver for parallel computing (HPC). Solve large scale problems at high Reynolds (Re) numbers. ethodology Variational ultiscale (VS) approach with LES modeling (VS LES) [Bazilevs et al., CAE, 2007], [Codina et al., CAE, 2007], [Wall et al, CAE 2010]. Backward Differentiation Formulas (BDF) for the time discretization to define a semi implicit scheme (VS LES/BDF). Scalable solver for HPC with ultigrid (L) preconditioner.

4 Introduction VS LES/BDF Implementation Results Conclusions The incompressible Navier Stokes equations Navier Stokes equations for incompressible flows (Newtonian) ρ u t + ρ(u )u f σ(u, p) = 0 in Ω (0, T ), u = 0 in Ω (0, T ), with suitable initial and boundary conditions on Γ D Ω; u and p are the velocity and the pressure, ρ the density, f the external forces, σ(u, p) = pi + 2µɛ(u) the stress tensor, µ the dynamic viscosity, and ɛ(u) = 1 2 ( u + ( u)t ) the strain tensor. Weak formulation The weak formulation of the Navier Stokes equations reads: find U = {u, p} V g : ( ) w, ρ u t + (w, ρ(u )u) + ( w, µ( u + u T )) ( w, p) + (q, u) = (w, f ) for all W = {w, q} V 0, t (0, T ), (1) with the function spaces V g = {u [ H 1 (Ω) ] d : u ΓD = g}, V 0 = {u [ H 1 (Ω) ] d : u ΓD = 0}, Q = L 2 (Ω), V g = V g Q, and V 0 = V 0 Q.

5 Introduction VS LES/BDF Implementation Results Conclusions Spatial approximation: Finite Elements and VS LES modelling Finite Elements discretization We define X h r := { v h C 0 (Ω) : v h K P r, for all K T h } as the Finite Elements function space of degree r 1 over Ω triangulated with a mesh T h of tetrahedrons; h K is the diameter of the element K T h. ultiscale direct sum decomposition The space V (V g or V 0 ) is decomposed into the coarse and fine scales subspaces: V = V h V, V h is the coarse scale function space associated to the Finite Elements discretization Xr h and V is an infinite dimensional function space representing the fine scales. For the weak formulation (1), we apply the decomposition to the trial and test functions: w = w h + w, q = q h + q, u = u h + u, p = p h + p.

6 Introduction VS LES/BDF Implementation Results Conclusions Spatial approximation: Finite Elements and VS LES modelling VS LES modeling [Bazilevs et al., CAE, 2007] The fine scale velocity and pressure components are chosen as: u τ (u h ) r (u h, p h ), p τ C (u h ) r C (u h ); r (u h, p h ) and r C (u h ) are the residuals of the momentum and continuity equations: r (u h, p h ) = ρ uh t + ρuh u h + p h µ u h f, r C (u h ) = u h. The stabilization parameters τ and τ C are set as: τ = τ (u h ) = ( σ 2 ρ 2 t 2 + ρ2 h 2 K ) 1/2 u h 2 + µ2 C hk 4 p, (2) τ C = τ C (u h ) = h 2 K τ (u h ), (3) with σ a constant equal to the order of the time discretization chosen with time step t and C p = 60 2 r 2 a constant related to an inverse inequality.

7 Introduction VS LES/BDF Implementation Results Conclusions Spatial approximation: Finite Elements and VS LES modelling Semi discrete VS LES formulation (weak residual) find U h = {u h, p h } V h g : A(W h, U h ) F (W h ) = 0 where: for all W h = {w h, q h } V h 0, t (0, T ), A(W h, U h ) := A NS (W h, U h ) + A VS (W h, U h ) F (W h ) := (w h, f ), ( ) A NS (W h, U h ) := w h, ρ uh + ( w h, ρ(u h )u h) t + ( w h, µ( u h + ( u h ) T ) ) ( w h, p h) + ( q h, u h), A VS (W h, U h ) := ( ρu h w h + q h, τ (u h ) r (u h, p h ) ) ( w h, τ C (u h ) r C (u h ) ) + ( ρu h ( u h ) T, τ (u h ) r (u h, p h ) ) VS SUPG and VS LES terms ( w h, τ (u h ) r (u h, p h ) τ (u h ) r (u h, p h ) ).

8 Introduction VS LES/BDF Implementation Results Conclusions Time discretization: implicit BDF schemes Backward Differentiation Formulas (BDF) We partition [0, T ] into N t subintervals of size t = T N t with t n = n t for n = 0,..., N t; u h n u h (t n) and pn h p h (t n). Depending the order σ of the BDF scheme, the approximation of the time derivative of the velocity reads: u h ασuh n+1 u h n,bdfσ, t t where for BDF schemes of orders σ = 1, 2, 3 we have: u h n if n 1, for σ = 1 (BDF1), u h n,bdfσ = 2u h n 1 2 uh n 1 if n 2, for σ = 2 (BDF2), 3u h n 3 2 uh n uh n 2 if n 3, for σ = 3 (BDF3), 1, for σ = 1 (BDF1), 3 α σ =, for σ = 2 (BDF2), 2 11, for σ = 3 (BDF3). 6

9 Introduction VS LES/BDF Implementation Results Conclusions Time discretization: implicit BDF schemes The fully implicit BDF scheme for VS LES modeling For a BDF scheme of order σ: find u h n+1 Vg h and pn+1 h Q h : ( ) w h, ρ ασuh n+1 uh n,bdfσ + ( ) w h, ρ (u h t n+1 )u h n+1 + ( w h, µ ( u h n+1 + ( u h n+1) )) T ( ( ) w h, pn+1) h + q h, u h n+1 + ( ρ u h n+1 w h + q h, τ (u h n+1) r (u h n+1, pn+1) ) h ( w h, τ C (u h n+1) r C (u h n+1) ) + ( ρ u h n+1 ( w h ) T, τ (u h n+1) r (u h n+1, p h n+1) ) ( w h, τ (u h n+1) r (u h n+1, p h n+1) τ (u h n+1) r (u h n+1, p h n+1) ) = ( w h, f n+1 ), given u h n,..., u h n+1 σ, with f n+1 = f (t n+1). The problem is nonlinear both in u h n+1 and p h n+1. for all w h V h 0 and q h Q h, n σ 1,

10 Introduction VS LES/BDF Implementation Results Conclusions Time discretization: the semi implicit BDF scheme (VS LES/BDF) Extrapolation with Newton Gregory backward polynomials The variables u h n+1 and p h n+1 are linearized by means of extrapolation with Newton Gregory backward polynomials. We consider the following extrapolations of orders σ= 1, 2, 3 for the velocity and pressure variables at the discrete time t n+1: u h n if n 0, for σ = 1 (BDF1), u h n+1,σ = 2 u h n u h n 1 if n 1, for σ = 2 (BDF2), 3 u h n 3 u h n 1 + u h n 2 if n 2, for σ = 3 (BDF3) pn h if n 0, for σ = 1 (BDF1), pn+1,σ h = 2 pn h pn 1 h if n 1, for σ = 2 (BDF2), 3 pn h 3 pn 1 h + pn 2 h if n 2, for σ = 3 (BDF3). The extrapolations of u h n+1 and p h n+1 induce similar extrapolations on the residuals and stabilization parameters.

11 Introduction VS LES/BDF Implementation Results Conclusions Time discretization: the semi implicit BDF scheme The semi implicit BDF scheme for VS LES modeling (VS LES/BDF) For a BDF scheme of order σ: find u h n+1 Vg h and pn+1 h Q h : ( w h, ρ ασuh n+1 uh n,bdfσ t ( w h, p h n+1 ( w h, τ n+1,σ C ( + ) + (w h, ρ ( ) ( u h n+1,σ )u h n+1 + w h, µ ( u h n+1 + ( u h n+1) T ) ) ) ( + q h, u h n+1) + (ρ u h n+1,σ w h + q h, τ n+1,σ r n+1,σ (u h n+1, pn+1) ) h r C (u h n+1) ) + ( ρ u h n+1,σ ( w h ) T, τ n+1,σ r n+1,σ (u h n+1, pn+1) ) h ( w h, τ n+1,σ r n+1,σ τ n+1,σ r n+1,σ (u h n+1, pn+1) ) h ( ) w h, τ n+1,σ given u h n,..., u h n+1 σ. w h, τ n+1,σ r n+1,σ r n+1,σ The problem is linear both in u h n+1 and p h n+1. τ n+1,σ u ρ α h n+1 σ t ) (u h n+1, pn+1) h τ n+1,σ ρ uh n,bdf t = ( w h, f n+1 ), for all w h V h 0 and q h Q h, n σ 1,

12 Introduction VS LES/BDF Implementation Results Conclusions Time discretization: the semi implicit BDF scheme Extrapolated stabilization parameters τ n+1,σ := τ n+1,σ C = ( σ 2 ρ 2 t 2 hk 2 τ n+1,σ + ρ2 hk 2. u h n+1,σ ) 1/2 2 + µ2 C p, h 4 K Extrapolated residuals r n+1,σ (u h n+1, p h n+1) := ρ r n+1,σ ( α σu h n+1 uh n,bdfσ t + p h n+1 µ u h n+1 f n+1, := r n+1,σ (u h n+1,σ, pn+1,σ), h ) + ρ u h n+1,σ u h n+1 r n+1,σ (u h n+1, p h n+1) := ρu h n+1,σ u h n+1 + p h n+1 µ u h n+1 f n+1. The extrapolation of the stabilization parameters and residuals is based on the extrapolated variables u h n+1,σ and p h n+1,σ.

13 Introduction VS LES/BDF Implementation Results Conclusions Time discretization: the semi implicit BDF scheme Remarks The fully discrete semi implicit formulation (VS LES/BDF) yields a linear problem in the variables u h n+1 and p h n+1 to be solved only once at each time step t n. At the discrete level, assembling and solving the linear system is done only once for each time step t n (fewer terms than with Newton linearization). BDF schemes with extrapolation yields stable time discretizations of the Navier Stokes equations at the continuous level (under suitable boundary conditions). The stability in respect of the time discretization for the VS LES/BDF scheme is not guaranteed a priori (for all t). In practice, stability in respect of the time discretization is also obtained for relatively large time steps t.

14 Introduction VS LES/BDF Implementation Results Conclusions Finite Elements library LifeV Numerical implementation The implementation of the VS LES/BDF mehod for the incompressible Navier Stokes equations is carried out in the Finite Elements Library LifeV. LifeV Finite Elements Library for the numerical approximation of PDEs Research code oriented to the development and test of new numerical methods and algorithms C++, object oriented, parallel LifeV relies on Trilinos Library/packages odular design; use of the Expression Template Assembly (ETA) package Developers: CCS EPFL, E(C)2 Emory, OX Polimi, REO, ESTIE INRIA Distributed under LGPL,

15 Introduction VS LES/BDF Implementation Results Conclusions ultigrid (L) preconditioner Solver We use the GRES method through the Belos package of Trilinos to solve the linear system. The stopping criterion is based on the relative residual, with tolerance tol = The right preconditioning strategy is used. L Preconditioner We use the ultigrid preconditioners available from the L package of Trilinos. The preconditioner is built on the system matrix [Gee et. al, Sandia National Labs., 2006]. Default parameters setting: NSSA, nonsymmetric smoothed aggregation variant for highly nonsymmetric operators; max levels = 3, cycle applications = 3, pde equations = 4; smoother: Gauss-Seidel, 3 sweeps; aggregation: type = Uncoupled-IS.

16 Introduction VS LES/BDF Implementation Results Flow past a squared cylinder Benchmark problem Flow past a squared cylinder at Re = 22,000 [Koobus and Farhat, 2004] (experimental data available). Γout Γin Lout 2H D y z Lin x Ls Space and time discretizations FE: P1 P1 1,323,056 DOFs; P2 P2 9,209,040 DOFs. Different choices of BDF order (σ = 1 or 2) and t. Conclusions

17 Introduction VS LES/BDF Implementation Flow past a squared cylinder Vortex structures Lambda 2 criterion Results Conclusions

18 Introduction VS LES/BDF Implementation Results Conclusions Flow past a squared cylinder Comparison of drag coefficient C D vs. time t for different discretizations FE P1 P1 and P2 P2; BDF orders σ = 1 and 2. 3 t = s 3 t = s t = s 3.2 t = s CD [-] CD [-] CD [-] CD [-] Time [s] 3 t = s CD [-] Time [s] t = s 3 t = s CD [-] Time [s] 3.2 t = s Time [s] Time [s] Time [s] P1 P1, BDF 1 P1 P1, BDF 2 P2 P2, BDF 2

19 Introduction VS LES/BDF Implementation Results Conclusions Flow past a squared cylinder Comparison of results for different discretizations and literature FE P1 P1 and P2 P2; BDF orders σ = 1 and 2. Numerical Results FE t BDF σ C D rms(c D ) rms(c L ) Strouhal s s P1 P s s s s s P2 P s s Literature (LES) LES method C D rms(c D ) rms(c L ) Strouhal VS F.V. [Koobus., 2004] Smagorinsky [Rodi, 1997] Dynamic LES [Sohankar, 2000]

20 Introduction VS LES/BDF Implementation Results Conclusions Flow past a squared cylinder Scalability results of the solver Linear solver based on GRES and L preconditioner; simulations performed using FE P2 P2, BDF 2, t = s Obtained Ideal 10 2 Obtained Ideal Time [s] Preconditioner assembly time vs. number of CPUs Number of CPUs Time [s] Number of CPUs Time to solve the linear system vs. number of CPUs Time to perform a time step vs. number of CPUs Time [s] Obtained Ideal Iterations Number of GRES iterations vs. number of CPUs Number of CPUs Number of CPUs Computations carried out with Piz Dora, a Cray XC40 supercomputer at the Swiss National Supercomputing Center (CSCS).

21 Introduction VS LES/BDF Implementation Results Conclusions Conclusions We considered a semi implicit scheme based on BDF formulas and extrapolation for the time discretization of the Navier Stokes equation with VS LES modeling (VS LES/BDF); spatial discretization was performed with low order Finite Elements method. We solved problems in a parallel computing framework (HPC), for which we showed the scalability of the solver with ultigrid preconditioning. We applied the method to internal and external flow problems at high Reynolds numbers. The numerical tests showed that the discretization based on the VS LES/BDF method is efficient, versatile, accurate, and stable for wide ranges of the time steps.

22 Introduction VS LES/BDF Implementation Results Conclusions References Y. Bazilevs, V.. Calo, J.A. Cottrell, T.J.R. Hughes, A. Reali, and G. Scovazzi. Variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows. Computer ethods Applied echanics and Engineering, 197: , O. Colomés, S. Badia, R. Codina, and J. Principe. Assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows. Computer ethods in Applied echanics and Engineering, 285:32 63, D. Forti and L. Dedè. Semi implicit BDF time discretization of the Navier Stokes equations with VS LES modeling in a High Performance Computing framework. ATHICSE report, , B. Koobus and C. Farhat. A variational multiscale method for the large eddy simulation of compressible turbulent flows on unstructured meshes. Application to vortex shedding. Computer ethods Applied echanics and Engineering, 193: , W. Rodi, J. Ferziger,. Breuer, and. Pourquié. Status of large eddy simulations: results of a workshop. ASE Journal of Fluids Engineering, 119: , 1997.