Artificial Compressibility Method and 知彼知己者 百戰不殆 Lattice Boltzmann Method If you know your enemies and know yourself, you can win a hundred battles without a single loss Similarities and Differences Taku Ohwada (大和田 拓) Department of Aeronautics & Astronautics, Kyoto University (京都大学大学院工学研究科航空宇宙工学専攻) Collaborators : Prof. Pietro Asinari, Mr. Daisuke Yabusaki May 4, 2011, Spring School on the lattice Boltzmann Method Beijing Computational Science Research Center May 2-6 1 0. What is a good numerical method ? • Performance • Cost (CPU) • Education (Human CPU) 2 Many paths to the summit 1. INTRODUCTION 4 Kinetic methods for fluid-dynamic equations Gas Kinetic Scheme Lattice Boltzmann Method Why Kinetic ? Boltzmann Eq. Euler, Navier-Stokes The path is INDIRECT ! 5 An indirect method is not always your best choice. 6 Extraction of essence Subtraction rather than addition 7 Why Kinetic ? Case of compressible flows Kinetic gadget fM (i ui ) 2 fM exp( ) 3/ 2 (2RT ) 2RT f M yields the flux for the Euler equations. u 1 2 u p f M d [ (e u 2 / 2) p ]u 2 / 2 The discontinuities at cell-interfaces produce numerical dissipation, which suppresses spurious oscillations around shock waves……….. Shock Capturing ! Riemann Problem Riemann Problem Riemann Problem Kinetic Flux Splitting f f t x Characteristics : x t Const. f 0 1 2 f d / 2 Prerequisite of Gas Kinetic Scheme is Taylor expansion !!!! Experiment using undergraduate students. 11 Gallery of Undergraduate Students ‘ Works Euler Blasius flow Navier-Stokes: Any asymptotic method is NOT employed. Pressure distribution along the wall Present 13 Case of incompressible flows Gas kinetic Scheme Lax-Wendroff M 0 No kinetic ingredient !!!! 14 Case of incompressible flows LBM f (t , xi ) ( 1,2,...,Q) f (t 1, xi)= f (t , xi V ) g (h(t , xi V )) f (t , xi V ) h u, v, P f (t , xi ) u(t , xi ), v(t , xi ), P(t , xi ) 15 f (t 1, xi)= f (t , xi V ) g (h(t , xi V )) f (t , xi V ) 1 f (t 1, xi)= g (h(t , xi V )) LBM -> Lattice Kinetic Scheme (LKS) No kinetic ingredient !! LKS -> Lattice Scheme (LK) -> ACM 16 Incompressible case: More difficult than compressible case !!! ui 0, xi ui ui P 2u i uj t x j xi x 2j u j ui 2P 2 xi xi x j 17 1 A LBM ! Poisson Free… • Poisson Free !!! • 2nd order accurate BB • Small time step t ~ (x) 2 • Parallel computation !!! Bouzidi, Firdaouss, Lallemand (2001) Ginzburg, d’Humières (2003) 18 LBM solves INSE via Artificial Compressibility Equations Prof. Asinari ’s morning lecture Chapman-Enskog expansion Hilbert expansion (diffusive scale) P ui 3(x) 0 t xi ui ui P 2ui uj 2 t x j xi x j 2 LBM 19 Artificial Compressibility Method (ACM) P ui k 0 t xi (Chorin,1967) (Témam, 1969) ui ui 2 ui P uj t x j xi x 2j usually k 0.1~10 LBM k 3(x) 2 20 Considering the fact that the lattice Boltzmann method starts with the kinetic theory and has been derived to conserve high-order isotropy, the lattice Boltzmann method should be more accurate than the artificial compressibility method in capturing pressure waves. He, Doolen, Clark (JCP2002) ACM: LBM: Macroscopic (356 papers) Kinetic (4053 papers) 21 Devil’s Project LBM-ACM LBM Chapman-Enskog Expansion ACM • Lattice Structure • Collocated Grid LBM ACM Finite Difference (Finite Volume) Kinetic 22 2. Numerical Computation of ACM Cartesian Grid D2Q9 Time Step Finite Difference Space : Central Difference Time : Semi-Implicit 23 ui t P t 0 x 24 P 1 u v ( x) 0 2 t x x u u u P 2u 2u u v ( 2 ) 2 t x y x x y v v v P 2v 2v u v ( 2 ) 2 t x y y x y h (i ) ( j ) h(( i 1) , j ) h(( i 1) , j ) ( x , y ) ~ x h(i , j ) x 2 h (i ) ( j ) h(i , ( j 1) ) 2h(i , j ) h(i , ( j 1) ) ( x , y ) ~ yy h(i , j ) 2 y 2 25 Basic form of ACM P 1 u v ( x) 0 t 2 x x u u u P 1 2u 2u u v ( ) t x y x Re x 2 y 2 v v v P 1 2v 2v u v ( ) t x y y Re x 2 y 2 uijn u(nt, i , j ) uijn 1 uijn t n 1 n vij vij t 1 n u u v u P ( xx yy )uij Re n ij n x ij n ij n y ij n x ij 1 u v v v P ( xx yy )vijn Re n ij n x ij Pijn1 Pijn t n ij n y ij n y ij 2 u v 1 n 1 x ij n 1 y ij 0 26 4th order accurate div u for compact stencil xu ~ xu yv ~ yv 2 6 2 6 xu y v O( ) yv ~ yv D(u, v) xu 2 6 yyyv O( 4 ) xxxu xxyv O( 2 ) 2 xu ~ xu xxxu O( 4 ) 2 6 2 6 xx y v O( 4 ) yy xu O( 4 ) xx y v y v 2 6 yy xu xu y v O( 4 ) Test Problems • Generalized Taylor-Green Vortex (2D, 3D) • Circular Couette Flow (2D) • Flow Past a Cylinder in a Channel (2D) • Lid-driven Cavity Flow (2D, 3D) • Flow Past a Sphere in Uniform Flow (3D) 28 2D Generalized Taylor-Green Periodic Boundary 29 Time history of L1 error 1 Re 0.001 30 Convergence Rate (t=100) 31 3D Generalized Taylor-Green (3D-GTG) 32 Circular Couette Flow R 5R 33 Error of Velocity Uθ (ACM) 34 Error of Velocity Uθ (MRT) 35 Error of Pressure P (ACM) 36 Error of Pressure P (MRT) 37 Comparison Uθ P ACM MRT ACM MRT 38 Convergence rate 39 The Flow Past a Cylinder M. Schäfer, S. Turek, (1996) Non-Slip Boundary Poiseuille Flow 2.1D D 2D 0th-order extrapolation Non-Slip Boundary 40 t=100 |div u| (Re=100) 41 |div u| (Re=100) t=100 42 Re=100 (unsteady) 43 * M. Schäfer, S. Turek, (1996) LBM: Mussa, Asinari, Luo, JCP 228 (2009) 44 Adaptive Mesh Refinement (Re=100) D: the diameter of the cylinder Poiseuille Flow D 0th-order extrapolation Simple Interpolation 45 t=100 Velocity u 46 t=100 Velocity v 47 t=100 Pressure 48 2D Lid-driven Cavity Flow Top boundary 49 2D Lid-driven Cavity Flow CPU Time (129×129, 100000step) (intel Corei7, openMP) ACM MRT 55.01 sec 405.51 sec ×7.37 356 papers 4053 papers 50 3D Lid-driven Cavity Flow Grid : 100³ Lo D. C. , Murugesan K. , Young D. L. , Int. J. Numer. Meth. Fluids (2005) 51 Flow past a sphere Re=300 52 Conclusion Comparisons of LBM and ACM • Taylor-Green Vortex Flow • Circular Couette Flow • Flow past a Cylinder • Lid-driven Cavity Flow LBM – ACM is NOT decisively positive !!! Performance of ACM in 3D problems • Lid-driven Cavity Flow • Flow past a Cylinder ACM is capable of practical 3D simulation 53 54 Intermission Re=2000 55 Re=500 56 3. Theory of ACM 57 Convergence of the solution of ACE P ui k 0 t xi ui ui 1 uk 2ui P uj ui 2 t x j 2 xk xi x j to the solution of INSE in the limit of k 0 . Témam (1969) How fast ? A mathematical analysis shows (Moise and Ziane, Journal of Dynamics and Differential Equations, Vol. 13, No. 2, 2001) | uik uiNS |H1 Ck 1/ 2 Numerical observations show that the error is O(k) !! 59 Diffusive mode of ACE solution P ui k 0 t xi 0 k 1 ui ui P 2ui uj 2 t x j xi x j Assumption ui ui ui ~ ~ ~ O(1) x j t P P ~ ~ O(1) x j t 60 ui u ku k u 0 i 1 i 2 2 i P P kP k P 0 1 2 ui0 0, xi 0 i 2 0 0 i 2 j ui1 Oseen type P 0 , xi t u u P 1 u 0 u uj uj 0 t x j x j xi x 1 i 0 i INSE O(1) u u P 0 u uj 0 t x j xi x 0 i 2 1 i 1 2 1 i 2 j O(k ) The intrinsic error of ACM O(k ) k ~ (x) ~ t 2 Acoustic mode of ACE solution P ui k 0 t xi k ui ui P 2ui uj 2 t x j xi x j P ui 0 t xi ui P 0 t xi 2 2P 2 P c t 2 x 2j c 1 k Characteristic time ~ Characteristic length / Speed of wave 1/ c k 62 Initial data for ACE = Initial data for INSE (ui (t 0), P(t 0)) (u (t 0), P (t 0)) 0 i wi ui ui0 , q P P0 0 wi (t 0) 0, q(t 0) 0 wi P 0 q k( ) 0 t t xi 0 2 wi w u w wi q 0 i i i uj wj wj t x j x j x j xi x 2j P 0 t acts as an initial impact !!! q P 0 t t wi 0 t Magnitude of acoustic mode q P 0 ~ O(1) t t t k q ~q wi P 0 q k( ) 0 t t xi q ~ O( k ) wi ~ O(k ) 64 q ui (0,.) ui0 (0,.) ui (0,.) ui0 (0,.) ku i1 (0,.) P (0,.) P 0 (0,.) P (0,.) P 0 (0,.) kP 1 (0,.) q 4. Improvement of ACM 66 4.1 Suppression of acoustic mode q wi k 0 t xi wi q 2wi 2 t xi x j t / k Q q / k Wi wi / k 2Q 2Q 3Q 2 k 2 2 x j x j 1. Bulk viscosity q wi k 0 t xi wi q 2wi w j 2 t xi x j xi x j 2 Q 2Q 3Q 2 k ( ) 2 2 x j x j 67 2. Dashpot wi q k( q) 0 t xi wi q 2wi 2 t xi x j 2Q 2Q Q 3Q (1 k ) 2 k k 2 2 x j x j Dashpot 68 1. Bulk viscosity k P ui 0 t xi ui ui 2 ui P uj t x j xi x 2j k P ui 0 t xi ui ui 2 ui P ui uj t x j xi x 2j xi xi 2. Dashpot k P ui 0 t xi ui ui 2 ui P uj t x j xi x 2j k( u P P) i 0 t xi ui ui 2 ui P uj t x j xi x 2j 69 Time history of L1 error 1 Re 0.001 1 70 4.2 Suppression of Checkerboard Instability 1 Acoustic mode killer 2 Checkerboard killer 3 stability 71 v Staggered grid P u u v u, v, P u, v, P u, v, P u, v, P Collocated grid 72 2D-LID Re=5000 P-contour 73 P-contour ACM (s=0,2) (256×256) C.-H. Bruneau* (2048×2048) *C.-H. Bruneau & M. Saad, Comput. & Fluids 35, 326-348 (2006) 74 Stream line Pressure 75 Stream line Pressure 76 4.3 Enhancer of stability 1 Acoustic mode killer 2 Checkerboard killer 3 stability 77 Linear Stability Analysis Pressure mode Velocity modes S=0 S=2 Checkerboard killer ~ k The normalized wave number 78 S=0 S=2 79 Stability in Lid-Driven ACM ACM MRT LBGK Grid (s=0) (s=2) 32×32 × ○ × × 64×64 × ○ × × 96×96 ○ ○ ○ × 128×128 ○ ○ ○ × 256×256 ○ ○ ○ ○ (χ=0.0378) (χ=0.0002) 80 Richardson Extrapolation in the Mach Numbers Diffusive mode Linear Combination 81 High order accurate ACM Spatial discretization 4th order accuracy (5point central finite difference) Time-marching 2nd order accuracy (Semi-implicit Runge-Kutta) 82 4th order accurate div u for compact stencil xu ~ xu yv ~ yv 2 6 2 6 xu y v O( ) xu ~ xu D(u, v) xu 2 6 yyyv O( 4 ) xxxu xxyv O( 2 ) 2 yv ~ yv xxxu O( 4 ) 2 2 6 6 xx y v O( 4 ) yy xu O( 4 ) xx y v y v 2 6 yy xu xu y v O( 4 ) ui0 0, xi nd order 4th 2 order accuracy accuracy u u P 0 u uj 0 t x j xi x 0 i 0 i 2 0 0 i 2 j O(1) O( ) ui1 P 0 xi t 2 0 1 2 1 1 ui1 u u ui P 1 0 i i uj uj 0 2 t x j x j xi x j ui2 P1 xi t u u P 2 u 0 u 1 u uj uj u j t x j x j xi x x j 2 i 0 i O( ) 4 2 i 2 2 1 i 2 j 1 i 3D Generalized Taylor-Green (3D-GTG) 85 Curved Solid Boundary LBM e.g. Interpolation Bounce-Back Bouzidi, Firdaouss, Lallemand (2001) Ginzburg, d’Humières (2003) ACM: Macroscopic data Interpolation or extrapolation 86 y N Fluid Quadratic extrapolation W B P W E B Solid Body S x P E x at B 87 y N W B P Fluid Quadratic extrapolation E W B Solid Body P E x at B S x at P 88 Why not ACM? Since the ACM does not employ any kinetic theory gadget, it is much easier than the LBM. Up to now, any decisive inferiority of ACM to LBM has not been found. Conversely, superiority of ACM over LBM has been found in some fundamental test problems. Therefore, it is highly recommended to master ACM before learning LBM. 89 Richardson extrapolation = 累遍増約術 Its usefulness for practical computations can hardly be overestimated. Birkoff & Rota, Ordinary differential equations (1978). Lewis Fry Richardson, ``Approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam ‘’, Phil. Trans. Royal Soc. London, Series A 210: 307–357 (1910). 谢谢您为您的关注 91 Finite Volume ACM 93 94 95 96 Excitation of acoustic mode P ui 0 t xi ui u ku k u k P P kP k P ui ui P 2ui uj 2 t x j xi x j 0 i 0 1 i 2 1 2 0 The initial data for (ui The initial data for 2 i 2 , P0 ) must be compatible with INSE. (uin , Pn ) (n 1,2,3,...) ui1 P 0 , xi t 0 1 2 1 1 ui1 u u ui P 1 0 i i uj uj 0 2 t x j x j xi x j troublesome