Fast Solvers for Models of Fluid FlowALott/research/palott_poster.pdf · 2008. 11. 5. · Fast...
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Fast Solvers for Models of Fluid Flow P. Aaron Lott Applied Mathematics and Scientific Computation Background Computational Methods for Fluid Flow Need to efficiently compute steady flow states to enable Implicit time stepping strategies Improved stability analysis Classification of flow bifurcations Fluid Models Incompressible Navier Stokes ∂ u ∂ t - ν ∇ 2 u +( u ·∇) u + ∇p = f in Ω, ∇· u = 0 in Ω. Advection-Diffusion -∇ 2 u +( w ·∇)u = g Viscous and Inertial forces occur on disparate scales lead to sharp flow features which: require fine numerical grid resolution cause poorly conditioned non-symmetric system. Spatial Discretization Spectral Element Method On each element, the solution is expressed via a nodal basis u N e (x , y )= N +1 i =1 N +1 j =1 u ij π i (x )π j (y ). (1) Figure: Simulation domain Ω (left) is divided into elements (middle). In each element grid points based on Gauss-Legendre-Lobatto nodes are chosen (right). Spectral Basis Functions Figure: 4th Order 2D Lagrangian nodal basis functions π i ⊗ π j based on the Gauss-Labotto-Legendre points. Fluid Simulation Layout Time step (BE) x n+1 = x n +ΔtF (t n+1 , x n+1 ) Nonlinear Solver (Newton) x k +1 = x k +Δx k Linear Solver (GMRES) AΔx k = b Preconditioner (DD) AP -1 P Δx k = b Domain Decomposition System ¯ P 1 II 0 ... 0 ¯ P 1 I Γ 0 ¯ P 2 II 0 ... ¯ P 2 I Γ . . . . . . . . . . . . . . . 0 0 ... ¯ P E II ¯ P E I Γ 0 0 ... 0 ¯ P S u I 1 u I 2 . . . u I E u Γ = ˆ b I 1 ˆ b I 2 . . . ˆ b I E g Γ ¯ P S = ∑ E e=1 ( ¯ P e ΓΓ - ¯ P e ΓI ¯ P e II -1 ¯ P e I Γ ) represents the Schur complement of the system. The interface u Γ is obtained via an iterative solve. Constant Wind Approximation When the “wind” w is constant on each element, then element interiors can be obtained via Fast Diagonalization and P -1 = A -1 . Otherwise using a constant wind approximation on each element P -1 ≈ A -1 . Figure: Illustration of a constant wind approximation ¯ P e -1 = ˜ M (V y ⊗ V x )(Λ y ⊗ I + I ⊗ Λ x ) -1 (V -1 y ⊗ V -1 x ) ˜ M Test Case: Constant Wind, Pc=400 w =(0, 200) Figure: Steady flow with constant wind exhibiting boundary layer at y = 1 using SEM N=16 & E=4x4. Interface Solver Convergence Table: Iteration count (E=4 × 4) Iterations Iterations N - R-R 4 240 44 8 108 42 16 103 43 Table: Iteration count (N=4) Iterations Iterations E - R-R 4 × 4 240 44 8 × 8 175 42 16 × 16 143 50 Robin-Robin preconditioned interface solve (R-R) is invariant to the number of points in the discretization and convergences in significantly fewer steps than the non-preconditioned system (-). Test Case: Recirculating Wind, Pc=400 w = 200(y (1 - x 2 ), -x (1 - y 2 )) Figure: Computed solution of steady flow with recirculating wind using SEM N=4 & E=12x12. Figure: Comparison of Outer iterations when inner iterations are varied. Convergence Properties for Refined Meshes Table: Iteration Count (E=4 × 4) Number of N Outer Iterations 5 52 7 56 9 55 11 53 13 51 Table: Iteration count (N=4) Number of E Outer Iterations 10 × 10 37 11 × 11 38 12 × 12 38 13 × 13 38 Summary & Future Directions Summary Improved simulation efficiency for steady Advection-Diffusion equation Future Directions Improve wind approximation on each element Coarse Grid Preconditioner to allow for more elements Use Preconditioner in Navier-Stokes simulations Apply to realistic fluid simulations References P. A. Lott, ”Fast Solvers for Models of Fluid Flow”, Ph.D. Thesis University of Maryland College Park, 2008 P. A. Lott and H. Elman, ”Matrix-free preconditioner for the steady advection-diffusion equation with spectral element discretization”, in preparation, 2008 , ”Matrix-free Block preconditioner for the Navier-Stokes equations”, in preparation, 2008 Symposium on Fluids Science and Turbulence [email protected] www.math.umd.edu/∼palott
Transcript of Fast Solvers for Models of Fluid FlowALott/research/palott_poster.pdf · 2008. 11. 5. · Fast...
Fast Solvers for Models of Fluid Flow P. Aaron LottApplied Mathematics andScientific Computation
Background
Computational Methods for Fluid FlowNeed to efficiently compute steady flow states toenable! Implicit time stepping strategies! Improved stability analysis! Classification of flow bifurcations
Fluid Models
Incompressible Navier Stokes
∂"u∂t − ν∇2"u + ("u ·∇)"u +∇p = f in Ω,
∇ · "u = 0 in Ω.
Advection-Diffusion
−∇2u + ("w ·∇)u = gViscous and Inertial forces occur on disparatescales lead to sharp flow features which:! require fine numerical grid resolution! cause poorly conditioned non-symmetric system.
Spatial Discretization
Spectral Element MethodOn each element, the solution is expressed via anodal basis
uNe (x , y) =N+1∑
i=1
N+1∑
j=1uijπi(x)πj(y). (1)
Figure: Simulation domain Ω (left) is divided into elements(middle). In each element grid points based onGauss-Legendre-Lobatto nodes are chosen (right).
Spectral Basis Functions
Figure: 4th Order 2D Lagrangian nodal basis functions πi ⊗ πjbased on the Gauss-Labotto-Legendre points.
Fluid Simulation Layout
Time step (BE) xn+1 = xn + ∆tF (tn+1, xn+1)
Nonlinear Solver (Newton) xk+1 = xk + ∆xkLinear Solver (GMRES) A∆xk = b
Preconditioner (DD) AP−1P∆xk = b
Domain Decomposition System
P1II 0 . . . 0 P1IΓ0 P2II 0 . . . P2IΓ... . . . . . . . . . ...0 0 . . . PEII P
EIΓ
0 0 . . . 0 PS
uI1uI2...uIEuΓ
=
bI1bI2...bIEgΓ
PS =∑Ee=1(PeΓΓ − PeΓI P
eII−1 PeIΓ) represents the
Schur complement of the system. The interface uΓis obtained via an iterative solve.
Constant Wind Approximation
When the “wind” "w isconstant on each element,then element interiors canbe obtained via FastDiagonalization andP−1 = A−1.
Otherwise using aconstant windapproximation on eachelement P−1 ≈ A−1.
Figure: Illustration of a constantwind approximation
Pe−1 = M(Vy ⊗Vx)(Λy ⊗ I + I⊗Λx)−1(V−1y ⊗V−1x )M
Test Case: Constant Wind, Pc=400
"w = (0,200)
Figure: Steady flow with constant wind exhibiting boundarylayer at y = 1 using SEM N=16 & E=4x4.
Interface Solver Convergence
Table: Iteration count (E=4× 4)Iterations Iterations
N - R-R4 240 448 108 4216 103 43
Table: Iteration count (N=4)Iterations Iterations
E - R-R4× 4 240 448× 8 175 42
16× 16 143 50
Robin-Robin preconditioned interface solve (R-R) isinvariant to the number of points in thediscretization and convergences in significantlyfewer steps than the non-preconditioned system (-).
Test Case: Recirculating Wind, Pc=400
"w = 200(y(1− x2),−x(1− y2))
Figure: Computed solution ofsteady flow with recirculatingwind using SEM N=4 &E=12x12.
Figure: Comparison of Outeriterations when inner iterationsare varied.
Convergence Properties for Refined Meshes
Table: Iteration Count (E=4× 4)Number of
N Outer Iterations5 527 569 5511 5313 51
Table: Iteration count (N=4)Number of
E Outer Iterations10× 10 3711× 11 3812× 12 3813× 13 38
Summary & Future Directions
Summary! Improved simulation efficiency for steadyAdvection-Diffusion equation
Future Directions! Improve wind approximation on each element! Coarse Grid Preconditioner to allow for moreelements
! Use Preconditioner in Navier-Stokes simulations! Apply to realistic fluid simulations
References
! P. A. Lott, ”Fast Solvers for Models of Fluid Flow”, Ph.D.Thesis University of Maryland College Park, 2008
! P. A. Lott and H. Elman, ”Matrix-free preconditioner for thesteady advection-diffusion equation with spectral elementdiscretization”, in preparation, 2008
! , ”Matrix-free Block preconditioner for theNavier-Stokes equations”, in preparation, 2008
Symposium on Fluids Science and Turbulence [email protected] www.math.umd.edu/∼palott
