Solution methods for the Unsteady Incompressible Navier-Stokes ...
Structure-Preserving B- spline Methods for the Incompressible Navier -Stokes Equations
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Transcript of Structure-Preserving B- spline Methods for the Incompressible Navier -Stokes Equations
Structure-Preserving B-spline Methods for the Incompressible Navier-Stokes Equations
John Andrew EvansInstitute for Computational Engineering and Sciences, UT Austin
Stabilized and Multiscale Methods in CFDSpring 2013
MotivationSo why do we need another flow solver?
Incompressibility endows the Navier-Stokes equations with important physical structure:
• Mass balance• Momentum balance• Energy balance• Enstrophy balance• Helicity balance
However, most methods only satisfy the incompressibility constraint in an approximate sense.
Such methods do not preserve structure and lack robustness.
Motivation
Consider a two-dimensional Taylor-Green vortex. The vortex is a smooth steady state solution of the Euler equations.
MotivationConservative methods which only weakly satisfy incompressibility do not preserve this steady state and blow up in the absence of artificial numerical dissipation.
2-D Conservative Taylor-Hood ElementQ2/Q1 and h = 1/8
MotivationMethods which exactly satisfy incompressibility are stable in the Euler limit and robust with respect to Reynold’s number.
Increasing Reynold’s number
2-D Conservative Structure-preserving B-splinesk’ = 1 and h = 1/8
Motivation
Due to the preceding discussion, we seek new discretizations that:
• Satisfy the divergence-free constraint exactly.
• Harbor local stability and approximation properties.
• Possess spectral-like stability and approximation properties.
• Extend to geometrically complex domains.
Structure-Preserving B-splines seem to fit the bill.
The Stokes Complex
The classical L2 de Rham complex is as follows:
From the above complex, we can derive the following smoothed complex with the same cohomology structure:
° ⏐ →⏐ Ψ grad⏐ →⏐ ⏐ Φ curl⏐ →⏐ V div⏐ →⏐ Q ⏐ →⏐ 0
Ψ := H 1
V :=H1 Q :=L2
The Stokes Complex
° ⏐ →⏐ Ψ grad⏐ →⏐ ⏐ Φ curl⏐ →⏐ V div⏐ →⏐ Q ⏐ →⏐ 0
Scalar Potentials Vector
PotentialsFlow Pressures
Flow Velocities
grad⏐ →⏐ ⏐ curl⏐ →⏐ div⏐ →⏐
The smoothed complex corresponds to viscous flow, so we henceforth refer to it as the Stokes complex.
The Stokes Complex
For simply-connected domains with connected boundary, the Stokes complex is exact.
• grad operator maps onto space of curl-free functions
• curl operator maps onto space of div-free functions
• div operator maps onto entire space of flow pressures
gradΨ = φ∈Φ :curlφ =0{ }
curlΦ= v∈V :divv=0{ }
divV=Q
The Stokes Complex
grad⏐ →⏐ ⏐ curl⏐ →⏐ div⏐ →⏐
grad(ψ )⋅ds=ψ (γ(1))γ∫ −ψ (γ(0))
curl(φ)⋅da= φ⋅ds∂S∫
S∫
div(v)dV = v⋅da∂V∫
V∫
Gradient Theorem:
Curl Theorem:
Divergence Theorem:
The Discrete Stokes Subcomplex
Now, suppose we have a discrete Stokes subcomplex.
Then a Galerkin discretization utilizing the subcomplex:• Does not have spurious pressure modes, and• Returns a divergence-free velocity field.
° ⏐ →⏐ Ψhgrad⏐ →⏐ ⏐ Φh
curl⏐ →⏐ Vhdiv⏐ →⏐ Qh ⏐ →⏐ 0
Discrete Scalar Potentials
Discrete Vector Potentials
Discrete Flow Pressures
Discrete Flow Velocities
The Discrete Stokes Subcomplex
Proposition. If vh ∈V h satisfies
div vh, qh( )L2 =0, ∀qh ∈Qh
then div vh =0.
Proof. Let qh =div vh ∈Qh.
⇒ div vh L2
2 = 0
The Discrete Stokes Subcomplex
The Discrete Stokes Subcomplex
Structure-Preserving B-splines
Review of Univariate (1-D) B-splines:Knot vector on (0,1) and k-degree
B-spline basis on (0,1) by recursion:
Ξ = {0, 0, 0, 0.2, 0.4, 0.6, 0.8, 0.8, 1, 1, 1}, k = 2
Knots w/multiplicity
Start w/ piecewise constants
Bootstraprecursively
to k
k=2
• Open knot vectors:• Multiplicity of first and last knots is k+1• Basis is interpolatory at these locations
• Non-uniform knot spacing allowed• Continuity at interior knot a function of knot repetition
Review of Univariate (1-D) B-splines:
Structure-Preserving B-splines
Review of Univariate (1-D) B-splines:• Derivatives of B-splines are B-splines
k=4 k=3
onto
Structure-Preserving B-splines
Review of Univariate (1-D) B-splines:• We form curves in physical space by taking weighted
sums.
- control points - knots
0
1
2
3
4
5
Quadratic basis:
“control mesh”
Structure-Preserving B-splines
• Multivariate B-splines are built through tensor-products• Multivariate B-splines inherit all of the aforementioned
properties of univariate B-splines
In what follows, we denote the space of n-dimensional tensor-product B-splines as
polynomial degree in direction i
i th continuity vector
Review of Multivariate B-splines:
Structure-Preserving B-splines
• We form surfaces and volumes using weighted sums of multivariate B-splines (or rational B-splines) as before.
Review of Multivariate B-splines:
Control mesh Mesh
Structure-Preserving B-splines
Define for the unit square:
In the context of fluid flow:
and it is easily shown that:
Sh :=Sa1 ,a2
k1 ,k2
R h :=Sa1 ,a2−1k1 ,k2−1 ×Sa1−1,a2
k1−1,k2
Wh :=Sa1−1,a2−1k1−1,k2−1
Sh = Space of Stream Φunctions
R h = Space of Φlow V elocities
Wh = Space of Pressures
° ⏐ →⏐ Shcurl⏐ →⏐ Rh
div⏐ →⏐ Wh ⏐ →⏐ 0
Two-dimensional Structure-Preserving B-splines
Structure-Preserving B-splines
Two-dimensional Structure-Preserving B-splines: Mapped Domains
On mapped domains, the Piola transform is utilized to map flow velocities. Pressures are mapped using an integral preserving transform.
Structure-Preserving B-splines
We associate the degrees of freedom of structure-preserving B-splines with the control mesh. Notably, we associate:
Sh :=control points
R h :=control faces
Wh :=control cells
Structure-Preserving B-splines
Two-dimensional Structure-Preserving B-splines
Two-dimensional Structure-Preserving B-splines, k1 = k2 = 2:
Structure-Preserving B-splines
Structure-Preserving B-splines
Two-dimensional Structure-Preserving B-splines, k1 = k2 = 2:
Structure-Preserving B-splines
Two-dimensional Structure-Preserving B-splines, k1 = k2 = 2:
Structure-Preserving B-splines
Two-dimensional Structure-Preserving B-splines, k1 = k2 = 2:
Structure-Preserving B-splines
Two-dimensional Structure-Preserving B-splines, k1 = k2 = 2:
Structure-Preserving B-splines
Two-dimensional Structure-Preserving B-splines, k1 = k2 = 2:
Structure-Preserving B-splines
Two-dimensional Structure-Preserving B-splines, k1 = k2 = 2:
Structure-Preserving B-splines
Two-dimensional Structure-Preserving B-splines, k1 = k2 = 2:
Define for the unit cube:
It is easily shown that:
Sh :=Sa1,a2 ,a3
k1 ,k2 ,k3
Nh :=Sa1−1,a2 ,a3
k1−1,k2 ,k3 ×Sa1,a2−1,a3
k1,k2−1,k3 ×Sa1,a2 ,a3−1k1 ,k2 ,k3−1
R h :=Sa1,a2−1,a3−1k1,k2−1,k3−1 ×Sa1−1,a2 ,a3−1
k1−1,k2 ,k3−1 ×Sa1−1,a2−1,a3
k1−1,k2−1,k3
Wh :=Sa1−1,a2−1,a3−1k1−1,k2−1,k3−1
° ⏐ →⏐ Shgrad⏐ →⏐ ⏐ Nh
curl⏐ →⏐ Rhdiv⏐ →⏐ Wh ⏐ →⏐ 0
Structure-Preserving B-splines
Three-dimensional Structure-Preserving B-splines
Flow velocities: map w/ divergence-preserving transformation Flow pressures: map w/ integral-preserving transformationVector potentials: map w/ curl-conserving transformation
Sh := wh :whoΦ∈Sh{ }
Nh := vh : DΦ( )T vhoΦ( )∈Nh{ }
R h := uh :det DΦ( ) DΦ( )−1 uhoΦ( )∈R h{ }
Wh := ph :det(DΦ)phoΦ∈Wh{ }
Structure-Preserving B-splines
Three-dimensional Structure-Preserving B-splines
Sh := control points
Nh := control edγes
R h := control faces
Wh := control cells
We associate as before the degrees of freedom with the control mesh.
Structure-Preserving B-splines
Three-dimensional Structure-Preserving B-splines
Control points:Scalar potential DOF
Control edges:Vector potential DOF
Structure-Preserving B-splines
Three-dimensional Structure-Preserving B-splines
Structure-Preserving B-splinesWeak Enforcement of No-Slip BCs
Nitsche’s method is utilized to weakly enforce the no-slip condition in our discretizations. Our motivation is three-fold:
• Nitsche’s method is consistent and higher-order.• Nitsche’s method preserves symmetry and ellipticity.• Nitsche’s method is a consistent stabilization procedure.
Furthermore, with weak no-slip boundary conditions, a conforming discretization of the Euler equations is obtained in the limit of vanishing viscosity.
Structure-Preserving B-splinesWeak Enforcement of Tangential Continuity Between Patches
On multi-patch geometries, tangential continuity is enforced weakly between patches using a combination of the symmetric interior penalty method and upwinding.
Summary of Theoretical Results
• Well-posedness for small data
• Optimal velocity error estimates and suboptimal, by one order, pressure error estimates
• Conforming discretization of Euler flow obtained in limit of vanishing viscosity (via weak BCs)
• Robustness with respect to viscosity for small data
Steady Navier-Stokes Flow
Summary of Theoretical ResultsUnsteady Navier-Stokes Flow
• Existence and uniqueness (well-posedness)
• Optimal velocity error estimates in terms of the L2 norm for domains satisfying an elliptic regularity condition (local-in-time)
• Convergence to suitable weak solutions for periodic domains• Balance laws for momentum, energy, enstrophy, and helicity
• Balance law for angular momentum on cylindrical domains
Spectrum AnalysisSpectrum Analysis
−∇⋅2n∇su( )+∇p =l u in W
∇⋅u =0 in W w/ Periodic BCs
Consider the two-dimensional periodic Stokes eigenproblem:
We compare the discrete spectrum for a specified discretization with the exact spectrum. This analysis sheds light on a given discretization’s resolution properties.
Spectrum Analysis: Structure-Preserving B-splines
Spectrum Analysis
Spectrum Analysis: Taylor-Hood Elements
Spectrum Analysis
Spectrum Analysis: MAC Scheme
Spectrum Analysis
Selected Numerical ResultsSteady Navier-Stokes Flow:
Numerical Confirmation of Convergence Rates
2-D Manufactured Vortex Solution
Selected Numerical ResultsSteady Navier-Stokes Flow:
Numerical Confirmation of Convergence Rates
2-D Manufactured Vortex Solution
Selected Numerical ResultsSteady Navier-Stokes Flow:
Numerical Confirmation of Convergence Rates
2-D Manufactured Vortex Solution
Selected Numerical ResultsSteady Navier-Stokes Flow:
Numerical Confirmation of Convergence Rates
Re 0 1 10 100 1000 10000Energy 1.40e-2 1.40e-2 1.40e-2 1.40e-2 1.40e-2 1.40e-2
H1 error - u 1.40e-2 1.40e-2 1.40e-2 1.40e-2 1.40e-2 1.40e-2
L2 error - u 2.28e-4 2.28e-4 2.28e-4 2.28e-4 2.28e-4 2.28e-4
L2 error - p 3.49e-4 3.49e-4 1.98e-4 1.96e-4 1.96e-4 1.96e-4
Robustness with respect to Reynolds number k’ = 1 and h = 1/16
2-D Manufactured Vortex Solution
Selected Numerical ResultsSteady Navier-Stokes Flow:
Numerical Confirmation of Convergence Rates
Re 0 1 10 100 1000 10000
H1 error - u 6.77e-4 6.77e-4 7.11e-4 2.26e-3 2.16e-2 X
L2 error - u 6.54e-4 6.54e-6 6.79e-6 1.97e-5 1.86e-4 X
L2 error - p 1.96e-4 1.96e-4 1.96e-4 1.96e-4 1.96e-4 X
Instability of 2-D Taylor-Hood with respect to Reynolds number
Q2/Q1 and h = 1/16
2-D Manufactured Vortex Solution
Steady Navier-Stokes Flow:Lid-Driven Cavity Flow
H
H
U
Selected Numerical Results
Steady Navier-Stokes Flow:Lid-Driven Cavity Flow at Re = 1000
Selected Numerical Results
Steady Navier-Stokes Flow:Lid-Driven Cavity Flow at Re = 1000
Method umin vmin vmax
k’ = 1, h = 1/32 -0.40140 -0.39132 0.54261
k’ = 1, h = 1/64 -0.39399 -0.38229 0.53353
k’ = 1, h = 1/128 -0.39021 -0.37856 0.52884
k’ = 2, h = 1/64 -0.38874 -0.37715 0.52726
k’ = 3, h = 1/64 -0.38857 -0.37698 0.52696
Converged -0.38857 -0.37694 0.52707
Ghia, h = 1/156 -0.38289 -0.37095 0.51550
Selected Numerical Results
Unsteady Navier-Stokes Flow:Flow Over a Cylinder at Re = 100
L
H
U
D
Lout
Selected Numerical Results
Unsteady Navier-Stokes Flow:Flow Over a Cylinder at Re = 100
Patch 1
Patch 2
Patch 3
Patch 4 Patch 5
Selected Numerical Results
Unsteady Navier-Stokes Flow:Flow Over a Cylinder at Re = 100
Simulation Details:
- Full Space-Time Discretization- Method of Subgrid Vortices- Linears in Space and Time- D = 2, H = 32D, L = 64D- Time Step Size: 0.25- Trilinos Implementation
- GMRES w/ ILU Preconditioning- 3000 time-steps (shedding
initiated after 1000 time-steps)- Approximately 15,000 DOF/time
step
Selected Numerical Results
Unsteady Navier-Stokes Flow:Flow Over a Cylinder at Re = 100
Selected Numerical Results
Unsteady Navier-Stokes Flow:Flow Over a Cylinder at Re = 100
Quantity of Interest:
Strouhal Number: St = fD/U
Computed Strouhal Number: 0.163Accepted Strouhal Number: 0.164
Selected Numerical Results
Unsteady Navier-Stokes Flow:Three-Dimensional Taylor-Green Vortex Flow
Selected Numerical Results
3-D Periodic Flow
Simplest Model of Vortex Stretching
No External Forcing
Unsteady Navier-Stokes Flow:Three-Dimensional Taylor-Green Vortex Flow
Selected Numerical Results
Time Evolution of Dissipation Rate
Reproduced with permission from[Brachet et al. 1983]
Unsteady Navier-Stokes Flow:Three-Dimensional Taylor-Green Vortex Flow at Re = 200
Selected Numerical Results
Enstrophy isosurface at time corresponding to maximum dissipation
Unsteady Navier-Stokes Flow:Three-Dimensional Taylor-Green Vortex Flow at Re = 200
Selected Numerical Results
Convergence of dissipation rate time history with mesh refinement (k’ = 1)
Unsteady Navier-Stokes Flow:Three-Dimensional Taylor-Green Vortex Flow at Re = 200
Selected Numerical Results
Convergence of dissipation rate time history with degree elevation (h = 1/32)