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)