Compressible Navier-Stokes (Euler) Solver based on Deal.II Library

Lei Qiao

Northwestern Polytechnical University

Xi’an, China

Texas A&M University

College Station, Texas

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Fifth deal.II Users and Developers Workshop Texas A&M University, College Station, TX, USA

August 5, 2015

Outline

Motivation and Goal

Current work

Conclusion and TODOs

2

Motivation

This is what I typically need to compute

3

The pain pointMesh generation:

Time consuming Boring Experience depending

Solution: Mesh adaptation — let solver tell what is good mesh

4

Deal.II the library

• Mesh adaptation with tree data structure • FEM discretization • Parallelization and excellent scalability

5

GoalA N-S solver based on deal.II that: • Starts from initial value on coarse mesh • Converges to solution on a reasonable fine

mesh adaptively • With High-order capability

6

Current workGoverning Equation

Solving technique

Solver implementation

Solver verification

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Governing Equation

Compressible Navier-Stokes Equation

@Q(w)

@t+r · F(w) = S(w)

Use primitive variables as working var.

w = (uj , ⇢, p)T

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Governing Equation

Fc(w) =

0

@⇢u⌦ u+ Ip

⇢u(E + p)u

1

A =

0

@⇢uiuj + �ijp

⇢ui

(E + p)ui

1

A

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The Flux

F = Fc � Fv

Governing EquationThe Flux

F = Fc � Fv

WithT =

p

�R⇢

⌧ij = µ(@ui

@xj+

@uj

@xi) + ��ij

@uk

@xk

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Fv

(w) =1

Re⇤1

0

@⌧ij

0⌧ij

ui

+ @T

@xj

1

A

Governing Equation

Weak form:

w = (uj , ⇢, p)T

Z

⌦vl@Ql(w)

@t

+

Z

⌦vl@Fl,i(w)

@xi=

Z

⌦vlSl(w)

8v = vl 2 T

, Find w 2 S that satisfy On domain ⌦

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Governing Equation

Weak form:

Integrate by part to get boundary flux

Z

⌦vl@Ql(w)

@t

+

Z

@⌦vlFl,i(w)ni

�Z

⌦

@vl

@xiFl,i(w) =

Z

⌦vlSl(w)

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Governing EquationTreat boundary conditions and discontinuities on hanging node with this boundary flux

Compute numerical flux with Roe scheme[1]

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Governing Equation

Boundary conditions:

Far field: Riemann invariant

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Slip wall: Non-penetration

u · n = 0⇢out

= ⇢in

pout

= pin

Solving technique

BDF-1 (Implicit Euler) time integration

Newton linearization

Direct linear solver from Trilinos** Iterative solver doesn’t stable enough, only used for scalability test

For steady run, time step size is determined according to norm of Newton update[2]

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Solver implementation• Starts from step-33 of deal.II tutorial

• Change working vars. from conservative ones to primitive ones • viscous flux • Riemann boundary condition • Roe flux[2] to replace the too viscous Lax flux • Parallelize: learned from step-40

• Version control with Git • CMake project • Regression test suit

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Solver verification• Scalability • Manufactured solution[3]

• Subsonic • Supersonic

• Adaptive simulation over circle • Flow over foil NACA2412

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Scalability22,288 cells 91,312 DoFs

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GMRES solver from Trilinos with DD+ILUT precond.

Solver verification• Manufactured solution[3]

@Q(w)

@t+r · F(w) = S(w)

w = J(x, y) =

8>><

>>:

u(x, y)= u0+u

x

sin(a

ux

⇡x)+u

y

cos(a

uy

⇡y)+u

xy

cos(a

uxy

⇡xy)

v(x, y)= v0+v

x

cos(a

vx

⇡x)+v

y

sin(a

vy

⇡y) +v

xy

cos(a

vxy

⇡xy)

⇢(x, y)= ⇢0+⇢

x

sin(a

⇢x

⇡x)+⇢

y

cos(a

⇢y

⇡y)+⇢

xy

cos(a

⇢xy

⇡xy)

p(x, y)= p0+p

x

cos(a

px

⇡x)+p

y

sin(a

py

⇡y) +p

xy

sin(a

pxy

⇡xy)

S(x, y) =@Q(J)

@t

+r · F(J)

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SolverJh(x, y)

Solver verification• Manufactured solution[3]

w = J(x, y) =

8>><

>>:

u(x, y)= u0+u

x

sin(a

ux

⇡x)+u

y

cos(a

uy

⇡y)+u

xy

cos(a

uxy

⇡xy)

v(x, y)= v0+v

x

cos(a

vx

⇡x)+v

y

sin(a

vy

⇡y) +v

xy

cos(a

vxy

⇡xy)

⇢(x, y)= ⇢0+⇢

x

sin(a

⇢x

⇡x)+⇢

y

cos(a

⇢y

⇡y)+⇢

xy

cos(a

⇢xy

⇡xy)

p(x, y)= p0+p

x

cos(a

px

⇡x)+p

y

sin(a

py

⇡y) +p

xy

sin(a

pxy

⇡xy)

Choosing different Constants to get subsonic

or supersonic

case

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Solver verification• Manufactured solution

• Subsonic • convergence

order

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Solver verification• Manufactured solution

• Supersonic • convergence

order

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Solver verification• Adaptive simulation over circle

• C1 mapping on wall boundary • Subsonic: Mach = 0.1 • Almost inviscid: Cv=1e-6 just for stabilization • Converged Cd=0.00029 which ideal value is zero

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Solver verification• Adaptive simulation over cylinder

Converged density contour

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Solver verification• Adaptive simulation over cylinder

Converge history of density contour and mesh adaptation

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Solver verification• Flow over NACA2412 foil

• Subsonic: Mach=0.3

Converged pressure contour

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Solver verification• Flow over NACA2412 foil

• Subsonic: Mach=0.3

Oscillation may be caused by geometry non-smoothness

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Solver verification

• Almost inviscid: Cv=3e-6 just for stabilization • Converged Cd=0.00025 which ideal value is zero • Converged Cl=0.2576, Cm=0.05235

• As a reference, xFoil[4] gives • Cd=-0.00076 • Cl=0.2704 • Cm=0.0585

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• Flow over NACA2412 foil • Subsonic: Mach=0.3

Conclusion• A prototype NS solver based on deal.II is constructed • The solver could run in parallel but the solver doesn’t

scale perfectly • High order convergence is confirmed by MMS • Solving process could start from very coarsen mesh

and go on with mesh adaptation • The solver can give out reasonable aerodynamics data

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ToDos• Stable and efficient preconditioner for iterative solver

(Now my hope is on MDF ordering of ILU) • Describe wall boundary with NURBS geometry • Non-slip boundary condition • Anisotropic adaptation for boundary layer

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Reference1. J. Blazek. Computational Fluid Dynamics: Principles and

Applications (Second Edition). Elsevier Science, Oxford, second edition edition, 2005. ISBN 978-0-08- 044506-9.

2. J. Gatsis. Preconditioning Techniques for a Newton–Krylov Algorithm for the Compressible Navier–Stokes Equations. PhD thesis, University of Toronto, 2013.

3. C. J. Roy, C. C. Nelson, T. M. Smith, and C. C. Ober. Verification of Euler/Navier- stokes codes using the method of manufactured solutions. International Journal for Numerical Methods in Fluids, 44(6):599--620, 2004.

4. M. Drela, H. Youngren. http://web.mit.edu/drela/Public/web/xfoil/

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Thanks

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