Department of Mechanical Engineering, The University of British Columbia

A Higher Order Accurate Unstructured Finite VolumeA Higher-Order Accurate Unstructured Finite-Volume Newton-Krylov Algorithm for Inviscid Compressible Flows

Amir Nejat

Knowledge Diffusion Network

١٣٨۶مهرماه ٢٩دانشکده مهندسی هوافضا، دانشگاه صنعتی شريف،

Aircraft Design & Fuel Efficiency

ηη < 15%

η : Fuel consumption per seat per mile

767777 ηη <

ηη <

15%

20%777787 ηη < 20%

Design Process

Mission Specification

Experience Initial Design

Multi-DisciplinaryOptimization

Multi-Physics NumericalPDE S l OptimizationPDE Solvers

Optimized Design

Opening: Design Process CFD

CFD

1-MeshComplex GeometryAdaptation and Refinement2-AccuracyDiscretization (Truncation) errorModeling error 3 C3-ConvergenceStabilityResidual dropping orderTime & CostTime & Cost

Background: CFD CFD Algorithm

CFD - Overall Algorithm

Geometry & Solution domain Mesh generation package

Physics & Fluid flow equations

Meshed domain

Boundary & Initial conditions

Meshed domainResidual

Discretization of the fluid flow equations & Flux Computation and Integration

L t f li ti

Implicit method

Fluid flowSparse

Large system of linear equations

Jacobian matrix

simulationSparse

matrix solverPreconditioning

Background: CFD Algorithm Motivation

Motivation2

2 )(OyyUx

xU)y,x(UU ccordernd ΔΔΔ +

∂∂

+∂∂

+=−

22222 UUU ΔΔ ∂∂∂

Second-order methods:

22

2

2

222

2

22 y

yUyx

yxUx

xU)(O ΔΔΔΔΔ

∂∂

+∂∂

∂+

∂∂

=Truncation error:

The 2nd-order truncation error acts like a diffusive term and causes

two significant numerical problems:

1-It smears sharp gradients and spoils total pressure conservation (isentropic flows).

2-It produces parasitic error by adding extra diffusion to viscous regions.

Higher-order: More accurate simulation

Existing research shows higher-order structured discretization technique for aExisting research shows higher-order structured discretization technique for agiven level of accuracy is more efficient.

Higher order: Can be more efficient !?Higher-order: Can be more efficient !?

Background: Motivation Literature Review

Literature ReviewLiterature Review

Structured Structured-Implicit Unstructured Unstructured-Implicit

Qualitative Illustration of Research on Solver Development

Second-order ♣♣♣♣♣♣♣♣♣

♣♣♣♣

♣♣♣♣♣♣

♣♣♣

Higher-order ♣♣♣

♣♣ ♣ ?

Trend:1- Increasing the efficiency using convergence acceleration techniques

such as implicit methods (Newton-Krylov).

2- Enhancing the accuracy using higher-order discretization scheme.

Background: Literature Review Contribution

Objective

• Developing an Efficient Higher-Order Accurate Unstructured Finite Volume Algorithm for Inviscid

Compressible Fluid Flow.

Objective: Contribution Model Problem

Model ProblemThe unsteady (2D) Euler equations which model compressible inviscidfluid flows, are conservation equations for mass, momentum, and energy.

Aerodynamic application: lift, wave drag and induced drag

0 FdA Udvdtd

=+ ∫∫ (1)dt cscv

⎥⎤

⎢⎡

uρρ

⎥⎤

⎢⎡

+n

ˆPuρ

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣

=

Evu

Uρρ

⎥⎥⎥⎥

⎦⎢⎢⎢⎢

⎣ +++

=

n

yn

xn

u)PE(n̂PvunPuu

Fρρ, (2)

⎦⎣ ⎦⎣ n

yxn nvnuu ˆˆ += 2/)v(u )1/(PE 22 ++−= ργ,

Theory: Model Problem Implicit Time Advance

Implicit Time AdvanceApplying implicit time integration and linearization of the governing equations in time leads to implicit time advance formula:

⇒=+ ))U(Rdt

dU( 0

R∂1

)Rt

UU( nnn

011

=+− +

+

Δ(3)

)UU()UR(RR nnnnn −

∂∂

+= ++ 11

RU)RI( ∂+ δ UUU nn+1δ

(4)

(5) ,RU)Ut

( −=∂

+ δΔ

UUU nn −=δ

U: Solution Vector

(5)

R: Residual Vector

∂R/∂U: Jacobian matrix

Eq. 5 is a system of linear equations arising from discretization of governing equations over unstructured domain.

Theory: Implicit Time Advance Linear System Solver

Linear System SolverGMRES (Generalized Minimal Residual, Saad 1986)

*GMRES algorithm, among other Krylov techniques, only needs matrix vector d t ( t i f i l t ti )products (matrix-free implementation).

*It is developed for non-symmetric matrices. *It predicts the best solution update if the linearization is carried out accurately.

To enhance the convergence performance of the GMRES solver, it is necessary to apply preconditioning:

MAbMx)AMbAx ≈=>−= −( 1

)n(ILUMLUM

MAbMx)AMbAx

≅=

≈> ,(

M is an approximation to matrix A which has simpler structure. ILU: Incomplete Lower-Upper factorizationp pp

Technique: Linear System Solver Reconstruction

Reconstruction

• Defining the Kth-order polynomial for each control volume.

• Finding the polynomial coefficients using the averages of the neighboring control volumes.

• This polynomial is constructed based on some constraints h t i t

yUxU)y,x(UU cc)K(

R +∂∂

+∂∂

+= ΔΔ

such as mean constraint.

yyUyx

yxUx

xU

yx

+∂∂

+∂∂

∂+

∂∂

∂∂

22

2

2

222

2

2 ΔΔΔΔ

...yyUyx

yxUyx

yxUx

xU

+∂∂

+∂∂

∂+

∂∂∂

+∂∂

6226

3

3

32

2

32

2

33

3

3 ΔΔΔΔΔΔ(6) CV

CV

)K(R U)y,x(U =∫ (7)

Technique: Reconstruction Monotonicity

Monotonicity

Limiting

Limitingg

Technique: Monotonicity Higher-Order Limiter

Higher-Order Limiter

part]Order-σ[Higherpart]σ][Linearσ)[(1ConstP O dHi h ++−+= φ (8)part]Orderσ[Higherpart] σ][Linearσ)[(1Const.P Order-High +++= φ

[ ]. :

.S ,. ,/ )S) ( tanh( →<

==−−=σφφ

φφφσ

0

00

00208021

(8)(9)

1.0 : =≥ σφφ 0

Technique: Higher-Order Limiter Flux Evaluation

Flux Evaluation• Discretization scheme :

Solution reconstruction: Kth-order accurate least-square t ti d (Olli i G h 1997)reconstruction procedure (Ollivier-Gooch 1997).

Flux formulation: Roe’s flux difference splitting (1981).

)(~1))()((1)( UUAUFUFUUF )(2

))()((2

),(),( LRRLRLRL UUAUFUFUUF −−+=

λΛΛ ~Diag~X~~X~A~ == − , 1

(10)

• Integration scheme : Gauss quadrature integration techniquewith the proper number of points.p p p

∫=iCV

i nds.FR (11)

Gauss quadrature for interior control volumes. Technique: Flux Evaluation 1st-Order Jacobian Matrix

1st-Order Jacobian Matrix

∑∑ ==kN,ikNi

facesii )ln̂)(U,U(Fdsn̂FR (12)

)ln̂(U

)U,U(FUR)N,i(J

kN,i

kN

kNi

kN

ik ∂

∂=

∂∂

= (13-1)

)ln̂(U

)U,U(FUR)i,i(J

kN,ii

kNi

i

i ∑ ∂

∂=

∂∂

= (13-2)

Technique: 1st-Order Jacobian Matrix Solution Strategy

Solution StrategySolution Strategy

Strategy: Solution Strategy Solution Procedure

Solution ProcedureSolution Procedure• Start up Process :

Before switching to Newton-GMERS Iteration, several pre-implicit iterations have been performed in the form of defect correction, using Eq. (5).

RUUR

tI

−=∂∂

+Δ

δ)(

R∂

(5)

solver.linear ILU(1)-GMRESby solvedis systemResultant

Order) (First UR

∂∂

• Newton-GMRES (matrix-free) iteration :At this stage, infinite time step is taken, and GMRES-ILU(4) is used to g , p , ( )solve the linear system at each Newton iteration.

RUR∂ δ)( (12)ε )U(R)vU(RvR −+

≅∂ (13)RU

U−=

∂δ)( (12) ε

v.U

≅∂

(13)

Procedure: Solution Procedure Results

ResultsSupersonic Vortex, Annulus-Meshes p ,

427 CVs 1703 CVs

108 CV108 CVs

6811 CVs 27389 CVs

Results: Supersonic Vortex Mach Contours Density Error Error Convergence Error versus CPU Time

Mach Contours-Supersonic Vortex, M=2.0

Density Error-Supersonic Vortex, M=2.0

Error Convergence-Supersonic Vortex, M=2.0

Density Error versus CPU Time / Supersonic Vortex, M=2.0M 2.0

Results: Error versus CPU Time Subsonic flow over NACA 0012 Airfoil Subsonic Convergence

Subsonic Flow over NACA 0012, M=0.63, AoA=2.0 deg.

4958CV 2nd-Order

4th Order3rd Order 4th-Order3rd-Order

Convergence history-Subsonic Case

Order Resid. Eval. Time (Sec) Work Units Newton Itr. Newton Work Units

2nd 126 26.88 349.1 3 136.1-39%

3rd 147 36.03 248.5 4 141.2-57%

4th 247 90 54 289 3 7 239 2-83%4th 247 90.54 289.3 7 239.2-83%

Results: Subsonic Convergence Transonic flow over NACA 0012 Airfoil Transonic Convergence

Transonic Flow over NACA 0012, M=0.80, AoA=1.25 deg.

4958CV 3rd-Order

Limiterφ LimiterσLimiter φ

Convergence history-Transonic Case

Order Resid. Eval. Time (Sec) Work Units Newton Itr. Newton Work Units

2nd 197 65.6 279 4 91-33%

3rd 241 106.7 281 5 119-42%

4th 450 311 4 590 10 221-37%4th 450 311.4 590 10 221-37%

Results: Transonic Convergence Transonic Mach Profile

Mach Profile-Transonic case

Order CL CD

2nd 0.337593 0.0220572

3rd 0.339392 0.0222634

4th 0.345111 0.0224720

AGARD / Structured (7488:192*39) 0.3474 0.0221

Results: Transonic Mach Profile Research Summary and Conclusion

Research Summary and Conclusion• An ILU preconditioned GMRES algorithm (matrix-free) has been used for

efficient higher-order computation of solution of Euler equations.• A start-up procedure is implemented using defect correction pre-iterations

before switching to Newton iterations. • As an over all performance assessment (including the start-up phase) the thirdAs an over all performance assessment (including the start up phase) the third

order solution is about 1.3 to 1.5 times, and the fourth order solution is about 3.5-5 times, more expensive than the second order solution with the developed solver technology.gy

• A modified Venkatakrishnan Limiter was implemented to address the convergence hampering issue, and to improve the accuracy of the limited reconstruction.eco s uc o .

• Using a good initial solution state, start up process and effective preconditioning are determining factors in Newton-GMRES solver performanceperformance.

• The possibility of benefits of higher-order discretization has been shown.

Closing: Research Summary and Conclusion Recommended Future Work

Recommended Future Work

• Improving the start-up procedure.

• Applying a more accurate preconditioning.pp y g p g

• E h i th b t f th t ti f di ti iti (li iti )• Enhancing the robustness of the reconstruction for discontinuities (limiting).

• Extension to 3D.

• Extension to viscous flows.

Closing: Recommended Future Work End

EndEnd

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