This document summarizes research on developing an efficient higher-order accurate unstructured finite volume algorithm for inviscid compressible fluid flows. The algorithm uses an ILU preconditioned GMRES method to solve the Euler equations on unstructured meshes. Higher-order solutions of up to fourth-order accuracy were obtained. Results show the third-order solution was 1.3-1.5 times more expensive than second-order, while fourth-order was 3.5-5 times more expensive, demonstrating the efficiency of the higher-order approach. Test cases included supersonic and transonic flows, with results agreeing well with structured solvers.

1. Department of Mechanical Engineering, The University of British Columbia
A Higher Order Accurate Unstructured Finite Volume
Higher-Order Finite-Volume
Newton-Krylov Algorithm for Inviscid Compressible Flows
Amir Nejat
Knowledge Diffusion Network
١٣٨۶ ‫داﻧﺸﮑﺪﻩ ﻣﻬﻨﺪﺳﯽ هﻮاﻓﻀﺎ، داﻧﺸﮕﺎﻩ ﺻﻨﻌﺘﯽ ﺷﺮﻳﻒ، ٩٢ﻣﻬﺮﻣﺎﻩ‬

2. Aircraft Design & Fuel Efficiency
η : Fuel consumption per seat per mile
η 777 < η 767 15%
η 787 < η 777 20%

3. Design Process
Mission Specification
Initial Design
Experience
Multi-Disciplinary
Multi-Physics Numerical
Optimization
PDE S l
Solvers
Optimized Design
Opening: Design Process CFD

4. CFD
1-Mesh
Complex Geometry
Adaptation and Refinement
2-Accuracy
Discretization (Truncation) error
Modeling error
3-Convergence
3C
Stability
Residual dropping order
Time & Cost
Background: CFD CFD Algorithm

5. CFD - Overall Algorithm
Mesh generation package
Geometry & Solution domain
Physics & Fluid flow equations
Meshed domain
Residual
Boundary & Initial conditions
Discretization of the fluid flow equations
& Flux Computation and Integration
Implicit method
L
Large system of li
t f linear equations
ti
Jacobian matrix
Fluid flow
Sparse
Preconditioning
matrix solver
simulation
Background: CFD Algorithm Motivation

6. Motivation
∂U ∂U
Δx + Δy + O( Δ )2
Second-order methods: U 2 nd −order= U ( xc , yc ) +
∂x ∂y
∂ 2U Δx 2 ∂ 2U ∂ 2U Δy 2
Truncation error: O( Δ ) = 2 ΔxΔy + 2
+
2
∂x 2 ∂x∂y ∂y 2
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 a
given level of accuracy is more efficient.
Higher-order:
Higher order: Can be more efficient !?
Background: Motivation Literature Review

7. Literature Review
Qualitative Illustration of Research on Solver Development
Structured Structured-Implicit Unstructured Unstructured-Implicit
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

8. Objective
• Developing an Efficient Higher-Order Accurate
Unstructured Finite Volume Algorithm for Inviscid
Compressible Fluid Flow.
Objective: Contribution Model Problem

9. Model Problem
The unsteady (2D) Euler equations which model compressible inviscid
fluid flows, are conservation equations for mass, momentum, and energy.
Aerodynamic application: lift, wave drag and induced drag
d
∫ Udv + ∫ FdA = 0 (1)
dt cv cs
⎡ρ⎤ ρun
⎡ ⎤
⎢ ρu ⎥ ⎢ ρuu + Pn ⎥ˆx
U =⎢ ⎥ , F =⎢ ⎥
n (2)
⎢ ρv ⎥ ⎢ ρvun + Pn y ⎥
ˆ
⎢⎥ ⎢ ⎥
( E + P )un ⎦
⎣E⎦ ⎣
u n = un x + vn y , E = P /( γ − 1 ) + ρ (u 2 + v 2 ) / 2
ˆ ˆ
Theory: Model Problem Implicit Time Advance

10. Implicit Time Advance
Applying implicit time integration and linearization of the governing
equations in time leads to implicit time advance formula:
U n +1 − U n
dU
+ R n +1 ) = 0
+ R( U ) ) = 0 ⇒ (
( (3)
Δt
dt
∂R n n+1
n +1
= Rn + ( ) (U −U n )
R (4)
∂U
∂R
I
)δU = − R , δU = U n+1 − U n
n
+
( (5)
Δt ∂U
U: Solution Vector
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

11. Linear System Solver
GMRES (Generalized Minimal Residual, Saad 1986)
*GMRES algorithm, among other Krylov techniques, only needs matrix vector
d t ( ti f
products (matrix-free i limplementation).
t ti )
*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:
−1
Ax = b − > ( AM ) Mx = b , A≈M
M = LU
M ≅ ILU ( n )
M is an approximation to matrix A which has simpler structure.
ILU: Incomplete Lower-Upper factorization
p pp
Technique: Linear System Solver Reconstruction

12. 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
such as mean constraint.
t it
h
∂U ∂U
Δx + Δy +
= U ( xc , yc ) +
(K)
UR
∂x ∂y
∂ 2U Δx 2 ∂ 2U ∂ 2U Δy 2
ΔxΔy + 2
+ +
∂x 2 2 ∂x∂y ∂y 2
∂ 3U Δx 3 ∂ 3U Δx 2 Δy ∂ 3U ΔxΔy 2 ∂ 3U Δy 3
+2 + +3 + ... ∫U R ( x , y ) = U CV
(K)
(6) (7)
∂x 6 ∂x ∂y 2 ∂x∂y ∂y 6
3 2
2 CV
Technique: Reconstruction Monotonicity

13. Monotonicity
Limiting
Limiting
g
Technique: Monotonicity Higher-Order Limiter

14. Higher-Order Limiter
PHi h -O d = Const + [(1 − σ)φ + σ][Linear part] + σ[Higher - Order part]
Const. (8)
High Order
σ = [ 1 − tanh( ( φ0 − φ )S ) ] / 2, φ0 = 0.8, S = 20. (9)
φ < φ0 : σ → 0.0
φ ≥ φ0 : σ = 1.0
Technique: Higher-Order Limiter Flux Evaluation

15. Flux Evaluation
• Discretization scheme :
Solution reconstruction: Kth-order accurate least-square
reconstruction procedure (Ollivier-Gooch 1997)
t ti d (Olli i G h 1997).
Flux formulation: Roe’s flux difference splitting (1981).
1 1~
F (U L ,U R ) = ( F (U L ) + F (U R )) − A (U R − U L ) (10)
2 ( L, R )
2
~ ~ ~~ ~
~
A = X −1 Λ X , Λ = Diag λ
Integration scheme : Gauss quadrature integration technique
•
with the proper number of p
pp points.
∫ F .nds
Ri = (11)
CVi
Gauss quadrature for interior control volumes.
Technique: Flux Evaluation 1st-Order Jacobian Matrix

16. 1st-Order Jacobian Matrix
∑ F nds = ∑ F ( U ,U
Ri = ˆ ˆ
)( nl )i ,N k
i i Nk (12)
faces
∂F ( U i ,U N k )
∂Ri
J ( i, Nk ) = = ˆ
( nl )i ,N k (13-1)
∂U N k ∂U N k
∂F ( U i ,U N k )
∂Ri
=∑
J ( i ,i ) = ˆ
( nl )i ,N k (13-2)
∂U i ∂U i
Technique: 1st-Order Jacobian Matrix Solution Strategy

17. Solution Strategy
Strategy: Solution Strategy Solution Procedure

18. Solution 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).
∂R
I
)δU = − R
+
( (5)
Δt ∂U
∂R
(First Order)
∂U
Resultant system is solved by GMRES - ILU(1) linear solver.
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.
R( U + εv ) − R( U )
∂R
∂R
.v ≅
)δU = − R (13)
( (12)
ε
∂U
∂U
Procedure: Solution Procedure Results

19. Results
Supersonic Vortex, Annulus-Meshes
p ,
427 CVs 1703 CVs
108 CV
CVs
6811 CVs 27389 CVs
Results: Supersonic Vortex Mach Contours Density Error Error Convergence Error versus CPU Time

20. Mach Contours-Supersonic Vortex, M=2.0

21. Density Error-Supersonic Vortex, M=2.0

22. Error Convergence-Supersonic Vortex, M=2.0

23. Density Error versus CPU Time / Supersonic Vortex,
M 2.0
M=2.0
Results: Error versus CPU Time Subsonic flow over NACA 0012 Airfoil Subsonic Convergence

24. Subsonic Flow over NACA 0012, M=0.63, AoA=2.0 deg.
4958CV 2nd-Order
3rd-Order
Order 4th-Order
Order

25. 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
90 54 289.3
289 3 7 239.2-83%
239 2-83%
Results: Subsonic Convergence Transonic flow over NACA 0012 Airfoil Transonic Convergence

26. Transonic Flow over NACA 0012, M=0.80, AoA=1.25 deg.
4958CV 3rd-Order
φ Limiter σ Limiter

27. 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
311 4 590 10 221-37%
Results: Transonic Convergence Transonic Mach Profile

28. 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

29. 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 third
start-up
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
performance
performance.
• The possibility of benefits of higher-order discretization has been shown.
Closing: Research Summary and Conclusion Recommended Future Work

30. Recommended Future Work
• Improving the start-up procedure.
• Applying a more accurate preconditioning.
pp y g p g
• E h i the robustness of the reconstruction f di
Enhancing th bt f th t ti for discontinuities (limiting).
ti iti (li iti )
• Extension to 3D.
• Extension to viscous flows.
Closing: Recommended Future Work End

31. End
Thank You for Your Attention