Department of Mechanical Engineering, The University of British Columbia


 A Higher Order Accurate Unstructured Finite Vo...
Aircraft Design & Fuel Efficiency




                  η : Fuel consumption per seat per mile
                  η 777 < η...
Design Process

                                    Mission Specification


                                        Initia...
CFD




   1-Mesh
   Complex Geometry
   Adaptation and Refinement
   2-Accuracy
   Discretization (Truncation) error
   M...
CFD - Overall Algorithm

                                                 Mesh generation package
  Geometry & Solution do...
Motivation
                                                             ∂U      ∂U
                                       ...
Literature Review

                Qualitative Illustration of Research on Solver Development

                         St...
Objective


        • Developing an Efficient Higher-Order Accurate
         Unstructured Finite Volume Algorithm for Invi...
Model Problem
            The unsteady (2D) Euler equations which model compressible inviscid
            fluid flows, are...
Implicit Time Advance
    Applying implicit time integration and linearization of the governing
    equations in time lead...
Linear System Solver
    GMRES (Generalized Minimal Residual, Saad 1986)
     *GMRES algorithm, among other Krylov techniq...
Reconstruction


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



                                         Limiting




                                         Limiting
  ...
Higher-Order Limiter




     PHi h -O d = Const + [(1 − σ)φ + σ][Linear part] + σ[Higher - Order part]
                  ...
Flux Evaluation
 • Discretization scheme :
      Solution reconstruction: Kth-order accurate least-square
      reconstruc...
1st-Order Jacobian Matrix




                                   ∑ F nds = ∑ F ( U ,U
                            Ri =    ...
Solution Strategy




Strategy: Solution Strategy Solution Procedure
Solution Procedure
       Start up Process :
   •
       Before switching to Newton-GMERS Iteration, several pre-implicit
...
Results
                    Supersonic Vortex, Annulus-Meshes
                      p              ,




                 ...
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 2.0
                            M=2.0



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




4958CV                                                        2nd-O...
Convergence history-Subsonic Case




                 Order Resid. Eval.   Time (Sec)   Work Units Newton Itr. Newton Wor...
Transonic Flow over NACA 0012, M=0.80, AoA=1.25 deg.




4958CV                                                   3rd-Orde...
Convergence history-Transonic Case




                 Order Resid. Eval.   Time (Sec)   Work Units Newton Itr. Newton Wo...
Mach Profile-Transonic case




                              Order                   CL            CD

                  ...
Research Summary and Conclusion
•    An ILU preconditioned GMRES algorithm (matrix-free) has been used for
     efficient ...
Recommended Future Work

•   Improving the start-up procedure.


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




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Dr. Amir Nejat

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

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Dr. Amir Nejat

  1. 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. 2. Aircraft Design & Fuel Efficiency η : Fuel consumption per seat per mile η 777 < η 767 15% η 787 < η 777 20%
  3. 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. 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. 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. 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. 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. 8. Objective • Developing an Efficient Higher-Order Accurate Unstructured Finite Volume Algorithm for Inviscid Compressible Fluid Flow. Objective: Contribution Model Problem
  9. 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. 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. 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. 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. 13. Monotonicity Limiting Limiting g Technique: Monotonicity Higher-Order Limiter
  14. 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. 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. 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. 17. Solution Strategy Strategy: Solution Strategy Solution Procedure
  18. 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. 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. 20. Mach Contours-Supersonic Vortex, M=2.0
  21. 21. Density Error-Supersonic Vortex, M=2.0
  22. 22. Error Convergence-Supersonic Vortex, M=2.0
  23. 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. 24. Subsonic Flow over NACA 0012, M=0.63, AoA=2.0 deg. 4958CV 2nd-Order 3rd-Order Order 4th-Order Order
  25. 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. 26. Transonic Flow over NACA 0012, M=0.80, AoA=1.25 deg. 4958CV 3rd-Order φ Limiter σ Limiter
  27. 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. 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. 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. 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. 31. End Thank You for Your Attention
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