Published on October 31, 2007
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
Aircraft Design & Fuel Efficiency η : Fuel consumption per seat per mile η 777 < η 767 15% η 787 < η 777 20%
Design Process Mission Specification Initial Design Experience Multi-Disciplinary Multi-Physics Numerical Optimization PDE S l Solvers Optimized Design Opening: Design Process CFD
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
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
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
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
Objective • Developing an Efficient Higher-Order Accurate Unstructured Finite Volume Algorithm for Inviscid Compressible Fluid Flow. Objective: Contribution Model Problem
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
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
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
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
Monotonicity Limiting Limiting g Technique: Monotonicity Higher-Order Limiter
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
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
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
Solution Strategy Strategy: Solution Strategy Solution Procedure
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
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
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 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 3rd-Order Order 4th-Order 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 90 54 289.3 289 3 7 239.2-83% 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 φ Limiter σ Limiter
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
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 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
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
End Thank You for Your Attention
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