Dr. Amir Nejat

50 %
50 %

Published on October 31, 2007

Author: knowdiff

Source: slideshare.net

Description

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

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

 User name: Comment:

Related pages

Top 12 Amir Nejat profiles | LinkedIn

View the profiles of professionals named Amir Nejat on LinkedIn. There are 12 professionals named Amir Nejat, who use LinkedIn to exchange information ...

View Amir Nejat’s professional profile on LinkedIn. LinkedIn is the world's largest business network, helping professionals like Amir Nejat discover ...

People Named Amir Nejat - Pipl - People Search

Information about Amir Nejat from California, Iran, Virginia and other places. Profile Photos, Address History, Phone Numbers, Relatives, Education ...

Amir Nejat - Google Scholar Citations

Amir Nejat. Associate Professor, ... Citations: 243: 184: h-index: 8: 7: i10-index: 6: 5: ... A Nejat, E Mirzakhalili, A Aliakbari, ...

Dr. Nejat - Doctor Amir Nejat - QEII HSC - HI Site Div. of ...

Dr. Nejat - Doctor Amir Nejat - QEII HSC - HI Site Div. of Cardiology2149 1796 Summer St , Halifax, Nova Scotia. Nova Scotia Doctor and Physician Directory

Najat Amir is on Facebook. Join Facebook to connect with Najat Amir and others you may know. Facebook gives people the power to share and makes the world...

Secret Societies and Freemasonry/Jamiyat Serri ...

Secret Societies and Freemasonry/Jamiyat Serri Vaframasoneri [Amir Nejat] on Amazon.com. *FREE* shipping on qualifying offers.