Development of a Consistent Discrete Adjoint Solver for the SU 2 Framework

Transcription

1 Development of a Consistent Discrete Adjoint Solver for the SU 2 Framework Tim Albring, Max Sagebaum, Nicolas Gauger Chair for Scientific Computing TU Kaiserslautern 16th Euro-AD Workshop, Jena December 8, 2014

2 Open-source multiphysics suite SU 2 Some key facts Developed at Stanford University (with global collaboration). Comprises complete self-contained optimization environment. Finite-Volume Discretization of (U)RANS equations and turbulence models using a highly modular C++ code-structure. Convergence Acceleration using Multigrid and local time-stepping methods. Albring Development of a Consistent Discrete Adjoint Solver 2/ 24

3 Demands for a (discrete) Adjoint Solver Automatic advancement with the flow solver: Number of commits per week to the SU 2 repository. Smooth integration into the optimization framework (reuse of existing routines). Robustness: Inheritance of convergence properties of the flow solver. Consistency: Resulting gradients should constitute as the exact derivatives based on the particular numerical methods used in the flow solver. Albring Development of a Consistent Discrete Adjoint Solver 3/ 24

4 Generic problem formulation Given the state vector y Y R N, the discrete design vector χ, the computational mesh u and the state equation c(y, u) = 0, the optimization problem can be formulated by the following generic problem description: min χ J(y(u(χ)), u(χ)) subject to R(y(u(χ)), u(χ)) = 0. Gradient computation of J is based on an implicitly defined state y. f can be e.g. C D, C L or C Mz. Usually, there are additional constraints. Albring Development of a Consistent Discrete Adjoint Solver 4/ 24

5 Aerodynamic Design Chain Components χ Mesh u J, y Flow Solver Deformation Mesh Deformation Surface Deformation using Hicks-Henne functions u surf (x) = n i=1 ) χ i sin (πx log 5 log t i Mesh Movement based on the Linear Elasticity Method Flow Solver JST-based centered spatial discretization Implicit Euler temporal discretization SA or SST turbulence model Albring Development of a Consistent Discrete Adjoint Solver 5/ 24

6 The Discrete Adjoint Solver Prerequisites For Newton-type methods, R(y, u) can be transformed into fixed-point equation y = G(y, u) by using G(y, u) := I P(y, u)r(y, u) with some suitable preconditioner P. Feasible solutions y = y(u) are then computed from the iteration y n+1 = G(y n, u). It is natural to assume that G is stationary only at feasible points, i.e. R(y, u) = 0 iff. y = G(y, u). But: For a complex flow solver, the explicit structure of P (and G) is unknown! Albring Development of a Consistent Discrete Adjoint Solver 6/ 24

7 The Discrete Adjoint Solver Lagrange viewpoint Mesh Deformation is included by adding a constraint M(χ) = u to the optimization problem. Lagrangian function: L(χ, y, u, ȳ, ū) = J(y, u) + ȳ T (G(y, u) y) + ū T (M(χ) u) }{{} =:N, Shifted Lagrangian KKT conditions give equations for ȳ, ū and dl/dχ: ȳ = J(y, u) + G T (y, u)ȳ = NT (y, ȳ, u) ū = J(y, u) + u u G T (y, u)ȳ = u NT (y, ȳ, u) dl dχ = d dχ MT (χ)ū E.J. Nielsen and M.A. Park. Using An Adjoint Approach to Eliminate Mesh Sensitivities in Computational Design., AIAA Journal, Vol. 40, No. 5, Albring Development of a Consistent Discrete Adjoint Solver 7/ 24

8 Comparison of Standard Form and Fixed-Point Form System of equations is solved using the iteration D n + R n y n = R n, y n+1 = y n + y n Vamshi M. Korivi. An Incremental Strategy for Calculating Consistent Discrete CFD Sensitivity Derivatives. Hampton, Langley Research Center, Albring Development of a Consistent Discrete Adjoint Solver 8/ 24

9 Comparison of Standard Form and Fixed-Point Form System of equations is solved using the iteration D n + R n y n = R n, y n+1 = y n + y n Approximation to Rn due to first order spatial discretizations only, inconsistent boundary conditions, approximate solutions of the linear system. Vamshi M. Korivi. An Incremental Strategy for Calculating Consistent Discrete CFD Sensitivity Derivatives. Hampton, Langley Research Center, Albring Development of a Consistent Discrete Adjoint Solver 8/ 24

10 Comparison of Standard Form and Fixed-Point Form System of equations is solved using the iteration D n + R n y n = R n, y n+1 = y n + y n Approximation to Rn due to first order spatial discretizations only, inconsistent boundary conditions, approximate solutions of the linear system. ( R ) T λ = J Vamshi M. Korivi. An Incremental Strategy for Calculating Consistent Discrete CFD Sensitivity Derivatives. Hampton, Langley Research Center, Albring Development of a Consistent Discrete Adjoint Solver 8/ 24

11 Comparison of Standard Form and Fixed-Point Form System of equations is solved using the iteration D n + R n y n = R n, y n+1 = y n + y n Approximation to Rn due to first order spatial discretizations only, inconsistent boundary conditions, approximate solutions of the linear system. ( R ) T λ = J No approximations allowed in this formulation to get correct (consistent) gradient. Vamshi M. Korivi. An Incremental Strategy for Calculating Consistent Discrete CFD Sensitivity Derivatives. Hampton, Langley Research Center, Albring Development of a Consistent Discrete Adjoint Solver 8/ 24

12 Comparison of Standard Form and Fixed-Point Form System of equations is solved using the iteration D n + R n y n = R n, y n+1 = y n + y n ( G := y D + R ) 1 R Approximation to Rn due to first order spatial discretizations only, inconsistent boundary conditions, approximate solutions of the linear system. ( ) G ȳ n+1 T = ȳ n + J Uses same approximative Jacobian, thus features convergence to consistent gradient with the same rate as state iteration. Vamshi M. Korivi. An Incremental Strategy for Calculating Consistent Discrete CFD Sensitivity Derivatives. Hampton, Langley Research Center, Albring Development of a Consistent Discrete Adjoint Solver 8/ 24

13 The Discrete Adjoint Solver Notes on the Implementation Based on Algorithmic Differentiation for the gradient computation: Uses a modified version of the Adept C++ software library (also prepared for dco and ADOL-C). Requires almost no modifications in the flow solver. Highly efficient in runtime: Time for adjoint solution Time for flow solution Actual value depends on the numerical methods used. Parallelized using AdjointMPI. Robin J. Hogan, Fast reverse-mode automatic differentiation using expression templates in C++, ACM Trans. Math. Softw., 40, 26:1-26:16, Albring Development of a Consistent Discrete Adjoint Solver 9/ 24

14 Performance Tuning In the AD-context 1 Solve linear system in Reverse Sweep using analytic formulation: u = W 1 v ū = 0 2 Assume a passive preconditioner P in the non-linear fixed-point solver: G(y, u) Reverse Mode R(y, u) = I P(y, u) Both methods can efficiently be combined. s = W T ū W = s u T v += s + P(y, u) R(y, u) Reasonable to drop this term since R(y k, u) goes to zero. Albring Development of a Consistent Discrete Adjoint Solver 10/ 24

15 Setting and Performance GMRES + LU-SGS Preconditioner (BCGSTAB also possible). Matrix stored using Block Compressed Row Storage. Linear System in Reverse Sweep solved using same solver with differentiated Matrix-Vector product. Multi-grid disabled. Flow Solver performance: 0.21s per Iteration (5 Iter. of Linear Solver), 85 MB peak memory Full Taping Active Prec. Passive Prec. Linear Iter Runtime Memory Relative performance of Adjoint Solver. Albring Development of a Consistent Discrete Adjoint Solver 11/ 24

16 Convergence - Adjoint Solver ρ rms = 10 4 Passive Preconditioner leads to small difference in gradient (error 0.05%). Much faster with reduced number of linear iterations (despite slightly higher number of outer iterations). (a) Convergence ȳ ρ (b) Convergence ū Albring Development of a Consistent Discrete Adjoint Solver 12/ 24

17 Convergence - Adjoint Solver ρ rms = 10 7 (c) Convergence ȳ ρ (d) Convergence ū Albring Development of a Consistent Discrete Adjoint Solver 13/ 24

18 Inviscid Flow Test case definition Baseline design: NACA0012 Flow Conditions Ma = 0.8 α = 1.25 Numerical Settings Mesh: Elements Multigrid: 3 Level W-Cycle, CFL = Hicks-Henne functions Problem statement: min χ C D (χ) subject to: C L (χ) C Mz (χ) Albring Development of a Consistent Discrete Adjoint Solver 14/ 24

19 Inviscid Flow Results Validation of gradient with Finite Differences resulted in l 2 -Norm distance of 0.068%. (e) Optimization History (f) Pressure Distribution C p Albring Development of a Consistent Discrete Adjoint Solver 15/ 24

20 Viscous Flow - SA Test case definition Baseline design: RAE2822 Flow conditions Ma = α = 2.31 Re = 6.5 Mio Numerical Settings Mesh: Elements 3 Level W-Cycle, CFL = Hicks-Henne functions Problem statement min χ C D (χ) subject to: C L (χ) = C Mz (χ) Area = Area initial. Albring Development of a Consistent Discrete Adjoint Solver 16/ 24

21 Viscous Flow - SA Validation (g) Drag Gradient (h) Lift Gradient Albring Development of a Consistent Discrete Adjoint Solver 17/ 24

22 Viscous Flow - SA Results - Optimization History (i) Discrete Adjoint (j) Continuous Adjoint Albring Development of a Consistent Discrete Adjoint Solver 18/ 24

23 Viscous Flow - SA Results - C p (k) Pressure Distribution Albring Development of a Consistent Discrete Adjoint Solver 19/ 24

24 Viscous Flow - SST Results (l) Optimization History (m) Pressure Distribution Albring Development of a Consistent Discrete Adjoint Solver 20/ 24

25 Viscous Flow Pressure Contours (n) Baseline (o) Optimized - Discrete Adjoint Albring Development of a Consistent Discrete Adjoint Solver 21/ 24

26 3D - Turbulent Flow Figure: Magnitude of ū surf model. for the Onera-M6 test case using the SST turbulence Albring Development of a Consistent Discrete Adjoint Solver 22/ 24

27 Summary & Outlook Summary Outlook Differentiation of the entire design chain of SU 2. Consistent gradients of complex flow models. Efficient and easy maintainable/extensible adjoint solver. Further performance optimizations. Application to 3D and unsteady flows. Release of the software suite to the public (early 2015). Extension to multi-disciplinary optimization (aeroacoustics, aero-structure coupling). Albring Development of a Consistent Discrete Adjoint Solver 23/ 24

28 Thank you for your attention! Albring Development of a Consistent Discrete Adjoint Solver 24/ 24