development of a consistent discrete adjoint solver for ... euroad workshop - tim albring... ·...

Development of a Consistent Discrete Adjoint Solverfor the SU2 Framework

Tim Albring, Max Sagebaum, Nicolas Gauger

Chair for Scientific ComputingTU Kaiserslautern

16th Euro-AD Workshop, JenaDecember 8, 2014

Open-source multiphysics suite SU2

Some key facts

Developed at Stanford University (with global collaboration).

Comprises complete self-contained optimization environment.

Finite-Volume Discretization of (U)RANS equations and turbulencemodels using a highly modular C++ code-structure.

Convergence Acceleration using Multigrid and local time-steppingmethods.

Albring Development of a Consistent Discrete Adjoint Solver 2/ 24

Demands for a (discrete) Adjoint Solver

Automatic advancement with the flow solver:

Number of commits per week to the SU2 repository.

Smooth integration into the optimization framework (reuse of existingroutines).

Robustness: Inheritance of convergence properties of the flow solver.

Consistency: Resulting gradients should constitute as the exactderivatives based on the particular numerical methods used in the flowsolver.


Generic problem formulation

Given the state vector y ∈ Y ⊂ RN , the discrete design vector χ, thecomputational mesh u and the state equation c(y , u) = 0, the optimizationproblem can be formulated by the following generic problem description:

minχ

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. CD , CL or CMz .

Usually, there are additional constraints.


Aerodynamic Design ChainComponents

Mesh

DeformationFlow Solver

χ u J, y

Mesh DeformationSurface Deformation usingHicks-Henne functions

∆usurf (x) =n∑

i=1

χi sin(πx

log 5log ti

)Mesh Movement based on theLinear Elasticity Method

Flow SolverJST-based centered spatialdiscretization

Implicit Euler temporaldiscretization

SA or SST turbulence model


The Discrete Adjoint SolverPrerequisites

For Newton-type methods, R(y , u) can be transformed into fixed-pointequation 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

yn+1 = G (yn, 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) isunknown!


The Discrete Adjoint SolverLagrange viewpoint

Mesh Deformation is included by adding a constraint M(χ) = u to theoptimization problem∗ .Lagrangian function:

L(χ, y , u, y , u) = J(y , u) + yT (G (y , u)︸︷︷︸=:N, Shifted Lagrangian

−y) + uT (M(χ)− u)

KKT conditions give equations for y , u and dL/dχ:

y =∂

∂yJ(y , u) +

∂

∂yGT (y , u)y =

∂

∂yNT (y , y , u)

u =∂

∂uJ(y , u) +

∂

∂uGT (y , u)y =

∂

∂uNT (y , y , u)

dL

dχ=

d

dχMT (χ)u

∗E.J. Nielsen and M.A. Park. ‘Using An Adjoint Approach to Eliminate Mesh Sensitivities in Computational Design.’, AIAA Journal, Vol.40, No. 5, 2006.


Comparison of Standard Form and Fixed-Point Form∗System of equations is solved using the iterationDn +

∂Rn

∂y

∆yn = −Rn, yn+1 = yn + ∆yn

∗Vamshi M. Korivi. ‘An Incremental Strategy for Calculating Consistent Discrete CFD Sensitivity Derivatives’. Hampton, LangleyResearch Center, 1992.



∂Rn

∂y

∆yn = −Rn, yn+1 = yn + ∆yn

Approximation to ∂Rn

∂y due to

first order spatialdiscretizations only,

inconsistent boundaryconditions,

approximate solutions ofthe linear system.




∂Rn

∂y

∆yn = −Rn, yn+1 = yn + ∆yn


∂y due to




(∂R∗

∂y

)T

λ =∂J∗

∂y




∂Rn

∂y

∆yn = −Rn, yn+1 = yn + ∆yn


∂y due to




(∂R∗

∂y

)T

λ =∂J∗

∂y

No approximations allowed inthis formulation to get correct(consistent) gradient.




∂Rn

∂y

∆yn = −Rn, yn+1 = yn + ∆yn


∂y due to




yn+1 =

(∂G∗

∂y

)T

yn +∂J∗

∂y

Uses same approximative Jaco-bian, thus features convergenceto consistent gradient with thesame rate as state iteration.

XG∗ := y∗ −(D∗ +

∂R∗

∂y

)−1

R∗



The Discrete Adjoint SolverNotes on the Implementation

Based on Algorithmic Differentiation for the gradient computation:

Uses a modified version of the Adept C++ software library∗ (alsoprepared 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≈ 1.2− 2.5

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, 2014.


Performance TuningIn the AD-context

1 Solve linear system in Reverse Sweep using analytic formulation:

2 Assume a passive preconditioner P in the non-linear fixed-point solver:

∂G (y , u)

∂y= I − P(y , u)

∂R(y , u)

∂y+∂P(y , u)

∂yR(y , u)

Reasonable to drop this term

since R(yk , u) goes to zero.

u = W−1v Reverse Mode

s = W−T u

W −= s · uT

v += s

u = 0

Both methods can efficiently be combined.


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 withdifferentiated Matrix-Vector product.

Multi-grid disabled.

Flow Solver performance: 0.21s per Iteration (5 Iter. of Linear Solver), 85MB peak memory

Full Taping Active Prec. Passive Prec.Linear Iter. - 5 2 5 2

Runtime 1.42 2.21 1.46 1.74 1.20Memory 30.04 24.42 23.52 21.91 21.76

Relative performance of Adjoint Solver.


Convergence - Adjoint Solverρrms = 10−4

Passive Preconditioner leads to small difference in gradient (error ≈ 0.05%).Much faster with reduced number of linear iterations (despite slightly highernumber of outer iterations).

(a) Convergence yρ (b) Convergence u


Convergence - Adjoint Solverρrms = 10−7

(c) Convergence yρ (d) Convergence u


Inviscid FlowTest case definition

Baseline design: NACA0012

Flow ConditionsMa∞ = 0.8

α = 1.25◦

Numerical Settings

Mesh: 10216 Elements

Multigrid: 3 Level W-Cycle,CFL = 4.0.

38 Hicks-Henne functions

Problem statement:

minχ

CD(χ)

subject to: CL(χ) ≥ 0.326

CMz(χ) ≥ 0.0336.


Inviscid FlowResults

Validation of gradient with Finite Differences resulted in l2-Norm distance of0.068%.

(e) Optimization History (f) Pressure Distribution Cp


Viscous Flow - SATest case definition

Baseline design: RAE2822

Flow conditionsMa∞ = 0.729

α = 2.31◦

Re = 6.5 Mio

Numerical Settings

Mesh: 59976 Elements

3 Level W-Cycle, CFL = 4.0

38 Hicks-Henne functions

Problem statement

minχ

CD(χ)

subject to: CL(χ) = 0.707

CMz(χ) ≥ −0.088

Area = Areainitial .


Viscous Flow - SAValidation

(g) Drag Gradient (h) Lift Gradient


Viscous Flow - SAResults - Optimization History

(i) Discrete Adjoint (j) Continuous Adjoint


Viscous Flow - SAResults - Cp

(k) Pressure Distribution


Viscous Flow - SSTResults

(l) Optimization History (m) Pressure Distribution


Viscous FlowPressure Contours

(n) Baseline (o) Optimized - Discrete Adjoint


3D - Turbulent Flow

Figure: Magnitude of usurf for the Onera-M6 test case using the SST turbulencemodel.


Summary & Outlook

Summary

Differentiation of the entire design chain of SU2.

Consistent gradients of complex flow models.

Efficient and easy maintainable/extensible adjoint solver.

OutlookFurther 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).


Thank you for your attention!


development of a consistent discrete adjoint solver for ... euroad workshop - tim albring... ·...

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