Adjoint-based Unsteady Airfoil Design Optimization with

Application to Dynamic StallKarthik Mani

Brian LockwoodDimitri Mavriplis

University of WyomingLaramie WY 82071

Outline

• Motivation and introduction• Analysis and sensitivity formulation• Validation• Results• Conclusions

MotivationDynamic stall

• Dynamic stall is an important problem in the unsteady aerodynamics of rotorcraft

• Retreating side susceptible to stall due to low dynamic pressure and high angle-of-attack.

• Can limit lifting capability and forward flight speed.

• Depending on mean angle-of-attack and amplitude of pitch can be categorized as no-stall, light-stall or deep-stall.

• Deep stall is most detrimental.

Motivation

• Deep stall characterized typically large scale separation initiated at leading edge.

• Vortex then rolls down the upper surface of the airfoil and is shed.

• Strong vortex rolling past the trailing edge produces a large nose down pitching moment spike

• Adversely impacts aeroelastics of rotor blade, pitch link loading etc.

• Many different approaches have been investigated to alleviate dynamic stall

• Leading edge slats, leading edge droop, active flow control have have shown reasonable success.

Motivation

• Most of methods have relied on engineering judgement to improve stall characteristics.

• Goal here is to remove intuition and replace it with mathematically rigorous procedure to improve airfoil design.

• Approach is to perform airfoil shape optimization using CFD and adjoint-based sensitivity (gradient) information.

NSU2D• 2D unstructured finite-volume RANS solver.

• 2nd –order accurate in space and time.

• Central differencing with Matrix dissipation.

• Full Navier-Stokes viscous discretization.

• One equation Spalart-Allmaras turbulence model.

• Rigid mesh rotation .

• Tension spring analogy to propagate surface deformations to interior.

• OpenMP parallelization.

Analysis Solver

Fully implicit discretization solved using Newton’s method at each time-step as:

Solution Procedure

• Preconditioned GMRES used for linear system

• Linear agglomeration multigrid for preconditioner

• Jacobi iterations as smoother for linear multigrid

Geometry Parameterization

)ln(/)5.0ln(

)(sin)( 4

i

i

Mi

mi

xm

xaxb

Hicks-Henne Sine bump function:

bump location

node location

node displacement

bump magnitude

10 xValid for:

Bump magnitudes a are the design variables D

Linearization for Gradient

Consider a time integrated objective function:

Linearize using chain rule with respect to design variables:

• Represents forward linearization of objective function w.r.t design variables.• Ideal for cases with only a single design variable.• Typical design problems have multiple design variables and single objective.• Transpose to obtain reverse or adjoint linearization.

Linearization for GradientTransposed or Adjoint linearization:

Expand summation in time:

• Linearization of objective w.r.t. flow state and mesh coordinates• Easy to compute since objective is scalar• Simply derivative of load routines• Results in vectors

Linearization of flow state w.r.t design variables is a matrix and cannot be easily obtained

Linearization for GradientConstraint equation to be satisfied:Implicit residual at each time-step.

Linearize w.r.t design variables using chain rule:

Transpose and rearrange to obtain convenient expression for flow variable sensitivity:

Linearization for Gradient

Substitute back into expanded adjoint linearization:

Define adjoint variable vector

Rerrange to get linear adjoint

system

Very similar to linear system in nonlinear analysis problemNote transposed flow jacobian

Linearization for Gradient• Adjoint system is linear.• Convergence is similar to linear systems arising in nonlinear analysis

problem• Both share the same flow Jacobian matrix with the exception of

transpose• RHS source is different• Same iterative methods used for analysis problem are employed for

adjoint solution (i.e. GMRES, multigrid etc.)

Once solved, substitute back:

Contribution to total gradient vector through series of matrix-vector products:

Repeat process sweeping back in time

Linearization for GradientObtaining full gradient vector is as follows:

• Solve flow problem (forward integration in time)• Write solution (state) to disk at each time-step while integrating• Adjoint solver starts at final time-step and sweeps back in time• Reads solution from disk during the process• One linear adjoint solution at each time-step going back in time

similar to one nonlinear solution at each time-step in the analysis problem going forward in time.

• Full gradient vector is assembled once 1st time-step is reached in the backward sweep.

• Cost of obtaining gradient vector is equal to cost of one analysis/flow solution in worst case scenario.

• Typically faster since adjoint is linear.

Linearization for GradientObtaining full gradient vector is as follows:

• Solve flow problem (forward integration in time)• Write solution (state) to disk at each time-step while integrating• Adjoint solver starts at final time-step and sweeps back in time• Reads solution from disk during the process• One linear adjoint solution at each time-step going back in time

similar to one nonlinear solution at each time-step in the analysis problem going forward in time.

• Full gradient vector is assembled once 1st time-step is reached in the backward sweep.

• Cost of obtaining gradient vector is equal to cost of one analysis/flow solution in worst case scenario.

• Typically faster since adjoint is linear.

Convergence Behavior

Convergence of flow and adjoint at a typical time-

step

Solver Validation

Test Case:• SC1095 Airfoil • Freestream Mach number 0.302• Reynolds Number 3.86 Million• Mean angle-of-attack = 9.92 degrees• Amplitude of Pitch = 9.90 degrees • Reduced frequency 0.148

• Frame 37109 from NASA TM 84245

• 80,000 grid nodes• 10,000 time steps of size dt = 0.02• 25 multigrid (5 levels) cycles• Covers 3 periods of pitch Computational mesh

for test case

Solver Validation

Lift Drag Moment

7 hours wall clock time on 8 cores with OpenMP parallelization

Gradient Verification

• Two step process to verify gradient vector.

• Forward linearization verified using Complex-step method

• Duality between transpose operations used to verify adjoint linearization against forward linearization.

• Complex step method is like finite-difference except no subtraction errors

• All floating point numbers and operators building up nonlinear function are defined as complex and a complex perturbation is introduced.

• Gradient can then be obtained from complex function output as imaginary part divided by the complex perturbation

Gradient Verification

Results• Same SC1095 Airfoil case chosen for optimization.• Goal is to reduce peak pitching moment in hysteresis cycle.• Maintain baseline lift profile.• Same flow conditions as before, M=0.302, kc=0.148, Re=3.86E6• 750 time-steps of size dt=0.08 used.• Corresponds to 1 cycle of pitch.• 25 multigrid cycles at each time-step.• Mesh with about 60,000 nodes.• 10 design variables (bumps). 5 on upper surface and 5 on lower surface.• LBFGS reduced hessian gradient-based optimizer from Nocedal et.al.

Idea from infinite norm. Large power targets highest value of moment in time

Weighted constraint on lift coefficient at each time-step.

Objective function defined as:

ResultsOptimization convergence:

Final/3rd stage of optimization.

ResultsOptimized airfoil comparison:

ResultsOptimized airfoil performance:

Results –Baseline SC1095

Results – Optimized Geometry

Conclusions• Adjoint capability for computing gradients in NSU2D developed.• Method provides discretely exact gradient information that can be

used in design optimization.• Cost of obtaining full gradient information approximately cost of

flow solution.• Removes guesswork and replaces it with mathematically rigorous

design method.• Method shows tremendous potential as a design tool.• Although results show good improvement in moment stall, it is point

design.• Airfoil will likely have unfavorable steady-state characteristics

influencing hover performance.• Infrastructure now exists to explore different geometry

parametrizations and objective functions.• Multi-objective optimizations need to be performed.

AcknowledgementsThis work was supported by ARMY SBIR Phase 1 contract

W911W6-11C-0037