July 11, 2006

Comparison of Exact and Approximate Adjoint for Aerodynamic Shape

Optimization

ICCFD 4 July 10-14, 2006, Ghent

Giampietro Carpentieri and Michel J.L. van Tooren Delft University of Technology

Barry Koren Centre for Mathematics and Computer science, Amsterdam

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Introduction

• Flow solver

• Adjoint solver

• Gradient computation

• Shape Optimization

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Median-dual discretization

Control volumefor node i

On each control volume/node ( + BC)

N nodes, semi-discrete form

Conserved variables vector

Residual vector ( )

DUAL OF THE MESH

MESH

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MUSCL reconstruction on each edge

Primitive variables reconstruction at edge mid-point:

Least-squares or Green-Gauss gradient

Numerical flux: Roe’s approximate Riemann solver

2nd order accuracy: evaluate flux with reconstructed variables

Venkatakrishnan’s limiter

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Adjoint equations

Shape parameter

Functional ( e.g. lift,drag )

State of system ( e.g. residuals)

Gradient/sensitivity computed as:

: adjoint variables, obtained from adjoint equation:

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Discrete Adjoint for MUSCL scheme

Reconstructed left and right states

Second order fluxes

Three vectors of length E, the number of edges(N is number of nodes)

Dummy matrix ( E x N )

Diagonal matrices, differentiated flux ( E x E )

Reconstruction matrices ( N x E )

Chain rule + transposition

Dependence residual vector on conservative variables:

To compute consider:

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At each time step linear system is solved iteratively :

Time marching flow/adjoint equations

Flow equations

Adjoint equations

Backward Euler scheme:

Symmetric Gauss-Seidel preconditioner (Matrix-free) is used

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Geometric sensitivities

Mesh coordinates

Mesh metrics

Boundary deformations

Limiter vector

Gradient vector

To compute consider:

Coordinates depend on shape parameter:

Residuals depend on coordinates:

Each term is computed using source code generated by Automatic Differentiation tool Tapenade

Chain rule

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Shape parameterization and mesh deformation

Chebyschev polynomials used to parameterize shape of airfoil

Mesh deformations computed with spring analogy solved by Jacobi iterations.

Boundary deformation implies mesh update

is stiffness of edge ij, inverse of edge length

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Shape optimization

Objective function, scaled drag coefficient

Relative maximum thickness constraint

Upper nose radius constraint

Lower nose radius constraint

Trailing edge angle constraint

Lift equality constraint

Minimize function with equality and inequality constraints and bounds on variables

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Optimization Algorithms

It is necessary to use constrained minimization techniques !

Unconstrained minimization techniques that treat constraints as penalty terms could be used. However, they are ill-conditioned and inaccurate.

Two algorithms used in this work:

Sequential Quadratic Programming (SQP)

Search direction found by solving sub-problem with quadratic objective and linearized constraints.Line search is performed using Lagrangian function.Hessian of Lagrangian updated by BFGS (or other) formulas.

Sequential Linear Programming (SLP)

Method of centers is used. Hypersphere fitting into linearized design space found by solving Linear Programming sub-problem. Design updated by moving in the center of the sphere.No second order information is collected.

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Exact and approximate Discrete Adjoint

• Edge-based assembly

• Exact Discrete Adjoint

• Approximations

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Edge-based assemblyMatrix-vector products with transposed residual Jacobians are assembled directly on edges similarly to the residuals assembly:

Two loops on the edges

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Differentiation of flux and reconstruction

Roe’s flux Jacobian

Reconstruction matrix

Five matrices (M) come from differentiation of .

Reconstruction contribution amounts to two transformation matrices and a diagonal matrix which contains limiter and gradient derivatives.

EXACT ADJOINT !

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Approximations

Approximation 1 neglect differentiation of limiter

Approximation 2 neglect differentiation of Roe matrix

Differentiation of limiter is complicated due to construction phase (muldi-dimensional limiter)

Differentiation of Roe matrix is very difficult, symbolic differentiation is used.

For both approximations, compared to exact adjoint, a relative error of 0.1-2.5% in computed gradient is found.

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Approximations

Approximation 3 neglect reconstruction operator

Ignoring reconstruction operator makes implementation of adjoint trivial. Only simple loop on edges is required. Error in computed gradient increases to 10-30% .

Two loops on the edges

One loop on the edges

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Optimization test cases

• NACA64A410 (SQP)

• RAE2822 (SQP)

• NACA0012 (SLP)

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NACA64A410 =0, Mach =0.75

Pressure contours

Drag (scaled) vs gradient iterations

Initial values

Lift

Relative max thickness

Upper nose radius

Lower nose radius

Trailing edge angle

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NACA64A410 =0, Mach =0.75

EXACT ( 19 gradients )

APPROX 1 ( 17 gradients )

APPROX 2 ( 34 gradients )

APPROX 3 ( stalled )

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NACA64A410 =0, Mach =0.75

Lift Thickness

Nose U. Nose L. TE Angle

EXACT -9.9x10-

6 < 10-8 -0.724 -0.126 -0.350

APPROX 1

4.6x10-

7

< 10-8 -0.782 -0.172 -0.032

APPROX 2

7.3x10-

6

< 10-8 -0.693 -4.1x10-7 -0.3

Constraint values show that airfoils satisfy design problem accurately

Lift constraint h = -9.9x10-6 means that final Lift coefficient is:

CL = (1 + h) CL0 = (1 - 9.9x10-6 ) CL0

Thickness constraint is always critical

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MESH 2 : 30000 nodes

MESH 1 : 12000 nodes

NACA64A410 =0, Mach =0.75 CHECK IF MESH 1 IS CAPABLE OF CAPTURING WEAK SHOCK

Mach number distributions do not change on second mesh

EXACT APPROX 1 APPROX 2

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NACA64A410 =0, Mach =0.75

Mach number

Three airfoils have differences in geometry of order 10-3

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RAE2822 =2, Mach =0.73

Lift

Relative max thickness

Upper nose radius

Lower nose radius

Trailing edge angle

Pressure contours

Drag (scaled) vs gradient iterations

Initial values

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RAE2822 =2, Mach =0.73

EXACT ( 15 gradients )

APPROX 1 ( 20 gradients )

APPROX 2 ( 10 gradients ) APPROX 3 ( stalled)

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RAE2822 =2, Mach =0.73

Lift Thickness

Nose U. Nose L. TE Angle

EXACT -1.3x10-

6 < 10-8 -0.416 -3.347 10-8

APPROX 1

-4.4x10-

6

< 10-8 -0.481 -1.964 9x10-8

APPROX 2

-6.2x10-

7

< 10-8 -0.589 -0.292 < 10-8

Constraint values show that airfoils satisfy design problem accurately

Thickness and trailing edge angle constraints are critical for the 3 airfoils

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RAE2822 =2, Mach =0.73

Differences in geometry of order of 10-3

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NACA0012 =2, Mach =0.75

Lift

Relative max thickness

Upper nose radius

Lower nose radius

Trailing edge angle

Pressure contours Drag (scaled) vs gradient

iterations

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NACA0012 =2, Mach =0.75

EXACT, APPROX 1, APPROX 2 APPROX 3

Lift Thickness

Nose U. Nose L. TE Angle

EXACT -4.8x10-

6 -0.012 -0.04 -4.17 -0.44

APPROX 1

1.5x10-

6

-0.013 -0.04 -4.12 -0.432

APPROX 2

3.6x10-

6

-0.013 -0.036 -4.22 -0.438

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NACA0012 =2, Mach =0.75

Differences in y-coordinates are of order 10-4 only

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Conclusions and future work

• Adjoint codes with approximation in the differentiation of fluxes and reconstruction operator, approximations 1 and 2, can be effective for shape optimization;

• When approximations are used, at least with SQP algorithm, the optimization can converge to different airfoils. The SLP algorithm has appeared to be insensitive to the approximations and converged to a unique airfoil;

• When the reconstruction operator is ignored, approximation 3, the adjoint code is not effective. The optimization with SQP and SLP algorithms stall and shock-waves are not removed completely from the airfoil.

July 11, 2006

Thank you for your interest