Aerodynamic Design Using VLMGradient Generation Using

ADIFOR (Automatic Differentiation in Fortran)

Santosh N. AbhyankarProf. K. Sudhakar

Brief Outline

• Why ADIFOR ?

• What is ADIFOR ?

• Where ADIFOR has been used ?

• Case studies in CASDE

Why ADIFOR ?

Gradient-based Optimization

• Gradients calculated to give search direction

• Accuracy of gradients affect:– Efficiency of the optimizer– Accuracy of the optimum solution

Different Ways to Calculate Gradients

• Numerical methods– Finite difference– Complex variable method– Adjoint method

• Analytical methods– Manual differentiation– Automated differentiation

dx

xfdxxfxf

)()()(

Finite Difference Vs Complex Variable

• CFL3D of NASA. Inviscil, Laminar, Turb., (Un)steady, Multi-blk, Accel, etcComplex 115 mts 75.2 MB

FD 36 mts 37.7 MB

41 mts 37.7 MB

Note : For several cases FDM required trying out several step sizes to get correct derivative. Factoring this in, it was seen that time taken on an average was more than 2 times for a single analysis.

Optimizer)(xf )(xh )(xg

General Flowchart of an Optimization Cycle

xAnalysis Functions

Say: f(x),h(x),g(x) = any

Complicated functions

Optimizer)( ii hxf )( ii hxh

)( ii hxg

Gradient Calculation using Forward Difference Method

x

)( ii hx Analysis Functions

Say: f(x),g(x),h(x) =

anyComplicated

function

Drawbacks of Numerical Gradients

• Approximate

• Round-off errors

• Computational requirements:

requires (n+1) evaluations of function f

• Difficulties with noisy functions

nx

f

x

f

x

f

,,21

Optimizer

Analysis Functions

Say: f(x) = Sin(x)

x

)(xf )(xh )(xg

Externally supplied Analytical Gradients:

= Cos(x)f h g

x

Facility to provide user-supplied Gradients

)(xf

Optimizer

F(x)/g(x)/h(x): Complex

Analysis Code

x

)(xf )(xh )(xg

Externally supplied Analytical Gradients:

= ?f h g

x

Facility to provide user-supplied Gradients

)(xf

User Supplied Gradients

Complex AnalysisCode in Fortran

Manually extractsequence of mathematical

operations

Code the complex derivative evaluator

in Fortran

Manually differentiatemathematical

functions - chain rule

FORTRANsource code

that can evaluategradients

User Supplied Gradients

Manually extractsequence of mathematical

operations

Use symbolic math packages to automate derivative evaluation

Code the complex derivative evaluator

in Fortran

Complex AnalysisCode in FORTARN

FORTRANsource code

that can evaluategradients

User Supplied Gradients

Parse and extract the sequence

of mathematical operations

Use symbolic math packages to automate derivative evaluation

Code the complex derivative evaluator

in Fortran

Complex AnalysisCode in FORTARN

FORTRANsource code

that can evaluategradients

Gradients by ADIFOR

Complex AnalysisCode in FORTARN

FORTRANsource code

that can evaluategradients

Automated Differentiation

Package

What is ADIFOR ?

AAutomatic utomatic DIDIfferentiation in fferentiation in FORFORtrantran{ADIFOR}{ADIFOR}

byMathematics and Computer Science

Division,Argonne National Laboratories,

NASA.

Initial Inputs to ADIFOR

• The top level routine which contains the functions

• The dependant and the independent variables

• The maximum number of independent variables

Functionality of ADIFOR• Consider

• The derivative of

is given by

22

21211 ),( xxxxf

21212 ),( xxxxf

2

1

2

2

1

2

3

2

2

2

1

1

3

1

2

1

1

3

2

1

6

2

2

2

x

x

x

x

x

x

fx

fx

f

x

fx

fx

f

f

f

f

f

)(xf

221213 32),( xxxxf

Functionality of ADIFOR …contd.

• For any set of functions say:

• ADIFOR generates a Jacobian:

)(xf

n

mm

n

x

f

x

f

x

f

x

f

J

1

1

1

1

SUBROUTINE test(x,f) double precision x(2),f(3) f(1) = x(1)**2 + x(2)**2 f(2) = x(1)*x(2) f(3) = 2.*x(1) + 3.*x(2)**2 return end

ADIFOR

subroutine g_test(g_p_, x, g_x, ldg_x, f, g_f, ldg_f) double precision x(2), f(3) integer g_pmax_ parameter (g_pmax_ = 2) integer g_i_, g_p_, ldg_f, ldg_x double precision d6_b, d4_v, d2_p, d1_p, d5_b, d4_b, d2_v, g_f(l *dg_f, 3), g_x(ldg_x, 2) integer g_ehfid intrinsic dble data g_ehfid /0/C call ehsfid(g_ehfid, 'test','g_subrout5.f')C if (g_p_ .gt. g_pmax_) then print *, 'Parameter g_p_ is greater than g_pmax_' stop endif

d2_v = x(1) * x(1) d2_p = 2.0d0 * x(1) d4_v = x(2) * x(2) d1_p = 2.0d0 * x(2) do g_i_ = 1, g_p_ g_f(g_i_, 1) = d1_p * g_x(g_i_, 2) + d2_p * g_x(g_i_, 1) enddo f(1) = d2_v + d4_vC-------- do g_i_ = 1, g_p_ g_f(g_i_, 2) = x(1) * g_x(g_i_, 2) + x(2) * g_x(g_i_, 1) enddo f(2) = x(1) * x(2)C--------

g_test contd.

d4_v = x(2) * x(2) d1_p = 2.0d0 * x(2) d4_b = dble(3.) d5_b = d4_b * d1_p d6_b = dble(2.) do g_i_ = 1, g_p_ g_f(g_i_, 3) = d5_b * g_x(g_i_, 2) + d6_b * g_x(g_i_, 1) enddo f(3) = dble(2.) * x(1) + dble(3.) * d4_vC-------- return end

g_test contd.

ADIFOR

Where ??

Applications of ADIFOR and ADICApplications of ADIFOR and ADIC

ADIFOR and ADIC have been applied to application codes from various domains of science and engineering. • Atmospheric Chemistry • On-Chip Interconnect Modeling • Mesoscale Weather Modeling • CFD Analysis of the High-Speed Civil Transport • Rotorcraft Flight • 3-D Groundwater Contaminant Transport • 3-D Grid Generation for the High-Speed Civil Transport • A Numerically Complicated Statistical Function -- the Log-Likelihood for log-F distribution (LLDRLF).

Mesoscale Weather Modeling :Temperature sensitivity as computed by Divided Difference using a second-order forward-difference formula

Mesoscale Weather Modeling:

Temperature sensitivity as computed by ADIFOR

Case Study at

CASDE

Optimization Problem • Minimize : induced drag (Cdi)• Subject to: CL = 0.2• Design variables: jig-twist() and angle of attack

at root (

has a linear variation from zero at root to at tip.

is constant over the entire wing semi-span.

cr

ct

The VLM Code600 lines (approx)

SUBROUTINE vlm(amach, cr, ct, bby2, sweep,twist,alp0,isym, ni_gr, nj_gr, cl, cd, cm)

CALL mesh(cr, ct, bby2, sweep, …,ni_gr, nj_gr)CALL matinv(aic, np_max, index, np)CALL setalp(r_p, beta, twist, bby2, alp0, alp, np)CALL mataxb(aic, alp, gama, np_max, np_max, np, np, 1)CALL mataxb(aiw, gama, w , np_max, np_max, np, np, 1)CALL loads(…,gama, w, str_lift, alift, cl, cd, cm)

The ADIFOR-generated derivative of VLM

subroutine g_vlm(g_p_, …, twist, g_twist,ldg_twist, alp0, g_alp0, ldg_alp0, isym, ni_gr, nj_gr, cl, g_cl,ldg_cl, cd, g_cd, ldg_cd, cm)

call mesh(cr, ct, bby2, sweep, …, ni_gr, nj_gr)call matinv(aic, np_max, index, np)call g_setalp(g_p_, r_p, beta, twist, g_twist, ldg_twist, bby2,

alp0, g_alp0, ldg_alp0, alp, g_alp, g_pmax_, np)call g_mataxb(g_p_, aic, alp, g_alp, g_pmax_, gama, g_gama, g_pmax_,

np_max, np_max, np, np, 1)call g_mataxb(g_p_, aiw, gama, g_gama, g_pmax_, w, g_w, g_pmax_, np_max, np_max, np, np, 1)call g_loads(g_p_, …, ni_gr, nj_gr, np, …, gama, g_gama, g_pmax_, w,

g_w, g_pmax_, str_lift, alift, g_alift, g_pmax_, cl, g_cl, ldg_cl, cd, g_cd, ldg_cd, cm)

Optimization ResultsJig-twist CL CDi

Starting

Values

2.0 5.4 0.4522 0.01104

Values at

Optimum

(FD)

-2.27439 3.782271 0.2 0.00191156

Values at

Optimum

(ADIFOR)

-2.29910 3.793176 0.2 0.00191155

Optimization ResultsJig-twist CL CDi

Starting

Values

2.0

-6.0

5.4

1.9

0.4522

-0.0538

0.01104

0.00121

Values at

Optimum

(FD)

-2.27439

-2.27439

3.782271

3.782271

0.2

0.2

0.00191156

0.00191156

Values at

Optimum

(ADIFOR)

-2.29910

-2.29910

3.793176

3.793176

0.2

0.2

0.00191155

0.00191155

Comparison of Time Takenfor Optimization

With

Finite Difference

With

ADIFOR

No. of function

Evaluations

25 15

Total time in

Seconds

35.41 21.45

Codes with CASDE

• Inviscid 3D Code for arbitrary configurations. Tried on ONERA M6. Optimised for memory and CPU time. – total subroutines : 93– total source lines : 5077

• Viscous laminar, 2D, Cartesian for simple configurations. Not optimized. More easily readable. Research code.– total subroutines : 35– total source lines : 2316

Limitations

• Strict ANSI Fortran 77 code.

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