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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.
Thank YouThank You