Aerodynamic Design Using VLM Gradient Generation Using ADIFOR Automatic Differentiation in Fortran S

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Aerodynamic Design Using VLMGradient Generation
Using ADIFOR (Automatic Differentiation in
Fortran)Santosh N. AbhyankarProf. K. Sudhakar
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Brief Outline

• Why ADIFOR ?
• What is ADIFOR ?
• Where ADIFOR has been used ?
• Case studies in CASDE

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Why ADIFOR ?
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Gradient-based Optimization

• Gradients calculated to give search direction
• Accuracy of gradients affect
• Efficiency of the optimizer
• Accuracy of the optimum solution

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Different Ways to Calculate Gradients

• Numerical methods
• Finite difference
• Complex variable method
• Adjoint method
• Analytical methods
• Manual differentiation
• Automated differentiation

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Finite Difference Vs Complex Variable

• CFL3D of NASA. Inviscil, Laminar, Turb.,
  (Un)steady, Multi-blk, Accel, etc
  
• Complex 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.

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Optimizer
Analysis Functions Say f(x),h(x), g(x) an
y
Complicated functions
General Flowchart of an Optimization Cycle
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Optimizer
Analysis Functions Say f(x),g(x), h(x) an
y
Complicated function
Gradient Calculation using Forward Difference Me
thod
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Drawbacks of Numerical Gradients

• Approximate
• Round-off errors
• Computational requirements
• requires (n1)
  evaluations of function f
  
• Difficulties with noisy functions

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Optimizer
Analysis Functions Say f(x) Sin(x)
Externally supplied Analytical Gradients
Cos(x)
Facility to provide user-supplied Gradients
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Optimizer
F(x)/g(x)/h(x) Complex Analysis Code
Externally supplied Analytical Gradients
?
Facility to provide user-supplied Gradients
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User Supplied Gradients
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User Supplied Gradients
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User Supplied Gradients
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Gradients by ADIFOR
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What is ADIFOR ?
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Automatic DIfferentiation in FORtran
ADIFOR by Mathematics and Computer Science Di
vision,
Argonne National Laboratories, NASA.
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Initial Inputs to ADIFOR

• The top level routine which contains the
  functions
  
• The dependant and the independent variables
• The maximum number of independent variables

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Functionality of ADIFOR

• Consider
• The derivative of
• is given by

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Functionality of ADIFOR contd.

• For any set of functions say
• ADIFOR generates a Jacobian

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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
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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
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g_test contd.
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_v C--
------ 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--
------
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g_test contd.
d4_v x(2) x(2) d1_p 2.0d0 x(2)
d4_b dble(3.) d5_b d4_b d
1_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_v
C-------- return end
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ADIFOR Where ??
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• Applications 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).

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

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Mesoscale Weather Modeling Temperature sensitivi
ty as computed by ADIFOR
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Case Study at CASDE
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Optimization Problem

• Minimize induced drag (Cdi)
• Subject to CL 0.2
• Design variables jig-twist(?) and angle of
  attack at root (a0)
  
• ? has a linear variation from zero at root to ?
  at tip.
  
• a0 is constant over the entire wing
  semi-span.
  

L
cr
ct
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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, swe
ep, ,ni_gr, nj_gr) CALL matinv(aic, np_max, inde
x, np) CALL setalp(r_p, beta, twist, bby2, alp0,
alp, np) CALL mataxb(aic, alp, gama, np_max, np_m
ax, np, np, 1) CALL mataxb(aiw, gama, w , np_max
, np_max, np, np, 1) CALL loads(,gama, w, str_li
ft, alift, cl, cd, cm)
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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, , n
i_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_ma
x, 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)
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Optimization Results
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Optimization Results
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Comparison of Time Takenfor Optimization
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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

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Limitations

• Strict ANSI Fortran 77 code.

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Thank You