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Brief Outline

Aerodynamic Design Using VLM Gradient Generation Using ADIFOR (Automatic Differentiation in Fortran) Santosh N. Abhyankar Prof. K. Sudhakar. Brief Outline. Why ADIFOR ? What is ADIFOR ? Where ADIFOR has been used ? Case studies in CASDE. Why ADIFOR ?. Gradient-based Optimization.

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Brief Outline

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

  2. Brief Outline • Why ADIFOR ? • What is ADIFOR ? • Where ADIFOR has been used ? • Case studies in CASDE

  3. Why ADIFOR ?

  4. Gradient-based Optimization • Gradients calculated to give search direction • Accuracy of gradients affect: • Efficiency of the optimizer • Accuracy of the optimum solution

  5. Different Ways to Calculate Gradients • Numerical methods • Finite difference • Complex variable method • Adjoint method • Analytical methods • Manual differentiation • Automated differentiation

  6. 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.

  7. Optimizer Analysis Functions Say: f(x),h(x), g(x) = any Complicated functions General Flowchart of an Optimization Cycle

  8. Optimizer Analysis Functions Say: f(x),g(x), h(x) = any Complicated function Gradient Calculation using Forward Difference Method

  9. Drawbacks of Numerical Gradients • Approximate • Round-off errors • Computational requirements: requires (n+1) evaluations of function f • Difficulties with noisy functions

  10. Optimizer Analysis Functions Say: f(x) = Sin(x) Externally supplied Analytical Gradients: = Cos(x) Facility to provide user-supplied Gradients

  11. Optimizer F(x)/g(x)/h(x): Complex Analysis Code Externally supplied Analytical Gradients: = ? Facility to provide user-supplied Gradients

  12. Complex Analysis Code in Fortran Manually extract sequence of mathematical operations Manually differentiate mathematical functions - chain rule FORTRAN source code that can evaluate gradients Code the complex derivative evaluator in Fortran User Supplied Gradients

  13. Complex Analysis Code in FORTARN Manually extract sequence of mathematical operations Use symbolic math packages to automate derivative evaluation FORTRAN source code that can evaluate gradients Code the complex derivative evaluator in Fortran User Supplied Gradients

  14. Parse and extract the sequence of mathematical operations Complex Analysis Code in FORTARN Use symbolic math packages to automate derivative evaluation FORTRAN source code that can evaluate gradients Code the complex derivative evaluator in Fortran User Supplied Gradients

  15. Complex Analysis Code in FORTARN Automated Differentiation Package FORTRAN source code that can evaluate gradients Gradients by ADIFOR

  16. What is ADIFOR ?

  17. Automatic DIfferentiation in FORtran {ADIFOR} by Mathematics and Computer Science Division, Argonne National Laboratories, NASA.

  18. Initial Inputs to ADIFOR • The top level routine which contains the functions • The dependant and the independent variables • The maximum number of independent variables

  19. Functionality of ADIFOR • Consider • The derivative of is given by

  20. Functionality of ADIFOR …contd. • For any set of functions say: • ADIFOR generates a Jacobian:

  21. 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

  22. 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

  23. 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--------

  24. 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_v • C-------- • return • end

  25. ADIFOR Where ??

  26. 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).

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

  28. Mesoscale Weather Modeling: Temperature sensitivity as computed by ADIFOR

  29. Case Study at CASDE

  30. 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

  31. 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)

  32. 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)

  33. Optimization Results

  34. Optimization Results

  35. Comparison of Time Takenfor Optimization

  36. 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

  37. Limitations • Strict ANSI Fortran 77 code.

  38. Thank You

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