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A Short Course in Applied Mathematics 2 February 2004 – 7 February 2004 N ∞M∞T Series Two Course Canisius College, Buffa

Computational Methods for Design Lecture 5 - Design and Optimization Problems John A. Burns C enter for O ptimal D esign A nd C ontrol I nterdisciplinary C enter for A pplied M athematics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0531

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A Short Course in Applied Mathematics 2 February 2004 – 7 February 2004 N ∞M∞T Series Two Course Canisius College, Buffa

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  1. Computational Methods for DesignLecture 5 - Design and Optimization ProblemsJohn A. BurnsCenterforOptimalDesignAndControlInterdisciplinaryCenterforAppliedMathematicsVirginia Polytechnic Institute and State UniversityBlacksburg, Virginia 24061-0531 A Short Course in Applied Mathematics 2 February 2004 – 7 February 2004 N∞M∞T Series Two Course Canisius College, Buffalo, NY

  2. OPTIMAL DESIGN PROBLEM: Find the parameter 1 < q0, to minimize the cost function q q q q (S) q q, Given data , 0 < x < 1the goal is to match by solving the following 1D Model Problem LET 1 < q <  and consider the boundary value problem

  3. q q q q q q (S) q, q q q Model Problem #1 SENSITIVITY The sensitivity equation fors(x, q) = qw(x , q)in the “physical” domain(q) = (0,q) is given by Can be made “rigorous” by the method of mappings. MORE ABOUT THIS NEAR THE END

  4. q q q q CONTINUOUS SENSITIVITY q q q q q DISCRETE SENSITIVITY h h h h q q q q q Typical Cost Function WHERE w( x , q )USUALLY SATISFIES A DIFFERENTIAL EQUATION AND q IS A PARAMETER (OR VECTOR OF PARAMETERS) THE CHAIN RULE PRODUCES OR (Reality) USING NUMERICAL SOLUTIONS

  5. h q TYPICAL APPROACHES TO COMPUTE q=q0 (I) BY FINITE DIFFERENCES h h q0 q q0 h q0 q (II) BYDISCRETESENSITIVITIES h h h h q0 q0 q0 q0 q0 Computing Gradients

  6. DISCRETE SENSITIVITIES FINITE DIFFERENCES • REQUIRES THE EXISTENCE OF THE • DISCRETE SENSITIVITY • REQUIRES 2 NON-LINEAR • SOLVES • IF SHAPE IS A DESIGN • VARIABLE, FD REQUIRES 2 • MESH GENERATIONS • IF SHAPE IS A DESIGN VARIABLE, • THE DISCRETE SENSITIVITY LEADS TO • MESH DERIVATIVES COMPUTATIONS WHAT IS THE “CONTINUOUS / HYBRID” SENSITIVITY EQUATION METHOD? --- SEM h h q0 q0 q0 Computing Gradients h, k APPROXIMATE

  7. w(x) w h(x) = Finite Element Approximation x x=0 x=q x=1 NUMERICAL APPROXIMATION h (S) h h h h q, q A Sensitivity Equation Method FORq> 1 ANDh=q/(N+1) CONSIDER (FORMAL) DISCRETE STATE EQUATION

  8. (S) h h q, q h q q h (S) h h h h q, q A Sensitivity Equation Method • IMPORTANT OBSERVATIONS • The sensitivity equations are linear • The sensitivity equation “solver” can be constructed independently of the forward solver -- SENSE™ • When done correctly “mesh gradients” are not required

  9. h q, q (S) h h q q s(x)= qw(x,q) (S) h s h,k(x) = Finite Element Approximation of x x=0 x=q x=1 2nd NUMERICAL APPROXIMATION h,k h,k q q A Sensitivity Equation Method FOR q> 1 ANDk= q/(M+1) CONSIDER (FORMAL)

  10. h h,k h q q q q h k h q q h k h k a trust region method should (might?) converge. When the error is small, then h q q h k R. G. Carter, “On the Global Convergence of Trust-Region Algorithms Using Inexact Gradient Information”, SIAM J. Num. Anal., Vol 28 (1991), 251-265. J. T. Borggaard, “The Sensitivity Equation Method for Optimal Design”, Ph.D. Thesis, Virginia Tech, Blacksburg, VA, 1995. J. T. Borggaard and J. A. Burns, “A PDE Sensitivity Equation Method for Optimal Aerodynamic Design”, Journal of Computational Physics, Vol.136 (1997), 366-384. Convergence Issues THEOREM. The finite element scheme is asymptotically consistent. IDEA:

  11. Convergence Issues N=16, M=32

  12. h q q h h NOT CONVERGENT Convergence Issues THE CASE k = h is often used, but may not be “good enough”

  13. THE CASE k = 2h offers flexibility and h 2h q q h convergence. Timing Issues But, what about timings? Approximately 96 .6% of cpu time spent in function evaluations Approximately 02 .4% of cpu time spent in gradient evaluations

  14. 480 CPU HRS ~3 WEEKS Mathematics Impacts “Practically” UNDERSTANDING THE PROPER MATHEMATICAL FRAMEWORK CAN BE EXPLOITED TO PRODUCE BETTER SCIENTIFIC COMPUTING TOOLS • A REAL JET ENGINE WITH 20 DESIGN VARIABLES • PREVIOUS ENGINEERING DESIGN METHODOLOGY REQUIRED 8400 CPU HRS ~ 1YEAR • USING A HYBRID SEM DEVELOPED AT VA TECH AS IMPLEMENTED BY AEROSOFT IN SENSE™REDUCED THE DESIGN CYCLE TIME FROM ... 8400 CPU HRS ~ 1YEAR TO NEW MATHEMATICSWAS THEENABLING TECHNOLOGY

  15. (DE) (SE) (DE) (SE) Special Structure of SE’s FIRST: SOLVE (DE) SECOND: SOLVE (SE)

  16. General Comments • THERE ARE MANY VARIATIONS THAT CAN IMPROVE THE BASIC IDEA • COMBINING AUTOMATIC DIFFERENTIATION AND SEM • SMOOTHING AND GRADIENT PROJECTIONS • ADAPTIVE GRID GENERATION • THE ORDER OF THINGS MATTER • DIFFERENTIATE-THEN-APPROXIMATE • DERIVE SENSITIVITY EQUATION BEFORE MAPPING TO A “COMPUTATIONAL DOMAIN” • DOES NOT REQUIRE MESH DERIVATIVES • REQUIRES A MORE SOPHISTICATED MATHEMATICAL FRAMEWORK • NEEDS A “DIFFERENT THEORY” J. A. Burns and L. G. Stanley, “A Note on the Use of Transformations in Sensitivity Computations for Elliptic Systems”, Journal of Mathematical & Computer Modeling, Vol. 33, pp. 101-114, 2001.

  17. OPTIMAL DESIGN PROBLEM: Find the parameter 0 < q0< 1, to minimize the cost function q (S) q q q q, q where MODEL PROBLEM #2 LET 1 < q <  and consider the boundary value problem DERIVE THE SENSITIVITY EQUATION

  18. MODEL PROBLEM #2

  19. (S) q , q (S) q, q q  1 x 0 q 0 1  = (0,1) (q) = (0,q) MODEL PROBLEM #2 The sensitivity equation for s(x, q) = qw(x , q) in the “physical” domain (q) = (0,q) is given by APPROXIMATIONS and CHANGE OF VARIABLES (METHOD OF MAPPINGS)  = T(x,q) = x/q

  20. “SOLVE” h h h h h h METHOD OF MAPPINGS S =T(x,q)  (q) x=M(,q)

  21. M(S) q q q -q2 q MODEL PROBLEM #2 Map (0,q) to (0,1) by  = T(x,q) = x/q and note that the inverse mapping M( ,q) = q maps (0,1) to (0, q). Define z( ,q) = w(M( ,q), q) = w(q , q) - transformed state p( , q) = q z( ,q) - sensitivity of the transformed state and r ( , q) = s(M( ,q), q) = s(q, q) - transformed sensitivity.

  22. M(S) M(S) q q q MODEL PROBLEM #2 To compute s(x, q) one has two choices Solve M( S) for r( , q) and transform back to get (1) s(x, q) = r( , q) = r(T(x,q), q) = r(x/q , q) Solve M(S) for p( , q) and transform back to get (2) s(x, q) = p(x/q , q) - z(x/q , q)[ M (x/q , q)]-1[qM (x/q , q)] MESH DERIVATIVE

  23. w(x) w h(x) = Finite Element Approximation x x=0 x=q x=1 NUMERICAL APPROXIMATION h (S) h h h q, q q, (S) q q MODEL PROBLEM #2 FOR q> 1 AND h=q/(N+1) CONSIDER (FORMAL) h h

  24. h h s(x)= qw(x,q) (S) h s h,k(x) = Finite Element Approximation of x x=0 x=q x=1 q, h,k 2nd NUMERICAL APPROXIMATION (S) h,k q q q q MODEL PROBLEM #2 FOR q> 1 ANDk= q/(M+1) CONSIDER (FORMAL)

  25. MODEL PROBLEM #2 ? WHAT HAPPENS ? Linear Finite Elements q = 1.5 q = 1.5 T w(x ,q ) z( ,q )

  26. MODEL PROBLEM #2 H1 - ERROR FOR w(x ,q )

  27. 0 N = 03 -0.05 N = 05 N = 09 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35  0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 MODEL PROBLEM #2 q [z( ,q)] = p( ,q ) M by (1) s(x ,q )

  28. 0.7 0.6 [z(x/q , q)] 0.5 [zh(x/q , q)] Finite Element Approximation of the Spatial Derivative 0.4 0.3 0.2 0.1 0 -0.1 -0.2 0 0.5 1 1.5 MODEL PROBLEM #2 (2) s(x, q) = p(x/q , q) - z(x/q , q)[ M (x/q , q)]-1[q M (x/q , q)]

  29. MODEL PROBLEM #2 s(x ,q ) M by (2) r( ,q ) THE HYBRID CONTINUOUS SENSITIVITY METHOD

  30. x 1  = 2 q q  = 1 q 0 q () q q q q q q q q 1D Interface Problem ELLIPTIC PROCESS MODEL - 2 MATERIALS CONTINUITY

  31. OPTIMAL DESIGN PROBLEM: Find the parameter 0 < q0< 1, to minimize the cost function q q q q q q q q q OR ... q q q 1D Interface Problem

  32. q q q q q q q q q q 1D Interface Problem THE SOLUTION AND SENSITIVITY IS GIVEN BY HOW SMOOTH ISs(x, q ) = q w(x , q)? s( · , q ) H1() ?

  33. 0.3 PLOT OF w(x, q) AT q = .5 PLOT OF SENSITIVITY s(x,q) AT q = .5 1 0.2 0.9 0.8 0.1 0.7 0.6 0 0.5 -0.1 0.4 0.3 -0.2 0.2 -0.3 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.4 -0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1D Interface Problem s( · , q ) H1()

  34. q q q q () q q q q (C) (J) 1D Interface Problem • HOWEVER, THE SENSITIVITY EQUATION IS GIVEN BY THE BOUNDARY VALUE PROBLEM • HOW DID WE DERIVE THIS SYSTEM? • WHAT DO WE MEAN BY A SOLUTION? • CAN THIS BE MADE RIGOROUS?

  35. q q LET q q q q q q q q TAKE THE TOTAL DERIVATIVE OF q q q q q q q q q q q q q- q q- q q+ q q+ q q q q q q q q q Formal Derivation

  36. CONTINUITY JUMP q q q q q q q q q q in [W (q)]’ ? () Formal Derivation LIKEWISE ... WEAKEST FORM OF THE ELLIPTIC PROBLEM

  37. Sensitivity Computations Petrov-Galerkin FE Method (Burns, Lin & STanley - 03)

  38. 2D Sensitivity Computations Petrov-Galerkin FE Method (Burns, Lin & STanley - 03) !! WORKS IN 2D !!

  39. 2D Sensitivity Computations Petrov-Galerkin FE Method (Burns, Lin & STanley - 03) !! WORKS IN 2D !! FOR COMPLEX GEOMETRY

  40. WHAT ABOUT NUMERICAL METHODS

  41. (IVP) x0 t0 Numerical Methods FORWARD DIFFERENCE

  42. x0 t0 Explicit Euler

  43. x0 t0 Implicit Euler Method BACKWARD DIFFERENCE

  44. x0 t0 Implicit Euler

  45. Numerical Methods Matter DIFFERENTIATE THEEQUATIONWITH RESPECT TOq

  46. Numerical Methods Matter INTERCHANGE THE ORDER OF DIFFERENTIATION

  47. Numerical Methods Matter

  48. SOME RUNS

  49. SOLUTION Numerical Methods Matter

  50. Numerical Methods Matter FORWARD EULER

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