1. 1VR&D1 INDUSTRIAL OPTIMIZATION: STATUS AND PROSPECTS G. Vanderplaats Vanderplaats Research & Development, Inc. 1767 S. 8th Street Colorado Springs, CO 80906 Ph. (719) 473-4611 Copyright VR&D 2004 www. vrand.com

2. 2VR&D2 Minimize F(X)Objective Function Subject to (Such That); Inequality Constraints Equality Constraints Side Constraints F(X), g j (X) and h k (X) May be Linear, Nonlinear, Explicit, Implicit, but Should be Continuous with Continuous First Derivatives GENERAL OPTIMIZATION PROBLEM STATEMENT

3. 3VR&D3 Given X q Update the Design by X q = X q-1 +   S q X q-1 +  X Note that this is Very Close to the Traditional Design Process of Beginning with a Design and Modifying it THE OPTIMIZATION PROCESS

4. 4VR&D4 1948: SIMPLEX Method for Linear Programming 1950’s: Various Random Methods. Gradient Based Methods Developed in the Late 1950’s 1960’s:Sequential Unconstrained Minimization Techniques, Sequential Linear Programming, Feasible Directions Methods 1970’s: Enhanced Feasible Directions Methods, Multiplier Methods, Reduced Gradient Methods, Response Surface Approximations 1980’s: Variable Metric Methods, Sequential Quadratic Programming Methods OPTIMIZATION ALGORITHMS

5. 5VR&D5 1990’s: Genetic Algorithms, Simulated Annealing, New Interest in Sequential Unconstrained Minimization Techniques 2000’s: Particle Swarming, Advanced Sequential Unconstrained Minimization Techniques Largest Known Test Example –500,000 Variables With 500,000 Active Constraints Largest Known Real Structural Optimization Problem –250,000+ Thickness Variables with Frequency Constraints –2,000,000+ Topology Variables OPTIMIZATION ALGORITHMS

6. 6VR&D6 OPTIMIZATION PROBLEM SIZE 0 100 1,000 10,000 196019701980199020002010 # Des. Var. Year 100,000 BIGDOT 500,000 VARIABLES

7. 7VR&D7 STRUCTURAL OPTIMIZATION 1960 - Schmit combined optimization and analysis –2Variables; 1/2 hour on IBM 653 1973 - Schmit et al introduced physics based approximations 1986 - Vanderplaats et al developed 2nd generation approximations 1975 - 1989 Optimization added to commercial structural analysis programs 1984 - 2000 General purpose engineering optimization software Optimization software used by engineers is usually created by engineers

8. 8VR&D8

9. 9 9 1975 OPTIMIZATION WORKS

10. 10VR&D10 1975 OPTIMIZATION WORKS

11. 11VR&D11 1976: A Two Hour Optimization Study OPTIMIZATION WORKS

12. 12VR&D12 1978: Today Called “Response Surface Method” OPTIMIZATION WORKS

13. 13VR&D13 It Has Been Working For Many Years –The Above Examples are 25-30 Years Old! The Aircraft Example was a 1 Man Month Study, Verified by a One Year, $250,000 Study by a Commercial Aircraft Company The Aircraft Take-off Example Solved a Ph.D. Problem that Took Over a Year and Got the Wrong Answer The Airfoil Example Produced a Design Almost Identical to a Multi Year Wind Tunnel Study It is Not Debatable that Optimization is Useful OPTIMIZATION WORKS

14. 14VR&D14 Use Approximations to Avoid Many Calls to the FEA –Optimizer Never Actually Calls the Finite Element Analysis MODERN STRUCTURAL OPTIMIZATION

15. 15VR&D15 Criteria –Find a Very Good Optimum Quickly –Use as Few Full Finite Element Analyses as Possible Basis for Criteria –Each Analysis Requires a Full Finite Element Solution This Can be Very Expensive Cost –About 10-15 Times the Cost of One Analysis This Estimate Assumes Analytic Gradients are Calculated It Also Assumes 2 nd Generation Approximation Techniques are Used THE COST OF STRUCTURAL OPTIMIZATION

16. 16VR&D16 Modern Structural Optimization Converts the Original Design Problem to an Approximate Form Before Calling the Optimizer –Optimizer Calls Approximate Analysis Many Times –Usually About Ten Detailed Finite Element Analyses are Needed 95% of CPU Time is Analysis and Gradient (Sensitivity) Calculations Finite Element Models of the Order of 1,000,000 Degrees of Freedom are Becoming Common Problems in Excess of 250,000 Design Variables Have Been Solved by the GENESIS Program STRUCTURAL OPTIMIZATION

17. 17VR&D17 Rocket Curved Stiffened Panel Minimize mass of the aluminum curved stiffened panel Eight design variables: –Thickness of skin and stiffeners –Stiffener web height –Stiffener flange widths Frequency constraint > 45 Hz (Initially = 23 HZ)

18. 18VR&D18 Panel Optimization Results Frequency constraint is satisfied 30% mass reduction

19. 19VR&D19 Spinning Disk Axi-symmetric structure w.r.t. the Y axis Centrifugal load resulting from a 12 Hz rotation Two material structure –Outer part is aluminum –Inner part is steel

20. 20VR&D20 Spinning Disk Results 26%26% Mass reduction Lowest natural frequency increased Maximum stress reduced

21. 21VR&D21 Shape Optimization of a Pin Pin must carry a specified load Nonlinear contact problem solved using ABAQUS Three materials: pin, adhesive, solid base

22. 22VR&D22 Shape Optimization of a Pin Minimize maximum stress in the solid base Constraints: displacement, stress Nine shape design variables

23. 23VR&D23 Shape Optimization of a Pin 11%Maximum stress reduction: 11% Improved stress distribution Small changes in the initial shape

24. 24VR&D24 Optimize Heat Sink Shape (for PC processor) Minimize:Mass Subject To: –min heat dissipation into the air –max t O in thyristor –max t O in chassis Analysis: Flux2D - FE based package for the analysis of electromagnetic and thermal devices and processes VisualDOC/FLUX2D

25. 25VR&D25 Initial design Final design Design variables: height of the base height and width of fins Result: 47% mass reduction all constraints satisfied Initial design was chosen infeasible for demo Final design looks like normal heat sink in PC VisualDOC/FLUX2D

26. 26VR&D26 CORE + COIL - COIL GAP C-Shaped Magnetic Circuit FLUX2D Model

27. 27VR&D27 CORE GAP X Y Flux Density in GAP of Initial Design - Initial geometry gives a non-uniform magnetic field (flux) in the air gap - Optimize the geometry of the gap to give a prescribed point flux, or uniform flux along the length of the gap

28. 28VR&D28 Pt. 1 Pt. 2 Pt. 3 Pt. 4 Pt. 5 Pt. 6 GAP Change the Y coordinates of points 1-6 and X coordinates of points 2-5 in order to produce a uniform flux density of 0.6 Tesla within the gap. Note: Symmetry Imposed Case 3: Optimum Flux Density in GAP Minimize the sum of the squares of the error (SSE) at 200 points CORE

29. 29VR&D29 Flux Density Variation in GAP of Optimized Designs Designing all X and Y coordinates produces the flattest flux density as shown in case 3 above

30. 30VR&D30 Transport Aircraft Wing Multilevel Optimization –System level: configuration design variables –Disciplinary level: aerodynamic analysis and structural analysis / Optimization Multidisciplinary Optimization with both aerodynamics & structures components Maximize the range for constant gross weight

31. 31VR&D31 Disciplinary Issues complexInteraction between system level and structural sub-optimization is complex Must converge on loads and displacements –Changes in aerodynamic shape at the system level affect the structural geometry Aerodynamic loads deform the structure Structural deformations affect aerodynamic loads

32. 32VR&D32 Transport Aircraft Wing initial final

33. 33VR&D33 Airfoil Optimization NACA 4-digit airfoil Design variables: –maximum mean line camber as fraction of chord (m) –chordwise position of maximum camber (p) –maximum thickness as fraction of chord (t) –Angle of attack (  ) Maximize ratio of Lift/Drag. Use GAMBIT/FLUENT for geometric/flow modeling.

34. 34VR&D34 Optimization Results Pressure Distribution Initial design Final design

35. 35VR&D35 Equivalent Material Properties Reduce the size of the Heat Exchanger FE model by replacing air fin shell elements with equivalent anisotropic solid elements (not able to run the modal analysis on available computers) Match the frequency and displacement responses Analysis: Genesis - FE analysis and Optimization code

36. 36VR&D36 Equivalent Material... Original Configuration Equivalent Configuration Overall mode shapes not changed 3% error in frequencies 5% error in displacements 1 million DOF reduction in FE model size 1st twist mode Diagonal elements of 6x6 material property matrix [G] were adjusted:

37. 37VR&D37 Number of Elements=60,704 Number of Design Variables = 60,704 Number of Design Variables = 7936 Traditional Results Casting Results Design variables reduced by 87% Manufacturing Constraints No constraints added

38. 38VR&D38 Topometry Optimization Example: Where to Reinforce? Objective: –Maximize Natural frequencies Constraints: –Mass Design Variables: 34,560 –Each Element thickness Added Mass (Kg) Increased Frequency (Hz) Maximizig First Torsion Frequency Maximizig First Bending Frequency Maximizig Average of two Frequencis 2.644.816.424.24 7.327.569.896.41 15.069.6612.1512.58

39. 39VR&D39 COMMERCIAL SOFTWARE CompanyWeb Address General Optimization Structural Optimization Altair Engineeringwww.altair.comHyperOptOptiStruct Ansyswww.ansys.com- - -Ansys-CADOE Engineous Softwarewww.engineous.comiSIGHT- - - MSC Softwarewww.mscsoftware.com- - -MSC.Nastran Noesiswww.noesissolutions.comOptimus- - - Opttekwww.optteck.comOptQuest- - - Oculus Technologieswww.oculustech.comCO- - - Phoenix Integrationwww.phoenix-int.comModel Center- - - UGS PLMwww.ugs.com- - -NX.Nastran Vanderplaats R&Dwww.vrand.comVisualDOC, DOT, BIGDOT GENESIS

40. 40VR&D40 FUTURE PROSPECTS Schmit – 1980s –“I believe optimization has a future because people think they can make money on it” Just as Spreadsheets are Routinely Used by Accounts Just as Word Processors are Routinely Used by Secretaries So Should Optimization be Used by Engineers Optimization Will be Widely Used when Management Understands the Enormous Benefit Progress Is Made One Retirement at a Time

41. 41VR&D41 SUMMARY Optimization Technology is Well Developed For General Applications –We Can Couple Almost Any Analysis With Optimization For Structural Optimization –Technology is Very Advanced –Find an Optimum Using Only About 10 Finite Element Analyses Optimization is the Most Powerful Design Improvement Tool Available Today