1. Engineering Optimization – Concepts and Applications Engineering Optimization Concepts and Applications WB 1440 ? F Fred van Keulen Matthijs Langelaar CLA H21.1 A.vanKeulen@tudelft.nl

2. Engineering Optimization – Concepts and Applications Contents ● Sensitivity analysis: – Brief recap discrete / SA approach – Adjoint method – Continuum sensitivities ● Topology optimization ● Closure

3. Engineering Optimization – Concepts and Applications Structural optimization E, t R r L h

4. Engineering Optimization – Concepts and Applications Topology optimization ● Topology:  (place),  (study): ~ the way parts of an object are connected to each other ● More general than shape optimization! No a priori assumptions needed about shape F ?

5. Engineering Optimization – Concepts and Applications Heuristic Methods ● Hard-kill ● Soft-kill ● Etc. etc.

6. Engineering Optimization – Concepts and Applications Milestones ● Michell, 1904: “Michell truss” Structures with optimal stiffness for a given weight Suzuki & Kikuchi (1991) ● Bendsoe & Kikuchi, 1988: Homogenization method

7. Engineering Optimization – Concepts and Applications “Classical” Approaches ● Various approaches: – Ground structure approach (truss sizing) AiAi titi – Sheet thickness optimization (thickness sizing) – Homogenization approach (microstructure sizing) h i, b i

8. Engineering Optimization – Concepts and Applications Compliance minimization ● Classical problem: for a given amount of material, find the stiffest structure – Save material costs (bridge, building) – Improve dynamic performance (automotive, machines) – Save fuel costs (aerospace) ● Stiffest structure = structure with minimal compliance: Linear elasticity: Compliance:

9. Engineering Optimization – Concepts and Applications Compliance minimization (2) ● Optimal solution has infinitely fine porous microstructure: impractical ● Remedy 1: restrict solution to pure solid/void designs + Manufacturable -Mesh refinement leads to more detailed solutions ● Remedy 2: restrict minimal member sizes

10. Engineering Optimization – Concepts and Applications Compliance minimization (3) ● Conventional approach: – Assign density variables to every element – Young’s modulus depends on density: SIMP (Solid Isotropic Material with Penalization) ii – Solve optimization problem:

11. Engineering Optimization – Concepts and Applications SIMP ● SIMP approach uses penalization to make intermediate densities unattractive: – Lower stiffness/weight ratio – Forces design to solid/void solution p = 1; C = 184p = 1.5; C = 210 p = 2; C = 220p = 3; C = 229

12. Engineering Optimization – Concepts and Applications Mesh independence / checkerboard filtering ● Problems: – Checkered solid/void patterns have artificially high stiffness (unrealistic) – Solution dependent on mesh size ● Heuristic solution: spatial filtering – Filtering of sensitivities or density values – Filter radius determines minimum member size r

13. Engineering Optimization – Concepts and Applications Solution procedure ● Compliance minimization problem: ● Solved by: – Constrained optimization algorithms (convex approximation methods: SLP, MMA) – Optimality criteria methods (heuristic)

14. Engineering Optimization – Concepts and Applications F Do it yourself! ● See www.topopt.dtu.dk!www.topopt.dtu.dk – Online optimization – Matlab programs

15. Engineering Optimization – Concepts and Applications Recent progress in other applications ● Topology optimization techniques also (being) developed for: – Multi-material designs, shells, 3D structures – Compliant mechanism design (large displacements) – Thermal actuator design (MEMS) – Crashworthiness design – PZT actuator design – Shape memory alloy actuator design

16. Engineering Optimization – Concepts and Applications Compliant mechanisms ● Precise, frictionless motion, single structure (no joints) ● Lu et al, 2003 ● Wang et al, 2005

17. Engineering Optimization – Concepts and Applications Mechanism design ● Inverter design, Kawamoto/Bendsoe/Sigmund, 2004

18. Engineering Optimization – Concepts and Applications Thermal actuator (Sigmund, 2000)

19. Engineering Optimization – Concepts and Applications Flow networks ● Flow network optimization (Klarbring et al, 2003) Ground structure approach Minimize dissipation / pressure drop

20. Engineering Optimization – Concepts and Applications Limit analysis ● Gilbert & Tyas, 2003; high-performance ground structure approach 116,288,875 potential members 6½ hours 13,263,825 potential members 42 minutes

21. Engineering Optimization – Concepts and Applications Element Connectivity Parameterization ● Topology defined by elements connected with zero-length links ● Stiffness of links controlled by design variables  ● Elements maintain original properties! ● To reduce number of DOFs, condensation is applied Original element properties U outer U inner k L i (  i )

22. Engineering Optimization – Concepts and Applications Advantages of ECP ● No material model interpolation required ● Straightforward sensitivity analysis ECP Density-based Compliant elements Extreme distortion F1F1 F2F2 Reference 1 ) Yoon and Kim, 2005 ● No numerical instabilities due to excessive distortion of weak elements 1 :

23. Engineering Optimization – Concepts and Applications Shape memory alloy actuator ● SMA: active material, actuation under temperature change SMA design domain

24. Engineering Optimization – Concepts and Applications Level sets ● Topology optimization using an implicit boundary definition as the zero-level contour of a level-set function

25. Engineering Optimization – Concepts and Applications Geometrically nonlinear mechanism Compliant gripper Element- density field

26. Engineering Optimization – Concepts and Applications Topology optimization summary ● Very versatile optimization technique: enormous variety of shapes possible ● Recent development: area of active research – Improvements (accuracy, efficiency) – Extensions (nonlinearities, multiple physics, …) ● Try it yourself: topology optimization Matlab program topopt.m available on Blackboard