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Engineering Optimization

Concepts and Applications WB 1440. F. ?. Engineering Optimization. Fred van Keulen Matthijs Langelaar CLA H21.1 A.vanKeulen@tudelft.nl. Contents. Sensitivity analysis: Brief recap discrete / SA approach Adjoint method Continuum sensitivities Topology optimization Closure. E , n.

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Engineering Optimization

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

  2. Contents • Sensitivity analysis: • Brief recap discrete / SA approach • Adjoint method • Continuum sensitivities • Topology optimization • Closure

  3. E, n L t R h r Structural optimization

  4. ? F Topology optimization • Topology: topos (place), logos (study): ~ the way parts of an object are connected to each other • More general than shape optimization! No a prioriassumptions needed about shape

  5. Heuristic Methods • Hard-kill • Soft-kill • Etc. etc.

  6. Bendsoe & Kikuchi, 1988:Homogenization method Suzuki & Kikuchi (1991) Milestones • Michell, 1904: “Michell truss”Structures with optimal stiffness for a given weight

  7. Sheet thickness optimization (thickness sizing) ti • Homogenization approach (microstructure sizing) hi, bi “Classical” Approaches Ai • Various approaches: • Ground structure approach (truss sizing)

  8. Stiffest structure = structure with minimal compliance: Linear elasticity: Compliance: 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)

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

  10. Solve optimization problem: ri Compliance minimization (3) • Conventional approach: • Assign density variables to every element • Young’s modulus depends on density:SIMP (Solid Isotropic Material with Penalization)

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

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

  13. Solution procedure • Compliance minimization problem: • Solved by: • Constrained optimization algorithms (convex approximation methods: SLP, MMA) • Optimality criteria methods (heuristic)

  14. F Do it yourself! • See www.topopt.dtu.dk! • Online optimization • Matlab programs

  15. 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. Compliant mechanisms • Precise, frictionless motion, single structure (no joints) • Lu et al, 2003 • Wang et al, 2005

  17. Mechanism design • Inverter design, Kawamoto/Bendsoe/Sigmund, 2004

  18. Thermal actuator (Sigmund, 2000)

  19. To reduce number of DOFs, condensation is applied Uouter Original element properties Uinner kLi(g i) Element Connectivity Parameterization • Topology defined by elements connected with zero-length links • Stiffness of links controlled by design variables g • Elements maintain original properties!

  20. No numerical instabilities due to excessive distortion of weak elements 1: Extremedistortion Reference Density-based ECP F1 F2 1) Yoon and Kim, 2005 Compliant elements Advantages of ECP • No material model interpolation required • Straightforward sensitivity analysis

  21. SMAdesigndomain Shape memory alloy actuator • SMA: active material, actuation under temperature change

  22. Level sets • Topology optimization using an implicit boundary definition as the zero-level contour of a level-set function

  23. Geometrically nonlinear mechanism Compliant gripper Element-density field

  24. 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

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