TOPOLOGY OPTIMIZATION THEORY, METHODS AND APPLICATIONS INTRODUCTION TO COMPUTER EXERCISES JAKOB SØNDERGAARD JENSEN

COMPUTER EXERCISES (the backbone of the course) • Wednesday: Basic exercises with the code top.m (alternatively top88.m) – Problem 1-3: Basic exercises (getting to know the code) – Problem 9: Robust topology optimization – Problem 12: Three dimensions • Thursday & Friday: Exercises with mathematical programming – Problem 4: The method of moving asymptotes (MMA) – Problem 5: Mechanism synthesis using MMA – Problem 6: Interior point methods (fmincon) – Problem 8: min-max optimization using MMA – Problem 10: SAND optimization formulation – Problem 11: Multiple outputs • Weekend: Catch up on missing obligatory exercises AND/OR do voluntary exercises of interest

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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

COMPUTER EXERCISES • Monday: Optimization in dynamics – Problem 7: Optimization with harmonic loads – Problem 13: Alternative measures of dynamic compliance – Problem 14: Problem of your choice from Bendsøe & Sigmund (2004)

⇒ Poster presentation of exercise work on tuesday 3

DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

MATLAB CODES Exercises based on the 99-line code written by Ole Sigmund:

CODE CAN BE DOWNLOADED FROM WWW.TOPOPT.DTU.DK 4

DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

EXERCISES Usage:

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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

EXERCISES

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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

EXERCISES

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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

ALTERNATIVE 88-LINE CODE (for experienced FE/Matlab people)

Usage:

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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

Filter type

88- VERSUS 99-LINE CODE Pros and cons of using 88-line version: √ Dramatically faster for large problems (speed quick-fix for 99-line code will be provided) √ Access to a density filter in addition to the sensitivity filter ÷ More difficult to read ÷ More difficult to modify

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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

99-LINE CODE SPEED QUICK-FIX Main problem: assembly of sparse matrix

REPLACE BY: (can be downloaded on the course homepage WWW.TOPOPT.DTU.DK/DCA MM)

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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

OTHER EXTENSIONS

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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

FILTERS 99-line code: sensitivity filter 88-line code: sensitivity filter (ft=1) density filter (ft=2) Sensitivity filter: spatial smoothing of sensitivities to avoid checkerboards and mesh dependency - mathematically inconsistent but works well in practice

Density filter: spatial smoothing of design variables + consistent update of sensitivities 12

DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

FILTERS Implementation of sensitivity filter in 99-line code:

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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

OPTIMALITY CRITERION Design sensitivities:

Heuristic design updates:

Adjusted until volume constraint fulfilled 14

DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

OPTIMALITY CRITERIA Implementation:

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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

PRACTICALITIES • Form groups of two by the computers • Get to know people ! (hook up with somebody you do not know already and preferably someone with a different background) • Get User IDs and passwords from me (non-DTU students)

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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

COFFEE CARDS • Coffee cards will be handed out together with User IDs and passwords • Prepaid 100 DKR = 25 cups of coffee, espresso, cappucino, tea, etc. • IMPORTANT: return coffee cards at the end of course coffee

exercises

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DTU Mechanical Engineering, Technical University of Denmark Jakob Søndergaard Jensen

Door closes at 10pm on weekdays Open 7am-5pm in the weekend