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
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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
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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
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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
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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