1 Evolutionary Structural Optimisation Lectures notes modified
from Alicia Kim, University of Bath, UK and Mike Xie RMIT
Australia
Slide 3
2 KKT Conditions for Topology Optimisation
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3 KKT Conditions (contd) Strain energy density should be
constant throughout the design domain Similar to fully-stressed
design. Need to compute strain energy density Finite Element
Analysis
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4 Evolutionary Structural Optimisation (ESO) Fully-stressed
design von Mises stress as design sensitivity. Total strain energy
= hydrostatic + deviatoric (deviatoric component usually dominant
in most continuum) Von Mises stress represents the deviatoric
component of strain energy. Removes low stress material and adds
material around high stress regions descent method Design
variables: finite elements (binary discrete)
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5 ESO Algorithm 1.Define the maximum design domain, loads and
boundary conditions. 2.Define evolutionary rate, ER, e.g. ER =
0.01, and an intial rejection ratio, RR, e.g. RR=0.3. 3.Discretise
the design domain with a finite element mesh. 4.Finite element
analysis. 5.Remove low stress elements, 6.Increase the rejection
ratio RR=RR+ER 7.Continue removing material and increasing
rejection ratio until a fully stressed design is achieved 8.Examine
the evolutionary history and select an optimum topology that
satisfy all the design criteria.
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An example (An apple hanging on a tree?) Gravity An object
hanging in the air under gravity loading The finite element
mesh
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Stress distribution of a square apple.
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Evolution of the object.
Slide 10
Comparison of stress distributions. movie
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Problems ESO What is common and what is different between SIMP
based topology optimization and ESO? What do you perceive as the
pros and cons of ESO compared to SIMP? Use ESO to design the MBB
beam by modifying the 99 line topology optimization program.
Compare to the solution produced by top.m 10
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11 Chequerboard Formation Numerical instability due to
discretisation. Closely linked to mesh dependency. Piecewise linear
displacement field vs. piecewise constant design update
Slide 13
Topology Optimisation using Level-Set Function Design update is
achieved by moving the boundary points based on their sensitivities
Normal velocity of the boundary points are proportional to the
sensitivities (ESO concept) Move inwards to remove material if
sensitivities are low Move outwards to add material if
sensitivities are high Move limit is usually imposed (within an
element size) to ensure stability of algorithm Holes are usually
inserted where sensitivities are low (often by using topological
derivatives, proportional to strain energy) Iteration continued
until near constant strain energy/stress is reached. 12
Slide 14
Numerical Examples 13
Slide 15
Thermoelastic problems Both temperature and mechanical loadings
FE Heat Analysis to determine the temperature distribution
Thermoelastic FEA to determine stress distribution due to
temperature Then ESO using these stress values 477 720 24 Design
Domain P Uniform Temperature
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Plate with clamped sides and central load 15
Slide 17
Group ESO Group a set of finite elements Modification is
applied to the entire set Applicable to configuration optimisation
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