Transcript StructOpt
Evolutionary Structural
Optimisation
Lectures notes modified
from Alicia Kim, University of
Bath, UK and Mike Xie RMIT
Australia
1
KKT Conditions for Topology Optimisation
Minimize
C u T Ku uet keue
e
Subject to
V V0 0
Lagrangian function,
L C V V0
L C
V
k
u T
u ve 0
e e
e
e
Let the strain energy of a solid element (e 1) s e ue k0ue
and (no SIMP) with the strain energy proportional to density
k
uT
u u T k0u s e
e
Therefore,
se
1
ve
Uniform strain energy density
2
KKT Conditions (cont’d)
se
1
1v e
Strain energy density should be constant throughout the
design domain
Similar to fully-stressed
design.
Need to compute strain energy density
Finite Element Analysis
3
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)
4
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,
e
RR
max
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.
5
An example
(An apple hanging on a tree?)
Gravity
An object hanging in the air
under gravity loading
The finite element mesh
Stress distribution of a “square apple”
.
Evolution of the object
.
Comparison of stress distributions
.
movie
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
Chequerboard Formation
Numerical instability due to discretisation.
Closely linked to mesh dependency.
Piecewise linear displacement field vs. piecewise constant design
update
11
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
Numerical Examples
13
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
720
477
Design Domain
24
P
Uniform
Temperature
Plate with clamped sides and central load
T = 0C
T = 3C
T = 5C
T = 7C
15
Group ESO
Group a set of finite elements
Modification is applied to the entire set
Applicable to configuration optimisation
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Example: Aircraft Spoiler
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Example: Optimum Spoiler Configuration
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