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 = 0C
T = 3C
T = 5C
T = 7C
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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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