1. Evolutionary Structural Optimisation

2. KKT Conditions for Topology Optimisation

3. KKT Conditions (cont’d)

4. KKT Conditions (cont’d)
Strain energy density should be constant throughout the design domain
This condition is true if strain energy density is evenly distributed in a design.
Similar to fully-stressed design.
Need to compute strain energy density
 Finite Element Analysis

5. 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)
High computational cost.
Other design requirements can been incorporated by replacing von Mises stress with other design sensitivities – 0th order method.

6. ESO Algorithm
Define the maximum design domain, loads and boundary conditions.
Define evolutionary rate, ER, e.g. ER = 0.01.
Discretise the design domain by generating finite element mesh.
Finite element analysis.
Remove low stress elements,
Continue removing material until a fully stressed design is achieved
Examine the evolutionary history and select an optimum topology that satisfy all the design criteria.

7. Cherry
ESO solution
Fixed
Gravitational Load
Initial design domain

8. Michell Structure Solution

9. ESO: Michell Structure

10. ESO: Long Cantilevered Beam

11. 2.5D Optimisation
Reducing thickness relative to sensitivity values rather than removing/adding the whole thickness
Load Case 1: 2N/mm
Roller support
Non-Design Domain
Fixed support
Load Case 2: 2N/mm
Mesh Size 14436, less the 268 elements removed from mouth of spanner

12. Spanner

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
477
720 24
Design Domain
Uniform
Temperature P

14. Plate with clamped sides and central load

15. Group ESO Group a set of finite elements
Modification is applied to the entire set
Applicable to configuration optimisation

16. Example: Aircraft Spoiler

17. Example: Optimum Spoiler Configuration

18. Multiple Criteria
Using weighted average of sensitivities as removal/addition criteria P
Maximise first mode frequency & Minimise mean compliance
AR 1.5
Mesh 45 x 30

19. Optimum Solutions (70% volume)
wstiff:wfreq = 1.0:0.0
wstiff:wfreq = 0.7:0.3
wstiff:wfreq = 0.5:0.5
wstiff:wfreq = 0.3:0.7
wstiff:wfreq = 0.0:1.0

20. Chequerboard Formation
Numerical instability due to discretisation.
Closely linked to mesh dependency.
Piecewise linear displacement field vs. piecewise constant design update

21. Smooth boundary: Level-set function
Topology optimisation based on moving smooth boundary
Smooth boundary is represented by level-set function
Level-set function is good at merging boundaries and guarantees realistic structures
Artificially high sensitivities at nodes are reduced, and piecewise linear update  numerically more stable
Manipulate G implicitly through f

22. 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.

23. Numerical Examples