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1 Evolutionary Structural Optimisation Lectures notes modified from Alicia Kim, University of Bath, UK and Mike Xie RMIT Australia

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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.

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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

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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

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Numerical Examples 13

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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

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Group ESO Group a set of finite elements Modification is applied to the entire set Applicable to configuration optimisation 16

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Example: Aircraft Spoiler 17

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Example: Optimum Spoiler Configuration 18