4 KKT Conditions (cont’d) Strain energy density should be constant throughout the design domainThis 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 methodDesign 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 AlgorithmDefine 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 achievedExamine the evolutionary history and select an optimum topology that satisfy all the design criteria.
11 2.5D OptimisationReducing thickness relative to sensitivity values rather than removing/adding the whole thicknessLoad Case 1: 2N/mmRoller supportNon-Design DomainFixed supportLoad Case 2: 2N/mmMesh Size 14436, less the 268 elements removed from mouth of spanner
13 Thermoelastic problems Both temperature and mechanical loadingsFE Heat Analysis to determine the temperature distributionThermoelastic FEA to determine stress distribution due to temperatureThen ESO using these stress values47772024Design DomainUniformTemperatureP
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 boundarySmooth boundary is represented by level-set functionLevel-set function is good at merging boundaries and guarantees realistic structuresArtificially high sensitivities at nodes are reduced, and piecewise linear update numerically more stableManipulate G implicitly through f
22 Topology Optimisation using Level-Set Function Design update is achieved by moving the boundary points based on their sensitivitiesNormal velocity of the boundary points are proportional to the sensitivities (ESO concept)Move inwards to remove material if sensitivities are lowMove outwards to add material if sensitivities are highMove limit is usually imposed (within an element size) to ensure stability of algorithmHoles 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.