StructOpt - Lec01

4Evolutionary 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)

In fully stressed design of an isotropic material, we usually use the von Mises stress in order to resize the element. It can be viewed as a stand in to the elastic energy, except that it represents only the deviatoric part (shape change, but not volume change) of the energy.

Fully stress design will remove material that has low stresses and add material near regions with high stresses. So ESO can be viewed as a version of fully stressed design.42

KKT Conditions for Topology Optimisation

We start by writing the compliance minimization problem with a volume constraint and form the Langrangian function for the problem. We differentiate it with respect to the densities of the elements as the Karush-Kuhn-Tucker (KKT) condition dictates. Recall that the densities are really the volume fraction of the element, so that a value of 0.3 would mean that the element is only 30% solid.

The derivative of the compliance with respect to the density has been derived in a previous lecture. The derivative of the volume with respect to the density of a single element is the volume of the element.

Unlike the SIMP method, we assume that the elastic constants are proportional to the density, so that the elastic energy in the element is equal to the density times the elastic energy se in a solid element with the same displacements. That implies that the derivative of the compliance is the negative of the elastic energy in the element.

This finally leads to the final optimality condition that the elastic energy density in all the elements should be the same at the optimum.

23KKT Conditions (contd)Strain energy density should be constant throughout the design domainSimilar to fully-stressed design.Need to compute strain energy density Finite Element Analysis

The condition that the strain energy density should be constant throughout the domain is similar to fully stressed design. It requires us to calculate the strain energy density, and we typically do that with finite element analysis.35ESO AlgorithmDefine the maximum design domain, loads and boundary conditions.Define evolutionary rate, ER, e.g. ER = 0.01, and an intial rejection ratio, RR, e.g. RR=0.3.Discretise the design domain with a finite element mesh. Finite element analysis.Remove low stress elements,Increase the rejection ratio RR=RR+ERContinue removing material and increasing rejection ratio until a fully stressed design is achievedExamine the evolutionary history and select an optimum topology that satisfy all the design criteria.

This summary of ESO starts by definition of design domain loads and boundary conditions. This is the common step to any topology optimization algorithm. The second step is to define an evolutionary rate, ER, which controls how fast we remove elements. We also define the initial removal ratio that is the threshold for removing elements.

Next we discretize the domain and perform finite-element analysis that produces the von Mises stress in each element. We now remove all the elements that have stresses lower than the removal ratio times the maximum stress. We can iterate several times between steps 4 and 5 until we cannot remove any more elements, and then we go to step 6 increase the rejection ratio and go back to step 4.

We continue iterating on steps 4-6 until a fully stressed design is obtained. However, it is possible that by that time we reduced the volume beyond the volume requirement, or the stresses are too high, so we may need to go back to an intermediate design that satisfies better the constraints.5An example (An apple hanging on a tree?)Gravity

An object hanging in the airunder gravity loadingThe finite element mesh

This example is taken from a 2009 presentation by Mike Xie who considers it the design of an apple. It is some times referred to as a cherry instead.

We start with a square domain and identify the fixed region in the middle top that is the stem that the objects hangs from.6Stress distribution of a square apple.

The stress distribution in the square apple identifies the four corners as regions to remove material from.7Evolution of the object.

Evolution reduced to four steps.8Comparison of stress distributions.

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Obviously we have not reached uniform stress distribution yet.9Problems ESOWhat 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.m101011Chequerboard FormationNumerical instability due to discretisation.Closely linked to mesh dependency.

Piecewise linear displacement field vs. piecewise constant design update

In ESO as in SIMP you can run into checkerboard patterns, and it is possible that it is due to the displacement field varying more smoothly than the design field.We have discussed the filter method as a way to prevent the checkerboard pattern, but it is also suppressed by level set methods that are becoming popular now. These are described on the next slide.11Topology Optimisation using Level-Set FunctionDesign 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.12

Level set method define the topology by boundary functions called level set functions because the terminology comes from iso-level contours. So instead of removing individual elements we move the boundary depending on the sensitivity (e.g. stresses or strain energy) at the boundary. We move inwards if the sensitivities are low and move outwards if they are high.

We may introduce holes inside if there are regions of low sensitivity inside.12Numerical Examples13

Examples of topologies obtained with level set methods.13Thermoelastic problemsBoth temperature and mechanical loadings FE Heat Analysis to determine the temperature distributionThermoelastic FEA to determine stress distribution due to temperatureThen ESO using these stress values47772024Design DomainP UniformTemperaturePlate with clamped sides and central load15

Group ESOGroup a set of finite elementsModification is applied to the entire setApplicable to configuration optimisation16

Example: Aircraft Spoiler17

Example: Optimum Spoiler Configuration18

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(T = 3(C

(T = 7(C

(T = 5(C