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Topology optimisation deals with the optimum layout of material within a given design space. When performing structural optimisation, the general formulation is the minimisation of the objective function f(x)

minimise f(x) objective function

such that the constraints are fulfilled:

gj (x)≤ 0; j=1,mg Inequality constraints

hk=0; k=1,mh Equality constraints

xil ≤ xi ≤ xiu; i=1,n Side constraints, upper and lower bounds

More descriptively, the optimisation goal can be described for example as

 Develop a structure with minimum weight with given loadings and support conditions, with the constraint of the deflection not exceeding a given value (otherwise the optimisation algorithm would develop a structure with zero weight), or

 Develop a structure with minimum compliance (maximum stiffness) with given loadings and support conditions, with the constraint of only a given ratio of the design space, the so-called volume fraction, to be filled with material - otherwise the optimisation algorithm would fill up all of the design space with material.

The Homogenization Method is a gradient based optimisation method as an addendum to the Finite Element Method, with the basic idea of subdividing the design space into small domains (pixels or voxels). The initial process of dividing the design space into areas with material and areas without material (0-1 problem) is re-formulated into a continuum problem. The optimisation problem is then to find an optimal structure described by pixels with density 1 (full material) and pixels density 0(no material) plus intermediate values between 0 and 1(„porous“ material). The material layout generated by the optimisation algorithm is a design proposal, which can then be interpretated and futher developed by the designer.

Fig. 2 shows a high beam and an optimised structure under a point load (Ramm 1996). The basic idea is, the optimised structure shows a very homogeneous stress distribution and carries the same load with less material.

A MATLAB code for the application of the homogenization method to structural optimisation is presented in (Lochner, Schumacher 2014). The studies carried out in is contribution use the commercial software Altair OptiStruct, which also uses the homogenization method.

Fig. 2 Typical structure (left) and optimised structure (right) with homogenuous stress conditions