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variance, which can be used to influence the delicacy of the structure, for example (Fig. 2.3, pp. 112f.). Since topology optimisation starts with a continu ous body, its results can be verified using what is known as continuum mechanics, in which a stress state is represented by principal stress trajectories that indicate the directions of ten sile and compression stresses. Principal stress trajectories are always perpendicular to one another and therefore intersect at an angle of 90 degrees. If as a result of topology optimisa tion, the structure follows these principal stress trajectories, a primary structure subject to either tensile or compression stress is created in line with the principles of efficiency. During the process of topology optimisation, the nature of the design space changes, and the structural solution moves from an initially isotropic space to one that is concentrated along paths. The tension and compression load paths tend toward orthogonality. These paths are similar to, but not the same as, principle stress trajectories, as they are discrete and not a solution on a continuum. The comparison of a fixed, vertical member under lateral wind load and the stages in the P/2
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topology optimisation process show an opti mum match between the analytical and the opti misation results. These patterns were used, for instance, in the development of the superstruc ture of the office tower at 100 Mount Street in Sydney, Australia (Fig. 2.6). The primary objective of topology optimisation in the design process is to develop the most material-efficient structural solution possible. However, it is currently difficult for optimisation to be carried out for more than one stress state, though in reality a structure is exposed to many, sometimes conflicting, stresses. This results in conflicting optimisation objectives and evalua tion criteria, such as minimum deformation, maximum rigidity, ductility, natural frequencies and stability parameters. The topology optimi sation process generates a number of design alternatives, which provide vital data for the discussion about the relationship between form and structure that should ideally take place between architects and engineers. Discrete topology optimisation In contrast to continuous topology optimisation, discrete topology optimisation is based on dis crete members. The Michell truss provides a clear example of this type of analytical optimisation. The calculus of variations, a prerequisite for the mathematical solution of optimisation prob lems, was developed in the mid-18th century. It was later applied to structural questions, in particular in the work of Scottish physicist James Clerk Maxwell (1831–1879; see “Maxwell’s theorem of optimal load path”, p. 89f.) and Aus tralian mechanical engineer Anthony George Maldon Michell (1870 –1958). In his 1904 publica tion “The limits of economy of material in framestructures”, Michell developed a particularly interesting frame structure of minimal weight for given support and load conditions (single load) and a defined design space. The individual frame members follow the principal stress lines, resulting in a “balloon-shaped” support [3]. Michell structures can be derived from continu ous topology optimisation by applying a single load to a continuum with a defined external geom