Keywords: topology optimization, layout optimization, generalized shape optimization, discretized topologies, analytical benchmarks, industrial applications, algorithm verification, checkerboard control, singular topologies, SIMP method, level set method, ESO.

After outlining the reasons for the unusual popularity of this field, the two main branches of structural topology optimization are outlined. In layout optimization, the topology, geometry and member sizes of grid-like or low volume fraction structures, (such as trusses, grillages or shell-grids) or fibre-reinforced continua are optimized simultaneously. In generalized shape optimization, the topology and shape of internal boundaries and the shape of the external boundary of perforated or composite continua are optimally chosen.

In reviewing the early history of discretized structural topology optimization, the vital role of three Danish researchers (Niels Olhoff, Martin Bendsoe and Ole Sigmund) and two Chinese scientists (Gengdong Cheng and Ming Zhou) is pointed out. Then better known methods of numerical topology optimization (SIMP, homogenization, optimal microstructures, level set-topological derivatives, global methods, and 'sudden death' methods e.g. ESO) are briefly reviewed, and their advantages and disadvantages explained.

The early history of exact (analytically derived) optimal topologies is also examined, starting with the classical truss theory of Michell over a century ago. After half a century of inactivity in this field, Michell trusses were re-discovered by Cox and Hemp and the theory of optimal grillage topology ('optimal flexure fields') developed by the author's research group. This was followed by a general theory of optimal topologies (called 'layout theory') by Prager and the author in 1977. More recent extensions of truss topology optimization are also summarized.

In the next part of the review lecture, computational difficulties (checkerboard error, singular topologies and non-global optima) are discussed, and methods for quality control (algorithm verification) examined. In this context, the importance of exact analytical benchmarks is pointed out. Recent developments in multi-disciplinary (multi-physics) topology optimization are also briefly summarized, with particular attention to Ole Sigmund's pioneering contributions in acoustics, electro-magnetics and optics. Additional multi-disciplinary applications by others are also examined.

Finally, a number of practical applications in the car and aerospace industries, resulting in significant savings, are shown. At the end of the paper future perspectives of topology optimization are briefly examined.

Until recently, the layout of man-made objects was based on human intuition and past experience. This has changed with the emergence of structural topology optimization. However, rigorous methods for large systems would require much greater computational capabilities than those available at present. For this reason, heuristic methods are used extensively in industrial design. It is hoped that greatly increased computational capabilities and improved topology optimization methods will overcome these problems in the future.

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