Keywords: structural topology optimisation, Michell, optimality criteria, optimal layout theory, multiple load conditions, multiple constraints, probabilistic design.

Structural topology optimisation (STO) is a relatively new, but rapidly expanding and extremely popular field of structural mechanics. Various theoretical aspects, as well as a great variety of numerical methods and applications, are discussed extensively in international journals and at conferences. The high level of interest in this field arises from the substantial savings that can be achieved by topology optimisation in industrial applications. Moreover, STO has interesting theoretical implications for mathematics, mechanics, multi-physics and computer science.

Exact analytical solutions constitute important benchmarks in structural topology optimisation because numerical methods in this field have many sources of error.

The aim of this review paper is threefold. First, it gives a brief review of the early history of exact structural topology optimisation, including Michell's [1] classical paper, and the optimal layout theory by Prager and Rozvany [2], explaining the theoretical foundations of the latter. Secondly, it examines a recent controversy about the limited range of validity of Michell’s truss theory, and ways of modifying this with a view to making it valid for a broader class of boundary conditions. Finally, it discusses extensions of the optimal layout theory for a wide range of problems, including those with multiple load conditions and multiple constraints including probabilistic formulation. This includes the description of the types of optimal regions in least-weight truss solutions and a review of various new ways of overcoming certain difficulties in obtaining exact analytical solutions.

[1]

A.G.M. Michell, "The limits of economy of material in frame structures", Phil Mag, 8, 589-597, 1904. doi:10.1080/14786440409463229

[2]

W. Prager, G.I.N. Rozvany, "Optimal layout of grillages", J Struct Mech, 5, 1-18, 1977. doi:10.1080/03601217708907301

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