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A b s t r a c tThe causes of singular structural topologies, which prevent most iterative computational algorithms from reaching the global optimal solution, are explained in the light of the theory of exact optimal layouts. This theory is also used for deriving eight fundamental characteristics of singular topologies. The above findings are illustrated with case studies of exact optimal layouts for a single load and for two load conditions with stress constraints. I n t r o d u c t i o nSingular topologies constitute one of the major computational obstacles in layout optimization. The aim of this note is to explain the causes and fundamental characteristics of this class of optimal layouts, with a view to developing later iterative algorithms for handling them.In topology optimization, we start off with a highly connected structure termed ground structure (Dorn et al. 1964) and eliminate all non-optimal members from the design. In geometrical optimization, the coordinates of the joints are optimized for a given topology. We can combine the two operations by choosing a ground structure consisting of an infinite number of members, which is then termed structural universe.This procedure gives the exact optimal topology and geometry for a structure and is termed exact layout optimization (e.g. Prager and Rozvany 1977). The corresponding structural configurations are termed exact optimal layouts, whilst those starting with a ground structure having a finite number of members are referred to as approximate or discretized optimal layouts.The theory of exact optimal layouts can be used for explaining basic characteristics of singular topologies. It is often claimed that exact layouts, usually consisting of a dense network of intersecting members, have no practical significance. In actual fact, exact solutions (e.g. Fig. la) can be used for constructing highly efficient discrete optimal layouts by connecting a finite number of nodes in the former by straight members (Fig. lb). More rigorous methods for optimal layouts with a finite number of members were developed by Rozvany and Prager (1976) and Prager (e.g. 1978). C a u s e s of s i n g u l a r t o p o l o g i e s a n d h i s t o r i c a l rem a r k sWhenever the feasible set is nonconvex, it is possible that an iterative solution algorithm finishes up at a local optimum. In the case of singular topologies, the feasible set for a problem with n design variables consists of an n-dimensional region, and some connected k-dimensional hyperplane segIn) discretized layout based on the former ments (with k < n). The fact that the global optimum is located in one of these very narrow "spikes" of the feasible set, makes it very difficult for an iterative procedure to reach it. The above properties of singular topologies were pointed out by Kirsch (1990) as well as Cheng and Jiang (1992) in the context of two-parameter problems.We illustrate singular topologies with an example of a three-bar truss (Fig. 2a) under one load condition (after Kitsch 1990, p. 135). If one variab...

Using optimality criteria for trusses with displacement constraints, a truss layout is optimized for a point load and a given displacement in different directions, with a view to demonstrating some unexpected features of non-self-adjoint problems.

In Part I of this study, earlier results are briefly reviewed and then general optimality criteria derived for exact leastweight plane truss layouts with combined stress and displacement constraints. Whilst these are necessary conditions for a local minimum with respect to any topology, Part II discusses analytical solutions within a given two-bar topology for a vertical support and a point load. The latter results are used for verifying the general theory in Part I.

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