The aim of this article is to initiate an exchange of ideas on symmetry and non-uniqueness in topology optimization. These concepts are discussed in the context of 2D trusses and grillages, but could be extended to other structures and design constraints, including 3D problems and numerical solutions. The treatment of the subject is pitched at the background of engineering researchers, and principles of mechanics are given preference to those of pure mathematics. The author hopes to provide some new insights into fundamental properties of exact optimal topologies. Combining elements of the optimal layout theory (of Prager and the author) with those of linear programming, it is concluded that for the considered problems the optimal topology is in general unique and symmetric if the loads, domain boundaries and supports are symmetric. However, in some special cases the number of optimal solutions may be infinite, and some of these may be non-symmetric. The deeper reasons for the above findings are explained in the light of the above layout theory.

On symmetry and non-uniqueness in exact topology optimization

This paper attempts to synthesize the works carried out in the field of optimization in civil engineering structural design and is intended to help practising engineers and researchers. The paper is broadly classified according to the various types of structural systems and suitable grouping is done within this framework. Promising optimization methods, approximation concepts and structural design strategies are identified in the final part of the paper. The references are arranged in B chronological fashion.

A survey on optimal design of civil engineering structural systems

A new approach to solving a large class of factorable nonlinear programming (NLP) problems to global optimality is presented in this paper. Unlike the traditional strategy of partitioning the decision-variable space employed in many branch-and-bound methods, the proposed approach approximates the NLP problem by a reverse-convex programming (RCP) problem to a controlled precision, with the latter then solved by an enumerative search. To establish the theoretical guarantees of the method, the notion of "RCP regularity" is introduced and it is proven that enumeration is guaranteed to yield a global optimum when the RCP problem is regular. An extended RCP algorithmic framework is then presented and its performance is examined for a small set of test problems.

Extended reverse-convex programming: an approximate enumeration approach to global optimization

Reverse convex programming (RCP) represents an important class of global optimization problems consisting of concave cost and inequality constraint functions. While useful in many practical scenarios due to the frequent appearance of concave models, a more powerful, though somewhat abstractly recognized, characteristic of the RCP problem is its ability to approximate a very general class of nonconvex nonlinear programming (NLP) problems to arbitrary precision. The goal of the present work is to make this abstract idea concrete by formalizing an extended RCP framework with a nearly algorithmic procedure to approximate the general NLP problem by an RCP one. Furthermore, an active-set RCP algorithm, which may be seen as an improved and modernized version of Ueing's method, is proposed and described in detail. Some preliminary results are presented for several NLP problems to demonstrate the potential of the proposed framework together with its shortcomings.

G. I. N. Rozvany

Papers

Concave programming and piece-wise linear programming