On symmetry and non-uniqueness in exact topology optimization

Lévy was a rather versatile engineer, physicist, mathematician and inventor. During the French−Prussian war (1870−71), Lévy was also in charge of cannon manufacture for the French artillery. Later he was professor of theoretical and astro-mechanics.

This was actually the only technical book Prager kept in his apartment in Savognin, Switzerland, after he moved there from Brown University. In fact, he published only joint papers with the author during the last decade of his life (apart from a short note nominally co-authored by his son). Moreover, in a book on layout theory, authored by Save and Prager (1985), but in fact assembled from Prager’s notes by Save after Prager’s tragic death in 1980, the present author is the most cited researcher (22 publications on layout theory are cited).

Lewinski and Telega (2001) also point out that the modified formulation \(\inf [ {\int {| {\sigma_1}|+| {\sigma_2}|} dxdy}]\) for Michell trusses was used before Strang and Kohn (1983) in the author’s first book (Rozvany 1976, p. 48). The latter was actually meant to be only a small educational example for the layout theory, considering a type of plane stress problem. It is not quite valid for all Michell structures, because it does not cover non-orthogonal truss layouts, like the ones in Fig. 24 of this article. Much of the literature on truss topology optimization (e.g. Strang and Kohn 1983) is based on the assumption that the optimal solution consists of members along the lines of principal directions. On the other hand, in the layout theory (Rozvany 1976; Prager and Rozvany 1977a), one starts off with a ground structure (structural universe) having members in all possible directions, and proves that (due to the inequality in (2) herein) the members may in general only run in the two principal directions. Exceptions are S-regions, where all directions are equally principal. Some other cases of non-orthogonality in Michell structures are discussed elsewhere (Rozvany 1997).

The type of problems considered are defined in Section 2.10, which includes the requirement that they have some feasible solution(s).

A solution is feasible, if it satisfies all constraints. A solution is basic, if it satisfies the same number of equalities as the number of variables. Some of these can be original equalities (e.g. equilibrium conditions), and some of them inequalities satisfied as equalities (e.g. non-negativity constraints). For an example, see the Appendix. Not all basic solutions are feasible (Strang 1980).

The terms ‘design’ and ‘symmetric topology optimization problem’ were defined in Section 3, par. (a) and (h).

Skew-symmetric topology optimization problems are defined in Section 3, par (i).

The distinction between ‘layout’ and ‘design’ is explained is Section 3, par. (a).

A ‘worst case compliance constraint’ means that the compliance must not exceed a specified value for any of the load conditions. In that case, the load condition (worst case) with the highest compliance value determines the design.