New analytical benchmarks for topology optimization and their implications. Part I: bi-symmetric trusses with two point loads between supports

Abstract

Extending some results by Sokół and Lewiński (Struct Multidisc Optim 42:835-853, 2010), the optimal topology of bi-symmetric trusses with two symmetrically placed pointloads is determined analytically, and verified by highly accurate numerical calculations. It is rather remarkable that Michell's best-known classical solution had to wait over hundred years for its generalization from one to two point loads. Some implications of these solutions, including properties of so-called O-regions, are also discussed.

Topology optimization in structural mechanics

“…The problem similar to that of Fig. 1 with the feasible domain being the full plane has been recently solved in Sokół and Rozvany [54]. The problem related to Fig.…”

“…The problem related to Fig. 1 but with one roller has been solved in Sokół and Lewiński [55,56] and Sokół and Rozvany [54]. New Michell's structures for different permissible stresses in tension and compression with taking into account the cost of supports have been put forward recently by Rozvany and Sokół [57].…”

Exact least-volume trusses for two symmetric point loads and unequal permissible stresses in tension and compression

“…It is rather rare to have such comfortable situations as in Figs. 5 and 6 of Sokół and Rozvany (2012), for which we knew the exact solutions. In such cases we can adjust the rectangular domain very precisely and this is the reason for the high 'densities' of ground structures applied in the second column in Table 1 of the above paper.…”

“…The word 'density' was used unfortunately there because the size of the base cell was the same for all cases in Fig. 5 of Sokół and Rozvany (2012). From this point of view we should say that the density was constant, only the size of ground structure was adjusted (for computational efficiency).…”

“…5 or 6 of Sokół and Rozvany (2012) due to different rectangular domains. Of course, if the design domain is fixed and we apply different ground structures for it, the denser mesh and therefore greater number of potential bars usually gives a better result.…”