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

Topology optimization is an important topic in structural mechanics. One common application is to obtain the optimal distribution of material that maximizes the stiffness of the solution (minimize the compliance). However, as an iterative process, topology optimization of large and complex structures is computationally intensive. The problem becomes even more complicated if the manufacturing constraints are taken into account in the optimization process. In this paper, a novel growth method based on principal stress lines (PSLs) is presented for topology optimization. The PSLs are traced in the design domain along the direction of principal stresses, in which the materials would be located to define the geometry and topology of the structure. Consequently, the optimization problem is converted into a geometric design problem. Compared to previous methods, the computation based on PSLs is fast, and the designer can have explicit control over the number of structural members. In addition, the manufacturing constraints can easily be incorporated. Multiple test cases are given to illustrate the presented method. The PSL-based method is promising for building practical designing tools for various structural applications.

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A structural topology design method based on principal stress line

The present paper is the first part of the four-part work on Michell cantilevers transmitting a given point load to a given segment of a straight-line support, the feasible domain being a part of the half-plane contained between two fixed half-lines. The axial stress σ in the optimal cantilevers is assumed to be bounded by −σ
 C
 ≤σ≤σ
 T
 , where σ
 C
 and σ
 T
 represent the allowable compressive and tensile stresses, respectively. The work provides generalization of the results of the article of Lewinski et al. (Int J Mech Sci 36:375–398, 1994a) to the case of σ
 T
 ≠σ
 C
 . The present, first part of the work concerns the analytical formation of the Hencky nets or the lines of fibres filling up the interior of the optimal cantilevers corresponding to an arbitrary position of the point of application of the given concentrated force.

Michell cantilevers constructed within trapezoidal domains—Part I: geometry of Hencky nets

Optimal design with thousands of variables is a great challenge in engineering calculations. In this paper beside the short history of optimality criteria methods, a solution technique is introduced for the topology optimization of elastic disks under single parametric static loading. Different boundary conditions and thousands of design variables are applied. Due to a simple mesh construction technique, the checker-board pattern is avoided. The Michell-type problem is investigated minimizing the weight of the structure subjected to a compliance condition. The numerical procedure is based on an iterative formula that is formed by the use of the first-order optimality condition of the Lagrangian function. The application is illustrated by numerical examples. The effect of the different loading conditions is studied for the Michell-type topologies as well.

New Type of Optimal Topologies by Iterative Method

In this paper, a systematic exploration of Michell layouts for various combinations of line supports is continued. The solutions discussed include those for reentrant corners, a line support and two free edges, line supports forming any convex polygon, supports consisting of two symmetric curves, and finally, rectangular domains bounded by one line support and three free edges.

Exact Michell layouts for various combinations of line supports-Part II

It is often stated, even in standard references, that in classical Michell trusses (i.e. least-weight trusses for one load condition with a stress or compliance constraint) a pair of intersecting compression and tensile bars must always be orthogonal. The aim of this brief note is to show that there are important exceptions to this rule and that the modification of this restriction enables us to obtain new classes of solutions.

G. I. N. Rozvany

Papers

Partial relaxation of the orthogonality requirement for classical Michell trusses