Improving Solutions to the Truss Topology Design Problem with Alternating Convex Optimization

In the design of a mechanical structure, we’re interested in finding a general layout of the structure’s supporting frame that best supports some anticipated design loads. Often, the design of such supporting frames are addressed in ad-hoc ways, such as by complicated domain-specific heuristics [TTG05, DG01]. Our project targets the problem of designing a truss–a collection of bars rigidly attached together that represents the layout of a mechanical structure–subject to constraints expressing limits on quantities like the total cost of materials. Examples of a truss include a construction crane, a railroad bridge, and the Eiffel tower. All other things being equal, it is desirable for a truss to be stiff in the sense that it does not deflect much under loads placed on the joints of the truss deemed typical for the truss’s operation. Because the truss’s deflection is not a scalar quantity, it is common in truss optimization problems to use the elastic stored energy of the truss deflected under a particular set of load forces as a proxy for the stiffness, as we do in this paper [Ric13]. Given a graph whose edges are bars in the truss and whose nodes correspond to fixed bar attachment points in physical space, one optimization problem that can reasonably be formed is to find bar cross-sectional areas so that under some set of loads the elastic stored energy is minimized. In the literature this class of problems is referred to as truss topology design (TTD), so called because it aims to optimize a truss with a certain basic graph connectivity (topology) [Fre04]. One shortcoming with solving the TTD problem is that in this method the locations of the nodes and thus those of the bars are fixed. Not only do these solutions look unnatural compared to structures used in practice they are also poorer structures under the metric of elastic stored energy when we compare them with arbtitrary trusses with similar topology. In this paper we describe a method using alternating convex programming to locally solve the TTD problem where in addition to solving for the bar cross-sectional areas, we solve for displacements to the original node positions. This way, our algorithm iteratively improves upon a original truss topology as an approximation to optimizing the mechanical structure.