An improved ground structure method for large- scale truss topology optimization problems
Tomasz Grzegorz Sokół
AbstractThe ground structure methods became popular and important tools in truss topology optimization due to their robustness and reliability. Contrary to other methods the positions of nodes of the ground structure is frozen which requires a relatively dense cloud of nodes with a huge number of possible connections, i.e. potential bars. This inevitably leads to a large-scale optimization problem which is hard to solve in a direct way, however, to overcome this drawback the adaptive ground structure methods can be applied. The main advantage of the ground structure methods results from the fact that the topology optimization problem can be written in the form of linear programming, which from definition is convex and free of local minima, thus enabling to find the globally optimal topology. It is well known that topologically optimal discretized trusses tend to Michell structures with infinite number of infinitesimal bars. In spite of some inherent limitations the theory of Michell structures plays an important role in structural topology optimization, by enabling the derivation of exact analytical solutions for the least-weight trusses capable of transmitting the applied loads to the given supports within limits on stresses in tension and compression. Thus the exact solutions derived by means of this theory may serve as valuable benchmarks for any structural topology optimization method. In general, the exact analytical solutions are very hard to obtain since they require in advance a good prediction of the optimal layouts. Fortunately, the Michell structures can effectively be approximated numerically using trusses of large but finite number of bars. In this paper a new method of solving large-scale linear programming problems related to Michell trusses is proposed. The method is an extension of the adaptive ground structure methods developed recently by the author. In the present version both bars and nodes can be switched between active and inactive states in subsequent iterations allowing significant reduction of the problem size. Thus, the numerical results can be attained for denser ground structures giving better approximation of exact solutions to be found. The proposed method makes use of both primal and dual formulation of truss topology optimization problem and can be regarded as a specific combination of the interior point and active set methods. Both methods combined together provide an unprecedented opportunity for solving huge optimization problems with the number of design variables of billions or more. The proposed method enabled to obtain new important solutions which extend the class of known Michell trusses to 3D space and multiple load conditions. The new results clearly indicates that the optimal 3D trusses form shell-like structures composed of lattice surfaces
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