Optimal design of bar structures based on virtual elements concept

Dariusz Bojczuk

Summary

In this presentation, at first we will discuss sensitivity derivatives associated with variation of topological parameters for bar structures. These parameters correspond to introduction of load carrying bars or supports that can be called virtual elements. As the sensitivity derivatives for the vanishing topological parameter corresponding to the unchanged structure provide an assessment of variation of the objective function (functional) and constraints, they can be used to formulation of conditions of structure modification by introduction of virtual elements. However, apart from the infinitesimal topology modifications also finite modifications can be taken into account.
In the case of trusses, the algorithms of simultaneous optimal design of topology, configuration and cross-sectional areas are considered. In the basic case, the problem of cost minimization with constraint imposed on global stiffness is analyzed. However, the presented approach is also successfully applied to the problem of the cost minimization with constraints imposed on stresses and on buckling loads. The main feature of this approach is application of the virtual bars, which can join existing nodes so far non-connected.
Here, two alternative methods of optimal design of trusses are presented. The first method corresponds to the optimization procedure composed of two mutually interacted stages. Initially, optimal topologies are determined taking into account virtual bars. In the next stage, configuration optimization is performed for the all optimal topologies and finally the best solution is chosen. In the second method, optimization is carried on simultaneously with respect to topological parameters, which represent forces in virtual bars and with respect to configuration parameters, which correspond to positions of nodes. Here, identically as in the first method, cross-sectional areas of bars are determined directly depending on bar forces. The considerations are illustrated by some numerical examples.
In the case of beam and frame structures, the problem of maximization of buckling load and the problem of maximization of natural vibration frequency under condition imposed on global cost is discussed. Cross-sectional areas of bar structures and number of supports, their positions and stiffnesses are selected as design parameters. Using conditions of topology modification by introduction of virtual supports, algorithms of optimization of bar structures with their supports are formulated and applied for analysis of some optimization problems. Illustrative examples confirm applicability of the proposed approach.