Study on Topology Optimization With Stress Constraints

Abstract The paper studies the relation between topology optimization and size optimization of truss structures. The goal of the optimization is to minimize the structural weight under stress constraints and side constraints on member cross sectional areas. The limiting stress concept is defined and the computational formula of limiting stress for truss structures calculated by a finite element method is given. Based on the limiting stress concept the continuity of the stress function at zero cross sectional area is carefully examined which enables us to understand the dilemma of defining the stress function in the closed interval up to zero cross sectional area. By considering the relation between the limiting stress and allowable stress the feasibility of stress constraints is discussed and the difficulty of adding a new bar to the truss or deleting an existing one is better understood. We have also shown that for topology optimization of truss structures the feasible design domain in the design space is a connected domain with possible degenerate subregions as long as upper bounds on cross sectional areas are large enough. To overcome this difficulty we introduce a quality function and replace the stress constraint by a new constraint which has the same feasibility and is continuous in the closed interval up to zero cross sectional area. In this way the formulations of structural topology optimization and size optimization are unified and an optimization algorithm which works only for problems with continuous constraint functions can be applied to optimize the topology of the truss structures. Simple examples are presented to show the possibility of automatically and rationally removing or adding bars and hence treating topological optimization in the same way as sizing optimization. Finally, further possible research on these topics is addressed.