Topology optimization of structural systems based on a nonlinear beam finite element model

A method for the topology optimization of structures composed of nonlinear beam elements based on a hysteretic finite element modeling approach is presented. In the context of the optimal design of structures composed of truss or beam elements, studies reported in the literature have mostly considered linear elastic material behavior. However, certain applications require consideration of the nonlinear response of the structural system to the given external forces. Particular to the methodology suggested in this paper is a hysteretic beam finite element model in which the inelastic deformations are governed by principles of mechanics in conjunction with first-order nonlinear ordinary differential equations. The nonlinear ordinary differential equations determine the onset of inelastic deformations and the approximation of the signum function in the differential equation with the hyperbolic tangent function permits the derivation of analytical sensitivities. The objective of the optimization problem is to minimize the volume of the system such that a system-level displacement satisfies a specified constraint value. This design problem is analogous to that of seismic design where inelastic deformations are permitted, yet sufficient stiffness is required to limit the overall displacement of the system to a specified threshold. The utility of the method is demonstrated through numerical examples for the design of two structural frames and a comparison with the solution from the topology optimization assuming linear elastic material behavior. The comparisons show that the nonlinear design is either comprised of a lower volume for a given level of performance, or offers better performance for a given volume in comparison with optimized linear design.