Gradient-Based Weight Minimization of Nonlinear Truss Structures With Displacement, Stress, and Stability Constraints

Abstract

This paper presents an effective and robust computational method of gradient-based methodology for weight minimization of geometrically nonlinear structures, considering 3D trusses as exemplary case study. The optimization framework can accommodate multiple different constraints: (i) bounds on the cross-sectional area of each design element, (ii) prescribed ranges for displacements and stresses, and (iii) nonlinear stability for geometries such as arches and domes. For large structures, this results in numerous optimization variables and constraints, including the highly nonlinear (ii) and (iii). Such constraints are evaluated consistently and simultaneously by combining path-following finite element analysis and null vector method. Typically, the gradient of the nonlinear structural response is approximated numerically, which is computationally intensive and can introduce inaccuracies deteriorating the optimization process. In contrast, this work derives a fully analytical gradient evaluation for nonlinear deformation, stress, and stability constraints. This is implemented directly within the finite element solver, enhancing robustness and computational efficiency of the optimization. Validation examples range from simple benchmarks to large structures.