Optimal shape and stress control of geometrically nonlinear structures with exact gradient with respect to the actuation inputs

This paper presents an efficient and robust optimization methodology for stress and shape control of actuated geometrically nonlinear elastic structures, applied to 3D trusses. The actuation inputs, modeled as prescribed strains, serve as the optimization variables. The objective is to minimize total actuation while satisfying several constraints: (i) actuation bounds in each actuated element and (ii) target ranges for nodal displacements and element stresses. Optimizing large nonlinear structures is computationally intensive. While gradient-based methods typically converge faster than gradient-free ones, their main bottleneck lies in numerical gradient evaluation, requiring multiple time-consuming nonlinear structural analyses (finite differences) with inaccuracies that may slow down convergence. The novelty of the proposal is an implicit differentiation approach to quickly compute the exact gradient of the nonlinear finite element solution with respect to the actuation inputs. This is implemented within the structural solver and leverages the already factorized tangent stiffness matrix to make the gradient cost negligible. As a result, the number of structural analyses and overall optimization time are significantly reduced.