Physics-informed optimizing discrete loss framework for nonlinear computational mechanics using automatic differentiation

This work proposes a framework for optimizing a discrete loss for nonlinear partial differential equations (PDEs) in computational mechanics. The framework is applied to the static bending analysis of functionally graded graphene origami-enabled auxetic metamaterial beams on Winkler-Pasternak foundations in thermal environments. The framework incorporates three high-accuracy discretization methods, including differential quadrature, Taylor/Fornberg finite differences, and the B-spline collocation method. The resulting physics-informed discrete loss is minimized by first-order, Gauss-Newton, and Levenberg-Marquardt optimizers, with automatic differentiation enabling efficient computation of the loss gradient and the residual Jacobian. An adaptive random-perturbation restart strategy is introduced to improve the robustness of the first-order solver on nonconvex problems. The framework is verified on canonical benchmarks and primarily assessed on the nonlinear bending problem governed by strongly coupled nonlinear PDEs based on Timoshenko beam theory and von Kármán geometric nonlinearity. Systematic comparisons on the nonlinear bending problems show that the framework achieves accurate solutions, with the Levenberg-Marquardt optimizer providing the best efficiency among the optimizers within the framework. A parametric study quantifies the effects of foundation and graphene origami parameters on transverse deflection, providing guidance for the lightweight design of auxetic metamaterial structures.