A second-order relaxed Newton framework for material parameters identification and stress-constrained structural optimization

This work presents the development and implementation of a second-order optimization algorithm based on a relaxed Newton method for the minimization of nonlinear scalar objective functions with multiple design variables. The proposed strategy combines a modified Newton scheme with a customized backtracking formulation, which explicitly enforces the Armijo sufficient decrease condition to improve convergence and stability in highly nonlinear problems and reduce computational cost. A refinement parameter is introduced to generalize the step-size sequence into a continuous form, providing a relaxation of the standard backtracking scheme. Unlike standard tuning parameters, it acts directly on the mathematical structure of the line-search procedure and remains independent of the specific physical problem, constitutive model, or objective function. The methodology is assessed through three case studies of increasing complexity, namely calibration of the Johnson–Cook plasticity model, calibration of the Gurson– Tvergaard–Needleman ductile damage model, and stress-constrained shape optimization of a hollow plate. The results demonstrate accurate parameter identification, robustness to initial guesses, and effective handling of both equality and inequality constraints through Lagrange multipliers and interior penalty functions. Comparisons with standard Newton and first-order methods, as well as Levenberg-Marquardt and Trust-Region algorithms, show better performance of the framework in terms of computational cost and number of iterations, providing an effective and computationally efficient optimization strategy for nonlinear finite element problems involving constitutive calibration and structural design.