Issue 78

A. Gallo et alii, Fracture and Structural Integrity, 78 (2026) 489-503; DOI: 10.3221/IGF-ESIS.78.28

algorithm to maintain a finer step resolution throughout the entire analysis or activated after a specified number of iterations by means of a conditional flag, ensuring it takes effect before the primary convergence tolerance is reached. In the following numerical examples, the first option was selected. After the achievement of convergence, it is necessary to verify whether the optimal solution is a global minimum rather than a local one. With this aim, the algorithm is restarted using different initial guesses. Consistent convergence to the same result of different runs with distinct starting points provides empirical evidence that the identified minimum is global. In this work, both initial guesses and numerical differentiation increment were varied to investigate the nature of the stationary point identified by the optimiser. Numerical derivatives were evaluated using forward finite difference formula. At each optimization iteration, the objective function was first evaluated at the current design point X k . The i -th gradient component was approximated as: ௜ ൫ ௞ ൯ൎ ൫ ௞ ൅ℎ ௜ ௜ ൯െ ൫ ௞ ൯ ℎ ௜ where ℎ ௜ is the variable-specific perturbation and ௜ is the i -th unit vector. The diagonal and mixed components of the Hessian matrix were respectively computed as: ଶ ௜ ଶ ൫ ௞ ൯ൎ ൫ ௞ ൅ 2 ℎ ௜ ௜ ൯െ 2 ൫ ௞ ൅ℎ ௜ ௜ ൯൅ ൫ ௞ ൯ ℎ ௜ ଶ and ଶ ௜ ௭ ൫ ௞ ൯ൎ ൫ ௞ ൅ℎ ௜ ௜ ൅ℎ ௭ ௭ ൯െ ൫ ௞ ൅ℎ ௜ ௜ ൯െ ൫ ௞ ൅ℎ ௭ ௭ ൯൅ ൫ ௞ ൯ ℎ ௜ ℎ ௭ Every function was evaluated through a dedicated FEM simulation using ANSYS APDL. The perturbation sizes were independently selected for each design variable to account for their different physical scales. The Hessian was assembled symmetrically by imposing H iz = H zi , and both the gradient and Hessian were recomputed at every accepted optimization point. The same optimization scheme was applied to the objective function in unconstrained problems and to the corresponding pseudo- objective or Lagrangian function for constrained problems. From a convergence standpoint, the introduction of the parameter does not alter the fundamental properties of the classical backtracking procedure. Indeed, since ∈ ሺ 0,1 ሿ , the resulting positive step size always satisfies 0 ൏ ൑ 1 , remaining within the admissible interval typically employed in line-search algorithms. Furthermore, because the Armijo sufficient decrease condition is explicitly enforced at every iteration, the monotonic reduction of the objective function and the convergence characteristics of traditional backtracking methods are preserved. This framework is particularly suitable for finite element optimization problems, where coarse step-size discretization may lead to premature stagnation or unnecessary objective function evaluations, reducing the number of iterations and the overall computational cost. The complete algorithm procedure is summarized in the flowchart reported in Fig. 1, where ∗ represents the step-size value at which the current iteration converges. Furthermore, a tolerance value of 10 -8 was implemented for step-size control. The numerical analyses, performed to test and validate the developed optimization algorithm, involve three representative case studies, each characterized by different class of material model and optimization objective. Specifically, the algorithm is applied to the calibration of Johnson-Cook (JC) constitutive plasticity model, the calibration of Gurson-Tvergaard Needleman ductile damage model, and a structural shape optimization problem. This selection allows for the assessment of the methodology across a broad range of optimization scenarios, from parameter identification in nonlinear constitutive laws to structural optimization involving distributed design variables. The calibration of constitutive and damage models is essential for ensuring the predictive capability of numerical simulations, especially when dealing with nonlinear material behaviour and failure mechanisms. Accurate parameter identification improves the agreement between experimental observations and numerical predictions, thereby enhancing the reliability of finite element (FE) analyses during both design and verification stages. In the present study, the JC plasticity model and the GTN damage model were calibrated using experimental reference data available in the literature for the magnesium alloy ZK60 [23,24]. This alloy has attracted considerable attention due to its high strength-to-weight ratio, favourable specific stiffness, and suitability for lightweight structural applications, especially in the aerospace, automotive, and transportation sectors. In this study, these literature results are assumed as benchmark data to evaluate the performance of the present optimization algorithm. Moreover, the calibration procedure is extended beyond the constitutive response to include ductile damage parameters through the GTN

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