Issue 78

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

Tab. 3, although with different calculation durations. For ψ equal to 0.9, 0.8 and 0.7, the computations required 149 minutes and 108 iterations, 162 minutes and 131 iterations, and 104 minutes and 83 iterations, respectively.

D ISCUSSION

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he results of numerical investigations confirm the adaptability and computationally efficiency of the proposed Relaxed Newton framework for different types of nonlinear optimization problems in computational mechanics. Although the three benchmark tests differ largely in terms of governing equations, number of variables, and constraint formulation, the algorithm consistently converged toward physically meaningful solutions with a limited number of iterations. For the Johnson–Cook model calibration, characterized by a limited number of parameters and a smooth objective function, the method accurately identified the reference parameters while ensuring stable convergence from non-optimal initial guesses. The calibration of the Gurson–Tvergaard–Needleman ductile damage model represented a significantly more challenging optimization problem due to the strong coupling among damage parameters, governing void nucleation, growth, and coalescence. The integration of the relaxed Newton algorithm with an internal penalty function allowed effective enforcement of equality and inequality constraints, enabling convergence toward physically consistent parameter sets even in high- dimensional design spaces. The framework was also successfully applied to a geometry-driven structural optimization problem involving mass minimization under stress constraints. In this case, the Lagrange multiplier method ensured accurate constraint satisfaction, while the developed algorithm efficiently managed the coupled update of geometric variables and multipliers. The outcomes confirmed that the methodology can be extended beyond material parameter identification to constrained structural optimization problems. To validate the effectiveness of proposed optimisation framework, a comparison was carried out for each test case with a second-order standard Newton scheme, using a fixed step size α equal to unity and without accounting for Wolfe conditions, the NLS, and the LM algorithm. For highly non-linear identification problems, the conventional Newton scheme showed rapid divergence, because of the absence of a suitable backtracking technique, whilst NLS and LM approaches resulted in higher computation cost and minimum reduction of the objective function. Conversely, the shape optimization problem involved a smoother objective function and a comparatively less robust constrained-to-unconstrained transition strategy. For this test, convergence was achieved using the standard Newton method, but the obtained solutions exhibited poorer constraint satisfaction and reduced optimization efficiency. Instead, the NLS and LM algorithms resulted in no significant reduction in the objective function, despite the lower computational cost of NLS and the higher computational burden of LM involved. The observed improvement in convergence efficiency of MBS can be attributed to the more accurate approximation of the locally optimal step length, reducing subjective intervention and improving reproducibility compared to manual or trial-and-error procedures. second-order optimization framework based on a Relaxed Newton Method was developed for nonlinear optimization problems in computational mechanics. The Modified Newton scheme combined with a customized backtracking strategy effectively controlled the step size and prevented divergence. The algorithm was first validated through the calibration of the Johnson–Cook plasticity model, showing accurate parameters identification and stable numerical behaviour. It was then applied to parameter identification of the Gurson–Tvergaard–Needleman ductile damage model, where constrained optimization and an internal penalty formulation ensured physical consistency and reliable calibration. Finally, the algorithm was extended to a mass optimization problem, demonstrating its applicability to geometry driven structural optimization under equality constraints. The numerical results showed that the developed optimization framework is a promising tool for solving a wide range of engineering optimization applications, in which standard first order or heuristic methods may suffer from poor convergence or lack of physical consistency. Although the computational costs associated with second-order methods and repeated finite element analyses remain a limiting factor, the improvements in stability, accuracy, and reproducibility make this approach suitable for complex material modelling and structural optimization tasks. Future developments may include the extension to large- scale problems, integration with explicit finite element formulations, and application to multi-physics or multi- objective optimization scenarios. A C ONCLUSIONS

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