Issue 78

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

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. K EYWORDS . Nonlinear optimization, Backtracking line search, Material model calibration, Relaxed Newton algorithm, Finite element method.

I NTRODUCTION

O

ptimization of scalar multivariable functions plays a crucial role in applied mathematics, mechanical design, and computational analysis, where the goal is to determine the set of design variables that minimize a given objective function, whether analytical, purely numerical or a combination of both. Typical applications include material model parameter identification, inverse analyses, topology and shape optimization, and structural design under mechanical constraints. The increasing complexity of numerical models, especially those based on finite element (FE) simulations, has increased the demand for optimization algorithms to provide accurate solutions while maintaining acceptable computational costs. Depending on the characteristics of the function and the availability of derivative information, different classes of optimization algorithms are usually categorized into zero-order, first-order, and second-order methods [7]. Zero-order methods rely exclusively on function evaluations, and no derivative information is required; although reliable, these approaches become generally inefficient for high-dimensional problems because they need exploring large search spaces [7,14,18]. First-order methods exploit gradient information but may exhibit slow or unstable convergence for complex nonlinear functions [1,4,7,9,10,13–18,25,27]. Second-order methods, such as the Newton and modified Newton algorithms, combine both gradient and Hessian information and therefore exhibit faster local convergence [7,14,18]. Nevertheless, the computational costs, associated with evaluations of second derivatives, increase nonlinearly with variables number, limiting their applicability to challenging large-scale problems [7,10,11,14,18]. In many engineering applications, optimization problems are often characterized by complex response surfaces, numerical noise, and strong coupling among design variables, mainly when objective functions are obtained from numerical simulations rather than closed-form analytical expressions [7,11,14,18]. Under these conditions, conventional optimization schemes may exhibit poor robustness, sensitivity to initial guesses, or premature stagnation. Since the standard Newton method with unitary step size may fail to converge for non-quadratic objective functions, a backtracking procedure based on Wolfe conditions, namely the Armijo rule and the curvature condition [19], is introduced to overcome this issue. The first Wolfe condition guarantees a sufficient decrease of the objective function, while the second ensures that the search direction remains descent. Such a conventional procedure generates admissible step lengths from a discrete geometric sequence, which may become excessively coarse for high-dimensional and strongly nonlinear objective functions. These limitations are particularly relevant in the calibration of complex constitutive material models, as the Gurson Tvergaard-Needleman (GTN) ductile damage model. The identification of GTN parameters is traditionally performed through inverse analyses, numerical fitting procedures, or semi-empirical approaches, which may be computationally expensive, problem-dependent, and sensitive to user-defined settings [1,2,5,6,8,12,20,26,28]. Within this context, the present work develops a relaxed Newton method [7,18] combined with a modified backtracking strategy (MBS) where an adaptive step size parameter is introduced to enhance stability and convergence efficiency. The satisfaction of both Wolfe conditions can become computationally demanding when dealing with complex nonlinear functions. To address this aspect, in the MBS only the Armijo rule was explicitly evaluated, while the curvature condition was indirectly controlled through a reduction factor. The MBS scheme aims to provide a consistent and systematic alternative capable of efficiently handling multivariable and highly nonlinear objective functions arising from material model parameter identification. The effectiveness and generality of the proposed framework are assessed through optimization problems characterized by different levels of nonlinearity, dimensionality, and functional complexity. These test cases included analytical and numerical objective functions exhibiting pronounced non-convexity, strong coupling among variables, and ill-conditioned Hessian matrices features that are known to be challenging for conventional gradient-based and Newton-type optimization schemes.

490