**2. Contributions**

The finite element method plays a key role in the numerical simulation of fatigue and fracture. Alshoaibi and Fageehi [1] developed an adaptive FEM-based approach for simulating the crack advance and the fatigue life in rectangular cross-section plates subjected to tension and bending. The numerical models considered a spider web mesh centred at the crack tip, and the stress intensity factors were computed using displacement extrapolation methods. Li and Xie [2], also basing their research on the finite element method, developed an algorithm to optimize the tooth surface contact stress in spur gears, considering tooth profile deviations, meshing errors, and lead crowning modifications. The problem was addressed using a three-dimensional model of one of the engaged teeth, by combining an optimized area of high refinement level with a non-refined area connected via multi-point constraint.

The analysis of critical engineering components by combining the finite element method with advanced fatigue methods was another line of research. Concli et al. [3] developed a critical-plane approach in conjunction with three-dimensional FEM models to study the early crack propagation stage in teeth subjected to bending fatigue. The models, created from extruded meshes and capable of accounting for boundary effects, allowed the determination of crack direction and the critical region associated with the crack nucleation. Sánchez et al. [4] determined the load-bearing capacity of tubular beams made of aluminium by applying the theory of critical distances and linear-elastic twodimensional finite element models. The proposed methodology has been successfully validated for cantilever beams with circumferential U-shaped notches, leading to errors in the predicted load-bearing capacity lower than ±20%.

Computational tools can play an important role in optimization problems. The paper by Khan et al. [5] presents a simulation-based optimization methodology for mold design and the prediction of reliability in mechanical components with minimum level of casting

**Citation:** Branco, R.; Berto, F.; Wu, S. Computational Methods for Fatigue and Fracture. *Metals* **2022**, *12*, 739. https://doi.org/10.3390/ met12050739

Received: 11 April 2022 Accepted: 24 April 2022 Published: 27 April 2022

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defects. The reliability was computed using classical strength–stress models and probability distribution functions. Woo et al. [6] proposed a parametric accelerated life testing approach to improve the fatigue life of mechanical components subjected to impact loading. The concept was tested in a domestic refrigerator hinge kit system, leading to a new design and an extended fatigue life. These promising results make the proposed parametric accelerated life testing approach applicable to metallic parts of other machines, namely cams, gears, crankshafts, and dies, to mention just a few.

In the context of fatigue crack propagation, the extended finite element method (XFEM) allows the alleviation of the shortcomings of the finite element method, namely with regard to the modelling of cracks and material interfaces. These advantages were explored by Fageehi [7], who studied the fatigue crack growth under mixed-mode loading in modified four-point bending beams and cracked plates with three holes. The fatigue assessment was conducted using fracture mechanics, stress-life methods, and strain-life methods. The proposed methodology was capable of simulating the crack paths, calculating the mixedmode stress intensity factors at the crack front, and estimating the fatigue life for different geometrical configurations.
