PSI - Issue 79

João Alves et al. / Procedia Structural Integrity 79 (2026) 326–334

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overcome this new barrier, Murakami developed a new equation inspired by the Paris law, which is represented by equation 4 (Murakami et al., 2021; Murakami & Endo, 2023). =c( −1) (4) Where da/dN is the crack growth rate; C, m, and n are constant universal values (C = 10 − 4 , m = 2, n = 1)(Murakami & Endo, 2023), which, according to the authors (Murakami & Endo, 2023), only depend on the defect characteristics. In equation 4 , σ is the applied stress amplitude; σ w is the fatigue limit, which decreases cycle by cycle with crack growth stress; and, a, is the crack size in terms of √ defect in μm. However, according to (Morgado et al., 2025) the proposed universal constants showed that some modifications were needed to adapt to the use of nanotomography as a defect acquisition technique, and in a recent study developed by (Alves et al., 2025) a new value for C and m was proposed for a Ti-10Ta alloy. Yet, to obtain a value of C and m , that fits into the experimental data, it is necessary the use of optimisation algorithms, capable of finding a solution to an intended input. One of these algorithms, which is worth to mention, is Nelder-Mead, that is widely used for unconstrained optimisation, without needing gradient calculation, and it works by adjusting a simplex, that by each iteration moves closer to the function’s minimum . Basically, this algorithm uses n+1 vertices in a n-dimensional space, to explore the cost function and find the points that minimize it. Being composed by the following steps (Gao & Han, 2012; Lagarias et al., 2012): a. Initialization: It starts with an initial guess, and the rest of the points (n) are created by perturbing the initial guess. This combination of points leads to the appearance of a centroid (c), represented by equation 5, which works as a reference point, and results from the average of all vertices of the simplex. = 1 ×∑ = 1 (5) b. Reflection: Starts with the identification of the vertex with the highest value of the cost function. After identifying the worst value ( ), the point is reflected in the opposite direction using equation 6. If the reflected Point ( ) reduces the error, in comparison with the second worst point ( ) , (f( )< f ( )) , it is accepted, and the worst point is replaced by a new point ( +1 ). is the step length. = + ( + ) (6) c. Expansion: In the case that the reflected point is better than the best Point ( (f( )< f ( 1 )) ), the algorithm attempts to expand by further moving along the reflection direction based on the equation 7. If the expansion point ( ), further reduces the loss ( (f( )< f ( )) ), it replaces the reflected point ( ). is is the step length. = + ( − ) (7) d. Contraction: However, in the situation where the second worst point ( ) performs better than the reflected point ( ) ( (f( )< f ( )) . The algorithm attempts to contract the simplex to search closer to the centroid, leading to an outsider contraction, or in the case that the reflected point performed worse than the worst point ( (f( )< f ( )) , the simplex contracts directly to the centroid, both represented respectively by equation 8. is is the step length. { = + ( − ) = + ( − ) (8) e. Shrinkage: If neither of the phases improve the reduction of the loss function, all the points are moved closer to the best point ( 1 ) , through the equation 9. is the step length.