PSI - Issue 72

504 Nenad Vidanović et al. / Procedia Structural Integrity 72 (2025) 499– 506 was R = 0. As a final step, mathematical techniques had to be employed to seek optimised values of C & which would meet the number of cycles and crack extension length as observed in the workshop, and then, based on provided values, residual life of the HPT case was calculated for both models. Determination of the initial set of input parameters C & , which had to be optimised, was based on the user defined approach in the design of experiments (DoE) submodule. All combinations of C & , for measured crack length (0.41), have to give solutions for (2 807). Sixteen different combinations from literature of C & values were simulated for Inconel 718 and only two sets nearly satisfied demands for crack length and number of cycles. For Model 1, initial set from the literature: = 2.2793 ∙ 10 − 12 √ and = 2.733 , was chosen. For Model 2, using central composite design (CCD) type, combination from the literature: = 2.5897 ∙ 10 − 13 √ and = 2.9 had additionally been refined within [8.00 ∙ 10 − 14 ; 7.00 ∙ 10 − 13 ] and [2.5; 3] domains in which initial set: = 5.00 ∙ 10 − 13 √ and = 2.662 , was chosen. Based on these two calculated initial sets, the search space had to be explore most efficiently and accurately enough within the predefined boundaries (Table 2): Table 2. Initial values and lower and upper bounds for Paris’ coefficien ts Model 1 Model 2 Parameter C [MPa√mm] M C [MPa√mm] m Initial values 2.2793 ∙ 10−12 2.733 5.0 ∙ 10−13 2.662 Lower bound 1.3 ∙ 10−12 2.5 4.0 ∙ 10−13 2.5 Upper bound 3.3 ∙ 10−12 3 6.0 ∙ 10−13 2.7 Using CCD, defined with rotatable design type and enhanced template type, the search space modelling was conducted. Generated numerical experiments were approximated with Kriging response surface type (Forrester et al. (2008)), and then – based on conducted meta-modelling – optimisation process was run driven by the MOGA method (Multi-Objective Genetic Algorithm), as a hybrid variant of NSGA-II (Non-dominated Sorted Genetic Algorithm) based on controlled elitism concept (Deb et al. (2002)). Optimisation goal for both models was defined as – Obtain observed number of cycles of service loading that corresponds to the predefined crack length :  Objective: Seek target N = 2 807; Objective Importance – higher (0.999),  w.r.t.: N = 2 807; Constraint Importance – higher (0.999); Constraint Handling – strict,  Objective: Seek target a = 0.41; Objective Importance – lower (0.333). Optimisation Model 1 converged after 92/1 000 iterations, while optimisation Model 2 converged after 105/1 000 iterations. 3. Results and discussion Paris’ coefficients provided by the proposed fr amework are given in the following table (Table 3): Table 3. Optimised C & values with corresponding a and N predicted and verified values Model 1 Model 2 Parameter C 2.8022 ∙ 10−12 5.3073 ∙ 10−13 Parameter m 2.6242 2.6692 a N a N Predicted 0.41004 2 807 0.41636 2 807 Verified 0.41397 2 815 0.41614 2 803 Discrepancies, |Δ| [%] 0.96 0.28 0.05 0.14 Along with C & values, Table 3 presents predicted and verified values for a and N , and calculated discrepancies, which suggest that search space is obviously well performed and approximated. As it can be seen, obtained values for exponent are similar for both models, and the difference between them is |Δ| = 1.71%. What is more, obta ined