126 R. Callens et al. Table 13.6 Results of the optimization problem Simulation model Surrogate Full simulation XFEM 0, 004 0, 005 PFF −0, 023 0, 00075 Table 13.7 Original and updated material properties for XFEM and PFF with corresponding difference in the optimum Parameter XFEM PFF Difference − init. opt. init. opt. − Fracture toughness (J/mm2) 4, 94 19, 76 4, 94 18, 34 −7, 19% Young’s modulus (Mpa) 2890 1195, 48 2890 1298, 78 +8, 64% Length scale factor (mm) − − 0, 34 0, 26 − ˆe 2 tot (-) 0, 294 0, 004 0, 919 −0, 023 − (X), where subscripts ( s) and ( e) indicate respectively the simulated and experimental values. Weights α and β are defined such that the initial value at the beginning of the optimization is 1/2 for each of both terms in the summation. This is done to achieve an equal weighting for both contributions. Consequently, the total goal function variation can also be expected to range between 1 (initialization) and 0 (full convergence). When solving the minimization problem, a large number of simulations is required, resulting in a large computation time (see Tables 13.2 and 13.4). Therefore, in this paper, the minimization is done using surrogate models for the goal function as defined in equation 13.5. For the training of the surrogate model, a supervised learning approach is applied that actively selects the most informative additional training samples, as well as the most appropriate surrogate modelling strategies. In this case, batches of 8 samples are used per iteration. At each iteration, the active learning algorithm selects between several response surfaces with varying basis. New batches of samples are generated until convergence of one of the surrogate models is sufficient. Both resulting surrogate models for the XFEM and PFF goal functions are cubic surfaces, with residuals in all the sample points smaller than 1· 10−12. The minimization is then performed on the surrogate models, reaching convergence at small goal function values, as indicated in the first column of Table 13.6. These small values show that the models reach good correspondence with the experimental data. Still, it can be noted that for PFF, there is a negative value, caused by approximation errors of the surrogate in the optimum. In order to estimate this error, full simulations are performed at the final optimization step. These results, reported in the second column of Table 13.6, show that the goal functions remain small. It can be concluded that the surrogate models produced a useful optimum, that can be used as starting point for the validation in the next section. The initial and optimized material properties are summarized in Table 13.7. From this table, based on relative difference of less than 10%, it is concluded that the material properties agree well for both simulation methods. 13.5 Model Validation After the parametric model updating, the model validation is performed to assess the residual model form error in both modelling approaches. For the validation, two criteria are used: the force displacement curve and the full field DIC strains in the loading directioneyy. First the force displacement curves from both simulations are compared to the experiments. This is visualized in Fig. 13.5. The force displacement curve can be cut in two parts, before and after the maximal force. In the first part there is linear elastic behavior and in the second part the crack is propagating. When comparing the simulations with the experiments some conclusions are made. In the first part, the simulations are similar to the experiments, with a small model form uncertainty. The maximal force is almost the same, this due to the model updating sequence. In the second part, during the crack propagation, the experiments give higher forces then the simulations. In this part, the model form uncertainty is comparably large for both simulations. When comparing both simulation models, they have almost the same model form
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