424 A. Kuczkowiak et al. Fig. 42.4 Real mode shapes 0 10 20 30 40 50 60 10 0 0 0.5 0.5 1.5 1.5 nz = 894863 2.5 2.5 x 104 x 104 3 3 0 1 1 2 2 20 30 40 50 60 nz = 366 Fig. 42.5 Position of non-zero terms of matrice ŒA with different size of FE model (left: lowdofs, right: large dofs), without renumbering When dealing with very simple models, as it is the case for this study, heuristic approaches are sometimes faster and more reliable than optimization algorithms. Here, for each step of horizon of uncertainty, the cost function is assessed at each boundary of the domain of uncertainty. In this particular case, two parameters are supposed to be uncertain: the domain of uncertainty is simply a square and thus the cost function has to be assessed four times (one time at each corner). The nominal design is first analyzed in order to assess the robustness of the expansion to lack of knowledge in E1 and E5: at each horizon of uncertainty, the cost function is assessed four times, represented by the points and then the robustness curve is constructed by taking the maximum of the cost function (cf. Fig. 42.6). Then, an investigation is performed based on a prior design space, depicted in Fig. 42.7, in order to seek the most robust design, that is to say the one which maximizes the robustness function (Eq. (42.13)). In other words and concerning this particular case, the robust design minimizes the impact of lack of knowledge in the FE model (E1 and E5) on expansion errors. As above-mentioned, E2 and E4 are the model parameters we choose to calibrate: 24 designs are analyzed and the Fig. 42.8 expresses their respective robustness. As expected, some designs are more robust than other ones. The robust design, i.e., the curve in black with filled black circle, is obtained after taking the minimum of sc at each O˛ step. This curve does not match with a particular fixed design but rather indicates the minimum error expected when the uncertain parameters are
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