PSI - Issue 28

Adrian Loghin et al. / Procedia Structural Integrity 28 (2020) 2304–2311 A. Loghin et al. / Structural Integrity Procedia 00 (2020) 000–000

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Fig. 5: { X, Y } DOE space used for the response surface calibration. Nominal location: { X, Y } = { 45.9, 10.4 }

technique is used to demonstrate the framework of setting up response surface models for both the crack path and the mode I SIF solutions ( K I ). Such surrogate models, once verified against the equivalent FEM (parent) model, can be used e ffi ciently in probabilistic applications to predict non-planar straight-through crack propagation paths and RUL within the prespecified problem bounds. The first step of the surrogate modeling is to define the solution domain in which the model can operate. Two geometric parameters are used in the surrogate model development: { X, Y } representing position of center of the hole relative to the notch. The nominal hole location is { 45.9, 10.4 } . These parameters are perturbed by ± 0 . 5mm, so that the sub-problem domain is determined by X ∈ [45 . 4 , 46 . 4] and Y ∈ [9 . 9 , 10 . 9]. The next step is generating the parent model solutions within this sub-problem domain. In this case, the parent model is the 3D FE model described in Section 2.1. The problem domain is filled with finite number of points for which the model response is generated. The two sets of responses generated from this model are: (1) crack path in terms of crack tip { x, y } coordinates and (2) the K I levels at the crack tip for each incremental crack length. Latin Hypercube Sampling (LHS) Method is used to fill in the problem space. The LHS method is freely available in lhs library of R programming language (Carnell (2019)). This sampling method uses the S-optimality technique, which maximizes the mean distance between the design points so that the points are as dispersed as possible (Carnell (2019)). The design matrix generated by the LHS method is shown in Figure 5 and it contains two sets of LHS samples: (a) 27 design points for RBF interpolation model calibration shown as blue dots and (b) 10 additional design points for model verification plotted with maroon triangles. For each of these instances a 3D FEM fatigue crack growth simulation is performed using an updated specimen geometry, the same mesh setup throughout the entire process and same increment size to, again, minimize meshing influence in the predicted solutions. The RBF model used in this development is a part of the freely available Python SciPy Package (Virtanen, et al. (2020)). The package provides di ff erent kernel functions including Gaussian, inverse, linear and cubic. Once the RBF option is picked, SciPy application constructs an approximation function by linearly summing up the weighted RBF functions for each calibration point. The weights of these functions can be determined by representing the equation in a matrix form and solving for the unknown weights. Further details of such techniques are described by Wilna (2008). With the parent solutions generated for the 27 calibration design points, three di ff erent surrogate models were set up using cubic kernel RBF models. The first two models predict the crack tip { x, y } coordinates for the given hole center { X, Y } coordinate set, while the third surrogate model predicts the K I values as a function of the hole center { X, Y } coordinate set and the crack length. Model verification can be performed both qualitatively and quantitatively. The correlation plots of the verification for the crack tip X and Y coordinates in Figure 6 and the K I solutions in Figure 7 represent the qualitative verification of the surrogate model. As far as a quantitative measure, the R-score values of the models on the verification points came out greater than 0.9999 for all three models.