Appl. Sci. 2022 , 12 , 1684
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Figure 4. Application of the model to interpret measured data. A least-square fit is used to find the effective material parameters and displacement fields that provide the best explanation for the observed data. Note that this diagram only displays the principal data flow between separate logical modules needed to apply the model to measured data. The optimization algorithm (“Optimizer”) implements the main control flow (including, e.g., termination conditions when the desired level of precision is needed), utilizing the bending model to evaluate its cost function. To achieve this, the model (16), denoted as “Bending model” in Figure 4, is used to pre- dict the deformed surface using a mesh-free Ritz–Galerkin approach (the implementation used here follows largely the simple textbook approach as explained, for example, in [23] and would most likely provide room for improvements). It should be noted that, although this step is only shown as an arrow in Figure 4, it requires an iterative algorithm to calculate the parameters defining the surface (“Functional surface description” in Figure 4). This prediction step is seen as a function → F mapping the parameters mentioned previously to a vector of the expected surface heights at given sampling points. The final step of this mapping (“Virtual sampling” to generate “Predicted heights at sample points” in Figure 4) is a simple evaluation of the functional surface representation defined by the parameters at the known sampling points. The displacement field is numerically represented by the vector of coefficients → q of a polynomial approximation. The surface load consists of the weight of the board and the reaction forces at the points where the corrugated board is in contact with the underground. The board weight is assumed to be equally distributed in the xy -plane, neglecting the fact that sloped areas contain more board per projected area. This approximation greatly simplifies the problem and is consistent with the small deformation assumption needed for the linearity of Kirchhoff theory. The contact forces are determined at discrete points distributed in a regular grid across the board. An elastic contact model is used, i.e., the contact forces compress the board by a small amount, following Hooke’s Law. However, the contact forces result in nonlinear behavior, since they cannot become negative but stay equal to zero once the board lifts of the ground at a specific point (the contact force is proportional to − w · H ( − w ) where H ( · ) denotes the nonlinear Heaviside step function and w the displacement). This means that the model must be solved as a non-linear system. In practice, this function → F (which, as defined above, maps the effective proper- ties of the board to the surface form) has no analytic representation, and evaluating it represents a numerical workload noticeable on a typical desktop computer. However,
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