Appl. Sci. 2022 , 12 , 1684
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defining this function allows calculation of the initially unknown parameter as solution to a minimization problem:
→ F E
1 , E 4 , G ∗ , → q − → m
2 2
argmin E 1 , E 4 , G ∗ , → q
(27)
This problem can be solved using a gradient descent algorithm, where the gradient can, if necessary, be approximated numerically using a simple difference quotient. This approach allows use of (16) directly, where parameters describing both the base materials and the detailed structure of the corrugation have been aggressively summarized to a few effective constants, without any need to further analyze the detailed structure of the board. This also includes the interpolation functions f k ( z ) as introduced in (6) and their integrals κ k as defined in (9): only the weighted sums of the displacements in the individual sheets are used; therefore, the mere existence of the interpolation functions or the coefficients κ k is sufficient for the numerical application. A detailed analysis of a specific corrugation structure could of course be used to reconstruct the interpolation functions in order to gain additional insight into inner effects within the corrugated board; this is, however, not the aim of this work. In fact, due to the normalizations and substitutions applied, the displacement fields calculated internally by the numerical implementation are represented in units of measurement which are implicitly defined by the properties of the analyzed corrugated board and would therefore be hard to analyze. 4. Comparison with Measured Surfaces Samples taken from the production process of a commercial corrugated board plant were measured with optical triangulation sensors. The shape of the surface was then approximated based on the proposed model, where the displacement functions were limited to quadratic polynomials in machine direction and cubic polynomials in cross- direction. Since the polynomial parameters describing the displacement functions need to be determined from the measurement data, the aim is to choose the lowest possible order without seriously reducing the quality of the fit. Needlessly increasing the number of degrees of freedom could lead to overfitting and increase the impact of measurement noise on the result; moreover, the number of measurement points required would be increased. Therefore, the entire model presented in this article is focused on reducing the number of parameters. Two examples (size of the boards 1884 × 770 mm and 1430 × 1040 mm, respectively) from ongoing research are presented in Figure 5. While boards were obtained from a commercial production plant, they would have been considered waste due to the amount of warp present. The surface defined by the model (displayed as continuous, colored surface in the plots) shows the same shape as the measured data points (black circles in the plots) on a larger scale, while smoothing local variations. The interpolation capability of the model is shown by the fact that only a 6 × 6 grid of measured data points (highlighted as red crosses in Figure 5) was used as input to the fitting algorithm, while the other data points were reserved to verify the results. Some slight systematic deviation from the data can be seen near the edges of the board, which are most likely caused by the numerical solver for the differential Equation (16). The histograms in Figure 5 show the deviation between the measured points and the surface defined by the proposed model, with most of the data points only deviating by a few millimeters (95% of the measured data points are within 2.6 mm and 2.9 mm, respectively, of the reconstructed surfaces in the examples) from the model surface. Since, as seen in the 3D plots, the model surface reproduces the large-scale deformation of the corrugated board—in other words, the warp—the remaining deviations can be seen as being part of the noise when analyzing or controlling warp.
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