## 3 Applying the Model on Measured Surface Data  
The primary intended use for the proposed model is the post-processing of measured surface data. This allows both the interpolation of the surface and the separation of warp and other surface defects which are not caused by warp. The basic idea, as shown in Figure 4, is to fit the parameters \(\widehat{E}_{1/4},\ \widehat{G}^{*}\), the surface load \(p\), and the effective displacement field \((u,v)^{T}(x,y)\)--together called \"displacement and stiffness parameters\" in Figure 4--so that the surface geometry predicted by the model matches the surface height, measured at defined sample points and stored in order in a measurement vector \(\stackrel{{\rightarrow}}{{m}}\) (\"Measured data at sample points\"). While the location of those sample points in the \(xy\)-plane must be known, the method proposed here does not depend on a specific sample pattern (e.g., rectangular grid), although the location will influence the numerical precision. The outer loop (consisting of everything in Figure 4 outside of the grey area) follows the general principle of regression by quadratic error minimization, where an optimizer tries to find an input to an abstract function block (the grey area in Figure 4) in order to minimize the quadratic difference between the output of this function block and some external value (in this case, the measured data). To achieve this, the model (16), denoted as \"Bending model\" in Figure 4, is used to predict 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 \(\overset{\rightarrow}{F}\) 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 \(\overset{\rightarrow}{q}\) 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 [PERSON] 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 \(-w\cdot H(-w)\) where \(H(\cdot)\) 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 \(\overset{\rightarrow}{F}\) (which, as defined above, maps the effective properties 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,  
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.