## 1 Introduction
[12], [PERSON] et al. [13], [PERSON] [14], [PERSON] and [PERSON] [15]. The stretching and bending of the metal sheet lead to a thinning of the sheet that is typically measured by means of gauges [16] or ultrasonic probes [17]. Moreover, the sheet thinning is expressed as a unique value that is the average thickness of the stretched sheet in comparison with that of the initial plane sheet. Instead, the estimation of the thickness distribution inside the stretched sheet is important to evaluate the quality of the stretched product and its strain uniformity [18; 19]. In fact, a uniform distribution of the thickness inside the stretched sheet indicates a better result of the forming process because the product shows a similar strength in each of its sections. It derives from the stress and the strain distribution in the deformed sheet [20]. Moreover, the thickness distribution may be considered as the manufacturing signature of the forming process, i.e., the imprint left by the process on the sheet. In the literature, the papers are focused on different methods to measure the thickness of sheet or thin films, such as the optical instruments [21], a micrometer [22], and so on. Moreover, in the literature, the finite element simulations of the forming process achieve a value of the stretched sheet that is calculated as the distance between two nodes of the used mesh. A distribution of the thickness inside the simulated stretched sheet may give more important information about the quality of the process and enrich the simulation model. This is the motivation of the present work. The aim of the present work was to develop a numerical model of a forming process in which to evaluate the thickness distribution of a sheet by means of a geometric algorithm, once it was stretched numerically through a finite element model using conventional displacement elements for plane strain, axisymmetric cases. For this reason, it is possible to consider only the inner and outer profiles of the stretched sheet that were discretized by a set of points; they were the nodes of the elements used to model the sheet. The geometric algorithm calculates the distance between each couple of faced points, perpendicular to the profiles. In this way, the geometric algorithm enables the calculation of the minimum value of the thickness in each section of the stretched sheet, while the traditional method that calculates the distance between two opposite nodes of the same element provides an overestimate. The algorithm may evaluate a great number of thicknesses along the stretched profile of the sheet. To validate the developed numerical model, some experimental tests were carried out. Some sheets were stretched, and the points of the obtained profiles were measured by a coordinate measuring machine (CMM). CMM is widely used to inspect mechanical parts by means of a contact probe that exercises 0.1 N pressure on the part. A knowledge-based system was developed by [PERSON] et al. [23] to automatically configure the touch probe used for the measurement. The coordinate measuring machine is a general-purpose device suitable to control free form surfaces, as demonstrated by [PERSON] et al. [24], even if it is needed to accurately choose the fixturing system, as discussed by [PERSON] et al. [25]. CMM was used in this study to acquire a set of points on the external and internal surfaces of each stretched sheet for different steps of the forming process. The geometric algorithm was applied to profiles of the stretched sheet obtained by the experimental approach too. A good agreement was found between numerical and experimental results. The geometric algorithm was developed in the MATLAB(r) software package. Once the sheet thickness trend has been obtained, which represents the manufacturing signature of the forming process inside a stretched sheet, it may be used inside a tolerance analysis approach to estimate the geometric deviations of assemblies constituted by sheets, such as those of automotive or aerospace applications.