## 5 Cross-Section Model Based on Polygonal Discretization
On the other hand, a large or huge number of fibers leads to high computational costs, which until recently was a computation barrier for the practical use of fiber computational models to nonlinear analyses of 3D framed structures. The proposed model aims to give almost identical computational accuracy, solving all of the above computational problems of the classical fiber models. It uses a different approach for the cross-sectional discretization and for the concrete compressive-stress integration over the compressive zone. The discretization of the cross-section initially concerns its polygonization and subsequently concerns the use of the derived polygons for the purposes of the above-mentioned stress integration. The whole process is performed in the following two stages. * In the first stage, the cross-section geometric polygonization is performed separately for its cover and its core area. Practically, this means that each one of these two areas is suitably divided into trapezoids or triangles with their bases parallel to one centroidal axis, e.g., \(y\). This polygonization process uses the very fast algorithm of [PERSON] and Clapworth [21], which can handle any polygonal areas of random geometry, excluding any internal parts (such as holes in the case of RC cross-sections). An example of discretizing such random polygonal areas into trapezoids or triangles is shown in Figure 10, where each different color represents a separate area. * In the second stage of the process, control points are placed uniformly and with appropriate density on the perimeter and the inside of each trapezoid or triangle in order to create a grid of points to be used for Delaunay triangulation. Next, a Voronoi diagram is computed that defines the influence area of each control point. Figure 11 depicts the process of polygonising a random RC-cross-section shape, placing control points in it followed by Delaunay triangulation, and the final result of the Voronoi diagram of each separate polygonal area. Cyan and yellow colors refer to core and cover of the cross-section, respectively. Figure 10: Discretization example of random polygonal geometric areas. (**a**) Original area, (**b**) Discretized area. As will be described in the following sections, the Delaunay triangulation will be used for calculating the concrete-compressive force \(F_{c}\), due to the distribution of the concrete-compressive stresses over the cross-section compressive area, while the Voronoi diagrams will be used for the computation of the cross-sectional stiffness. The above-mentioned inelastic constitutive stress-strain law is assigned to control points of a specific trapezoid of a material (concrete cover or confined core). Therefore, the inelastic response at each control point is calculated in each iteration of the main model algorithm, according to its constitutive law. However, in order to calculate the stiffness of the cross-section, it is also necessary to know the influence area of each control point, \(cp\). This influence area, \(A_{lp,i}\), is obtained from the calculated Voronoi diagram of all the points of each trapezoid (Figures 11e,f and 12). Following the classic definition of the Voronoi diagram (Figure 12), it should be recalled that for a given plain area with a given set of scattered points in it, which are considered control points, the diagram is essentially a discretization of the whole area into a set of individual influence areas, one for each control point of the set. A unique characteristic of a particular influence area is that all of the points contained within it are closer to its particular control point than to any other point of the given control-point set. Thus, each Voronoi area can be considered the influence area of its control point. Figure 11: Discretization steps of an RC cross-section: (**a**) cross-section, (**b**) polygonization (trapezoids and rectangles), (**c**) control points, (**d**) Delaunay triangulation, (**e**) Voronoi diagrams per polygonal area, and (**f**) final result.