Volume Optimization of Solid Waste Landfill Using Voronoi Diagram Geometry

Abstract The loose material, falling freely on the flat area, creates a geometric object similar to the roof. This object, which is in fact an embankment, the authors described and examined in previous works. They stated that the top view of such an object is a flat diagram of Voronoi. In view of the need to enlarge the landfill by combining the existing disjoint embankments, the authors were encouraged to apply the results obtained in practice. They also provided a detailed methodology for creating such embankments. In particular, they used the standard version of AutoCAD software and built a 3D model of the landfill based on solid operations. This parametric approach to design enables the rapid determination of the volume and surface area and fits in the BIM technology of designing engineering objects.


Introduction
The roof geometry and Voronoi diagrams found applications in many elds ranging from photogrammetry with (semi) automatic reconstruction of city models (Bernd et al.;, Dikaiakou et al., 2003Flamanc et al., 2003;Laycock and Day, 2003;Swift et al., 2003), reconstruction of roads from satellite images (Akel, 2005; Thom, 2005), in cartography (Haunert and Sester, 2004;Wang et al., 2011), in the analysis of the morphology of materials with grain structure (Akel, 2005), in medicine for the representation, reconstruction and visualization of human organs ( Barequet et al., 2003), in the topographic analysis of the area, computer graphics in the eld of object shape recognition and analysis (Lawlor, 2004), in transport of readymixed concrete (Koźniewski et al., 2017), and in designing of earthwork organization (Koźniewski, 2018).. The solution presented in this paper is a new example of applying the above-mentioned concepts in the practical eld of designing the shape of embankments.
A direct, practical reason for dealing with the problem solved at work was the need to enlarge the land ll without signi cantly increasing the area (base) occupied by the embankment.
The limited possibilities of expanding the waste disposal area, and more precisely lack of such possibilities, prompted the management of the Municipal Waste Utilization Plant to seek new solutions by reconstructing the geometry of the existing quarters in order to nd additional optimal volume reserves for the repository. The authors of the present paper were commissioned to perform this reconstruction. The scope of work included the construction (forming) of the optimal 3D geometric model of the waste storage facility within the existing quarters with the determination of its capacity (absorption), horizontal areas (i.e. area with a small minimum inclination of 2%), surface of access roads and slopes (with a high slope of 1:0.58), after the adoption (in the assumption): The study did not cover any other aspects of the construction of the quarters, such as, for example, slope stability analysis, leachate removal system, post-process waste disposal system, etc.

Geometry of roofs (straight skeletons) and Voronoi diagrams in application to create embankments
As a theoretical basis, the research results of Koźniewski (Koźniewski, 2004a(Koźniewski, , 2004b(Koźniewski, , 2016(Koźniewski, , 2018 regarding roof geometry and Koźniewski et al. (Koźniewski et al., 2013) concerning, among others, applications of Voronoi diagrams for embankment modeling were adopted. The source setting of roof geometry lies in the area of teaching classical descriptive geometry (Grochowski, 2018) and roof geometry (Koźniewski, 2016). The authors of this paper found a new look through the algorithmization and computer implementations of roof geometries (straight skeletons) in (Aichholzer et al., 1995;Aichholzer and Aurenhammer, 1996), and the Voronoi diagrams in (De Berg et al., 2000) and in countless publications not quoted here. The method described in (Koźniewski et al., 2013), repeated and presented in this paper in the form of drawings and photographs, allows to design and model embankments from homogeneous material of any basis. The solid obtained by means of this method has an optimal volume assuming a predetermined slope angle. In addition, the same inclination angle ensures optimum cross-section area (Koźniewski et al., 2017). A virtual example of such an object is shown in Figures 1, 2 and 3. Figure 4 shows a photograph of a laboratory model.

Construction of the model of solid waste land ll -case study
In the design of the model, the authors adopted the assumptions discussed in the introduction to this article.

3.1:
As base contour, the line from the geodetic eld map of the quarters in the study marked by "4A" (Karmolińska -Słotkowska, 2013) was assumed. This line was obtained from the connections of three separate (up to now) polygons. The volume of the original component eld 4A was calculated at 260,000 m . In the case of municipal waste, it equals about 226,000 Mg of waste for storage (Karmolińska -Słotkowska, 2013). In the designed model, it was assumed that the eld base would consist of three disjoint polygons that were in contact with one side and the way of exploitation assumed the formation of three blocks with di erent crown heights (Figures 5, 6). Since such a design signi cantly limited the maximum volume of this eld, the proposal was made to connect the base during exploitation, liquidate three solids in favor of one ( Figure 7) and calculate how much this change will increase the volume of this solid.

3.2:
Based on the contour of the line from the geodetic map and accepted in accordance with the contract, at the client's request, a fairly large gradient slope (1:0.58), the authors of this study (E. Koźniewski and M. Orłowski) constructed a 3D model (Figures 8, 9) solid waste land ll and they made the necessary calculations (Table 1) and drawings of pro le (vertical) sections, which horizontal lines were marked in Figures 10, 11. The volume of the designed storage site is about 655,461 m . This gives more than two and a half times the increase in storage capacity in rela- tion to the size of 260,000 m obtained in the study (Karmolińska -Słotkowska, 2013). The volume of the slope waste land ll, due to the assumed slope 1:0.58, and the theory of Voronoi diagrams (Koźniewski et al., 2013), taking into account the deliberations in paragraph 3.3 of this paper, is the maximum for a given base.  of sides of the polygon while maintaining the normality of the string of inscribed polygons. By the normal sequence of polygons inscribed in the curve, the authors adopted such a sequence that the limit of the length of the sides of polygons with the largest length is zero. In the case of our design, we rst inscribed the 44-gon in the base contour, next the 84-gon. The authors created models of embankments E and E , with inclination angle of 45 degrees, for 44-gon and 84-gon, respectively: vol (E44) = 424332. 8[m ], vol (E84) = 425436.9 [m ]. The quotient q (1)  described in percentage points was less than 0.3%. So the authors decided that adopting the 44-sided polygon was enough. This polygon was used to build the nal model of a solid waste land ll (Figures 8,9,12).

3.4:
The use of standard AutoCAD software, compatible with other CAD programs, allows further processing of geometric objects in other programs and inclusion in a larger BIM system. The obtained objects are then available in parametric form and can be dynamically updated, for example during the actual creation of a land ll, monitoring the capacity of the temporary embankment, forecasting free space, etc.

Conclusions
1: The adopted land ll model was based on a polygon inscribed in the contour, which allowed to use the structure induced by the Voronoi diagram for the polygon. In the general case, with a su ciently large number of sides of the polygon, the di erence in volume between the solid built above the straight skeleton and the Voronoi diagram is small. Then one can solve the geometric roof instead of the geometric embankment, which greatly simpli es the construction. 2: Due to the polygonal nature of the base, large concave angles (close to the at angle), the proposed land ll, it was possible to successfully use the roof structure instead of the Voronoi diagram. Then, the part of the dihedral and the part of the surface of the cone inscribed in this dihedral limited by the tangents form "well-tting" surfaces. 3: The presented solution has allowed to increase the volume of the repository 2.5 times. 4: The solution in the AutoCAD environment allows for use it in BIM technology.