# Analysis of Lightweight Structure Mesh Topology of Geodesic Domes
###### Abstract  
This paper presents two methods of shaping the mesh topology of lightweight structures as spherical domes. The two given methods of dividing the initial face of the polyhedra determine the obtained structures, which differ in the way of connecting the nodal points. These points were obtained by applying the algorithm for calculating spherical coordinates presented in the paper, which were then converted to the Cartesian system using transformation formulas. Two models of dome structures are presented, based on a 4608-hedron according to the first division method, and on a 4704-hedron, using the second proposed method with numerical analysis. Thus, the novelty of this paper is an implementation of the formulas and algorithms from geodesic domes based on the regular dodecahedron to the regular octahedron, which has not been presented so far. The choice of the shape of the structure has impacts on sustainable development, dictated by structural and visual considerations, leading to the design of a light structure with low consumption of construction material (steel), which can undoubtedly be helpful when making the final structure shape. In addition, according to this research, it can be concluded that using the first method to create a geodesic dome mesh is more straightforward, safer, and requires less design experience. geodesic dome; single-layer strut structure; numerical simulation; topology; spherical coordinates; sustainable development; Cartesian coordinates; mesh +
Footnote †: journal: Applied  
Domes  
[PERSON], [PERSON], [PERSON], T.  
spatial structural forms. The following sets of the process of shaping geodesic dome structures include, among other things, optimization of the obtained structures. Thanks to this process, we receive a covering in the form of a geodesic dome, highlighting the advantages of this type of spatial form of structure. This is primarily about economic considerations, determined by the use of a small amount of construction material, which in turn leads to an increasing interest in this type of construction solution among architects and builders. Many scientific studies are focusing on the optimization of geodesic domes. The use of optimization algorithms concerning the shaped dome structures makes it possible to obtain covers with optimal cross-sections, sizes, and shapes, which consequently reduces the weight of the analyzed structure. Gradient-based optimization methods were applied in the paper of [PERSON] and [PERSON] [26]. The optimal design of reticulated shell structures using metaheuristic algorithms was studied by [PERSON] [27; 28], [PERSON] and [PERSON] [29], and [PERSON] and [PERSON] [30]. The development of a metaheuristic algorithm was presented by [PERSON] et al. [31] as [PERSON]'s algorithm. [PERSON] et al. [32] also presented another approach known as the Jaya algorithm. Many other papers aimed at optimizing structures in the form of domains, including [PERSON] and [PERSON] [33], as well as [PERSON] et al. [34]. Creating the structural form of a geodesic dome involves dividing its initial regular polyhedron into smaller faces by approximating each of its edges. The polyhedron that was the source of [PERSON]'s patent [1] was the icosahedron, and it is on its basis (as well as on the dodecahedron) that most studies on spherical coverings in the form of geodesic domes are based. The feature distinguishing these two polyhedra from other Platonic solids is the higher symmetry and repeatability of elements that can be achieved compared to meshes based on other polyhedra. However, it should be emphasized here that the remaining polyhedra, including the regular octahedron, are also the basis for shaping the topology of geodesic dome grids. The aspect of sustainable development is becoming more important over the years. In this case, the structures must be designed with due care and knowledge. Therefore, lightweight structures [35; 36; 37; 38; 39; 40] perfectly fit into the idea of sustainable development.