## 2 Material and Methods
### Nodal Connectivity Optimization Design Model  
The primary focus at findings of earlier available research data related to stochastic open or closed cells is the type of deformation for most lattices, or whether it would be stretch or bending-dominated under the loading. Still, despite numerous research findings and results, a wide specter of applied materials and designs caused stiffness and strength, in correlation with relative density, to fall far below the assigned assumptions. According to the findings of the reference [26] regarding a stochastic cellular structure, the topology of the low-connectivity cells causes the edges of the cells to bend when the structure is loaded. Even when the cells are closed, the deformation primarily bends constituents since the thin cell faces buckle easily. Figure 2 shows a typical open-cell stochastic lattice made from struts connected in joints or nodes. Previously, most significant studies have focused on connectivity phenomena [8; 15], defining the number of struts that meet in the node. Unlike them, if the focus moves from the variable nodal connectivity degree to the nodal formation definition, it creates space to decrease the number of nodal imperfections in lattice composition. By this, the first sub-hypothesis takes the significance of nodal tolerance definition in getting valid formation. It implies introducing a controlled mechanism for nodal formation based on tolerance value to ensure lattice correctness since no authors have dealt with lattice validity. This assumption returns the process to Voronoi tessellation and investigates whether all struts and nodes make the structure valid. If the loading primarily bends constituents, causing the thin cell faces to buckle easily, whole volume tessellation must be reviewed. Voronoi tessellation populates volume geometry, ensuring a series of random points inside the defined boundaries using two major parameters definition: the number of points to add and a random cell seed for insertion. The tessellation causes polygon's corners where topological formation retains sharpness, and as a response, it piles up a more significant number of nodes, making topology arrangement more complicated. This means that tessellation causes excess points concentrated at a relatively small curve distance, which is considered to have an irregular topology arrangement. Figure 3 shows the interior of a structure, where the blue arrow indicates a distance between two points, obtained by measuring distances between each point in the cluster that is less than allowed to form a regular node. Node formation based on this unallowed distance causes shape irregularity, leading to the appearance of notches and creating stress concentrations. Therefore, the proposed design model ensures an algorithmic routine to prevent the appearance of these irregularities, eliminate notches and, thus, errors in node formation, and improve not only a mechanical response but also the visual appeal of the lattice. Now, nodal connectivity is considered through the winged-edge data structure, using edges to control and keep track of overall topological formation. Edges are used to make line discontinuity and achieve point recognition. Upon the points (nodal) recognized, the  
Figure 2: A typical stochastic open cellular structure with shown constituent elements: nodes and struts. tolerance value for defining minimal nodal distance is introduced to prevent points from piling up. The goal is to cull inconvenient points for regular topology formation, resulting in retaining a suitable point using tolerance value (Figure 4). It is crucial to underline that tolerance value correlates with strut thickness and must be adjusted to be not much greater or lesser than the thickness value. Besides strut thickness, lattice volume impacts the tolerance value. The algorithm tracks the winged-edge structure and, based on the defined tolerance, seeks two or more points having a distance equal to or lesser than the tolerance value to merge them into a single point.