# Fine-Grained Boundary Conditions in Field-Based Routing
## 5 Advanced Fine-Grained Boundary Conditions
### Advanced Fine-Grained Boundary Conditions
#### 5.2.1 Outer Boundary Conditions
To accomplish this, we introduce the concept of virtual nodes, as depicted in Figure 2. Virtual nodes do not physically exist in the network but serve as a means to enable real outer boundary nodes to function as network boundaries, while also reflecting traffic load, similar to general mesh nodes. We explain how to arrange virtual nodes for implementing the Robin boundary conditions and discuss their effects. All mesh nodes inspect the deployment of their neighbors and calculate potential values by placing a few virtual nodes based on the inspection results. To illustrate, we consider the scenarios shown in Figure 2. In the first case, neighbors are present only in the second quadrant. Initially, a mesh node identifies itself as an outer boundary node based on the proposed outer boundary node definition. Subsequently, it arranges virtual nodes as if they were non-existent but virtually within its  
Figure 2: Proposed finer-grained outer boundary conditions with virtual nodes. No neighbors in (1) three; (2) two; (3) only one consecutive quadrants. transmission range. The highest fixed potential value is assigned to these virtual nodes. The outer boundary node then calculates potential values with the same considerations as general mesh nodes, leading to packet movement to the network interior direction from the outermost edges of the network, as indicated in Figure 2. The second and third cases follow the same principles as the first case, with the only distinction for the placement of virtual nodes, depending on the arrangement of neighbors. As depicted in the figure, virtual nodes are positioned along the boundary in the quadrants opposite to where actual mesh nodes are deployed. However, the conditions and methods illustrated in the figure for arranging virtual nodes along the boundary in the opposite quadrants are just one specific example, and they can be flexibly adjusted at the discretion of the network operator. In this paper, we assume a simple virtual node deployment rule, where two virtual nodes are positioned counterclockwise along the boundary, midway between the starting and ending quadrants of the contiguous quadrants where actual mesh nodes are not deployed. In more detail, the first virtual node is positioned by adding one-third of the angular difference, counterclockwise, between the starting angle of the starting quadrant and the ending angle of the ending quadrant of contiguous quadrants where actual nodes are not deployed, starting from the starting angle of the starting quadrant. The position of the second virtual node is determined by subtracting one-third of the angular difference, counterclockwise, between the starting angle of the starting quadrant and the ending angle of the ending quadrant. Lastly, in cases where there are no neighboring nodes in all four quadrants, there is no viable solution, regardless of whether virtual nodes are placed or not. Therefore, in such scenarios, the node is directly defined as an outer boundary node, and the highest potential value is assigned to it. This situation is not illustrated in the figure. The boundary conditions, involving virtual nodes, can be applied to various network topologies with irregular node deployment patterns in real-world environments, as well as simple grid-like topologies. By strategically placing virtual nodes within the transmission range of calculating mesh nodes, the proposed boundary conditions allow each mesh node to calculate potential values without any exceptions, making them highly suitable for real network environments. Note that the existing boundary conditions, which do not consider virtual nodes, could not support the calculation of potential values in all cases, especially in scenarios involving network holes. More detailed information, including the reason for this limitation of the inability to calculate potential values in certain areas, is discussed in the following section.