# Fine-Grained Boundary Conditions in Field-Based Routing
## 4 Problem Statement  
In this section, we primarily discuss the limitations posed by existing boundary conditions in field-based routing protocols. This discussion provides a foundation for us to elaborate on the driving factors behind the development of our proposed approach. To provide a concise overview of the existing boundary conditions, consider a scenario where a packet is sent from a mesh node with a high potential value to a gateway with the lowest potential value through a few intermediate mesh nodes, as depicted in Figure 1. As is typical in related research [12; 14], the outermost nodes within the network are designated as outer boundary nodes, while mesh gateways are classified as inner boundary nodes. The potential values of all the boundary nodes are set as fixed values, with the highest potential value to outer boundary nodes and the lowest potential value assigned to inner boundary nodes. The boundary conditions exemplified by a fixed value in (1) can be expressed as follows:  
\[\phi(B)=1,\;\phi(D)=0 \tag{1}\]where \(\phi(n)\) is a potential value at node \(n\), \(B\) represents a set of outer boundary nodes, and \(D\) indicates a set of inner boundary nodes as destinations of packets. These boundary conditions, applied in previous related research, are referred to as coarse-grained boundary conditions in this paper. These coarse-grained boundary conditions correspond to Dirichlet boundary conditions [15], which assign a single fixed potential value to each point on the boundary. The limitations of the Dirichlet-based coarse-grained boundary conditions can be described as follows. First, we explore existing outer boundary conditions. Traditionally, outer boundary nodes have been implicitly considered a subset of general mesh nodes located at the outermost edges of the given network. However, the challenge lies in defining the outermost edges, particularly in network topologies with irregular node deployments, as opposed to grid regular network topologies. To illustrate this challenge, consider a straightforward extreme example involving a grid regular network topology with a hole at its center. Here, the hole is defined as regions where no active mesh nodes are present. According to the existing implicit definition of outer boundary nodes, the mesh nodes surrounding the central hole would not be qualified as outer boundary nodes. However, it is more reasonable to categorize these mesh nodes surrounding the hole as the outer boundary nodes since their location is not fundamentally different from that of mesh nodes located at the outermost edges of the network in terms of neighbor node deployment. This highlights the need for a clear rule to determine outer boundary nodes based on neighbor node deployment rather than just relying on the physical location of the mesh nodes. Secondly, under the existing Dirichlet-based outer boundary conditions, all outer boundary nodes are uniformly assigned the highest potential value, regardless of their surroundings. This uniformity implies that outer boundary nodes tend not to exchange packets among themselves due to their identical potential values. Consequently, this situation may lead to congestion issues because these nodes consistently forward packets into the interior of the network. On the other hand, regarding inner boundary conditions, all gateways are assigned the same lowest potential value, regardless of the load levels in their vicinity. This approach hinders effective load balancing since it does not consider the load levels when determining the potential value. Given the importance of load balancing as a key property of routing protocols [16], it is crucial to explore new inner boundary conditions that can better support this aspect. However, when considering new inner boundary conditions, it is also essential to consider that inner boundary nodes play a crucial role as drainage points for achieving convergence in the routing field.