# 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  
The limitations discussed in the previous section regarding outer boundary conditions can be classified into two categories: first, the absence of clear definitions for outer boundary nodes; and second, the inability to account for congestion caused by the identical fixed potential values assigned to boundary nodes. These challenges can be addressed by providing clear definitions for boundary nodes and incorporating new boundary conditions that allow for the assignment of different potential values based on congestion levels, even at the boundary nodes. Firstly, we establish a clear definition for outer boundary nodes to categorize them consistently without any ambiguity, as described in the previous section. This definition enables each mesh node to determine its eligibility as an outer boundary node based on neighbor node deployment rather than just relying on the physical location of the mesh node. To inspect the deployment of neighbor nodes, we adopt a four-quadrant technique, a widely used method in mathematics and geometry for dividing a space around each mesh node into smaller parts. In this context, quadrants are formally defined \begin{table}
\begin{tabular}{l l} \hline \hline
**Well-Known Boundary Conditions** & **Definition of Mathematical Expression** \\ \hline Dirichlet Boundary Conditions & \(\phi(n)=f(n),\forall n\in\partial\Omega\) \\ Neumann Boundary Conditions & \(\frac{\partial\phi(n)}{\partial n}=f(n),\forall n\in\partial\Omega\) \\ Robin Boundary Conditions & \(\alpha\phi(n)+\beta\frac{\partial\phi(n)}{\partial n}=f(n),\forall n\in \partial\Omega\) \\ \hline \hline \end{tabular} \(\Omega\): given domain/\(\partial\Omega\): boundary in given domain/\(n\): node/\(\alpha\), \(\beta\), \(f\): functions (regarded as constant functions in this paper for simplicity unless otherwise specified).  
\end{table} 
Table 1: Definition of well-known boundary conditions. as regional parts obtained by dividing a space into four areas by two orthogonal coordinate axes. Specifically, quadrant 1 corresponds to the top-right part, quadrant 2 to the top-left part, quadrant 3 to the bottom-left part, and quadrant 4 to the bottom-right part. It considers the challenge presented by the example of a network with a hole discussed in the previous section. Let \(N\) represent the set of all mesh nodes in a given network, and let \(Q_{n,k}\) denote the \(k\)-th quadrant within the transmission range of mesh node \(n\in N\). We assume that each quadrant includes the axis that is first encountered counterclockwise to avoid ambiguity. Additionally, \(a\mod b\) represents the remainder when \(a\) is divided by \(b\), and \(\{\ \}\) indicates an empty set. With this notation, a general mesh node is qualified as an outer boundary node if one of the following four conditions is satisfied. The designation of an outer boundary node totally depends on the number of consecutive quadrants without neighboring nodes within the transmission range. This rule is formulated separately for four cases and can be expressed mathematically as shown below, where \(i\in\{0,1,2,3\}\):  
1. No neighbors in four consecutive quadrants: \(Q_{n,(i\mod 4)+1}\), \(Q_{n,((i+1)\mod 4)+1}\), \(Q_{n,((i+2)\mod 4)+1}\), and \(Q_{n,((i+3)\mod 4)+1}=\{\}\)
2. No neighbors in three consecutive quadrants: \(Q_{n,(i\mod 4)+1}\), \(Q_{n,((i+1)\mod 4)+1}\), and \(Q_{n,((i+2)\mod 4)+1}=\{\}\)
3. No neighbors in two consecutive quadrants: \(Q_{n,(i\mod 4)+1}\) and \(Q_{n,((i+1)\mod 4)+1}=\{\}\)
4. No neighbors in only one consecutive quadrant: \(Q_{n,(i\mod 4)+1}=\{\}\)  
Once all mesh nodes have determined their status as either general mesh nodes or outer boundary nodes based on the rule mentioned above, the next crucial task is to implement the proposed Robin-based boundary conditions. These conditions allow for varying potential values for the outer boundary nodes.