# Fine-Grained Boundary Conditions in Field-Based Routing
## 6 Performance Evaluation
### Simulation Results
#### 6.3.1 Outer Boundary Conditions
As shown in Figure 6, the protocol with fine-grained Robin boundary conditions exhibits superior performance to the protocol with coarse-grained Dirichlet boundary conditions, achieving an aggregate throughput exceeding 200 Kbps and increasing the packet delivery ratio by 17% on average. Additionally, the proposed scheme, corresponding to the former, demonstrates approximately three times better performance in standard deviation compared to the latter, corresponding to the conventional scheme in terms of both the overall packet delivery and aggregate throughput. This indicates that the protocol with coarse-grained Dirichlet boundary conditions performs poorly in networks with holes because the holes obstruct the diffusion of potential. As a result, the routing field is constructed unstably. In contrast, the protocol using fine-grained boundary conditions exhibits particular strength in networks that include holes as it can stably achieve potential field convergence in any scenario. Figure 4: Overall packet delivery ratio under different outer boundary conditions with respect to the ratio of broken links. Figure 5: Non-grid regular topology with holes. Lastly, we evaluate our proposed scheme in a practical Google WiFi network topology [20] with irregular node positions, as shown in Figure 7. This real-world environment naturally contains network holes, making the convergence of a routing field challenging. The protocol with the proposed Robin boundary conditions consistently outperforms the protocol with the existing Dirichlet-based boundary conditions in terms of the overall packet delivery ratio and aggregate throughput, as shown in Figure 8. Specifically, significant improvements are observed over 1000 times in terms of both the overall packet delivery ratio and aggregate throughput on average, with standard deviations showing 16% and 35% smaller results in the same metrics, respectively. The existing Dirichlet-based boundary conditions fail to construct a normal routing field because they cannot form the necessary triangles required for potential value calculation. This limitation of the existing boundary conditions is effectively addressed by the proposed Robin boundary conditions. Figure 6: Performance according to kind of outer boundary conditions in non-grid regular topology with holes. (**a**) Overall packet delivery ratio; (**b**) Aggregate throughput. Figure 8: Performance according to the kind of outer boundary conditions in the Google WiFi network topology. (**a**) Overall packet delivery ratio; (**b**) Aggregate throughput. Figure 7: Google WiFi network topology. Notably, a performance gap between the proposed Robin boundary conditions and the existing Dirichlet boundary conditions is substantial in both packet delivery ratio and throughput, as evident from the exponential scale on the y-axis. This highlights the significant weakness of the existing Dirichlet boundary conditions in real-world scenarios with irregular node deployments and multiple naturally occurring holes, while the proposed Robin boundary conditions demonstrate robustness in such contexts.