# Fine-Grained Boundary Conditions in Field-Based Routing
## 1 Introduction  
Industrial wireless mesh networks (WMNs) have gained significant attention in recent times due to their cost-effectiveness and environment adaptation capability [1; 2]. According to several market reports [3; 4], such industrial WMNs are also expected to have substantial growth potential as a cost-effective and practical wireless network solution for efficient communication in various industries such as manufacturing, logistics, and warehouses. However, similar to many other emerging network technologies, industrial WMNs face challenges in areas such as routing, resource management, and security. Among these, routing is a critical issue since it plays a crucial role in ensuring the reliable and efficient delivery of data across the network [5]. As industrial WMNs continue to evolve to support a wider range of service scenarios, the development of more efficient routing protocols becomes essential to achieve better data transmission, considering the unique characteristics of these networks [6]. Various routing protocols have been introduced for these networks, such as shortest-path routing, back-pressure routing, and combination-type routing. As one of the combination-type routing protocols, field-based routing has emerged as a promising solution, flexibly balancing proximity to gateways with congestion awareness. Interestingly, field-based routing draws inspiration from physics, employing well-established differential equations such as [PERSON]'s or [PERSON]'s equations in physics [7], thus positioning itself as a physics-inspired routing protocol. The strength of this approach lies in its capacity to leverage centuries of validated scientific knowledge. In the case of field-based routing, this approach aids in deriving a novel routing metricthat flexibly considers proximity to gateways and congestion when transmitting packets to gateways. The routing metric is derived by adapting the principles described by [PERSON]'s or [PERSON]'s equations to interpretations customized for network algorithms. These differential equations originally describe the interactions governing the forces of attraction and repulsion between positive and negative charges in the physical world, which has been validated over a long period. In the context of field-based routing, solving these differential equations corresponds to assigning a potential value at each node. A routing field is constructed using these potential values across all nodes. Therefore, solving these equations to determine our routing metric is essential, and it is closely intertwined with boundary conditions. Boundary conditions are restrictions applied to the equations, directly impacting the resulting solution. Well-known boundary conditions in physics include Dirichlet, Neumann, and Robin boundary conditions. In this context, it becomes evident that boundary conditions applied to boundary nodes play a crucial role within the networking domain. This is because these conditions directly shape the routing field, subsequently having a substantial influence on routing performance. Nevertheless, previous research has often underestimated the significance of boundary conditions, predominantly considering primarily simple and coarse-grained boundary conditions. This motivates us to carefully explore the boundary condition problem as a new possibility to achieve better routing performance. Our main contribution in this paper is to propose novel fine-grained boundary conditions tailored for field-based routing, with the goal of facilitating practical field-based routing in diverse industrial environments. These fine-grained boundary conditions can be categorized into two types: outer boundary conditions applied to a subset of general mesh nodes, typically located at the outermost edges of the network, referred to as outer boundary nodes, and inner boundary conditions applied to gateways, referred to as inner boundary nodes. The proposed outer boundary conditions introduce virtual nodes for adaptability.