Finite Urban Module Method as a Tool for Modelling and Calculation of Urban Planning Systems

Abstract

The paper presents the Finite Urban Module Method (FUMM) – a fundamentally new hybrid approach to mathematical modelling and calculation of urban planning systems based on the rigorous adaptation of the finite element method (FEM) to spatial planning problems. The relevance of this work is driven by the absence of a unified mathematical framework capable of simultaneously describing planar urban zones, linear infrastructure networks and point-based centres of gravity within a single system of equations with explicit handling of regulatory constraints. The urban system is represented through a hybrid discrete space in which planar urban modules (blocks, districts, functional zones) are described as 2D-elements, transport and engineering networks as 1D-elements, and point objects and agents (gravity centres, transit stops, demand sources) as 0D-elements. Local operators are derived for each element type: the stiffness matrix, the accumulation matrix and the load vector, with explicit urban interpretation of the coefficients (spatial conductivity, network capacity, density, demand). The concept of a functional module compatibility matrix is introduced to algebraize urban planning regulations and ensure mathematically rigorous coupling between adjacent urban zones: full compatibility implies a strong interface connection, incompatibility implies a barrier element or zero coupling. The block structure of the global system of equations is examined in detail, covering Dirichlet, Neumann, Robin and saddle-point boundary conditions for problems with regulatory constraints, as well as solution methods for linear and nonlinear systems including Krylov iterative methods, multilevel AMG solvers and direct sparse factorisation. Dynamic modelling via IMEX schemes for stiff problems and stochastic analysis via non-intrusive and intrusive polynomial chaos approaches are described. A comparative analysis of FUMM against cellular automata, agent-based and conventional transport models confirms the systematic advantages of the proposed approach: rigorous variational formulation, adaptive discretisation, scalable solvers and natural alignment with current regulatory standards. Compliance of FUMM with DBN B.2.2-12:2019 is demonstrated, and practical applicability for scenario analysis, optimisation tasks and automated urban indicator mapping is established.