Numerical simulation of beam network models

Network models are used as efficient representation of materials with complex, interconnected locally one-dimensional structures. They typically accurately capture the mechanical properties of a material, while substantially reducing computational cost by avoiding full three-dimensional resolution. Applications include the simulation of fiber-based materials, porous media, and biological systems such as vascular networks. This article focuses on two representative problems: a stationary formulation describing the elastic deformation of beam networks, and a time-dependent formulation modeling elastic wave propagation in such materials. We propose a two-level additive domain decomposition method to efficiently solve the linear system associated with the stationary problem, as well as the linear systems that arise at each time step of the time-dependent problem through implicit time discretization. We present a rigorous convergence analysis of the domain decomposition method when used as a preconditioner, quantifying the convergence rate with respect to network connectivity and heterogeneity. The efficiency and robustness of the proposed approach are demonstrated through numerical simulations of the mechanical properties of commercial-grade paperboard.