Lucia Swoboda | Searcharxiv

arXiv2026

A Hybridizable Discontinuous Galerkin Method for Wave Propagation in Elastic Beam Networks

This paper studies the numerical solution of elastic wave propagation on networks, modeled by elastodynamic equations posed on each edge, coupled at the nodes through suitable transmission conditions. We propose and analyze a hybridizable discontinuous Galerkin method that exploits the network structure to reduce the global problem at each time step to a linear system whose size depends only on the number of network nodes and not on the polynomial degree of the discretization. Combining it with an energy-conservative implicit time discretization, we derive a priori error estimates of optimal order in space and time. The implicit time discretization avoids the severe CFL restriction caused by the large variation in fiber segment lengths. To efficiently solve the resulting, typically ill-conditioned global system, we introduce a two-level overlapping additive Schwarz preconditioner. Under suitable assumptions on the network, we establish uniform convergence of the resulting preconditioned conjugate gradient method. Numerical experiments confirm the theoretical findings.

math.NA↗

arXiv2025

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.

math.NA↗