Arbitrary order approximations at constant cost for Timoshenko beam network models

TL;DR

This paper introduces a hybridizable discontinuous Galerkin method for Timoshenko beam network models that achieves arbitrary order convergence at constant cost, with proven uniform preconditioned iterative solver convergence.

Contribution

It develops a novel discretization approach for beam networks that maintains fixed system size regardless of convergence order and provides a robust preconditioning strategy.

Findings

Achieves arbitrary order convergence without increasing system size

Proves uniform convergence of the preconditioned iterative solver

Numerical experiments confirm theoretical results

Abstract

This paper considers the numerical solution of Timoshenko beam network models, comprised of Timoshenko beam equations on each edge of the network, which are coupled at the nodes of the network using rigid joint conditions. Through hybridization, we can equivalently reformulate the problem as a symmetric positive definite system of linear equations posed on the network nodes. This is possible since the nodes, where the beam equations are coupled, are zero-dimensional objects. To discretize the beam network model, we propose a hybridizable discontinuous Galerkin method that can achieve arbitrary orders of convergence under mesh refinement without increasing the size of the global system matrix. As a preconditioner for the typically very poorly conditioned global system matrix, we employ a two-level overlapping additive Schwarz method. We prove uniform convergence of the corresponding preconditioned conjugate gradient method under appropriate connectivity assumptions on the network. Numerical experiments support the theoretical findings of this work.