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.

翻译：本文研究了铁木辛柯梁网络模型的数值求解方法。该模型由网络中每条边上的铁木辛柯梁方程构成，并通过刚性节点条件在网络节点处耦合。通过杂交化处理，我们可以将原问题等价地重构为定义在网络节点上的对称正定线性方程组系统。这种重构之所以可行，是因为梁方程耦合所在的节点是零维对象。为离散化梁网络模型，我们提出了一种可杂交间断伽辽金方法，该方法能在网格细化下实现任意阶收敛，且无需增大全局系统矩阵的规模。针对通常条件数极差的全局系统矩阵，我们采用了一种两级重叠加性施瓦茨方法作为预条件子。在网络满足适当连通性假设的前提下，我们证明了相应预条件共轭梯度法的一致收敛性。数值实验验证了本文的理论结果。