用于弹性梁网络中波传播的可杂交间断伽辽金方法
A Hybridizable Discontinuous Galerkin Method for Wave Propagation in Elastic Beam Networks
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中文总结 AI 辅助
本文针对弹性梁网络的波传播问题，提出可杂交间断伽辽金方法，结合能量守恒隐式时间离散化，引入两层重叠加性施瓦茨预条件子，经数值实验验证了理论结果。
中文摘要 AI 辅助
本文研究网络上弹性波传播的数值解，该网络由各边的弹性动力学方程建模，通过合适的传输条件在节点处耦合。我们提出并分析一种可杂交间断伽辽金（Hybridizable Discontinuous Galerkin）方法，该方法利用网络结构将每一步的全局问题简化为线性系统，其规模仅取决于网络节点数，与离散化的多项式阶数无关。结合能量守恒的隐式时间离散化，我们推导了空间和时间上最优阶的先验误差估计。隐式时间离散化避免了因纤维段长度的巨大差异导致的严重CFL限制。为高效求解所得的通常病态全局系统，我们引入了两层重叠加性施瓦茨预条件子。在对网络的合适假设下，我们证明了所得预条件共轭梯度法的一致收敛性。数值实验证实了理论结果。
英文摘要
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.
