Arbitrary order approximations at constant cost for Timoshenko beam network models

TL;DR

The paper addresses the numerical solution of Timoshenko beam networks represented as graphs, where edges carry one-dimensional beam equations and nodes enforce rigid joints. A hybridizable discontinuous Galerkin (HDG) discretization yields a symmetric positive definite system on network nodes, enabling arbitrary-order convergence with almost constant global cost as the polynomial degree $p$ increases. A two-level overlapping Schwarz preconditioner is developed to ensure uniform convergence of the preconditioned conjugate gradient method, with spectral equivalence to a weighted graph Laplacian under connectivity assumptions. Numerical experiments on manufactured solutions and a large-scale paper network demonstrate the method's optimal convergence rates and scalability to networks with hundreds of thousands of edges.

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.

Figures (4)

• Figure 2.1: Fiber network model of paper at the millimeter scale.
• Figure 5.1: An artificial mesh $\mathcal{T}_H$ over a network (left) and a basis function $\varphi_i$ with the boundary of its support marked in red (right).
• Figure 6.1: All four plots show the $L^2$-errors of the primal variables. The top left, top right, and bottom left plots show the errors for the polynomial degrees $1$ (blue), $2$ (red), $5$ (black), and $6$ (magenta) as a function of the mesh size, for different choices of the stabilization parameter $\tau$. The bottom right plot shows the errors for the beam lengths $1$ (cyan), $2^{-1}$ (purple), $2^{-4}$ (gray), and $2^{-5}$ (brown) as a function of the polynomial degree.
• Figure 6.2: On the left an illustration of the network before (blue) and after (red) deformation is shown. On the right is a convergence plot of the preconditioned conjugate gradient method. There the relative residual is plotted as a function of the iteration number for constant material parameters (black) and realistic material parameters (orange).

Theorems & Definitions (22)

• Remark 2.1: Local formulation
• Remark 3.1: Tilde notation
• Lemma 3.2: Well-posedness of local solver
• proof
• Lemma 3.3: Properties of condensed problem
• proof
• Lemma 3.4: Well-posedness of discretized local solver
• proof
• Remark 3.5: Size of the global system of equations
• Lemma 3.6: Properties of condensed HDG problem