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Support-Graph Preconditioners for 2-Dimensional Trusses

Samuel I. Daitch, Daniel A. Spielman

TL;DR

The paper develops support-graph preconditioners for stiffness matrices of 2D trusses that are stiffly connected, using fretsaw extensions to bound spectral conditioning. It combines Schur-complement techniques, graph embeddings, and low-congestion augmentations to transform the original truss system into a near-tree extension that is fast to factor and apply as a preconditioner. The TrussSolver scheme achieves near-linear-time performance $O\left(n^{5/4} (\log^2 n \log\log n)^{3/4} \log(1/\epsilon)\right)$ to solve $A_{\mathcal{T}} x=b$ to relative error $\epsilon$ under geometric constraints on edge lengths, angles, and material properties. This yields scalable, practical solvers for large planar truss systems in elasticity.

Abstract

We use support theory, in particular the fretsaw extensions of Shklarski and Toledo, to design preconditioners for the stiffness matrices of 2-dimensional truss structures that are stiffly connected. Provided that all the lengths of the trusses are within constant factors of each other, that the angles at the corners of the triangles are bounded away from 0 and $π$, and that the elastic moduli and cross-sectional areas of all the truss elements are within constant factors of each other, our preconditioners allow us to solve linear equations in the stiffness matrices to accuracy $ε$ in time $O (n^{5/4} (\log^{2}n \log \log n)^{3/4} \log (1/ε))$.

Support-Graph Preconditioners for 2-Dimensional Trusses

TL;DR

The paper develops support-graph preconditioners for stiffness matrices of 2D trusses that are stiffly connected, using fretsaw extensions to bound spectral conditioning. It combines Schur-complement techniques, graph embeddings, and low-congestion augmentations to transform the original truss system into a near-tree extension that is fast to factor and apply as a preconditioner. The TrussSolver scheme achieves near-linear-time performance to solve to relative error under geometric constraints on edge lengths, angles, and material properties. This yields scalable, practical solvers for large planar truss systems in elasticity.

Abstract

We use support theory, in particular the fretsaw extensions of Shklarski and Toledo, to design preconditioners for the stiffness matrices of 2-dimensional truss structures that are stiffly connected. Provided that all the lengths of the trusses are within constant factors of each other, that the angles at the corners of the triangles are bounded away from 0 and , and that the elastic moduli and cross-sectional areas of all the truss elements are within constant factors of each other, our preconditioners allow us to solve linear equations in the stiffness matrices to accuracy in time .

Paper Structure

This paper contains 10 sections, 17 theorems, 51 equations, 3 figures.

Table of Contents

  1. Preconditioning
  2. Using a Larger Matrix
  3. Congestion-Dilation
  4. Trusses and Stiffness Matrices
  5. Fretsaw Extension
  6. Path Lemma
  7. Graph Embeddings
  8. Solving the Linear System
  9. Preconditioning Lemmas
  10. Augmented Spanning Tree

Key Result

Theorem 1.2

For positive semidefinite $A,B$, and vector $\boldsymbol{\mathit{b}}$, let $\boldsymbol{\mathit{x}}$ satisfy $A\boldsymbol{\mathit{x}}=\boldsymbol{\mathit{b}}$. Each iteration of the preconditioned conjugate gradient method multiplies one vector by $A$, solves one linear system in $B$, and performs

Figures (3)

  • Figure 1: The truss on the right is a fretsaw extension of the truss on the left, as given by the fretsaw algorithm. The vertex positions in the fretsaw extension are distorted slightly so as to be able to distinguish vertex copies in the same location. The subgraph F is shown as solid lines, while the rest of the trusses' connectivity graphs are shown as dotted lines. Note that the connectivity graph of the fretsaw extension has one edge not in F.
  • Figure 2: A truss path from $\boldsymbol{\mathit{v}}_0$ to $\boldsymbol{\mathit{v}}_{11}$, with triangles and vertices labeled appropriately.
  • Figure 3: The TrussSolver algorithm

Theorems & Definitions (39)

  • Definition 1.1
  • Theorem 1.2: Axelsson
  • Definition 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Lemma 1.6: Congestion-Dilation Lemma
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Main Result
  • ...and 29 more