Almost‐Rigidity of Frameworks

Abstract

We extend the mathematical theory of rigidity of frameworks (graphs embedded in d‐dimensional space) to consider nonlocal rigidity and flexibility properties. We provide conditions on a framework under which (I) as the framework flexes continuously it must remain inside a small ball, a property we call “almost‐rigidity”; (II) any other framework with the same edge lengths must lie outside a much larger ball; (III) if the framework deforms by some given amount, its edge lengths change by a minimum amount; (IV) there is a nearby framework that is prestress stable, and thus rigid. The conditions can be tested efficiently using semidefinite programming. The test is a slight extension of the test for prestress stability of a framework, and gives analytic expressions for the radii of the balls and the edge length changes. Examples illustrate how the theory may be applied in practice, and we provide an algorithm to test for rigidity or almost‐rigidity. We briefly discuss how the theory may be applied to tensegrities. © 2020 Wiley Periodicals LLC.

Epsilon local rigidity and numerical algebraic geometry

“…After the preparation of this paper, we learned of the related 2019 paper [21], which defines the almost rigidity of frameworks. This paper succeeds at extracting even more information from the rigidity matrix dg| p0 , specifically from its singular value decomposition and also existing tests for pre-stress stability [9,10] using semidefinite programming (SDP).…”

“…Secondly, our methods allow the radius ε > 0 be chosen arbitrarily, so it is possible to continue decreasing ε until the sphere actually meets the nearest flex. Given p 0 , the methods of [21] output a single radius η 1 , which comes with no claim of minimality. In addition, when our ε is decreased so much as to meet a continuous flex, Algorithm 2 can be applied to follow that flex by a parameter homotopy in ε, computing more nearby points sampled from the flex.…”

“…In addition, when our ε is decreased so much as to meet a continuous flex, Algorithm 2 can be applied to follow that flex by a parameter homotopy in ε, computing more nearby points sampled from the flex. While [21] makes one η 1 estimate based on the linearization dg of the system of equations g, our method stably computes real solutions g(x) = 0 to the full nonlinear system, and allows a search over all possible radii ε > 0. In particular, this allows the computation of radii ε 1 < ε 2 to replace the radii η 1 < η 2 of [21] with tighter bounds closer to the minimal ε 1 and maximal ε 2 .…”

“…While [21] makes one η 1 estimate based on the linearization dg of the system of equations g, our method stably computes real solutions g(x) = 0 to the full nonlinear system, and allows a search over all possible radii ε > 0. In particular, this allows the computation of radii ε 1 < ε 2 to replace the radii η 1 < η 2 of [21] with tighter bounds closer to the minimal ε 1 and maximal ε 2 .…”

Emergence of zero modes in disordered solids under periodic tiling

“…For instance, in second-order rigid systems, such as under-constrained networks that rigidify under tension, Eq. 1 cannot be used to describe the rigidity [4][5][6][7][8]. Another example where this constraint counting method fails is in systems with shear degrees of freedom or special symmetries (such as square or Kagome lattices), where the alignment of states of self-stress can lead to internal floppy modes that are not included in the Maxwell-Calladine count [9,10].…”