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A Constrained Optimization Approach for Constructing Rigid Bar Frameworks with Higher-order Rigidity

Xuenan Li, Christian D. Santangelo, Miranda Holmes-Cerfon

TL;DR

This work develops a constrained optimization framework for constructing bar frameworks that are rigid but not first-order rigid, by solving local edge-length optimization problems under fixed other-edge constraints. It establishes a formal link between KKT conditions and self-stresses, and between second-order sufficiency and prestress stability, providing guarantees that local optima yield prestress-stable frameworks. The authors extend the approach to stress design and to a bifurcation-based method for achieving third-order rigidity, including explicit 2D and 3D examples and a detailed procedure for constructing third-order rigid configurations. The method offers a systematic, symmetry-free pathway to higher-order rigidity with practical implications for designing lightweight, adaptable structures and metamaterials, while outlining open questions on multiple self-stresses and inequality-constrained extensions.

Abstract

We present a systematic approach for constructing bar frameworks that are rigid but not first-order rigid, using constrained optimization. We show that prestress stable (but not first-order rigid) frameworks arise as the solution to a simple optimization problem, which asks to maximize or minimize the length of one edge while keep the other edge lengths fixed. By starting with a random first-order rigid framework, we can thus design a wide variety of prestress stable frameworks, which, unlike many examples known in the literature, have no special symmetries. We then show how to incorporate a bifurcation method to design frameworks that are third-order rigid. Our results highlight connections between concepts in rigidity theory and constrained optimization, offering new insights into the construction and analysis of bar frameworks with higher-order rigidity.

A Constrained Optimization Approach for Constructing Rigid Bar Frameworks with Higher-order Rigidity

TL;DR

This work develops a constrained optimization framework for constructing bar frameworks that are rigid but not first-order rigid, by solving local edge-length optimization problems under fixed other-edge constraints. It establishes a formal link between KKT conditions and self-stresses, and between second-order sufficiency and prestress stability, providing guarantees that local optima yield prestress-stable frameworks. The authors extend the approach to stress design and to a bifurcation-based method for achieving third-order rigidity, including explicit 2D and 3D examples and a detailed procedure for constructing third-order rigid configurations. The method offers a systematic, symmetry-free pathway to higher-order rigidity with practical implications for designing lightweight, adaptable structures and metamaterials, while outlining open questions on multiple self-stresses and inequality-constrained extensions.

Abstract

We present a systematic approach for constructing bar frameworks that are rigid but not first-order rigid, using constrained optimization. We show that prestress stable (but not first-order rigid) frameworks arise as the solution to a simple optimization problem, which asks to maximize or minimize the length of one edge while keep the other edge lengths fixed. By starting with a random first-order rigid framework, we can thus design a wide variety of prestress stable frameworks, which, unlike many examples known in the literature, have no special symmetries. We then show how to incorporate a bifurcation method to design frameworks that are third-order rigid. Our results highlight connections between concepts in rigidity theory and constrained optimization, offering new insights into the construction and analysis of bar frameworks with higher-order rigidity.

Paper Structure

This paper contains 20 sections, 14 theorems, 57 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background information on rigidity and optimization
  3. A brief review of rigidity theory for bar frameworks
  4. A brief review of equality-constrained optimization
  5. The connection between rigidity and optimization
  6. The connection between solutions to $P_{k}^\pm$ and prestress stable frameworks
  7. Stress design
  8. Designing prestress stable frameworks using optimization
  9. Isostatic examples.
  10. 3d examples.
  11. Examples with additional linear constraints.
  12. Examples obtained by stress design.
  13. Creating third-order rigid frameworks
  14. Theory: a characterization of third-order rigidity via motion on a feasible set
  15. Algorithm to construct a third-order rigid framework
  16. ...and 5 more sections

Key Result

Theorem 2.12

Suppose $p^*$ is a local solution of $P_{k}^\pm$, and the LICQ holds. Then the KKT conditions hold for some Lagrange multiplier $\lambda^*$.

Figures (7)

  • Figure 1: Various prestress-stable bar frameworks that are not first-order rigid: the first row depicts well-known examples -- (a) from connelly1996second, (b)-(c) from grasegger2019graphs, (d) from roback2025tuning, and (e)-(f) from the sphere-packing data in holmes2016enumerating; the second row shows prestress-stable frameworks obtained by perturbing these well-known structures and then applying our constrained optimization approach.
  • Figure 2: Isostatic, prestress stable frameworks designed by our optimization method. (a),(c) show first-order rigid frameworks used as initial conditions for optimization. The red dashed edge is the chosen free edge. (b),(d) show the prestress stable frameworks found by maximizing (b) and minimizing (d) the free edges. The edge color indicates the sign of the self-stress: red means the edge is stretched ($\omega_k >0$) and blue means the edge is compressed ($\omega_k<0$). The arrows indicate the $\mathcal{T}^{\perp}$-flex.
  • Figure 3: 3d prestress stable frameworks designed by our optimization method. (a) and (c) come from the sphere packing data; (b) and (e) are a perturbed version of (a) and (c); (c) and (f) are the prestress stable bar frameworks obtained by our constrained optimization scheme. Colors are proportional to the self-stress: red means under compression and blue means under tension. The color intensity represents the magnitude of the stress, with darker colors indicating higher stress values and lighter colors indicating lower stress values.
  • Figure 4: Designing a prestress stable framework with midpoint constraints. (a) the initial condition, which is flexible; the dashed edge is the free edge whose length will be minimized; (b) the optimal state found by the constrained optimization approach; (c) and (d) are the two $\mathcal{T}^{\perp}$-flexes for the framework in (b). The color scheme in (b) is the same as the one in \ref{['fig:2D-prestress-example']}.
  • Figure 5: Designing stress ratios on target edges. (a)-(c) are over-constrained frameworks used as initial conditions for the optimization, and (d)-(f) are solutions to \ref{['eqn:stress-design']} with desired self-stresses. The color scheme is the same as the one in \ref{['fig:3d-opt-length']}.
  • ...and 2 more figures

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 31 more