Parametric Vertex Splitting for Thickness-Accommodating Quasi-Rigid-Foldable Polyhedra

2026 Mar

<= 10 min random random

Parametric Vertex Splitting for Thickness-Accommodating Quasi-Rigid-Foldable Polyhedra

failed research proposal

update: i got admitted!!

1. Project Summary

I am proposing to look at vertex splitting as a way to reduce stress concentration in quasi-rigid-foldable (QRF) polyhedra, especially when we start using thicker materials for applications such as the soles of climbing robots. Building on the work in “Origami Morphing Surfaces with Arrayed Quasi-Rigid-Foldable Polyhedrons” [1], the main question I want to answer is this:

Can we come up with a parametric approach to vertex splitting that avoids relying on a single vertex, while still keeping a high QRF rate, good shear resistance, and the zero-Poisson’s ratio property?

Over the summer, I plan to run a series of simulations that test how vertex-splitting distance and material thickness affect stress concentration at the vertex. I will reuse the exact same evaluation methods from [1] (QRF/NRF definition, fixed-boundary shear tests, and the zero-Poisson check via end-surface length change) so my results line up directly with the original designs. Since I only have Abaqus Learning Edition (which caps structural models at 1000 nodes), I will use a shell + hinge/connector model, which takes facets as shells and creases as hinge connectors. This is a standard approach in origami finite-element work (see [5] for example). By the end, I hope to show a more durable version of the QRF polyhedron that can handle different material thicknesses. That should help push these structures toward real-world use by balancing the usual trade-off between vertex stress and shear resistance, all while keeping the high QRF rate and zero-Poisson behavior.

2. Background & Motivation

Vertex splitting is a proven technique to accommodate thick panels to work in origami while maintaining the single-degree-of-freedom kinematics of degree-4 vertices without holes. [2] In the QRF polyhedra from [1], though, the asterisk pattern relies on degree-6 vertices, which end up giving lower shear resistance than the “minus” pattern because of all the extra creases.

Luckily, the generalized bow-tie pattern (the same as the asterisk in [1]) has been shown to be kinematically identical to a version that uses modified two-split degree-4 vertices [3]. So the split-vertex design is expected to inherit the original folding behavior, i.e. rigid-foldable (or at least quasi-rigid-foldable) and zero-Poisson.

Nevertheless, splitting introduces more creases and trapezoid panels, which could have an adverse effect on shear resistance. At the same time, other studies on non-rigid origami have shown that even tiny changes at the vertex (such as small cut-outs or slits) can dramatically change stiffness and nonlinear behavior [5]. Together, these motivates a focused study of how the split distance affects (i) vertex stress concentration and (ii) shear performance, especially once the material gets thicker.

3. Objectives

By the end of the project, I aim to:

  • Aim 1: Reproduce the baseline QRF model and all the evaluation steps from [1].
  • Aim 2: Figure out how vertex-split distance interacts with material thickness and vertex concentration.
  • Aim 3: See how different split distances change shear resistance.
  • Aim 4: Put together a practical framework for designing thickness-accommodating QRF polyhedra using vertex splitting.

Hypothesis: Shorter vertex-split distance will preserve shear resistance but greater vertex concentration; Longer vertex-split distance will decrease shear resistance but less vertex concentration.

4. Methods & Feasibility

Due to limited time, I will be only focusing on splitting the horizontal axis of the vertex in the asterisk pattern as it intersects the 6 creases which contain the highest compressive strain [1], where thicker material will most likely contact and interfere.

4.1 Parameters

I will keep the same α and β from the original paper (they control the concavity of the side faces), and I am adding two new ones: relative thickness τ and split ratio δ.

  • Relative thickness: τ = t/L, where t is panel thickness and L is the shortest crease length at the central vertex B (creases 5 or 6). I will treat the material as “thick” once τ ≳ 0.05, as Peng & Chirikjian suggested: “A pattern enters the thick-origami regime when panel thickness τ is not negligible compared to a characteristic in-plane length L (e.g., crease length or panel width).” [4]
  • Split ratio: δ = γ/t, where γ is the splitting distance. For easier interpretation I will also track η = γ/L (so η = δτ).

4.2 Specimen Choice

To stay consistent with [1], I will not model a whole array. Instead I will test single-unit specimens for compression/QRF/zero-Poisson and the same fixed-boundary shear specimens they used. This is sufficient because the paper already showed that unit-level properties predict how the array behaves.

4.3 Evaluation Metrics

I will reuse the same metrics from [1] wherever possible:

  • Compressive strain: ε = (h₀ – h)/h₀
  • QRF/NRF rate: based on change in total length of the horizontal “red” creases
  • Shear resistance: initial shear stiffness k_shear from the force–displacement curve, plus peak force before any instability
  • Zero-Poisson check: normalized length of a representative end-surface edge l(ε)/l₀

For vertex durability (since we cannot resolve true peak stresses with the 1000-node limit), I will use two simple proxies from the hinge connectors following the approach used in [5]:

  • Primary: total elastic strain energy summed over all creases connected to the split-vertex region
  • Secondary: peak hinge moment (or rotation) in those same creases

Lower values here mean less concentration at the vertex.

4.4 Simulation Procedure

Step 1: Build and run the baseline asterisk and minus patterns to confirm I match the trends in [1] (shear stiffness ordering, end-edge invariance, etc.). Step 2: Add the split-vertex geometry (horizontal split distance γ) to the asterisk pattern. Step 3: Run parameter sweeps over τ (thin to thick) and δ (including δ = 0), keeping α and β fixed at the original design point to start. For each combination I will do both compression and shear simulations. Step 4: Plot the trade-offs, check that QRF rate stays high and zero-Poisson still holds, then propose a simple guideline for picking δ as a function of τ.

4.5 Tools

Everything will be done in Abaqus/CAE with Python scripting for setup and post-processing, plus NumPy and Matplotlib for analysis.

5. Expected Outcomes & Impact

  • A clean, reproducible pipeline that matches [1] for QRF rate, shear stiffness, and zero-Poisson checks.
  • Clear maps showing how τ and δ affect vertex concentration, shear stiffness, and the kinematic properties.
  • A practical rule-of-thumb for choosing the split distance γ for any target thickness τ.

If it works, this should make QRF polyhedra more practical for thicker, tougher applications such as climbing-robot feet and morphing surfaces.

6. Timeline (12 weeks max)

  • Weeks 1–2: Build baseline models + Python post-processing scripts.
  • Week 3: Reproduce the original trends (Aim 1).
  • Week 4: Implement split-vertex variants and quick smoke tests.
  • Weeks 5–7: Full sweeps for compression and shear.
  • Week 8: Sensitivity checks on mesh and connector settings.
  • Weeks 9–10: Trade-off plots and guideline (Aim 4).
  • Weeks 11–12: Write-up, figures, and package everything on GitHub.

References

  1. J. Li, J. Bao, C. Ho, S. Li, and J. Xu, “Origami Morphing Surfaces with Arrayed Quasi‐Rigid‐Foldable Polyhedrons,” Advanced Science, vol. 11, no. 36, Jul. 2024, doi: https://doi.org/10.1002/advs.202402128.
  2. K. A. Tolman, R. J. Lang, S. P. Magleby, and L. L. Howell, “Split-Vertex Technique for Thickness-Accommodation in Origami-Based Mechanisms,” ASME 2017 IDETC/CIE, Aug. 2017, doi: https://doi.org/10.1115/DETC2017-68018.
  3. J. Farnham, T. C. Hull, and A. Rumbolt, “Rigid folding equations of degree-6 origami vertices,” Proceedings of the Royal Society A, vol. 478, no. 2260, Apr. 2022, doi: https://doi.org/10.1098/rspa.2022.0051.
  4. R. Peng and G. S. Chirikjian, “Thick-panel origami structures forming seamless surfaces,” Nature Communications, vol. 16, no. 1, Apr. 2025, doi: https://doi.org/10.1038/s41467-025-59141-2.
  5. M. Yang, S. W. Grey, F. Scarpa, and M. Schenk, “Large impact of small vertex cuts on the mechanics of origami bellows,” Extreme Mechanics Letters, vol. 60, 2023, 101950, doi: https://doi.org/10.1016/j.eml.2022.101950.