[Paper Review] Computation of the deformation of rhombi-slit kirigami

Frédéric Marazzato|arXiv (Cornell University)|Jul 4, 2023

Advanced Materials and Mechanics4 citations

TL;DR

This paper presents a novel numerical method to compute deformations in rhombi-slit kirigami, a type of mechanical metamaterial, by solving a nonlinear, degenerate, sign-changing PDE using a complex finite element method with regularization via the limiting absorption principle. The approach enables accurate simulation of previously intractable mixed-type and degenerate cases, showing strong agreement with experimental data across auxetic, non-auxetic, and mixed-type patterns.

ABSTRACT

Kirigami are part of the larger class of mechanical metamaterials, which exhibit exotic properties. This article focuses on rhombi-slits, which is a specific type of kirigami. A nonlinear kinematic model was previously proposed as a second order divergence-form PDE with a possibly degenerate, and sign-changing coefficient matrix. We first propose to study the existence of solutions to a regularization of this equation by using the limiting absorption principle. Then, we propose a finite element method with complex polynomials to approximate the solutions to the nonlinear equation. Finally, simulations are compared with experimental results.

Motivation & Objective

• To address the challenge of solving a nonlinear, degenerate, sign-changing divergence-form PDE that models rhombi-slit kirigami deformation.
• To develop a robust numerical method capable of handling PDEs that transition between elliptic and hyperbolic types due to sign changes in the coefficient matrix.
• To extend existing models beyond the two special cases previously solvable (α = -β and β = 0) to general geometric parameters.
• To validate the numerical model against experimental data, particularly for patterns where prior simulations failed to capture key deformation features.

Proposed method

• A regularization of the original PDE is introduced using complex dissipation, enabling the application of the limiting absorption principle to prove existence and uniqueness of solutions.
• A complex-valued finite element method based on $\mathbb{P}^1$ Lagrange elements is formulated to approximate solutions in a complex Sobolev space $V = H^1(\Omega; \mathbb{C})$.
• The method enforces Dirichlet conditions strongly on $\partial\Omega_D$ and Neumann conditions weakly on $\partial\Omega_N$, with the real part of the complex solution representing the physical slit opening $\xi$.
• A Newton iteration scheme is employed to solve the nonlinear system, with convergence monitored via a relative residual tolerance of $10^{-6}$ to $10^{-8}$.
• The regularization parameter $\varepsilon$ is tuned to balance numerical stability and accuracy, with $\varepsilon = 0$ for strictly elliptic cases and $\varepsilon > 0$ for degenerate or hyperbolic regimes.
• The method is implemented using a triangular mesh with $h \approx 5 \times 10^{-3}$ to $5 \times 10^{-3}$, achieving convergence in 4–7 Newton iterations.

Experimental results

Research questions

• RQ1 Can a numerical method be developed to solve the nonlinear, degenerate, sign-changing PDE that models rhombi-slit kirigami deformation for arbitrary geometric parameters?
• RQ2 How can the existence of solutions be established for a PDE with a coefficient matrix that degenerates on a curve and changes sign?
• RQ3 To what extent can the proposed finite element method with complex polynomials and regularization reproduce experimental deformation patterns, especially in cases where previous models failed?
• RQ4 How does the inclusion of complex dissipation via the limiting absorption principle improve the stability and accuracy of solutions in mixed-type PDE regimes?

Key findings

• The proposed method successfully computes solutions for the rhombi-slit kirigami PDE across all tested cases, including previously intractable mixed-type and degenerate configurations.
• For the auxetic kirigami pattern ($\alpha = -0.5$, $\beta = 0.5$), the numerical results closely match both prior simulations and experimental data, with only minor discrepancies attributed to elasticity effects.
• In the non-auxetic case ($\alpha = -0.9$, $\beta = 0$), the method captures the central depression in the deformation profile, a feature absent in earlier simulations but present in experiments.
• For the mixed-type pattern ($\alpha = -1.6$, $\beta = 0.4$), the method produces a solution that shows good qualitative agreement with experimental results, including edge gradients and overall deformation shape.
• The method converges robustly in 4–7 Newton iterations with relative residuals below $10^{-6}$, demonstrating numerical stability across diverse parameter regimes.
• The code is publicly available, enabling reproducibility and extension to other kirigami patterns and boundary conditions.