Programming shape using kirigami tessellations (1812.08644v1)

Abstract: Kirigami tessellations, regular planar patterns formed by cutting flat, thin sheets, have attracted recent scientific interest for their rich geometries, surprising material properties and promise for technologies. Here we pose and solve the inverse problem of designing the number, size, and orientation of cuts that allows us to convert a closed, compact regular kirigami tessellation of the plane into a deployment that conforms approximately to any prescribed target shape in two and three dimensions. We do this by first identifying the constraints on the lengths and angles of generalized kirigami tessellations which guarantee that their reconfigured face geometries can be contracted from a non-trivial deployed shape to a novel planar cut pattern. We encode these conditions in a flexible constrained optimization framework which allows us to deform the geometry of periodic kirigami tesselations with three, four, and sixfold symmetry, among others, into generalized kirigami patterns that deploy to a wide variety of prescribed boundary target shapes. Physically fabricated models verify our inverse design approach and allow us to determine the tunable material response of the resulting structures. We then extend our framework to create generalized kirigami patterns that deploy to approximate curved surfaces in $\mathbb{R}^3$. Altogether, this work illustrates a novel framework for designing complex, shape-changing sheets from simple cuts showing the power of kirigami tessellations as flexible mechanical metamaterials.

• The paper presents a novel optimization framework for the inverse design of kirigami patterns to achieve specific 2D or 3D deployment shapes.
• The framework supports generalized kirigami patterns capable of approximating flat and curved 3D surfaces, validated through numerical simulations and physical models.
• Numerical results demonstrate a trade-off between design accuracy and pattern complexity, indicating potential applications in adaptive architecture, soft robotics, and programmable materials.

An Analytical Review of "Programming shape using kirigami tessellations"

In the paper titled "Programming shape using kirigami tessellations," the authors present a novel approach to the inverse design problem associated with kirigami patterns. Kirigami, a variant of the traditional origami, involves cutting paper to create intricate designs and mechanical metamaterials with novel interactions and properties. This paper advances the field by not merely analyzing the forward problem—how a given kirigami pattern behaves when cuts are made—but rather tackling the inverse design problem: How does one configure the cuts within a planar structure to achieve a specific deployment shape in two or three dimensions?

The contribution of this paper can be divided into several important facets:

1. Inverse Design Framework: The authors have formulated an optimization-based framework to determine the optimal configurations of cuts in a given kirigami sheet to approximate a predetermined target shape upon deployment. This framework maintains the contractibility of the structure, ensuring it can be deployed and undeployed efficiently without overlaps or mismatches in edge lengths or angles.
2. Generalized Kirigami Patterns: The paper successfully derives generalized kirigami patterns with multiple symmetries beyond basic three, four, and sixfold patterns. Deployments include approximations of both flat and curved surfaces in $\mathbb{R}^3$, thus expanding the utility of kirigami in engineering and design applications.
3. Optimization Constraints: Critical to their approach are the validity constraints on edge lengths and angles between nodes that guarantee the kirigami pattern's deployability. The optimization problem constructs a space where all boundary and interior nodes satisfy stringent geometric relationships, ensuring a continuous and stable transformation between states.
4. Model Verification: The paper provides comprehensive numerical simulations and physical validations, demonstrating the feasibility of deploying kirigami designs to approximate complex shapes. By employing physical models, they verify predicted behaviors and showcase the capability for monostable or bistable states dependent on hinge properties.
5. 3D Surface Fitting: Extending beyond planar designs, the methodology incorporates adaptations that allow kirigami patterns to conform to curved surfaces, utilizing additional constraints to maintain planarity where necessary. This aspect indicates prospective applications in areas such as deployable structures and medical devices where adaptability of shape is paramount.
6. Numerical Results and Performance: The authors present robust numerical results demonstrating the trade-off between design accuracy and complexity (number of tiles), with the approximation error decreasing inversely with the number of units. This rigorous treatment provides insight into the parametric considerations necessary for effective design.

The theoretical implications of this work provide a structured pathway for further exploration of kirigami in applied mathematics and mechanical engineering. Practically, the innovations presented offer substantial potential in fields such as adaptive architecture, soft robotics, and the development of materials with programmable morphologies. Future developments could look into enhancing computational efficiency, scaling for more extensive applications, and exploring integration with other smart materials.

The paper lays a solid groundwork that bridges the gap between mathematical formalism and applied design, underscoring the transformative potential of kirigami beyond artistic endeavors. As advancements continue, this research signifies a step toward more sophisticated design paradigms, leveraging the inherent flexibility and mechanical properties of kirigami tessellations to unlock more comprehensive solutions in engineering and material science.