A Review of Chebyshev Nets.

Before continuing with our own work, we provide a brief review of a smooth Chebyshev net, defined as a smooth parameterized surface patch 𝐂:D⊂ℝ2→ℝ3, from a domain D in 2D with its orthogonal coordinate system (u,v) to a smooth surface in 3D, under the condition ∥∂𝐂/∂u(u,v)∥=∥∂𝐂/∂v(u,v)∥=1. Physically, this condition corresponds to inextensibility of the rods along the two initially perpendicular parameter directions. We regard 𝐂 as the deformation of a continuum planar domain of inextensible rods into a surface, during which crossing rods shear freely at joints. The shear angle function 0<ω(u,v)<π turns the unit tangent vector ∂𝐂/∂u into ∂𝐂/∂v.We focus on smooth Chebyshev nets that do not have singularities; i.e., no rod collapses onto another with ω∈{0,π}.

A smooth Chebyshev net exists locally around each point of a surface (18, 28), but a global obstruction constraining the curvature arises from the necessary Gauss equation (21)−K(u,v)sin⁡ω(u,v)=∂2∂u∂vω(u,v),[1]which enforces a strict coupling between shear and Gauss curvature, K(u,v). As rods shear, infinitesimal squares deform into rhombi, and the surface area element is dA=sinωdudv. Integrating Eq. 1 (with respect to u and v) over an axis-aligned rectangle (denoted as □) yields a seminal result due to Hazzidakis (ref. 29; see also ref. 21): The integrated Gauss curvature of each deformed rectangle depends only on its four interior angles 0<αi<π,∫𝐂(□)K(u,v)dA=2π−∑i=14αi,[2]so the maximum integrated Gauss curvature of every region enclosed by four rods is 2π. Due to this inability to encode regions with high curvature, finding a singularity-free smooth Chebyshev net that lies on, or approximates to, an entire arbitrary target shape is challenging and the topic of much ongoing theoretical (30⇓⇓⇓–34) and applied research (35⇓–37). The key idea is to carefully design the domain, which translates into the footprint of an elastic gridshell.

Footprint and Actuation Ansatzes from a Chebyshev Net.

We use smooth Chebyshev nets as educated guesses (ansatzes) for the design of elastic gridshells that provide solutions under the nontrivial inextensibility conditions, but neglect bending.

Formalizing an elastic gridshell, we define its footprint F𝐛d by an enclosing original boundary 𝐛, which cuts an infinite square grid of spacing d. A correspondence 𝐠:𝐛→𝐚 between positions along 𝐛 and an actuated boundary 𝐚 determines the location of the pinned edge points. The resulting actuated elastic gridshell is represented by 𝐆:F𝐛d→ℝ3. A specific Chebyshev net 𝐂 is regarded as a viable ansatz if, for each position b∈𝐛 of the original boundary that maps to an actuated position a∈𝐚, we have 𝐂(b)=𝐠(b)=a. Determining a viable ansatz for an elastic gridshell involves three steps. First, we specify an actuated boundary 𝐚 in the Chebyshev net surface. Second, we compute the original boundary in the domain by solving for the contour 𝐂(𝐛)=𝐚. Third, we fix d>0 to determine the footprint F𝐛d enclosed by 𝐛.

Our ansatz procedure does not consider the pinned boundary conditions on the actuated boundary, which lead to higher-order constraints from variational considerations of the bending energy (19). Therefore, we quantify deviations, as follows, to attest the accuracy of viable Chebyshev net ansatzes. We first restrict 𝐂 to points of the footprint F𝐛d (d sampling of the Chebyshev net). Then, for each footprint point, x∈F𝐛d, we define Δ(x)=∥𝐂(x)−𝐆(x)∥, the pointwise distance between an actuated elastic gridshell and a d-sampled viable ansatz.

Elastic Gridshells That Resemble Spherical Caps.

With the goal of designing hemispherical and spherical-cap gridshells, we consider Chebyshev’s hemisphere 𝐂h (22, 26, 38, 39) (see SI Appendix, S1. Hemispherical Ansatz for details on its construction). We then use 𝐂h as a viable ansatz by solving for the original boundary (shown as the outermost contour of Fig. 3B) that corresponds to an actuated boundary of the equator and fixing a unit cell spacing d=20 mm. This yields a footprint F𝐛d, which is then actuated into an elastic gridshell 𝐆h. In Fig. 3 C1 and C2, we quantify the deviation between the computed d sampling of 𝐂h (blue curves) and the elastic gridshells corresponding to 𝐆h obtained from both DER simulations (black curves) and experiments (red solid circles). A visual comparison is shown in Fig. 3 C1, while Fig. 3 C2 plots the data using spherical coordinates without longitudinal information (radius r, latitude ϕ); there is excellent agreement between theory, simulations, and experiments. Hereafter, we compare a viable ansatz only with a DER actuated elastic gridshell. Quantitatively, we find that there is ≲2% deviation between the d-sampled 𝐂h and the DER elastic gridshell 𝐆h (using Δ as introduced above).

To further affirm the strength of our ansatz protocol, we use the same 𝐂h to compute original boundaries of footprints for other spherical caps. Circles of latitude are used to slice the hemisphere by planes at varying heights (0≤z∗/ρ<1) and understood as actuated boundaries 𝐚z∗. Note that ρ is the fixed radius of the sphere and agrees with the radius of 𝐚z∗ only when z∗/ρ=0 (entire hemisphere). After fixing d=20 mm, the corresponding original boundaries 𝐛z∗ (Fig. 3B) yield a series of actuated elastic gridshells 𝐆z∗. In Fig. 3D, we quantify the resulting deviations. As z∗/ρ increases, the deviation decreases from 1.87% (z∗/ρ=0.3) to 0.17% (z∗/ρ=0.9). This agreement is surprising since the smaller footprints contain even fewer rods; a continuum Chebyshev net ansatz may continue to be valid even for very coarse footprints within the limits of our inextensibility assumptions, whose applicability is investigated in SI Appendix, S5. Inextensibility Limits of Elastic Gridshells. To emphasize the geometric nature of these results, we considered a common household strainer (Fig. 3E), which also forms a spherical cap from a network of plastic inextensible rods. Interestingly, its flattened shape (after being cut along its “actuated boundary” rim) shows excellent agreement with a corresponding “original boundary” from 𝐂h (Fig. 3F), thereby pointing to the generality of our framework.

Integrated Gauss Curvature per Unit Cell.

Even though the surface of a unit cell is ill-defined, it does have well-defined crossing angles at its joints. Using Hazzidakis’ result, Eq. 2, for each unit cell of an actuated gridshell, we define its integrated Gauss curvature as K□=2π−∑i=14αi. If the enclosing rods of an actuated unit cell were to lie along geodesics, then the Gauss–Bonnet theorem (21) would state that the integrated Gauss curvature of the enclosed region would be ∑i=14αi−2π, exactly the negative of our quantity K□. We will see that, generally, K□ carries the sign matching our intuition, implying that we do not interpret elastic gridshell rods as lying along geodesics of an underlying surface.

Next, we evaluate the validity of the notion of integrated Gauss curvature, K□, to design and describe elastic gridshells. We study viable ansatzes from surfaces of constant (i) positive, (ii) zero, and (iii) negative Gauss curvature, for which we use, respectively, (i) Chebyshev’s hemisphere 𝐂h, (ii) half a circular cylinder parameterized by helices 𝐂c, and (iii) half an analytic parameterization for a pseudosphere of revolution 𝐂p. The latter two are special cases of Chebyshev nets on surfaces of revolution (40) (SI Appendix, S2. Cylindrical and Pseudospherical Ansatzes).

In Fig. 4A, Left to Right, we show that the obtained elastic gridshells closely resemble their ansatz d samplings, with only small deviations. In Fig. 4B, we plot K□ vs. the centroid of its four vertices in space along the shown axis. We find that K□ matches our expectation, with actuated unit cells that visually resemble regions of positive, zero, and negative curvature having an integrated Gauss curvature that is correctly signed. Our chosen ansatzes do not incorporate bending, which impacts the shape of the actuated forms. Most prominently, Fig. 4A, Right shows 𝐆p protruding above its negatively curved Chebyshev ansatz 𝐂p immediately after leaving its actuated boundary, before dipping in the middle. This behavior is consistent with a change from positive to negative curvature and is reflected in the distribution of K□, which derives from the actuated elastic gridshell itself, not a choice of viable ansatz.

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Fig. 4.

Notion of integrated Gauss curvature. (A) Elastic gridshells (DER simulation; black solid lines) and their d-sampled Chebyshev ansatzes (blue solid lines), which are from Left to Right, respectively, a hemisphere, a cylinder, and a saddle (each with half-span L). (B) Distribution of K□, the integrated Gauss curvature of a unit cell obtained from Eq. 2, for three elastic gridshells. Each unit cell is positioned by its normalized centroid x¯=x/L. (C) Three indentation cases of an actuated hemispherical gridshell: (Left) inward and (Center) outward normal displacement at the north pole and (Right) inward displacement at π/4 latitude and longitude. The normalized displacement field is |𝐮|/δo. (D) Binned K□ for C (Left to Right). Averages (bins) and standard deviations (error bars) show the spatial heterogeneity (i.e., nonlocality) of the shearing response.

We therefore find K□ to be an excellent proxy for the curvature of actuated unit cells and its definition in terms of crossing angles at joints provides a way for us to rationalize the distribution of shearing across an elastic gridshell.

Nonlocal Response to Point Loading.

The quantity K□ not only describes the geometry of elastic gridshells, but also enables us to interpret and quantify their nonlocal response under point loads. Given the hyperbolic nature of the underlying Gauss equation, Eq. 1, the rods can be regarded as characteristics, regularized by bending energy, that act as “highways of deformation.” As such, we expect nonlocal behavior to be prevalent in elastic gridshells. In contrast, similar loading of a thin isotropic continuum shell leads to a deformation that is local to a small spherical cap (41⇓–43).

We subject an actuated hemispherical elastic gridshell 𝐆h to three types of point loading: (i and ii) inward (i) and outward (ii) normal displacement at the north pole and (iii) inward displacement at π/4 latitude and longitude. The magnitude of the indentation was chosen to be δo=0.1ρ. In Fig. 4C, we present the top view of these loaded elastic gridshells obtained with DER simulations and quantify the displacement from the unloaded, but actuated, state. Corresponding experimental elastic gridshells under manual indentation are shown in Movie S2. The response for all three loading cases of the hemispherical elastic gridshell is strikingly nonlocal. Case i introduces not only a local dimpled cap but also displacements at the midpoints and corners of the largest concentric curved rhombus, nearly reaching the boundary. Case ii causes the center pair of rods to approach their inextensibility limits, forcing displacements to reach down to the equator. Case iii introduces displacement in a series of concentric curved rhombi that span the system.

This nonlocal response can further be rationalized using K□. Movies S3–S5 present the time evolution of the spatial distribution of K□ for the above indentation cases i–iii. Each actuated unit cell is segmented by the centroid of its vertices along the axis shown in Fig. 4A, Left (total length 2L). In Fig. 4D, as a snapshot, we plot the spatial average of K□ vs. bins of size 0.4/L. The SD within a bin measures the spatial heterogeneity of the curvature carried by each quad. The unindented gridshell configuration is treated as reference. Upon indentation (in all three cases), the shear response is highly nonlocal, and the spatial distribution of K□ changes along its entire extent (Movies S3–S5). Interestingly, as indicated by horizontal lines in Movies S3–S5, the total integrated Gauss curvature (sum of all K□) remains constant throughout the indentation. This result reinforces that K□ is a valid reflection (beyond shearing) of integrated Gauss curvature. Moreover, a constant total K□ establishes that nonlocality in elastic gridshells is related to the regions enclosed by rods.