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Inverse Design of Tightly Woven Smart Fabrics

Einav Berin, Hillel Aharoni

Abstract

We present a geometric framework for the inverse design of smart woven fabrics composed of non-uniformly shrinking threads. A sufficiently tight weaving structure imposes strong local criteria on the material deformation and reduces the local geometry to a single scalar degree of freedom. Control over this degree of freedom can be achieved through simple calibration for each specific material system, via either mechanical experiments or numerical simulations. This reduction allows us to inverse-design a woven smart fabric, that conforms to an arbitrary target geometry when actuated, by solving a nonlinear hyperbolic partial differential equation. We validate this approach by deriving the thread-level actuation required for specific target geometries. We present both exact analytic solutions for symmetric shapes and a numerical optimization method for arbitrary freeform surfaces. These results confirm the practicality of our framework in achieving programmable, complex three-dimensional shaping.

Inverse Design of Tightly Woven Smart Fabrics

Abstract

We present a geometric framework for the inverse design of smart woven fabrics composed of non-uniformly shrinking threads. A sufficiently tight weaving structure imposes strong local criteria on the material deformation and reduces the local geometry to a single scalar degree of freedom. Control over this degree of freedom can be achieved through simple calibration for each specific material system, via either mechanical experiments or numerical simulations. This reduction allows us to inverse-design a woven smart fabric, that conforms to an arbitrary target geometry when actuated, by solving a nonlinear hyperbolic partial differential equation. We validate this approach by deriving the thread-level actuation required for specific target geometries. We present both exact analytic solutions for symmetric shapes and a numerical optimization method for arbitrary freeform surfaces. These results confirm the practicality of our framework in achieving programmable, complex three-dimensional shaping.
Paper Structure (6 sections, 23 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 6 sections, 23 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Supporting Information
  2. The Ropelength Problem
  3. Gaussian Curvature Formulae
  4. Geometric construction of a local solution for the Cauchy problem
  5. Inverse Problem Symmetric Solution
  6. Rectangular Parametrization Algorithm

Figures (4)

  • Figure 1: Illustration of a plain woven smart fabric. (A) A pre-actuated smart fabric. The fabric is flat and uniform (as shown by the identical zoomed-in unit cells). (B) When actuated, the spatially varying pre-programmed thread response deforms each unit cell differently, endowing the fabric with a non-Euclidean geometry that invokes its morphing into a non-planar surface. (C) Material coordinates $(u,v)$ aligned with warp and weft threads, respectively, are used to keep track of the fabric's geometry and shape. (D) The local tangent spacings of adjacent warp/weft threads define the metric scale factors $\sqrt{E}$ and $\sqrt{G}$. In a tightly woven fabric, shear is suppressed, and threads remain orthogonal.
  • Figure 2: Energy-minimizing configurations of a plain-weave unit cell for various prescribed thread rest lengths, for two physical thread models: (A) negligible bending modulus (orange), and (B) bending modulus of an elastic rod (blue). Each tapered line starts (narrow end) at the prescribed target lengths $(L_u, L_v)$, and ends (wide end) at the tangentially projected equilibrium lengths of the final configuration, namely the metric components $(\sqrt{E},\sqrt{G})$, respectively (all lengths are measured in units of the thread diameter). As seen in (A,B), an extended domain within the space of thread rest lengths (marked in gray) collapses onto a one-dimensional curve of unit-cell equilibrium metrics. (C) The tight-weave $E-G$ curves obtained for the two physical models in (A) and (B), fitted to the one-parameter form $\sqrt{E}=2\cos^{c}\alpha$, $\sqrt{G}=2\sin^{c}\alpha$ ($R^{2} = 99.5\%, 99.86\%$, respectively). The fitted parameters are $c=0.52\pm0.01$ and $c=0.50\pm0.01$, respectively. Also shown is the E-G curve of the ropelength-critical weave, obtained numerically by constrained optimization. As expected, it lies close to the negligible-bending curve.
  • Figure 3: Inverse design of woven smart-fabric surfaces of revolution. (A) A pre-actuated woven cylinder. (B-D) Actuating the smart fabric with different actuation profiles $\alpha(u)$ deforms the cylinder into (B) a sphere, (C) a pseudo-sphere, and (D) a vase-shaped surface of revolution. The actuation profile required to obtain each desired shape was solved using \ref{['eq: alpha-ODE']}.
  • Figure 4: Inverse design of a generic woven smart-fabric surface. (A) Target face geometry given as a triangular mesh. (B) An approximate solution of \ref{['eq: wovenfabric-metric']} obtained numerically using the adapted Corman-Crane algorithm (\ref{['appendix: Matlab']}). The algorithm outputs both the weaving domain and the actuation parameter $\alpha(u,v)$ at each point, needed for manufacturing the smart fabric. (C) Upon actuation, the initially planar fabric will adopt the geometry of the target face shape, with the shown thread pattern. (D) Histogram of metric elements confirming the numerical solution. The sharp distribution of the $F$ component at $0$ confirms orthogonality, while the sharp distribution of the $(1/c)$-norm of $(E,G)$ at $4$ confirms adherence to the imposed tight-weave $E-G$ curve \ref{['eq: wovenfabric-metric']}.