Inverse Discrete Elastic Rod

TL;DR

The paper tackles inverse design for slender elastic structures, introducing inverse-DER to reconstruct undeformed configurations from given deformations under general boundary conditions and external fields. By reformulating the inverse problem as static equilibrium in the reference configuration and leveraging the DER framework with inverse dynamics, the method achieves forward-simulation–level efficiency without optimization overhead. Key contributions include a general inverse simulation framework, a path-independent inverse-DER formulation with parallel transport, and validation on both slender rods and nets under gravity and magnetic actuation, as well as demonstrations of computational efficiency and energy behavior. The work enables rapid, high-fidelity inverse design for applications in fabrication, soft robotics, and architectural nets, with future potential to extend to shells, solids, and non-conservative forces.

Abstract

Inverse design of slender elastic structures underlies a wide range of applications in computer graphics, flexible electronics, biomedical devices, and soft robotics. Traditional optimization-based approaches, however, are often orders of magnitude slower than forward dynamic simulations and typically impose restrictive boundary conditions. In this work, we present an inverse discrete elastic rods (inverse-DER) method that enables efficient and accurate inverse design under general loading and boundary conditions. By reformulating the inverse problem as a static equilibrium in the reference configuration, our method attains computational efficiency comparable to forward simulations while preserving high fidelity. This framework allows rapid determination of undeformed geometries for elastic fabrication structures that naturally deform into desired target shapes upon actuation or loading. We validate the approach through both physical prototypes and forward simulations, demonstrating its accuracy, robustness, and potential for real-world design applications.

Figures (10)

• Figure 1: The forward and inverse simulation framework.
• Figure 2: The computational detail of the discrete 3D curve. (a) The parallel transport between the reference configuration and the current configuration. (b) The computation of the discrete reference twist.
• Figure 3: The three curve-discretized surfaces with various Gaussian curvatures. (a) Spherical curve. (b) Conical curve. (c) Hyperbolic curve.
• Figure 4: Inverse design of compressed curve-discritized surfaces. (a) The comparison between simulation and experimental results of UC and DC for the spherical curve. (b) The comparison between simulation and experimental results of UC and DC for the conical curve. (c) The comparison between simulation and experimental results of UC and DC for the hyperbolic curve. The red parts of two ends are clamped.
• Figure 5: Inverse design of helical curve under gravity and hyperbolic curve under magnetic field. (a) Undeformed shapes of helical curves subjected to varying gravitational strengths. (b) Undeformed shapes of hyperbolic curves subjected to different magnetic field intensities.