Rest Shape Optimization for Sag-Free Discrete Elastic Rods

TL;DR

This work addresses sagging in DER-based strand simulations by introducing a rest-shape optimization framework that tunes rest length, curvature, and twist to achieve static equilibrium under gravity. It formulates a kinetic-energy-based objective with a regularizer and box constraints, solved efficiently by Gauss-Newton with a penalty approach. The approach yields sag-free, stable equilibria across diverse strand geometries and loading scenarios, while offering insights into normative conditioning and solver choices. The results suggest practical pathways for more robust inverse-design of hair and cable-like materials, with potential extensions to anisotropic, inhomogeneous, and contact-rich systems.

Abstract

We propose a new rest shape optimization framework to achieve sag-free simulations of discrete elastic rods. To optimize rest shape parameters, we formulate a minimization problem based on the kinetic energy with a regularizer while imposing box constraints on these parameters to ensure the system's stability. Our method solves the resulting constrained minimization problem via the Gauss-Newton algorithm augmented with penalty methods. We demonstrate that the optimized rest shape parameters enable discrete elastic rods to achieve static equilibrium for a wide range of strand geometries and material parameters.

Figures (11)

• Figure 1: Evaluation with a single vertical strand. For each $c_{\mathrm{stretch}}$ trio: (left) simulation results with naive initialization, (middle) with rest shape optimization, and (right) with rest shape optimization and zero gravity during simulation. Our rest shape optimizer modifies the rest length more significantly for softer materials to completely cancel gravity and thus achieve static equilibrium.
• Figure 2: Convergence profile for GD, Newton, and GN (ours). Both Newton and GN quickly converge while GD requires many more iterations.
• Figure 3: Evaluation with a single horizontal strand. For each $c_{\mathrm{bend}}$ trio: (left) simulation results with naive initialization, (middle) with rest shape optimization, and (right) with rest shape optimization and zero gravity during simulation. Our optimizer modifies the rest curvature more significantly for softer strands to achieve static equilibrium (b), (c), and (d) while it fails for too soft material (a).
• Figure 4: Test with a horizontal strand and an additional load (as illustrated with an orange arrow) using our rest shape optimization. For each case, we show the simulation with gravity (left) and without gravity (right). To counteract the additional load, our rest shape optimizer modifies the rest curvature more significantly.
• Figure 5: Stress test with a horizontal strand discretized with $1,000$ vertices. Both approaches successfully achieve static equilibrium.