Optimizing Parameters for Static Equilibrium of Discrete Elastic Rods with Active-Set Cholesky

Abstract

We propose a parameter optimization method for achieving static equilibrium of discrete elastic rods. Our method simultaneously optimizes material stiffness and rest shape parameters under box constraints to exactly enforce zero net force while avoiding stability issues and violations of physical laws. For efficiency, we split our constrained optimization problem into primal and dual subproblems via the augmented Lagrangian method, while handling the dual subproblem via simple vector updates. To efficiently solve the box-constrained primal subproblem, we propose a new active-set Cholesky preconditioner. Our method surpasses prior work in generality, robustness, and speed.

Figures (8)

• Figure 1: Our method optimizes material stiffness and rest shape parameters to achieve static equilibrium for DER-based strands. It preserves the original hairstyle without sagging, demonstrates natural dynamic strand behaviors in response to prescribed motions of the root vertices, and eventually restores the strands to their initial configuration. Our parameter optimization required 713 seconds for 3.2k strands; forward simulation without collision handling took 25 seconds per frame.
• Figure 2: Coil-like strand test. Rest shape optimization achieves static equilibrium unlike naive initialization (a). 4D rest curvatures with unbanded/banded systems generate identical results (b), while the reduced 2D rest curvatures also produce equivalent results given the redundant force spaces (c).
• Figure 3: We vertically translate the root position (initially the leftmost endpoint) of a horizontal strand up and down. The leftmost column shows the initialized states for each method and the remaining columns from left to right show the results of forward simulation. The penalty method fails to precisely enforce box constraints, causing significant rest curvature changes and allowing the strand's tail to end up on the left side after the motion stops (b). By contrast, our active-set method (c) strictly enforces the box constraints, keeping the tail on the right side and better preserving the original shape than the naive initialization (a).
• Figure 4: Evaluation with a vertical strand. The first edge (black) is fixed so its stretching stiffness parameter is undefined. The white/green edges represent lower/higher stiffness parameters. (a) Rest shape only. (b) Rest shape and material stiffness with penalty. (c) Rest shape and material stiffness with ALM (ours). With the rest shape only optimization, material stiffness parameters are unchanged (edges stay white), failing to achieve static equilibrium with rest shape parameters within their box constraints (a). Optimizing the material stiffness parameters additionally stiffens the edges and thus reduces the necessary rest shape changes. Treating the zero net force constraint as a hard constraint enables more significant stiffness changes to achieve static equilibrium (c), compared to using a soft one (b).
• Figure 5: Evaluation with a horizontal strand. Material stiffness parameters for bending are undefined for the first and last vertices (black). The white and green vertices represent lower and higher stiffness parameters, respectively. The insets provide enlarged views for clarity. The rest shape only optimization fails to achieve static equilibrium (a). While simultaneously optimizing rest shape and material stiffness parameters better enforces zero forces, treating the zero net force constraint as a soft constraint still fails (b), but as a hard constraint it succeeds in achieving the horizontal static equilibrium (c).