Accelerated simulation and design optimization of elastic rod networks with a spline-based least-squares formulation

• Formulation of Bézier spline-based model for least squares solution of bigon arm Boundary Value Problems (BVPs). • Benchmark study of computation speed and robustness of spline-based solver compared to several standard BVP solvers. • Introduction of a novel unconstrained optimization approach for simultaneous solution and shape optimization of Elastic Rod Networks. The approach is validated with numerical experiments and physical prototypes. Multi-stable Elastic Rod Networks (ERNs) have applications in adaptive systems for structural and aerospace engineering and soft robotics. Due to the nonlinear mechanical behavior and ill-conditioned system Jacobian matrix near unstable equilibria, the simulation and design optimization of ERNs is a computationally challenging task. To address these issues, an efficient simulation and optimization methodology for multi-stable ERNs is proposed. The methodology is based on finding least-squares solutions to the Kirchhoff rod boundary value problem (BVP). We apply this methodology to a class of ERNs made up of bigons, individually bistable mechanical units with tunable curvature. A Bézier curve-based computational model is defined to model linear bigon assemblages, also referred to as bigon arms. The performance of the proposed solver is benchmarked against several conventional BVP solvers, showing a significant computational efficiency and robustness to noise. Building upon the proposed solver, a novel shape optimization framework is introduced. The framework optimizes the bigon arm shape to approximate a design curve by adding shape-specific objectives to the least-squares function. The methodology is demonstrated to enable efficient physics-based shape optimization of multi-stable ERNs to target curves and end planes, by numerical experiments and validated with physical prototypes.