An efficient numerical method based on a Sobolev gradient is presented to find the equilibrium shapes of an elastic rod resistant to bending and twisting. A Sobolev preconditioning operator is introduced based on the energy of the elastic rod that preconditions the standard conjugate gradient method and results in a speedy convergence. It outperforms the usual conjugate gradient and steepest descent methods without requiring expensive calculation of the dense Hessian associated with the total energy of the rod. The method has also been validated to predict the Michell’s instability in a closed rod with a twist.

中文翻译：

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基于梯度的弹性杆数值模拟方法

提出了一种基于 Sobolev 梯度的有效数值方法来寻找抗弯曲和扭曲的弹性杆的平衡形状。基于弹性杆的能量引入了 Sobolev 预处理算子，该算子对标准共轭梯度方法进行了预处理，并导致快速收敛。它优于通常的共轭梯度和最速下降方法，而无需对与杆的总能量相关的密集 Hessian 进行昂贵的计算。该方法也已被验证可以预测带有扭曲的封闭杆中的 Michell 不稳定性。