Ship Electromechanical Equipment Institute, Room 315, Mechanical and Electrical Building, Dalian Maritime University, No.1 Linghai Road, Ganjingzi District, Dalian, 116026, Liaoning, China.
Sci Rep. 2023 Mar 24;13(1):4867. doi: 10.1038/s41598-023-32064-y.
This study develops forces equilibrium differential equations for the geometric modeling of 1D flexible objects with surface constraints. These second-order equations are an extension of the Cosserat elastic rod theory and include both bending and torsion. Variables were established for the centerline and attitude in the Cartesian coordinate system of the cross section. This paper specifically investigates the case of a 1D flexible object constrained by a cylindrical surface. To solve this problem, a novel hybrid semi-analytical numerical method is proposed. In this process, a Hamiltonian function and an initial integral operator are introduced in a cylindrical coordinate system. The analytical solution, decoupled in polar coordinates, is then derived. The improved finite difference method was then used to obtain three cylindrical coordinates, which ensured numerical stability and efficiency. The results of a geometric shape simulation with differing boundary conditions demonstrate that this proposed method is capable of real-time modeling. As such, this technique could be a promising new tool for use in graphics simulations of elongated structures, such as DNA molecules, drill pipes, and submarine cables.
本研究针对具有曲面约束的一维柔性物体的几何建模开发了力平衡微分方程。这些二阶方程是科萨捷弹性杆理论的扩展，包括弯曲和扭转。变量在截面的笛卡尔坐标系中建立了中心线和姿态。本文专门研究了受圆柱面约束的一维柔性物体的情况。为了解决这个问题，提出了一种新的混合半解析数值方法。在这个过程中，在圆柱坐标系中引入了哈密顿函数和初始积分算子。然后推导出在极坐标中解耦的解析解。然后使用改进的有限差分法获得三个圆柱坐标，以确保数值稳定性和效率。具有不同边界条件的几何形状模拟结果表明，该方法能够实时建模。因此，该技术可能成为伸长结构（如 DNA 分子、钻杆和海底电缆）图形模拟的有前途的新工具。
