Physical modeling and geometric shape simulation for one-dimensional flexible objects with cylindrical surface constraints

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.