Elastic Rod Model And Numerical Algorithm Self-contact

Elastic rod is an important mechanic model which is widely used in engineering systems such as fibers, electric cables and macromolecules. Many problems relate to contact and self-contact problems such as nets, knots and some other super slender elastic rods with complex topological structure. A computational difficulty in simulating a slender elastic rod with self-contacts is the change of its topological structure at the contact point set, which is an important issue in the study of the numerical methods of self-contact elastic rod.The main topic of this thesis includes the dynamical models of a self-contact elastic rod and the numerical methods. The main results are as follows:(1) Simply introducing some basic concepts of geometry and mechanics that are associated with the elastic rod, then calculated the correlation coefficient of elastic rod. Finally, the article discussed the theory of the Cosserat model and the expressions of the Lagrange motion equations.(2) To build a discrete dynamical model, the center line of an elastic rod is discretized into N-equal nodes. We respectively analyzed the centerline elements and the direction element and computed the expression of Lagrange motion equation. In regard to the discrete direction elements of the elastic rod, we got Lagrange motion equations with angular velocity and Euler parameters by introducing angular velocity as state variables.(3) In order to avoid self-penetration of elastic rod, we introduced the contact force and the Coulomb friction model to analyze the contact friction. The contact force and the friction are substituted into the Lagrange equation in chapter2, then we calculated the simplified Lagrange motion equation based on finite element method. At the same time, we gave some numerical examples. At last, we analyzed the relationship between the contact coefficient and self-penetration distance of elastic rod.(4) The contact can be achieved smoothly by adjusting the intensity of the repulsion with electrostatic repulsion as penalty functions, so the difficulty of crossing and topological structural changes that brought by elastic rod can be solved. In order to improve the numerical approach, we introduced a quaternion expression of the Lagrange equation of motion and a test criterion of the contact set. At last, we contrasted self-contact and self-penetration of elastic rod before and after adding penalty function.