Elastic Rod Dynamics Model And Its Numerical Method

An elastic rod is an important model widely used in Science and Engineer studies. Many systems,such as undersea cabals,fibers and DNA,can be modeled as elastic rods. There have been many important models be constructed,such as Kirchhoff equations and SMK models.Numerical approaches for simulating them are also found in recent years. Modeling and numerical methods are discussed in this paper.The main results include:(1) A mathematical model is given,which is based on more general conditions such as the rod experiences bend,twist,shear,extension,external force and external moment. Non-orthogonal local coordinate system(d₁,d₂,d₃),which is different from the cross section coordinate frames(e₁,e₂,e₃),is constructed.In the local system,the relationships of related variables in dynamic rod such as bend,twist,shear,extension and external force can be represented well;the balance equations based on the cross section coordinate frame are derived.Compared with the Kirchhoff equations and the SMK equations,the balance equations are more general.Consider how to make the numerical computation and the sureness the initial boundary problem easier,second-order balance equations are given.(2) Using finite elements method,semi-discrete scheme and the rigid matrix of the equations are derived.Because the rigid matrix is three-dimensional,the semi-discrete scheme of the equations is not easy to solve.Two-dimensional group of numbers can be represented as vector or matrix in Matlab.Using this feature,the semi-discrete scheme of the equations can be represented as matrix differential equation,and so that the work of computation is simplified a lot.