A novel four-field mixed FE approximation for Kirchhoff rods using Cartan’s moving frames

Abstract

A four-field mixed variational principle is proposed for large deformation analysis of Kirchhoff rods with the lowest order C0 mixed FE approximations. The core idea behind the approach is to introduce a one-parameter family of points (the centerline) and a separate one-parameter family of orthonormal frames (the Cartan moving frame) that are specified independently. The curvature and torsion of the curve are then related to the relative rotation of neighboring frames. The relationship between the frame and the centerline is then enforced at the solution step using a Lagrange multiplier (which plays the role of section force). In case of Kirchhoff rods, the cross sectional orientation can be described using frames like the Frenet–Serret, which are defined only using the centerline, thereby demanding higher-order smoothness for the centerline approximation. Decoupling the frame from the position vector of the base curve leads to a description of torsion and curvature that is independent of the position information, thus allowing for simpler interpolations. The four-field mixed variational principle we propose has the frame, section force, extension strain and position vector as input arguments. We discretize the position vector using linear Lagrange shape functions, while the frames are interpolated as piecewise geodesics on the rotation group. Finite element approximations for extensional strain and section force are constructed using constant shape functions. Using these discrete approximations, a discrete mixed variational principle is laid out which is then numerically extremized. The discrete approximation is then applied to a few benchmark problems. Vis-á-vis most available approaches, our numerical studies reveal an impressive performance of the proposed method without numerical instabilities or locking.