On nonlinear geometric transformations of finite elements

The paper develops a systematic procedure for formulating finite elements on manifolds. The theoretical developments give rise to a modular computational framework for composing coordinate transformations and manifold parameterizations. The procedure is demonstrated with the Cosserat rod model furnishing a novel finite element formulation that rectifies the lack of objectivity of existing finite elements without violating the director constraints or compromising the symmetry of the tangent stiffness at equilibrium. The framework is element-independent, allowing its implementation as a wrapper to existing element libraries without modification of the element state determination procedures.