Energy-decaying and momentum-conserving schemes for transient simulations with mixed finite elements

Abstract

We present the four-field (of displacements, velocities, stresses and strains) and the three-field (of displacements, velocities and stresses) mixed variational formulations for structural dynamics, which can be used to derive in an elegant way the energy-decaying and momentum-conserving (EDMC) time-stepping schemes for transient simulations with mixed solid and structural finite elements. These mixed variational formulations are in this work applied for the derivation of EDMC schemes for the geometrically exact, recently proposed, mixed-hybrid, shell finite elements (with excellent performance), which combine mixed interpolations of the Hu–Washizu type or the Hellinger–Reissner type with the assumed natural strain concept. To this end, the first-order and the second-order accurate EDMC schemes are designed. The superior properties of the considered mixed-hybrid shell finite elements, which are reflected in statics by excellent convergence properties, ability to take very large solution steps, and low-sensitivity to mesh distortion, are in this manner extended for (long-term) transient simulations. Numerical examples, which illustrate transient simulations with the EDMC schemes and mixed-hybrid shell finite elements, show (among other effects) that the underlying structure of the shell motion is numerically preserved even for very large time steps.