Formulation and analysis of variational methods for time integration of linear elastodynamics

Abstract

A general framework for variational formulations of linear elastodynamics is introduced. Variational formulations which originate from the principle of minimum potential energy, the Hellinger-Reissner principle and the Hu-Washizu principle are considered. They are characterized by the weak enforcement of the velocity-displacement relation and of the initial displacement and velocity. Special cases are obtained by a priori enforcement of these conditions. The time integration algorithms obtained discretizing in the space and time domains are studied with reference to accuracy, stability and high-frequency behaviour. It is shown that some of these algorithms are very competitive with respect to classical finite difference methods. The only drawback of variational methods seems to be the large size of the set of simultaneous equations which they generate. Effective solution procedures, aiming at minimizing both the storage and the computational effort, will be discussed in a separate paper.