Condensed Galerkin Time-Element for Structural Dynamics with Adaptive Time-Stepping in Maximum Norm

Abstract

In this paper, the newly-developed condensed Galerkin element of polynomial degree [Formula: see text] for first-order initial-value problems is reformulated for and extended to the structural dynamic equations, yielding a high-performance dynamic element of one-step, unconditionally stable type with the conventional finite element (FE) convergence [Formula: see text] within the element and the extra-superconvergence [Formula: see text] at the element end-nodes. Further, based on the element energy projection (EEP) technique, a superconvergence EEP formula for the condensed element is proposed in the paper and mathematical analysis is presented to prove that the derived EEP solution gains a superconvergence [Formula: see text], which is an order higher than the FE solution of [Formula: see text] for all element degree [Formula: see text] and is hence qualified to serve as a point-wise error estimator. As a result, a simple, efficient, and reliable algorithm with adaptive time-stepping controlled by the maximum norm is proposed. Representative numerical examples are presented to verify the validity of the proposed theorems and to demonstrate the high performance of the proposed element and the associated adaptivity algorithm.