A weak form temporal quadrature element formulation for linear structural dynamics

Various time integration methods and time finite element methods have been developed to obtain the responses of structural dynamic problems, but the accuracy and computational efficiency of them are sometimes not satisfactory. The purpose of this paper is to present a more accurate and efficient formulation on the basis of the weak form quadrature element method to solve linear structural dynamic problems.

A variational principle for linear structural dynamics, which is inspired by Noble's work, is proposed to develop the weak form temporal quadrature element formulation. With Lobatto quadrature rule and the differential quadrature analog, a system of linear equations is obtained to solve the responses at sampling time points simultaneously. Computation for multi-elements can be carried out by a time-marching technique, using the end point results of the last element as the initial conditions for the next.

The weak form temporal quadrature element formulation is conditionally stable. The relation between the normalized length of element and the suggested number of integration points in one element is given by a simple formula. Results show that the present formulation is much more accurate than other time integration methods and its dissipative property is also illustrated.

The weak form temporal quadrature element formulation provides a choice with high accuracy and efficiency for solution of linear structural dynamic problems.