Development of a Stable Implicit Method Without Numerical Dissipation for Nonlinear Structural Dynamic Analysis

Abstract

This paper develops a self-starting implicit time integration method for nonlinear dynamic analysis. The method is single-step and is based on a non-dissipative second-order accurate approximation scheme. The method employs simple expressions for velocity and acceleration while internal forces are calculated at a specified point. This point results from a weighted combination of the relevant displacements and velocities at the bounds of the time step. In the linear case, the scheme is unconditionally stable, there is no amplitude decay for any time step value and the period elongation is the same as in the Newmark scheme. In nonlinear case, the scheme maintains the same numerical characteristics of dissipation and dispersion of the Newmark scheme while demonstrating a substantial improvement in stability. Summarizing, the presented scheme (i) is a single-step scheme with self-starting attribute; (ii) does not involve additional variables or artificial parameters chosen by the user; (iii) has at least second-order accuracy; (iv) shows to have very wide stability ranges in nonlinear analyses. It also proves that the computational effort per step required by the presented scheme is the same as the effort required in the widely used implicit single-step methods and a substantial reduction in the number of Newton–Raphson iterations in the steps is obtained. For comparison, methods with similar characteristics such as Newmark, generalized-[Formula: see text] and Bathe schemes are tested. Stiff nonlinear problems regarding elastic behavior and finite element structural models are analyzed to compare the considered solution methods.