Complex-time-step Newmark methods with controllable numerical dissipation

In this paper, unconditionally stable higher-order accurate time-step integration algorithms with controllable numerical dissipation are presented. The algorithms are based on the Newmark method with complex time steps. The ultimate spectral radius (μ), the sub-step locations (βj) and the weighting factors (αj) are the algorithmic parameters. For an algorithm that is (2n−1)th order accurate, the sub-step locations which may be complex, are shown to be the roots of an nth degree polynomial. The polynomial is given explicitly in terms of n and μ. The weighting factors are then obtained by solving a system of n simultaneous equations. It is further shown that the order of accuracy is increased by one for the non-dissipative algorithms with μ=1. The stability properties of the present algorithms are studied. It is shown that if the ultimate spectral radius is set between −1 and 1, the eigenvalues of the numerical amplification matrix are complex with magnitude less than or equal to unity. The algorithms are therefore unconditionally C-stable. When the ultimate spectral radius is set to 0 or 1, the algorithms are found to be equivalent to the first sub-diagonal and diagonal Pade approximations, respectively. The present algorithms are more general as the numerical dissipation is controllable and are very suitable for parallel computers. The accuracy of the excitation responses is found to be enhanced by the present complex-time-step procedure. To maintain high-order accuracy, the excitation may need some modifications. © 1998 John Wiley & Sons, Ltd.