Solving non‐linear problems by complex time step methods

Abstract

SUMMARYRecently, a new type of time step integration algorithms using complex time steps has been proposed. For linear problems, the algorithms are higher order accurate, unconditionally stable and have directly controllable numerical dissipation. Solutions with high accuracy can be generated using large time steps. In this paper, the algorithms are extended to solve non-linear problems. The pseudo-force approach is adopted in treating the non-linear terms. To maintain the solutions accuracy, the pseudo-force is reconstructed by interpolation. Special treatments are required to compute the excitation at the complex time steps. Several numerical examples are analysed. It is observed that the complex time step method can be computationally more e cient than the Newmark method when very accurate numerical solutions are required.

Numerical dissipation in time‐step integration algorithms for structural dynamic analysis

“…Fung36 proposed a sub‐stepping procedure to construct time‐step integration algorithms with arbitrary order of accuracy. When the sub‐steps chosen are complex, the complex time‐step method is obtained37–40.…”