Optimization of a class of composite method for structural dynamics

A class of composite method that implements the trapezoidal rule in the first few sub-steps and the backward difference formula in the last sub-step is studied in this paper. The optimal schemes of two sub-step, three sub-step and four sub-step methods, where the four sub-step composite scheme is developed for the first time, are proposed by optimizing their accuracy. Compared with several existing composite methods, the optimal schemes are also endowed with second-order accuracy, unconditional stability and strong numerical damping, and they can achieve higher amplitude and period accuracy under the same amount of calculation. Moreover, it follows that in the optimal schemes the more sub-steps the higher accuracy, so the optimal four sub-step method is highly recommended. Several test problems are used to validate the performance.