New Family of Explicit Structure-Dependent Integration Algorithms with Controllable Numerical Dispersion

Direct integration algorithms are effective methods to solve the temporally discretized differential equations of motion for structural dynamics. Numerous researchers have worked out various algorithms to achieve desirable properties of explicit expression, unconditional stability, and controllable numerical dissipation. However, studies involving the numerical dispersion of integration algorithms are limited. In this paper, a precorrected bilinear transformation from a continuous domain to a discrete domain associating with pole-matching based on the control theory is utilized to develop a new family of explicit structure-dependent integration algorithms, referred to as TL-$\phi$ algorithms. In contrast to the existing algorithms, the significant improvement of the proposed method is that it can control the amount of numerical dispersion by an additional parameter related to the critical frequency of the structure. Stability, energy dissipation, and numerical dispersion properties of the proposed algorithms for both linear and nonlinear systems are fully studied. It is shown that the proposed family of algorithms is unconditionally stable for linear systems while only conditionally stable for nonlinear systems. Though the numerical dissipation property of the TL-$\phi$ algorithms is quite similar to that of other well-developed methods, its ability to minimize the period errors when compared with other methods makes it beneficial to the accuracy of the numerical simulation of dynamic responses. Four numerical examples are used to investigate the improved performance of the new method, and the results show that the proposed algorithms can be potentially used to solve linear and nonlinear structural dynamic problems with desirable numerical dispersion performance.