Generalising direct time-stepping algorithms for transient problems in solid mechanics

A new family of single-step time integration methods, referred to as the generalized method, has been developed for solving general nonlinear transient problems. Most of the existing implicit and explicit single-step algorithms are shown to be special cases of the generalized method. The derivation of the generalized method starts with the Taylor series approximation of the displacement field with respect to time from a known state time station. The velocity and acceleration fields are obtained by taking the first and the second derivatives of the Taylor approximation of displacement field, respectively. Equilibrium is satisfied in a weighted-average sense over the time-step through the Galerkin procedure in weak form. The displacement, velocity and acceleration fields for the next time-step are updated using their truncated Taylor series expansions. The generalized method is simple to implement. Furthermore, the computation cost of advancing one step using this algorithm is the same as that of a single-step Collocation algorithm, such as the Hilber-Hughes-Taylor Alpha method, the Wilson method, among others. This paper focuses mainly on the philosophy of the generalization of time-stepping algorithms, the generalization process, and the potential for optimizing the generalized algorithm. The generalized method originally has nine integration parameters. Optimal values of the integration parameters can be obtained by imposing second order accuracy and unconditional stability, by minimizing the dissipation error and the dispersion errors, by controlling overshooting and algorithmic damping for higher modes, etc. A member of the family of generalized methods has been obtained and compared with other existing algorithms. It is found that the developed algorithm is superior to the others in all compared aspects.