Semi-explicit Unconditionally Stable Time Integration method based on Generalized-α technique

In structural dynamic analysis, various time integration techniques have been proposed. Generally, these algorithms discretize the time domain into a finite number of intervals and approximate the displacements, velocities, and accelerations via mathematical expressions at each time increment. Based on the structure of these approximations, time integration schemes are classified as explicit and implicit. Explicit schemes are much simpler and often march forward only through pure vector operations. On the other hand, implicit strategies require more computational efforts especially in nonlinear behaviors since they involve solving a system of simultaneous equations at each time step using iterative techniques. Although computationally more expensive, implicit schemes are unconditionally stable, meaning that the growth of solution errors at each time increment remains bounded. On the contrary, explicit techniques suffer from instabilities which manifest as unrealistic growth of amplitude of the responses. To overcome this issue, time step size should be chosen small enough to meet the stability criterion. In this paper, by gathering the advantages of both approach, a new semi-explicit unconditionally stable time integration method based on the well-known implicit Generalized-α (G-α) technique is proposed. To this end, first, the fundamental approximating relationships of the suggested method is introduced for a single degree of freedom system with the unknown integration parameters. Then, using the concept of amplification matrix, these unknown parameters are determined so that the method possesses the same characteristic equation as the G-α technique. This leads to a set of model-dependent integration parameters that are no longer scalar constants. Due to this kind of formulation, similar stability and accuracy behavior are observed when comparing the proposed method with the G-α technique, both analytically and numerically. After generalization of the proposed algorithm to the multi-degree of freedom systems, some numerical examples are solved and comparisons are also made with other similar time integration schemes. Findings reveal the merits of the proposed algorithm over the other well-known time stepping techniques.