Development of a Direct Time Integration Method Based on Quartic B-spline Collocation Method

Abstract

A new version of quartic B-spline direct time integration method for dynamic analysis of structures is presented. This procedure is derived based on uniform quartic B-spline piecewise polynomial approximations and collocation method, named Alpha Quartic B-spline time integration method. In this way, at first, the method is implemented to solve the governing differential equation of motion of single-degree-of-freedom systems, and later, the proposed method is generalized for multi-degree-of-freedom systems. Stability and accuracy analysis of the proposed algorithm have been investigated completely. In the proposed algorithm by using two collocation parameters α1 and α2, unconditional stability is achieved, but a local instability is created. The best values of these two parameters have been determined not only to maintain the stability, but also to ensure the desired accuracy. For accuracy analysis, dissipation and dispersion errors have been investigated for different cases of α’s. Finally, for the proposed method, a simple step-by-step algorithm was presented. The effectiveness and robustness of the proposed algorithm in solving linear dynamic problems are demonstrated in the numerical examples.