A multi-step transversal linearization (MTL) method in non-linear structural dynamics

An implicit family of multi-step transversal linearization (MTL) methods is proposed for efficient and numerically stable integration of nonlinear oscillators of interest in structural dynamics. The presently developed method is a multi-step extension and further generalization of the locally transversal linearization (LTL) method proposed earlier by Roy (Proceedings of the Academy of the Royal Society (London) 457 (2001) 539–566), Roy and Ramachandra (Journal of Sound and Vibration 41 (2001a) 653–679, International Journal for Numerical Methods in Engineering 51 (2001b) 203–224) and Roy (International Journal of Numerical Methods in Engineering 61 (2004) 764). The MTL-based linearization is achieved through a non-unique replacement of the nonlinear part of the vector field by a conditionally linear interpolating expansion of known accuracy, whose coefficients contain the discretized state variables defined at a set of grid points. In the process, the nonlinear part of the vector field becomes a conditionally determinable equivalent forcing function. The MTL-based linearized differential equations thus become explicitly integrable. Based on the linearized solution, a set of algebraic, constraint equations are so formed that transversal intersections of the linearized and nonlinearized solution manifolds occur at the multiple grid points. The discretized state vectors are thus found as the zeros of the constraint equations. Simple error estimates for the displacement and velocity vectors are provided and, in particular, it is shown that the formal accuracy of the MTL methods as a function of the time step-size depends only on the error of replacement of the nonlinear part of the vector field. Presently, only two different polynomial-based interpolation schemes are employed for transversal linearization, viz. the Taylor-like interpolation and the Lagrangian interpolation. While the Taylor-like interpolation leads to numerical ill-conditioning as the order of interpolation increases, the Lagrangian interpolation is shown to overcome this numerical problem. Finally, the family of MTL methods is illustrated through limited numerical results for a couple of harmonically driven workhorse oscillators, viz. the hardening Duffing and the Duffing–Holmes’ oscillators. Comparisons with results obtained via a sixth-order Runge–Kutta method with adaptive time step-sizes are also provided.