Extension of LMS formulations for L‐stable optimal integration methods with U0–V0 overshoot properties in structural dynamics: the level‐symmetric (LS) integration methods

Abstract

AbstractIn the present article a new family of linear multi‐step (LMS) optimal integration methods for stiff structural dynamic problems is synthesized. Developing the ideas following the original works in References (Int. J. Numer. Meth. Engng 2004; 59:597–668, Int. J. Numer. Meth. Engng 2004; 60:1699–1740, Int. J. Numer. Meth. Engng 2006; 66:1738–1790) of design of optimal and controllable dissipation, extensions are made in this paper to a class of optimal integration methods with maximum possible damping of high frequencies separately from the degree of damping of low frequencies. Generalized single step single solve (GSSSS) optimal algorithm recently developed and basic well‐known ‘weighted‐residual’ methods are compared with ‘level‐symmetric’ (LS) integration methods proposed. These LS methods are created as symmetric variants of extended three‐level (3L‐LMS) integration algorithm with direct use of dynamic equations to obtain algorithmically simple integration methods, which belong to [U0–V0] L‐stable optimal class without overshoots and have the maximum damping of high‐frequency modes. General formulas for L‐stable family of multi‐step LS‐N methods are obtained. Standard two‐level representation (2L‐LMS) of 3L‐LMS integration algorithm is also obtained, and L‐stable second‐order accurate LS‐BDF integration method for the stiff first‐order ordinary differential equations is proposed. Roots, dissipation and dispersion properties of LS‐1 integration method (second‐order accurate in displacement) and of other obtained LS‐2, LS‐3, LS‐4 methods (third‐order accurate in displacement) are analysed and demonstrated. Comparison with some up‐to‐date integration methods is considered in three numerical examples. Copyright © 2007 John Wiley & Sons, Ltd.