Direct time integration algorithm with controllable numerical dissipation for structural dynamics: Two-step Lambda method

In present paper the ability to control numerical dissipation is added to algorithm of two-step level-symmetric LS-2 integration method designed for numerical solution of stiff structural dynamic problems. LS integration methods were presented in original work of the author in [International Journal for Numerical Methods in Engineering 71 (2007) 1598–1632]. These methods are L-stable in sense of “Maximal Damping of High-frequency Accelerations (MDHA)”. Family of unconditionally stable implicit LS methods provides strong dissipation of high-frequency modes and has so-called U0–V0 overshoot properties (zero-order displacement and velocity overshoots). LS methods belong to generalized linear multi-step class of integration methods with second derivative (GLMSd2) and employ direct use of non-modified dynamic equations. Besides, LS methods show the identical rate of convergence in displacements, velocities and accelerations or, in other words, they are “symmetrically converging” (SC) algorithms in all three integration levels. In the current paper, two-step Lambda integration method is developed. This method also has U0–V0 and SC features and depends on a parameter λ such that as λ is varied from 0 to 1, the method is varied from LS-2 method with MDHA property of dissipation to Newmark method of average acceleration. Since latter method has no numerical damping, dissipation can be changed from maximum to zero. This provides wide functionality of Lambda method. Lambda integration method is unconditionally stable and has the second order of accuracy. Numerical dissipation, phase dispersion and high-frequency damping of this “direct time” two-step method are compared with similar properties of contemporary “weighted residual” one-step integration methods, such as Hilber–Hughes–Taylor (HHT-α) method and generalized single step single solve optimal (GSSSS-optimal) method.