A re-evaluation of overshooting in time integration schemes: The neglected effect of physical damping in the starting procedure

An important property of implicit time integration algorithms for structural dynamics is their tendency to “overshoot” the exact solution in the first few steps of the computed response due to high-frequency components in the initial excitations. The typical analysis technique for overshooting involves the study of asymptotic response of the algorithm's first step in the limiting high frequency case. This article finds that the prior analysis of overshooting in much of the engineering literature is incomplete in that it neglects the effect of physical damping. With physical damping included, first-order overshooting components enter into several well-known time integration algorithms which were previously thought to exhibit zero-order overshooting in displacement. The Newmark method, Wilson- (Formula presented.) method, Bazzi- (Formula presented.) method, HHT- (Formula presented.) method, WBZ- (Formula presented.) method, and three parameter optimal/generalized- (Formula presented.) method are analyzed, as well as the generalized single-step single-solve (GSSSS) framework which encompasses all of the prior schemes and other new and optimal algorithms and designs based upon the issues under consideration. The additional overshooting component is eliminated in the novel amended GSSSS V0 family (which is noteworthy and cannot be derived by conventional means), while the numerically dissipative schemes in the GSSSS U0 family (encompassing traditional methods such as the HHT- (Formula presented.) method, WBZ- (Formula presented.) method, and three parameter optimal/generalized- (Formula presented.) method among other new algorithm designs) are shown to be irremediable as the additional overshooting component from physical damping enters into the second step of the response, which is a wholly new finding. Numerical verifications of the overshooting analysis are performed for SDOF and MDOF structures with and without physical damping, and practical recommendations are given for the solution of MDOF problems on the basis of algorithm design and selection for various initial conditions.