Inconsistent Stability of Newmark's Method in Structural Dynamics Applications

Abstract

The stability of numerical time integrators, and of the physical systems to which they are applied, are normally studied independently. This conceals a very interesting phenomenon, here termed inconsistent stability, wherein a numerical time marching scheme predicts a stable response about an equilibrium configuration that is, in fact, unstable. In this paper, time integrator parameters leading to possible inconsistent stability are first found analytically for conservative systems (symmetric tangent stiffness matrices), then several structural arches with increasing complexity are used as numerical case studies. The intention of this work is to highlight the potential for this unexpected, and mostly unknown, behavior to researchers studying complex dynamical systems, especially through time marching of finite element models.

Extended framework of Hamilton's principle applied to Duffing oscillation

“…In fact, physically meaningful variable is J (not J ) as shown in Eqs. We would like to conclude this section with pointing out that unlike other numerical methods [48,49], the present method is non-iterative to analyze the forced-damped Duffing equation.…”

Highly precise time integration method for linear structural dynamic analysis

“…Consistent stability refers to that for a physically stable system, a numerically stable time integrator is expected, whereas for a physically unstable system, a numerically unstable time integrator is desirable. The expanded system (3) can be decomposed into a series of single‐DOF systems with the same form as where p and q are real numbers, which can be positive, zero, or negative.…”

“…In the following, physical and numerical stabilities are introduced briefly, and then, the consistent stabilities of the three employed schemes, ie, CDM ( = 1/2, = 0), TR ( = 1/2, = 1/4), and IDM ( = 11/20, = 3/10), are investigated on the basis of the work. 28 Physical stability…”

A robust composite time integration scheme for snap-through problems

“…Even though it leads to retrieving the correct stability in the case of stable systems, unconditional stability does not necessarily imply that the prediction is a good approximation of the true solution, but only that it remains bounded. Unconditionally stable integrators may, when applied to complex systems with multiple equilibria, predict trajectories that are non-physical and/or greatly underestimate the severity of the response [45]. Following similar methods with those used in [29,45], a bound on the time step size can be determined to enforce the condition that the algorithm will not predict a stable response for an unstable linear system.…”

“…Unconditionally stable integrators may, when applied to complex systems with multiple equilibria, predict trajectories that are non-physical and/or greatly underestimate the severity of the response [45]. Following similar methods with those used in [29,45], a bound on the time step size can be determined to enforce the condition that the algorithm will not predict a stable response for an unstable linear system. The maximal time steps for the methods compared in this paper are shown in Fig.…”