Computing failure boundaries by continuation of a two-point boundary value problem
Hinke M. Osinga
We propose a novel approach to investigating the parameter dependence of system behaviour for models that are subject to an external force. As a particular example we consider the analytical model of a tied rocking block on an elastic foundation, which exhibits dynamics equivalent to that of a planar, post-tensioned frame on a shake table; we are interested in predicting behaviour of models subject to an aperiodic external force (an earthquake), but in this paper we restrict to periodic external forcings only. The failure boundary separates initial conditions, given by the angle and angular velocity of the rocking block, for which trajectories starting from time 0 move past a given maximum angle (in absolute value) from those that remain in the admissible regime for arbitrarily long time integration. There are no methods to compute such a failure boundary directly. Instead, numerical studies have, so far, applied brute-force simulations over a grid of initial conditions.
This paper presents an efficient method, based on the fact that the failure boundary must consist of initial conditions that graze the maximum-angle line. We set up a two-point boundary value problem (2PBVP) that defines trajectory segments starting from a given initial condition and ending at the maximum angle (or its negative) with zero angular velocity. We use continuation to find a one-parameter family of initial conditions that solve this 2PBVP; more precisely, we compute two continuous curves of solutions, one for the positive and one for the negative maximum angle. The failure boundary is a piecewise-smooth curve composed of a finite subset of bounded segments from these two families. We describe properties of the failure boundary in detail and discuss how parameter variation can cause the admissible regime to split into two disjoint pieces.
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