A PDE approach for the invariant measure of stochastic oscillators with hysteresis

TL;DR

This work addresses invariant measures for a white-noise-driven three-dimensional bilinear elasto-plastic oscillator (BEPO) modeled as a stochastic variational inequality. It develops a PDE-based computation using backward Kolmogorov equations and a Lyapunov function to establish the existence of an invariant measure, and it provides a scalable finite-difference scheme to approximate the measure. Two concrete applications—threshold-crossing frequency and the probability of serviceability—demonstrate the method’s accuracy and efficiency relative to Monte Carlo simulations. The results show that the PDE approach yields reliable stationary statistics for high-dimensional, nonsmooth hysteretic systems and offers a practical alternative to trajectory-based simulations in structural dynamics.

Abstract

This paper presents a PDE approach as an alternative to Monte Carlo simulations for computing the invariant measure of a white-noise-driven bilinear oscillator with hysteresis. This model is widely used in engineering to represent highly nonlinear dynamics, such as the Bauschinger effect. The study extends the stochastic elasto-plastic framework of Bensoussan et al. [SIAM J. Numer. Anal. 47 (2009), pp. 3374--3396] from the two-dimensional elasto-perfectly-plastic oscillator to the three-dimensional bilinear elasto-plastic oscillator. By constructing an appropriate Lyapunov function, the existence of an invariant measure is established. This extension thus enables the modelling of richer hysteretic behavior and broadens the scope of PDE alternatives to Monte Carlo methods. Two applications demonstrate the method's efficiency: calculating the oscillator's threshold crossing frequency (providing an alternative to Rice's formula) and probability of serviceability.

Figures (5)

• Figure 1: Constitutive models. (a) An EPPO model that contains a linear mass, a dashpot, and a spring connected in series with a Coulomb friction-slip joint. (b) A BEPO model, with a spring connected in parallel based on the EPPO model. The two models are excited by a time-dependent random force $\sigma \dot{W}_{t}$, where $\dot{W}_t$ is a white noise, $\sigma$ is the noise intensity.
• Figure 2: Archetypal evolution of the restoring force $\mathbf{F}(t)$ versus $X(t)$. (a) The EPPO model. In the plastic phase, $\mathbf{F}(t)$ remains a constant. (b) The BEPO model. In the plastic phase, $\mathbf{F}(t)$ is a linear function.
• Figure 3: Discretization of $\mathcal{D}_{XY}$. At black triangles, the discretized equation is satisfied. At gray round points, homogeneous Neumann boundary conditions are used. The non-standard boundary conditions are employed at blue squares/diamonds and red squares.
• Figure 4: Frequency of threshold crossing per unit time interval for $a_{1} \in [-3,3]$.
• Figure 5: Probability of the serviceability in the limit state for $a_{2}\in[0,3.5]$.

Theorems & Definitions (16)

• Remark 2.1
• Theorem 3.1
• Remark 3.2
• Theorem 3.3
• Theorem 3.4
• Corollary 3.5
• Theorem 3.6
• Remark 3.7
• Remark 3.8
• Remark 4.1