Transient Fokker-Planck equation with SPH method application in seismic design

Abstract

In reliability theory, the knowledge of the time evolution of the probability density function (pdf) of the response of a diffusive random system may be required. The time evolution of this pdf is described by the transient Fokker-Planck-Kolmogorov (FPK) equation. This equation models the conservation of probability in the system state space. The transient FPK equation is an interesting tool for the study of reduced models of structures in the context of transient random excitations such as earthquakes. In this case, the deterministic approach can be competitive compared with classical Monte Carlo simulations, especially for extreme value problems. This paper presents a flexible method for the resolution of transient FPK equation using a Lagrangian method inspired by the mesh-free method: Smoothed Particle Hydrodynamics (SPH) method. Numerical implementation shows notable advantages of SPH method for solving FPK equation in an unbounded state-space compared with mesh-based method: (i) the conservation of total probability is explicitly written, (ii) no artefact is required in low density zones, (iii) the positivity of the pdf is ensured, (iv) the formalism offers straightforward extension to higher dimension systems, (v) the method is adapted for a large kind of initial conditions, even slightly dispersed distributions. In the paper, the method is illustrated with a particular interest for reliability problems in seismic applications. Probabilities of failure are calculated for strongly nonlinear systems (nonlinear viscous forces, hysteretic behaviour) subjected to transient excitations, for reasonable computation times compared with stochastic simulations.