Transient Fokker-Planck Equation solved with SPH

peer reviewedIn many engineering matters, systems are submitted to random excitations. Probabilistic theories aim at describing the properties of a system by means of statistical properties such as probability density function (pdf). For a deterministic system randomly excited, the evolution of its pdf is commonly described with Fokker-Planck-Kolmogorov equation (FPK). The FPK equation is a conservation equation of a hypothetical fluid, which represents physically the transport of probability. To solve this equation, Smoothed Particle Hydrodynamics (SPH) are used: the system is modelled with a conservation equation for the system and a transport equation for each particle. Numerical implementation shows the superiority of this method over many other mesh-based methods: (i) the conservation of total probability in the state space is explicitly written, (ii) no specific boundary conditions must be imposed if an adaptive smoothing length is chosen and if particles are initially regularly spread out, (iii) the positivity of the pdf is ensured. Furthermore, thanks to the moving particles, this method is adapted for a large kind of initial conditions (quasi-deterministic or even discontinuous). The FPK equation can be solved without any a priori knowledge of the stationary distribution; just a precise representation of the initial distribution is required