Transient Fokker-Planck-Kolmogorov equation solved with smoothed particle hydrodynamics method

in International Journal for Numerical Methods in Engineering (2013), 94(6), 535553

Probabilistic theories aim at describing the properties of systems subjected to random excitations by means of statistical characteristics such as the probability density function (pdf). The time ... [more ▼]

Probabilistic theories aim at describing the properties of systems subjected to random excitations by means of statistical characteristics such as the probability density function (pdf). The time evolution of the pdf of the response of a randomly excited deterministic system is commonly described with the transient Fokker-Planck-Kolmogorov equation (FPK). The FPK equation is a conservation equation of a hypothetical or abstract fluid, which models the transport of probability. This paper presents a generalized formalism for the resolution of the transient FPK equation using the well-known mesh-free Lagrangian method, Smoothed Particle Hydrodynamics (SPH). Numerical implementation shows notable advantages of this method in an unbounded state space: (i) the conservation of total probability in the state space is explicitly written, (ii) no artifact is required to manage far- eld boundary conditions , (iii) the positivity of the pdf is ensured and (iv) the extension to higher dimensions is straightforward. Furthermore, thanks to the moving particles, this method is adapted for a large kind of initial conditions, even slightly dispersed distributions. The FPK equation is solved without any a priori knowledge of the stationary distribution; just a precise representation of the initial distribution is required. [less ▲]

Efficient uncoupled stochastic analysis with non-proportional damping

in Journal of Sound & Vibration (2012), 331(2012), 52835291

The use of normal modes of vibration in the analysis of structures with nonproportional damping reduces the size of the resulting set of governing equations, but does not decouple them. A common practice ... [more ▼]

The use of normal modes of vibration in the analysis of structures with nonproportional damping reduces the size of the resulting set of governing equations, but does not decouple them. A common practice consists in decoupling the equations by disregarding the o -diagonal elements in the modal damping matrix. Recently, an approximation based on an asymptotic expansion of the modal transfer matrix has been proposed in a deterministic framework to partially account for o -diagonal terms, but still with a set of uncoupled equations. This paper aims at extending this method in a stochastic context. First the mathematical background is introduced and the method is illustrated with a simple example. Then its relevance is demonstrated within the context of the structural analysis of a large and realistic structure. [less ▲]

Transient Fokker-Planck Equation solved with SPH

in Proceedings of the 5th International Conference on Advanced Computational Methods in Engineering (2011)

In 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 ... [more ▼]

In 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. [less ▲]

Amortissement non-linéaires des structures par des câbles

Master's dissertation (2010)

Ce travail traite de l'interaction câble-structure et évalue la possibilité de pouvoir intégrer des câbles comme dispositifs anti-vandalisme ou comme dispositifs de confort sur des passerelles. LE cadre ... [more ▼]

Ce travail traite de l'interaction câble-structure et évalue la possibilité de pouvoir intégrer des câbles comme dispositifs anti-vandalisme ou comme dispositifs de confort sur des passerelles. LE cadre de cette étude et la passerelle de Limbourg, structure métallique hybride entre un arc et une poutre Virendeel. Un résumé de modélisation de l'action humaine sur des structures est proposé. L'action de vandales est modélisée par des sauts successifs en utilisant le modèle de Racic et Pavic. Après une description de la passerelle, des mesures réalisées sur celle-ci et des différents modèles éléments-finis exploités, l'interaction câble-structure est étudiée successivement à l'aide d'un modèle analytique, puis d'un modèle numérique. Le modèle analytique est le modèle de Ernst-Bleich. La structure est projetée dans son premier mode propre de vibration. Les simulations sont menées à l'aide de la méthode de Newmark non-linéaire. Les résultats obtenus avec ce modèle sont intéressants au regard de la dynamique non-linéaire, mais l'exacerbation de la sensibilité aux conditions initiales rend difficile toute interprétation ou prédiction. Le modèle numérique pour le câble est construit à l'aide d'éléments finis de barres non-linéaires. La structure est toujours projetée dans son premier mode propre. L'amortissement dans le câbles est intégré comme un amortissement proportionnel avec le modèle de Rayleigh. La dissipation dans le câble permet d'augmenter l'amortissement structurel équivalent de manière significative. Une étude paramétrique est menée pour dégager des paramètres pertinents afin d'utiliser des câbles comme éléments finis dissipatifs dans une structure. [less ▲]