[en] 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.
Disciplines :
Civil engineering Engineering, computing & technology: Multidisciplinary, general & others
Author, co-author :
Canor, Thomas ; Université de Liège - ULiège > Département ArGEnCo > Analyse sous actions aléatoires en génie civil
Denoël, Vincent ; Université de Liège - ULiège > Département ArGEnCo > Analyse sous actions aléatoires en génie civil
Language :
English
Title :
Transient Fokker-Planck-Kolmogorov equation solved with smoothed particle hydrodynamics method
Publication date :
May 2013
Journal title :
International Journal for Numerical Methods in Engineering
ISSN :
0029-5981
eISSN :
1097-0207
Publisher :
Wiley, Chichester, United Kingdom
Volume :
94
Issue :
6
Pages :
535–553
Peer reviewed :
Peer Reviewed verified by ORBi
Funders :
F.R.S.-FNRS - Fonds de la Recherche Scientifique [BE]
Lin YK, Cai GQ. Probabilistic Structural Dynamics. Advanced Theory and Applications, 2nded. McGraw-Hill: New-York, 2004.
Oksendal B. Stochastic Differential Equations, 3rded. Springer-Verlag: Heidelberg, 1992.
Risken H. The Fokker-Planck Equation. Methods of Solution and Applications, 2nded. Springer-Verlag: Berlin, 1996.
Stratonovitch RL. Topics in the Theory of Random Noise, 1sted., Vol.1. Gordon and Breach Science Publishers: New-York, 1963.
Wong E, Zakai M. On the relation between ordinary and stochastic differential equations. International Journal of Engineering Sciences 1965; 3:213-229.
Kroese DP, Taimre T, Botev ZI. Handbook of Monte Carlo Methods, Series in Probability and Statistics. Wiley: Hoboken, New-Jersey, 2011.
Grigoriu M. Stochastic Calculus. Applications in Science and Engineering, 1sted. Birkhauser: Boston, 2002.
Roberts JB, Spanos PD. Random Vibration and Statistical Linearization, Dover Edition. Dover Publications: New-York, 2003.
Proppe C, Pradlwarter HJ, Schuëller GI. Equivalent linearization and Monte-Carlo simulation in stochastic dynamics. Probabilistic Engineering Mechanics 2003; 18(1):1-15.
Soize C. Steady-state solution of Fokker-Planck equation in high dimension. Probabilistic Engineering Mechanics 1988; 3(4):196-206.
Chang JS, Cooper G. A practical difference scheme for Fokker-Planck equations. Journal of Computational Physics 1970; 6:1-16.
Pichler L, Masud A, Bergman LA. Numerical solution of the Fokker-Planck equation by finite difference and finite element methods - a comparative study. In Proceeding of the Third Conference on Computational Methods in Structural Dynamics and Earthquake Engineering - COMPDYN 2011, Corfu, Greece, May 2011; 2983-2999.
Spencer BF, Bergman LA. On the numerical solution of the Fokker-Planck equation for nonlinear stochastic systems. Nonlinear Dynamics 1993; 4:357-372.
Langley RS. A finite element method for thes tatistics of random nonlinear vibration. Journal of Sound and Vibration 1985; 101:41-54.
Wojtkiewicz SF, Bergman LA. Numerical solution of high dimensional Fokker-Planck equations. In 8th ASCE Specialty Conference on Probabilistic Mechanics and Structural Reliability, NotreDame, IN, USA, 2000.
Langtangen HP. A general numerical solution method for Fokker-Planck equations with applications to structural reliability. Probabilistic Engineering Mechanics 1991; 6:33-81.
Kumar M, Singla P, Junkins J, Chakravorty S. A multi-resolution meshless approach to steady state uncertainty determination in nonlinear dynamical systems. In Proceeding of the 38th IEEE Southeastern Symposium on Systems Theory, Cookeville, TN, USA, March 2006; 344-348.
Kumar M, Chakravorty S, Singla P, Junkins JL. The partition of unity finite element approach with hp-refinement for the stationary Fokker-Planck equation. Journal of Sound and Vibration 2009; 327(1-2):144-162.
Kumar M, Chakravorty S, Junkins JL. A semianalytic meshless approach to the transient Fokker-Planck equation. Probabilistic Engineering Mechanics 2010; 25(3):323-331.
Lozinski A, Chauviière C. A fast solver for Fokker-Planck equation applied to viscoelastic flows calculations: 2D FENE model. Journal of Computational Physics 2003; 189(2):607-625.
Chauvière C, Lozinski A. Simulation of dilute polymer solutions using a Fokker-Planck equation. Computers & Fluids 2004; 33(5-6):687-696.
Chinesta F, Chaidron G, Poitou A. On the solution of Fokker-Planck equations in steady recirculating flows involving short fiber suspensions. Journal of Non-Newtonian Fluid Mechanics 2003; 113(2-3):97-125.
Ammar A, Mokdad B, Chinesta F, Keunings R. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. Journal of Non-Newtonian Fluid Mechanics 2006; 139(3):153-176.
Ammar A, Mokdad B, Chinesta F, Keunings R. A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids: Part II: Transient simulation using space-time separated representations. Journal of Non-Newtonian Fluid Mechanics 2007; 144(2-3):98-121.
Chupin L. Fokker-Planck equation in bounded domain. Annales de l'Institut Fourier 2010; 60(1):217-255.
DerKiureghian A. Structural reliability methods for seismic safety assessment: a review. Engineering Structures 1996; 18(6):412-424.
Lucy LB. A numerical approach to the testing of the fission hypothesis. Astronomical Journal 1977; 82:1013-1024.
Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Notices of the Royal Astronomical Society 1977; 181:375-389.
Feldman J, Bonet J. Dynamical refinement and boundary contact forces in SPH with applications in fluid flow problems. International Journal for Numerical Methods in Engineering 2007; 72:295-324.
Morris JP. Simulating surface tension with Smoothed particle hydrodynamics. International Journal for Numerical Methods in Fluids 2000; 33(3):333-353.
Rabczuk T, Eibl J. Simulation of high velocity concrete fragmentation using sph/mlsph. International Journal for Numerical Methods in Engineering 2003; 56:1421-1444.
Rabczuk T, Belytschko T. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering 2007; 196(29-30):2777-2799.
Johnson GR, Stryk RA, Beissel SR. SPH for high velocity impact computations. Computer Methods in Applied Mechanics and Engineering 1996; 139(1-4):347-373.
Cleary PW, Monaghan JJ. Conduction modelling using Smoothed particle hydrodynamics. Journal of Computational Physics 1999; 148:227-264.
Jeong JH, Jhon MS, Halow JS, van Osdol J. Smoothed particle hydrodynamics : Application to heat conduction. Computers Physics Communication 2003; 153:71-84.
Randles PW, Libersky LD. Smoothed particle hydrodynamics: some recent improvements and applications. Computer Methods in Applied Mechanics and Engineering 1996; 139:375-408.
Monaghan JJ. Smoothed particle hydrodynamics. Reports on Progress in Physics 2005; 68:1703-1759.
Liu GR, Liu MB. Smoothed Particle Hydrodynamics. A Meshfree Particle Method. World Scientific Publ.: Singapore, 2003.
Liu M, Liu G. Smoothed particle hydrodynamics (SPH): an overview and recent developments. Archives of Computational Methods in Engineering 2010; 17(1):25-76.
Chaubal CV, Srinivasan A, Egecioglu O, van Leal LG. Smoothed particle hydrodynamics techniques for the solution of kinetic theory problems. Journal of Non-Newtonian Fluid Mechanics 1997; 70:125-154.
Chaubal CV, Leal LG. Smoothed particle hydrodynamics techniques for the solution of kinetic theory problems: Part 2. The effect of flow perturbations on the simple shear behavior of LCPs. Journal of Non-Newtonian Fluid Mechanics 1999; 82(1):25-55.
Ammar A, Chinesta F. A Particle Strategy for Solving the Fokker-Planck Equation Modelling the Fiber Orientation Distribution in Steady Recirculating Flows Involving Short Fiber Suspensions - Meshfree Methods for Partial Differential Equations II, volume 43 of Lecture Notes in Computational Science and Engineering. Springer Berlin Heidelberg: Heidelberg, 2005. 1-15.
Prulière E. Modélisation de la microstructuration dans les polymères chargés. application à la mise en forme. Ph.D. Thesis, Université Joseph Fourier, Grenoble, 2007.
Liu MB, Liu GR, Lam KY. Constructing smoothing functions in smoothed particle hydrodynamics with applications. Journal of Computational and Applied Mathematics 2003; 155:263-284.
Lacombe G. Analyse d'une équation de vitesse de diffusion. Comptes Rendus de l'Académie des Sciences, Mathématique 1999; 329:383-386.
Combis P, Fronteau J. A purely Lagrangian method for numerical integration of Fokker-Planck equation. Europhysics Letters 1986; 2:227-232.
Canor T, Denoël V. Transient Fokker-Planck equation solved with SPH. In Proceeding of the Fifth International Conference ACOMEN 2011, Liege, Belgium, November 2011.
Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes. The Art of Scientific Computing, 3rded. Cambridge University Press: Cambridge, 2007.
Swegle J, Hicks J, Attaway S. Smoothed particle hydrodynamics stability analysis. Journal of Computational Physics 1995; 116:123-134.
Belytschko T, Guo Y, KamLiu W, PingXiao S. A unified stability analysis of meshless particle methods. International Journal for Numerical Methods in Engineering 2000; 48(9):1359-1400.
Rabczuk T, Belytschko T, Xiao SP. Stable particle methods based on Lagrangian kernels. Computer Methods in Applied Mechanics and Engineering 2004; 193(12-14):1035-1063.
Vidal Y, Bonet J, Huerta A. Stabilized updated Lagrangian corrected SPH for explicit dynamic problems. International Journal for Numerical Methods in Engineering 2007; 69(13):2687-2710.
DiPaola M, Mendola L, Navarra G. Stochastic seismic analysis of structures with nonlinear viscous dampers. Journal of Structural Engineering 2007; 133(10):1475-1478.
Pekcan G, Mander JB, Chen SS. Fundamental considerations for the design of non-linear viscous dampers. Earthquake Engineering and Structural Dynamics 1999; 28(11):1405-1425.
Khalil HK. Nonlinear Systems, 3rded. Prentice Hall: Upper Saddle River, New-Jersey, 2002.
Ismail M, Ikhouane F, Rodellar J. The hysteresis Bouc-Wen model, a survey. Archives of Computational Methods in Engineering 2009; 16(2):161-188.
Wen YK. Method for random vibration of hysteretic systems. Journal of the Engineering Mechanics Division ASCE 1976; 102(2):249-263.
Ikhouane F, Manosa V, Rodellar J. Adaptive control of a hysteretic structural system. Automatica 2005; 41(2):225-231.
