Theoretical background and implementation of the finite element method for multi-dimensional Fokker–Planck equation analysis. (November 2017)

• Record Type: Journal Article
• Authors: Král, Radomil; Náprstek, Jiří
• Is Part Of: Advances in engineering software. Volume 113(2017)
• Page Start: 54
• Page End: 75
• Publication Date: 2017-11
• Subjects: Fokker–Planck equation -- Finite Element Method -- Simplex element -- Multi-dimensional problem -- Non-symmetric operator; ADES Advances in Engineering Software (journal name) -- CPU Central Processing Unit (time consumption) -- DOF Degree of freedom -- FDM Finite Difference Method, -- FEM Finite Element Method -- FE Finite Element (used when a particular element is addressed) -- FPE Fokker–sPlanck equation -- JPEM Journal of Probabilistic Engineering Mechanics -- MCD Method of Central Differences -- MD multi-dimensional (space) -- MDOF, n-DOF Multiple degree of freedom or n-Degree of freedom -- ODE ordinary differential equation (system) -- PDE partial differential equation -- PDF probability density function -- SDOF, TDOF Single or Two degrees of freedom (system) -- 2D, 3D, 4D two, three or four dimensional (space or problem) -- SM, EM System matrix or Evolution matrix -- SPMD Single Program-Multiple Data parallel computing; Computer-aided engineering -- Periodicals; Engineering -- Computer programs -- Periodicals; Engineering -- Software -- Periodicals; Periodicals; 620.0028553
• DOI: 10.1016/j.advengsoft.2017.02.013
• Languages: English
• ISSNs: 0965-9978

Abstract

Highlights: Overall computational strategy for stochastic analysis of nonlinear multi-DOF dynamic systems. Description of multi-dimensional Finite Element mesh generator adapted for Fokker-Planck equation solution. Numerical or closed-form spatial integration of simplex element in n-dimensional space. Probability space integration method for visualization of 2-D conditional probability. Abstract: Fokker–Planck equation is one of the most important tools for investigation of dynamic systems under random excitation. Finite Element Method represents very effective solution possibility particularly when transition processes are investigated or more detailed solution is needed. However, a number of specific problems must be overcome. They follow predominantly from the large multi-dimensionality of the Fokker–Planck equation, shape of the definition domain and usual requirements on the nature of the solution which are out of a conventional practice of the Finite Element employment. Unlike earlier studies it is coming to light that multi-dimensional simplex elements are the most suitable to be deployed. Moreover, new original algorithms for the multi-dimensional mesh generating were developed as well as original procedure of the governing differential and algebraic systems assembling and subsequent analysis. Finally, an illustrative example is presented together with aspects typical for the problem with large multi-dimensionality.