Finite element method analysis of Fokker–Plank equation in stationary and evolutionary versions
Introduction
Response of dynamical system subjected to an external excitation with random character in time can be investigated by several methods. Classical methods being based on spectral and correlation principles are effective in linear cases with additive Gaussian excitation only. Although their application can be considered even in more general cases, the efficiency should be always carefully premeditated. A solution procedure could shift out of the predetermined aim very easily and the result would be far from an original intention. The main reason are various hidden properties of these methods being based essentially on the superposition principle. Consequently, this modesty should be applied every time when multiplicative processes appear and mainly when non-linear systems are to be discussed.
Many uncertainties can be avoided using methods based on the theory of Markov processes. They are more general from the viewpoint of the type and structure of system which should be investigated. However they include certain conditions limiting admissible types of input processes. For instance, it is usual to presume that excitation processes are of Wiener type. In such a case the probability density function (PDF) can be described by means of the Fokker–Planck (FP) equation admitting an evolution of the PDF in time. If the relevant solution succeeds to be found, it can be taken as a natural extension of a deterministic result. It gives the full information about a response random character and enables to deduce also additional special attributes of the response, such as frequency and local stability.
Section snippets
Physical system and Fokker–Planck equation
A response of the mechanical system results from an external excitation being in general of deterministic and random character. This is commonly treated by means of the differential system of the first order in the normal form. Random effects are introduced separately in a form of certain linear combinations of input processes. Let us accept, that satisfactorily general formulation can be expressed as follows:
- wr(t)
– Gaussian white noises
Linear single degree of freedom system
Let us assume that an SDOF system, see Fig. 2, is excited by an additive noise and a multiplicative noise in the damping coefficient. The respective differential equation reads:Processes wb = wb(t), wa = wa(t) are supposed to be centered Gaussian white noises. Their densities are denoted: Kbb, Kab, Kaa. Drift and diffuse coefficients follows from Eq. (3):The FP equation can be
Non-linear system of Duffing type
The Duffing equation in basic or normal form under white noise additive and multiplicative excitations can be written as follows:Eq. (3) imply relevant drift and diffuse coefficients:Using Eqs. (13), (14) relevant FP equation can be evolved:In the absence of a multiplicative noise (Kss = K
Nonlinear system of Van der Pol type
The Van der Pol equation in basic or normal form being excited by additive and multiplicative random noises has a form:Using Eq. (3) drift and diffusion coefficients can be easily derived:FP equation corresponding to Van der Pol system (17) reads:Contrary to
Conclusion
The Fokker–Planck equation represents an important tool determined for probabilistic analysis of dynamic systems subjected to additive and multiplicative random excitation by Gaussian white noises. Possibilities to solve this equation using analytical or semi-analytical methods are limited. It seems that the finite element method is able to occupy an important position among other numerical methods considered for FP equation analysis. It is challenging that many aspects of FP equation remaining
Acknowledgment
The kind support of the Czech Science Foundation Project No. 103/09/0094, Grant Agency of the ASCR Project No. A200710902 and RVO 68378297 institutional support are gratefully acknowledged.
References (34)
Transient responses of dynamical systems with random uncertainties
Probabilist Eng Mech
(2001)- et al.
Petrov–Galerkin finite element solution for the first passage probability and moments of first passage time of the randomly accelerated free particle
J Comput Methods Appl Mech Eng
(1981) - et al.
Application of multi-scale finite element methods to the solution of the Fokker–Planck equation
J Comput Methods Appl Mech Eng
(2005) A consistent method for the solution to reduced FPK equation in statistical mechanics
Phys A: Stat Theor Phys
(1999)- et al.
Approximate solution of the Fokker–Planck–Kolmogorov equation
Probabilist Eng Mech
(2002) A finite element method for the statistics of nonlinear random vibration
J Sound Vib
(1985)A general numerical solution method for Fokker–Planck equations with applications to structural reliability
J Probabilist Eng Mech
(1991)- et al.
A multiscale-stabilized finite element method for the advection–diffusion equation
J Comput Methods Appl Mech Eng
(2004) - et al.
Response of stochastic dynamical systems driven by additive Gaussian and Poissonwhite noise: solution of a forward generalized Kolmogorov equation by a spectral finite difference method
J Comput Methods Appl Mech Eng
(1999) Exact stationary solution for a class of non-linear systems driven by non-normal delta-correlated processes
Int J Non-Linear Mech
(1995)
Response of dynamic systems to Poisson white noise
J Sound Vib
Stochastic integro-differential and differential equations of nonlinear systems excited by parametric poisson pulses
Int J Non-Linear Mech
Parallel processing in computational stochastic dynamics
J Probabilist Eng Mech
Multi-scale finite element modelling using bubble function method for a convection–diffusion problem
J Chem Eng Sci
Stochastic differential equations
Random vibrations of elastic systems (in Russian)
Stochastic differential systems-analysis and filtering
Cited by (28)
Separable Gaussian neural networks for high-dimensional nonlinear stochastic systems
2024, Probabilistic Engineering MechanicsA fully adaptive method for structural stochastic response analysis based on direct probability integral method
2022, Computer Methods in Applied Mechanics and EngineeringCitation Excerpt :The probability transformation method to obtain the probability density function (PDF) was introduced by Falsone and Settineri [17–19]. For more complex nonlinear systems, the statistical linearization [20], equivalent nonlinear system [21], path integral method [22,23], stochastic averaging method [24], Hamiltonian system method [25], and some other numerical techniques [26,27] were developed to resolve the corresponding problem. However, these methods may not be always applicable for the strong nonlinear system [4].
Study on stationary probability density of a stochastic tumor-immune model with simulation by ANN algorithm
2022, Chaos, Solitons and FractalsCitation Excerpt :Therefore, an inevitable task related to the stochastic tumor-immune model is how to solve FPE or AFPE analytically and numerically as well. As far as by now, some numerical methods (such as finite difference method [22,23], finite element method [24,25], variation method [26,27], path integral method [28], Galerkin method [29,30]) and simulation method (such as Monte-Carlo technique [31–35]) have been gradually developed to solve FPE or AFPE. Relatively speaking, the numerical methods highly rely on the grid discretization of the calculation domain, the sparsity and the dense of the grid, which will affect greatly the accuracy and calculation quantity [21].
New semi-analytical solutions of the time-fractional Fokker–Planck equation by the neural network method
2022, OptikCitation Excerpt :High accuracy methods can depict the anomalous diffusion phenomenon more accurately. Some efficient numerical methods have been developed, for example, finite difference method [7], finite element method [8] and path integral method [9–11]. Nevertheless, we often need the analytical solutions for theoretical analysis.
A model for describing the velocity of a particle in Brownian motion by Robotnov function based fractional operator
2020, Alexandria Engineering Journal