430 / 2022-08-02 20:47:36

Basis Reduction in the Finite Element Analysis of Fokker-Planck Equations

Fokker-Planck equation, finite element analysis, basis reduction, nonstationary probability density function

We consider the probability density function (pdf) describing the state of a nonlinear, single-degree-of-freedom (SDOF) oscillator subjected to Gaussian white noise excitation. This pdf is governed by the Fokker-Planck (FP) equations obtained systematically from the state-variable representation of the oscillator’s equation of motion. The FP equations are partial differential equations in the displacement and velocity state variables and time, and are readily discretized using a Galerkin finite element (FE) formulation over a suitable domain in the x x phase plane. This results in an equation of the form Mp(t)=-Kp(t), where p(t) is a vector of nodal values of the pdf and M and K are square matrices of corresponding dimensions.

Experience has shown that a practical FE mesh will have at least 100 elements in each dimension of the phase plane, resulting in the vector p having a length of roughly 10,000. The time integration needed to compute the nonstationary pdf starting from known (perhaps deterministic) initial conditions is therefore time-consuming, and may be unacceptable for some purposes, such as numerical design optimization. We attempt to improve the speed of these calculations by representing the FE problem on a reduced basis consisting of a subset of the eigenvectors of the matrix M, and compare the resulting solutions in terms of speed and accuracy.

In our examples we study a Duffing oscillator in detail, but the methods used are extensible to other nonlinearities, such as Van der Pol, and to systems with more degrees of freedom (for which unreduced problem size rapidly grows prohibitive).