Multi-dimensional Fokker-Planck equation analysis using the modified finite element method

13th International Conference on Motion and Vibration Control; 12th International Conference on Recent Advances in Structural Dynamics

Náprstek, J.^1 ; Král, R.^1

Institute of Theoretical and Applied Mechanics, V.v.i., Prosecká 76, Prague 9

CZ-190 00, Czech Republic^1

关键词: Aeroelastic system;    Arbitrary dimension;    Effective solution;    Finite element meshes;    Multidimensionality;    Single degree of freedom systems;    Specific problems;    Transition process;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/744/1/012177/pdf
DOI  :  10.1088/1742-6596/744/1/012177

来源: IOP

【 摘 要 】

The Fokker-Planck equation (FPE) is a frequently used tool for the solution of cross probability density function (PDF) of a dynamic system response excited by a vector of random processes. FEM represents a very effective solution possibility, particularly when transition processes are investigated or a more detailed solution is needed. Actual papers deal with single degree of freedom (SDOF) systems only. So the respective FPE includes two independent space variables only. Stepping over this limit into MDOF systems a number of specific problems related to a true multi-dimensionality must be overcome. Unlike earlier studies, multi-dimensional simplex elements in any arbitrary dimension should be deployed and rectangular (multi-brick) elements abandoned. Simple closed formulae of integration in multi-dimension domain have been derived. Another specific problem represents the generation of multi-dimensional finite element mesh. Assembling of system global matrices should be subjected to newly composed algorithms due to multi-dimensionality. The system matrices are quite full and no advantages following from their sparse character can be profited from, as is commonly used in conventional FEM applications in 2D/3D problems. After verification of partial algorithms, an illustrative example dealing with a 2DOF non-linear aeroelastic system in combination with random and deterministic excitations is discussed.