A data-based CR-FPK method for nonlinear structural dynamic systems. Issue 4 (July 2018)

• Record Type: Journal Article
• Authors: Li, Jie; Jiang, Zhongming
• Is Part Of: Theoretical & applied mechanics letters. Volume 8:Issue 4(2018)
• Page Start: 231
• Page End: 244
• Publication Date: 2018-07
• Subjects: Data-driven computational mechanics -- FPK equation -- Dimension reduction -- Cell renormalization -- Model-free; Mechanics, Applied -- Periodicals; Mechanics, Analytic -- Periodicals; Mechanics, Analytic; Mechanics, Applied; Periodicals; 620.1
• DOI: 10.1016/j.taml.2018.04.008
• Languages: English
• ISSNs: 2095-0349

Abstract

ABSTRACT: Stochastic dynamic analysis of the nonlinear system is an open research question which has drawn many scholars' attention for its importance and challenge. Fokker–Planck–Kolmogorov (FPK) equation is of great significance because of its theoretical strictness and computational accuracy. However, practical difficulties with the FPK method appear when the analysis of multi-degree-of-freedom (MDOF) with more general nonlinearity is required. In the present paper, by invoking the idea of equivalence of probability flux, the general high-dimensional FPK equation related to MDOF system is reduced to one-dimensional FPK equation. Then a cell renormalized method (CRM) which is based on the numerical reconstruction of the derived moments of FPK equation is introduced by coarsening the continuous state space into a discretized region of cells. Then the cell renormalized FPK (CR-FPK) equation is solved by difference method. Three numerical examples are illustrated and the effectiveness of proposed method is assessed and verified. Graphical Abstract: Stochastic dynamic analysis of the nonlinear system is an open research question which has drawn many scholars' attention for its importance and challenge. Fokker–Planck–Kolmogorov (FPK) equation is of great significance because of its theoretical strictness and computational accuracy. However, practical difficulties with the FPK method appear when the analysis of multi-degree-of-freedom (MDOF) with more general nonlinearity is required. In the present paper, by invoking the idea of equivalence of probability flux, the general high-dimensional FPK equation related to MDOF system is reduced to one-dimensional FPK equation. Then a cell renormalized method (CRM) which is based on the numerical reconstruction of the derived moments of FPK equation is introduced by coarsening the continuous state space into a discretized region of cells. Then the cell renormalized FPK (CR-FPK) equation is solved by difference method. Three numerical examples are illustrated and the effectiveness of proposed method is assessed and verified.