Homotopy analysis method for linear and nonlinear stochastic problems

Obtaining convergent solutions to stochastic problems with large random function remains challenging for engineering structural analysis. In this study, a procedure based on homotopy analysis method (HAM) is developed to solve linear and nonlinear stochastic problems. Based on least square approximation principle, the expectation of stochastic square residual error (ESRE) is proposed to determine the optimal convergence-control parameter for the homotopy-series solution of stochastic problems. Further, a stochastic finite element method based on HAM (SFEM-HAM) is used to study the stochastic vibration of engineering stochastic structures, heat conduction, and diffusion of chloride ions in concrete. The calculation accuracy and efficiency of the proposed, perturbation, polynomial chaos expansion, and Monte Carlo simulation methods are compared in four examples. The results of the study show that the convergent explicit homotopy-series solution of these stochastic problems can be obtained based on ESRE, HAM, and SFEM-HAM, regardless of the magnitude of the random fluctuation. The proposed method can achieve significantly accurate results, compared with the Monte Carlo simulation and perturbation methods, particularly for nonlinear stochastic problems.