A proposed technique of SFEM on solving ordinary random differential equation (original) (raw)
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Differential Equations and Nonlinear Mechanics, 2007
The stochastic finite element method (SFEM) is employed for solving stochastic one-dimension time-dependent differential equations with random coefficients. SFEM is used to have a fixed form of linear algebraic equations for polynomial chaos coefficients of the solution process. Four fixed forms are obtained in the cases of stochastic heat equation with stochastic heat capacity or heat conductivity coefficients and stochastic wave equation with stochastic mass density or elastic modulus coefficients. The relation between the exact deterministic solution and the mean of solution process is numerically studied.
MODIFIED SPECTRAL STOCHASTIC FINITE ELEMENT METHOD FOR SPATIAL DISTRIBUTION OF MATERIAL PROPERTIES
The present study deals with the application of the modified Spectral Stochastic Finite Element Method (SSFEM) in the static analysis of beam and plate bending with uncertain material properties subjected to deterministic load, A fundamental issue in SSFEM is the quantification of the uncertainty characterizing the structural parameters. The uncertainty is quantified by using the theory of stochastic functions such as random processes or random fields. The material property is modeled as a stochastic process. The Karhunen-Loeve expansion is used to represent this process in a computationally expedient manner by means of a set of random variables. Further, the well established deterministic finite element method is used to discretize the differential equations governing the structural response. A spectral expansion of the nodal random variables is introduced involving a basis in the space of random variables. The basis consists of the polynomial chaoses that are polynomials orthogonal with respect to the Gaussian probability measure. The derived results found in good agreement with data obtained by a Monte Carlo Simulation (MCS). Numerical examples are used to show that the modified approach explained in this paper has a good accuracy and efficiency.