Keywords: random fields, perturbation method, stochastic finite element method.

Stochastic computational techniques were implemented using various theoretical and computational approaches [2] (stochastic spectral methods, various Monte-Carlo simulation techniques as well as perturbation technique). The usage, efficiency and computational implementation algorithms strongly depend on the input random fields types, their correlations, interrelations between the first few probabilistic moments as well as numerical technique used to solve the basic deterministic problem. Some of those issues are discussed here in terms of Gaussian input included in the boundary value problem with random parameters being solved by the stochastic perturbation method. This method, applied frequently in its second order moment version, has well-known limitations on the input coefficients of variation and that is why the traditional approach is generalized now and implemented as nth order technique.

The basic idea of the stochastic perturbation approach consists in Taylor series expansion of all the parameters about their spatial expectations using some small parameter ; it yields in case of random function

where is the first variation of about its expected value . The expected value of this function can be derived as [1]

where is the ordinary probabilistic moment of th order [3] and where higher than th order components are neglected.

Therefore, the classical FEM approach for elastostatic problems is based on a solution for the following linear algebraic equations system:

is the solution vector, - the stiffness matrix and represents the external forces vector. In th order perturbation-based General Stochastic Finite Element Method (GSFEM) we solve equivalently from zeroth to nth order equations [1]

The zeroth order solution is determined from the first equation of this system. Inserted into the next equation, it returns first order solution etc. until nth order equation solution is completed. After all the solution vector components are determined, their expected values, variances and the other probabilistic moments can be computed. Numerical convergence of the GSFEM is discussed using simple MAPLE symbolic computations of up to tenth order perturbation-based approximations for the expected values and standard deviations in tension of the linear elastic bar discretised with 10 one-dimensional linear finite elements.

The results of the analysis show that convergence of perturbation technique depends on coefficient of variation of random input (alfab) and, in smaller range, on the perturbation parameter (eps). Even the expected values for a simply tensioned homogeneous bar computed in 2nd (Figure 1) and 10th (Figure 2) order approaches differ by about 10% at the tensioned edge. Further comparative numerical studies with the other stochastic methods are necessary - improved Monte-Carlo analyses of analogous engineering problems would be recommended to establish the most efficient perturbation orders for different input and perturbation parameters and the non-Gaussian variables [3].

1

M. Kaminski, "Stochastic perturbation approach in vibration analysis using Finite Difference Method", Journal of Sound and Vibration, 251(4): 651-670, 2001. doi:10.1006/jsvi.2001.3850

2

M. Kleiber, T.D. Hien, The Stochastic Finite Element Method. Wiley, 1992.

3

E. Vanmarcke, Random Fields. Analysis and Synthesis. MIT Press, 1983.

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