Stochastic Analysis of Elastostatic Problems with Material Uncertainty Using Spectral Cell Method

Document Type : Original Research

Author
P. Zakian
Department of Civil Engineering, Faculty of Engineering, Arak University
10.22034/23.5.10
Abstract
Nowadays, advances in numerical methods have led to model real-life physical problems effectively. One of the difficulties in modelling the real-life physical problems is the geometric creation, because the mesh definition for a complex geometry is hard. In order to overcome this issue, one can use the spectral cell method due to employing a Cartesian mesh even for a complex geometry, such that constant Jacobian is considered for cells in the mesh. Spectral cell method is a combination of the spectral element method and the fictious domain concept, which uses an adaptive integration employing the quadtree or octree partitioning for the cells intersecting arbitrary boundaries as well as the cells including nonuniform material distribution. The interpolation functions of Lobatto family of spectral elements are utilized in spectral cell method. The spectral cell method is an efficient numerical method to solve the governing equations of continuum structures with complicated geometries. On the other hand, uncertainty naturally exists in the parameters of an engineering system (e.g., elastic modulus) and the input of that system (e.g., loading). Thus, the effects of those uncertainties are important in the response calculation of the engineering system. There are two types of uncertainty: aleatoric and epistemic. Aleatoric uncertainty is defined as an intrinsic variability of certain quantities, while epistemic uncertainty is defined as a lack of knowledge about certain quantities. An alternative to a deterministic modelling is a stochastic modelling, but analysing such a stochastic model is harder than a deterministic model having deterministic material properties and configuration. This is because the behaviour of the stochastic model is inevitably stochastic. Traditionally, Monte-Carlo simulation analyses a stochastic model by generating numerous realizations of the stochastic problem, and then solves each one like a deterministic problem. Nevertheless, Monte-Carlo simulation needs very high computational cost, particularly for large-scale problems. A systematic technique for uncertainty quantification is the stochastic finite element method providing a variety of statistical information. However, the method is computationally expensive with respect to the finite element method, and thus there are many developments for stochastic methods. Consequently, this paper presents stochastic form of spectral cell method to solve elastostatic problems considering material uncertainties. Therefore, uncertainty quantification of an elastostatic problem with geometrically complex domain can be modelled more efficiently than the traditional stochastic finite element method. In the proposed method, Fredholm integral equation is discretised using spectral cell method to solve Karhunen-Loève expansion used for the random field decomposition. Also, this method uses fewer cells than the stochastic finite cell method, and does not require formation of the eigenfunctions. In addition, Karhunen-Loève and polynomial chaos expansions are used to decompose the random field and to consider the response variability, respectively. Simple mesh generation, desirable accuracy and computational cost are the main features of the present method. In this study, two benchmark numerical examples are provided to demonstrate the efficiency and capabilities of the proposed method in the solution of elastostatic problems. The results are compared to those of stochastic finite element method and stochastic spectral element method.

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Modares Civil Engineering journal
Volume 23, Issue 5
November and December 2023
Pages 139-149