Generalized stochastic cell-based smoothed finite element method (GS_CS-FEM) for solid mechanics G.R. Liu a , W. Zeng a,n , H. Nguyen-Xuan b,c a School of Aerospace Systems, University of Cincinnati, 2851 Woodside Dr, Cincinnati, OH 45221, USA b Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City 700000, Viet Nam c Department of Mechanics, Faculty of Mathematics & Computer Science, University of Science, VNU-HCMC, Nguyen Van Cu Street, District 5, Ho Chi Minh City 700000, Viet Nam article info Article history: Received 30 December 2011 Received in revised form 24 August 2012 Accepted 27 August 2012 Available online 29 September 2012 Keywords: Smoothed finite element method (S-FEM) Stochastic FEM General perturbation technique Monte Carlo simulation abstract The smoothed finite element method (S-FEM) was recently proposed to inject softening effects into and improve the accuracy of the standard FEM. In the S-FEM, the system stiffness matrix is obtained using strain smoothing technique over the smoothing domains associated with cells, nodes, edges or faces to establish models of desired properties. Stochastic FEM is regarded as an extension of the classical deterministic FEM to deal with the randomness of properties of input parameters in solid mechanics problems. In this paper, the cell-based S-FEM (or CS-FEM) is extended for stochastic analysis based on the generalized stochastic perturbation technique. Numerical examples are presented and obtained results are compared with the solution of Monte Carlo simulation. It is found that the present GS_CS-FEM method can improve the solution accuracy significantly for stochastic problems, in terms of the estimated means, variances and other probabilistic characteristics. & 2012 Elsevier B.V. All rights reserved. 1. Introduction For several decades, the finite element method (FEM) has become one of the most popular numerical tools in solving practical problems in aeronautical, mechanical and civil engineer- ing. Due to the complexity nature of the problems, lower order FEM is widely preferred. However, FEM using lower order elements has some inherent limitations leading to deficiency in many applications. For example, it exhibits an overestimation of stiffness matrix and underestimation of the internal strain energy. In addition, the element shape cannot be distorted too much due to the use of mapping in FEM [1]. To overcome these issues, the smoothed finite element method (S-FEM) was proposed recently by combining the strain smoothing technique often used in mesh-free methods [2] with the standard finite element method (FEM) [3,4]. The S-FEM may be formulated using the weakened weak (W2) formulation based on the G space theory [5]. It has been proven that a W2 model behaves ‘‘softer’’ than the corresponding standard Galerkin weak form model. Furthermore, upper bound solutions (to the exact solution for force-driving problems) can be also derived using W2 models with sufficient softness, such as when the NS-FEM is used. Therefore, we can now bound the solution from both sides using a standard weak form together with a proper W2 form [6], if so desired. In a more general form of meshfree setting, the smoothed point interpolation methods (S-PIMs) have also developed as typical W2 models. The S-PIM can be node-based (known as NS-PIM or LC-PIM) [7], edge-based (ES-PIM) [8], and cell-based (CS-PIM) [9]. The S-FEM can be, on the other hand, regarded as the linear form of S-PIM, and it is much simpler than S-PIM but reserving its major properties [10]. The S-FEM models can use the same FEM mesh, and work particularly well for lower order elements. Four different types of smoothing domains created leading to cell-based S-FEM (CS-FEM), node-based S-FEM (NS- FEM), edge-based S-FEM (ES-FEM), and face-based S-FEM (FS- FEM). Each of the four S-FEM models will have different features and properties [11], but they all have the following general properties: (1) There is no need to compute the derivatives of shape functions to form the stiffness matrix, because of the use of smoothing operation and field gradients are computed using shape functions directly; (2) No coordinate transformation or mapping is involved in S-FEM, and hence no restriction on the elements shape and hence mesh can be heavily distorted; (3) Many existing algorithms of FEM can be easily modified and applied to S-FEM with very little change; (4) The approximated accuracy and convergence rate can be improved with little increase in computational cost leading much higher computa- tional efficiency, compared to the standard FEM [3]. Randomness of parameters is a natural characteristic in many engineering systems, and should be properly dealt with in Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/finel Finite Elements in Analysis and Design 0168-874X/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.finel.2012.08.007 n Corresponding author. Tel.: þ41 513 477 5889. E-mail address: zengwe@mail.uc.edu (W. Zeng). Finite Elements in Analysis and Design 63 (2013) 51–61