PSI - Issue 5

Behzad V. Farahani et al. / Procedia Structural Integrity 5 (2017) 584–591 Behzad V. Farahani et al./ Structural Integrity Procedia 00 (2017) 000 – 000

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3.1. Shape parameter definition Many radial basis function methods include a free shape parameter that plays an substantial role for the accuracy of the method. Smaller values generally lead to higher accuracy. It means that it is challenging to get near the polynomial limit in practice. On the other hand, the optimal value, i.e., the value that produces the smallest error, is usually a positive value (Larsson & Fornberg 2005). Fasshauer et al. (Fasshauer & Zhang 2007) numerically investigated two applications of leave-one-out cross validation for the determination of RBF shape parameters; iterated approximate moving least squares approximation, and an RBF pseudo-spectral method. Fasshauer (Fasshauer 2002) studied the shape parameter supporting multiquadric radial basis functions through Newton iteration operator to solve non-linear elliptic PDEs, the shape parameter was reported as: = √ 2 (18) where N can be represented as the total number of points discretizing the problem domain or the number of points in a preferential spatial direction. With a similar scope of the present work, a computational method based on radial basis functions has been applied to the linear solution of thin plates (Ferreira et al. 2005). Moreover, it is possible to find some relevant works regarding RBF studies considering shape parameter concept. [see e.g. (C.M.C. Roque & Ferreira 2009; C. M. C. Roque & Ferreira 2009; Liu & Gu 2001; Roque et al. 2010)]. 4. Analysis and Results In this study, a static analysis of Mindlin plates in bending was taken into account. A simply supported isotropic thick square plate with side = 2 ( ) was considered in this analysis, as Fig. 1-a shows. The plate has a constant thickness of ℎ = 0.2 ( ) and it was submitted to a uniform transverse load on the mid-plane of = 1 ( ) . The modulus of elasticity is taken as E = 109201 (MPa), Poisson’s ratio is taken as ν = 0 . 3 and = 5⁄6 . Moreover, a regular × point cloud is used to discretize the domain [0, ] × [0, ] . For clarification purposes, a 10-by-10 regular nodal distribution is shown in Fig. 1-b. Besides, the maximum non-dimensional transverse displacement is set as (Reddy 1997): ̅ = 11 4 (19) where the flexural stiffness is obtained as 11 = ℎ 3 12(1 − 2 ) ⁄ . According to the convergence study performed by the authors, the optimal shape parameter used in the RBF formulation take the following form: = 1.04 0.04 √ ( ) + ( + ) (20) In which N represents the total number of points along one of the spatial directions and is a non-dimensional constant so, {∀ ∈ ℝ | > 1} . The boundary conditions for an arbitrary edge with simply supported are defined as (referring to Equations 9-12);  For border: = 0 ; = 0 and = 0.  For border: = 0 ; = 0 and = 0. The exact solution of maximum transverse displacement for thick Mindlin plates in bending was considered as ( = 0.00427 ) (Timoshenko & Woinowsky-Krieger 1959). Preliminarily, RBF ̅ results, based on Equation (19), were evaluated in the presence of the Fasshauer’s shape parameter, Equation ( 18), and compared to the exact solution as shown in Fig. 2-a. In addition, the relative error in terms of total number of nodes is shown in Fig. 2-b, where the