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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Higher-order theories can signify the kinematics better, may disregard shear correction factors and yield more accurate transverse shear stresses. Higher-order theories in the thickness direction were addressed in the literature see e.g. [(Kapania & Raciti 1989; Reddy 1997)]. In particula r, Carrera’s investigation (Carrera 1996; Carrera 1998) demonstrates interesting approaches of computing transverse and normal stresses in laminated composite or sandwich plates. Ferreira et al. (Ferreira et al. 2004) conducted proper a research on the RBFs combined with the higher-order theories for plates and beams of composite structures. 3. The RBF formulation Since the RBF method is a meshless method simple to understand and implement, it is slowly making its way into the toolbox of researchers and scientists dealing with the computational solution of partial differential equations. This special issue is a much-needed effort to collect the state-of-the-art in this field. Radial basis functions are generated by a single univariate basic function which is shifted to various center points and composed by a Euclidean norm to make it a radial function. Due to scalar nature of the constructed vector norm, this approach generalizes immediately to any space dimension, and thus radial basis functions can be used for multivariate approximation problems. Considering a set of nodes 1 , 2 , … ∈ Ω ⊂ ℝ , the radial basis functions centred at is defined as: ( ) ≡ (‖ − ‖) ∈ ℝ , = 1,2, … , (13) Being ‖. ‖ the Euclidean norm. Consider the generic boundary value problem with a domain Ω ⊂ ℝ with boundary Ω and linear differential operators and , a linear elliptic partial differential equation is written as; ( ) = ( ), ∈ Ω ⊂ ℝ ; | Ω = (14) The nodes along the boundary is defined as { , = 1, 2, … , } and in the interior as { , = +1 , … , } . The function ( ) can be approximated by: ( ) ≃ ̅ ( ) = ∑ =1 (15) where = { 1 , 2 , … , } are parameters to be determined after the collocation method is applied. A global collocation method was considered where the linear operators ℒ and ℬ acting in Ω \ Ω domain and boundary Ω define a set of global equations in the form: ( ) ( ) = ( ) [ ][ ] = [ ] (16) In which and denote domain and boundary nodes, respectively; and are some external conditions on the domain and boundary. The function represents a radial basis function. In this study, a piecewise smooth, compact radial function ( ) , Wendland type order 6 (W6), was considered (Schaback & Wendland 2001; Wendland 2005). ( ) = (1 − ) 8 [32( ) 3 + 25( ) 2 + 8 + 1] (17) where is the shape parameter and r presents the distance between points belonging the point distribution discretizing the problem domain. The RBF Wendland function (Schaback & Wendland 2001; Wendland 2005) accounts for the lowest degree guaranteeing that [ ] matrix is positive definite for all distinct node locations in the 2D and 3D. The shape parameter controls their ‘flatness’. In the case of the Wendland functions, their ‘order’ refers to their degree of smoothness. For the solution of the static problem and the Eigen problem, one can see (Ferreira et al. 2011).