Issue 29

F. Tornabene et alii, Frattura ed Integrità Strutturale, 29 (2014) 251-265; DOI: 10.3221/IGF-ESIS.29.22 253 found in the form     1 N j j j f x x      (1) These polynomials constitute a linear vector space N V and are used for the approximation (1) [7, 17]. A list of several basis functions is given in Tab. 1 and it should be used as a reference for the present manuscript. It should be mentioned that Tab. 1 presents the polynomials in their definition interval indicated by r . The functional approximation is performed using the Cartesian coordinates, as indicated by Eq. (1). The approximation can be done if the domain is discretized in N discrete points, such as k x for , , , 1 2   k N . Tab. 2 reports the most common grid point distributions that can be found in literature. It is assumed that every grid is defined in the interval   0,1 k   . In addition, some cases need extra points for the enforcement of the boundary conditions, known also as  -point technique [9, 12]). Hence, a grid distribution   0,1 k   without  -points can be defined as , for , 1, 2, ..., k k M N k N      (2) whereas considering  -points it takes the following aspect 1 2 1 1 0, , 1 , 1, , for 2, 2, 3, ..., 1 N N k k M N k M                     (3) The stretching formulation [7, 17] can be defined using the previous definitions (2)-(3), a grid distribution   0,1 k   (Tab. 2) and a non-zero constant  , as     2 3 1 3 2 , for 1, 2, ..., k k k k k N           (4) This technique takes the points in the   0,1 k   domain and stretches them into the domain   0,1 k   . The parameter  cannot take any value because for some entries the distribution   0,1 k   (see for more details [140]). Finally, the collocation points in dimensionless form   0,1 k   must be transformed into the physical interval   , k x a b  , thus, a general coordinate transformation [7] can be used as   , for 1, 2,..., k k b a x c a k N d c        (5) where   , k c d   is a dimensionless discretization. Using the so-called differentiation matrix procedure [7, 10, 11, 17], the approximation (1) for the one-dimensional case can be written in matrix form as  f A λ (6) where       1 2 , , , T N f x f x f x      f  is the vector of the unknown function values, λ is the vector of the unknown coefficients j  and the components of the coefficient matrix A are given by   ij j i A x   for , 1, 2, , i j N   . If a structural component is described in the interval   0, k x   , where  is the domain length, the transformation (5) corresponds to , for 1, 2,..., k k x k N     (7) Expression (1) can be derived for the general n -th order polynomial and the derivative is transferred to the functions   j x  , because the unknown coefficients j  do not depend on the variable x     1 , for 1, 2,..., 1 n n N j j n n j d x d f x n N dx dx        (8) Expression (8) can be written in matrix form as follows