Issue 29

F. Tornabene et alii, Frattura ed Integrità Strutturale, 29 (2014) 251-265; DOI: 10.3221/IGF-ESIS.29.22 252 Lagrange polynomials (PDQ) Lagrange trigonometric polynomials (HDQ)                     1 1 1 1, , , , 1,2,..., , j j j j N N k j j k k k j k L r l r r j N r r L r L r r r L r r r                                     1 1 1 1, , 0,2 , 1,2,..., sin 2 sin , sin 2 2 j j j j N N j k k j k k j k S r S r r j N r r S r r r r r S r S r                                     Jacobi polynomials (Jac) Legendre polynomials (Leg)                 , 1 1 1 2 ! 1 1 1,1 , 1,2,..., , , 1 j j j j j j j j d J r r r dr j r r r j N                                    2 1 1 2 ! 1,1 1,2,..., j j j j j j j d P r r j dr r j N             Chebyshev polynomials (I kind) (Cheb I) Chebyshev polynomials (II kind) (Cheb II)       cos arccos , 1,1 , 1,2,..., j j T r j r r j N                       sin 1 arccos , 1,1 , 1,2,..., sin arccos j j j r U r r j N r            Chebyshev polynomials (III kind) (Cheb III) Chebyshev polynomials (IV kind) (Cheb IV)         2 1 arccos cos 2 , 1,1 , 1,2,..., arccos cos 2 j j j r V r r j N r                                    2 1 arccos sin 2 , 1,1 , 1,2,..., arccos sin 2 j j j r W r r j N r                            Power or monomial polynomials (Power) Exponential polynomials (Exp)   , , , 1,2,..., j j j M r r r j N                 1 , , , 1,2,..., j r j j E r e r j N              Hermite polynomials (Her) Laguerre polynomials (Lague)       2 2 1 , , , 1,2,..., j j r r j j j d H r e e r j N dr                   1 , 0, , 1,2,..., ! j j r j j r j d G r r e r j N j e dr             Bernstein polynomials (Bern) Fourier polynomials (Fourier)           1 1 ! 1 1 ! ! 0,1 , 1,2,..., N j j j j N B r r r j N j r j N                      1 1 cos for even 2 1, 1 sin for odd 2 0,2 , 2,3,..., j j j j F r r j F r j F r r j r j N                                   Lobatto polynomials (Lob) Sinc function (Sinc)       1 , 1,1 , 1,2,..., j j j d A r P r r j N dr                        sin 1 1 0,1 , 1,2,..., j j j j N r r Sinc r N r r r j N                Table 1 : List of several basis functions j  and their definition interval used in structural mechanics applications. D IFFERENTIAL AND INTEGRAL QUADRATURE enerally an unknown sufficiently smooth function   f x can be approximated by a set of basis functions   j x  for 1, 2, , j N   , where N is the total number of collocation points in a closed definition interval. A polynomial set uniformly converges to the unknown function when the number of grid points tends to infinity and the unknown function is smooth in a closed interval. Hence, the approximate solution of a function   f x can be G