Issue 29

F. Tornabene et alii, Frattura ed Integrità Strutturale, 29 (2014) 251-265; DOI: 10.3221/IGF-ESIS.29.22 254             with for , 1, 2,..., i n j n n n n ij j i n x d x A x i j N dx       f A λ (9) where         1 2 , , , N T n n n n n n n x x x d f x d f x d f x dx dx dx           f is the vector which contains the derivative of the function   f x at all the discrete points i x . In order to perform the derivation, the matrix A should be invertible. This property depends on the basis functions   j x  and on the location of the grid points. The unknown parameters vector λ can be evaluated from Eq. (6) by simply inverting the matrix A , such as 1   λ A f (10) Subsequently, substituting expression (10) into (9), one obtains       1 n n n    f A A f D f (11) The differentiation matrix   n D is found as a matrix product between the inverse of the matrix A containing the values of the basis functions   j i x  in all the grid points and the derivatives of the same functions     n j i x  contained in the   n A matrix. All the steps presented above are valid for any derivative order n . In general, Eq. (11) takes the form               1 1 with for , 1, 2,..., i n N n n n n ij j ij ij n ij j x d f x D f x D i j N dx         A A (12) In conclusion, a generic n -th order derivative can be expressed by the following relation       1 i n N n ij j n j x d f x f x dx     (13) Although Eq. (12) is valid for any basis function and grid point distribution, the coefficient matrix A , since it is like the Vandermonde matrix, can become ill-conditioned when the number of grid points N is large. It is important to note that, when the Lagrange polynomials   j l x , Lagrange trigonometric polynomials   j S x or the Sinc function   j Sinc x are chosen as a basis of the linear vector space N V , the coefficient matrix A becomes an identity matrix  I A (14) This is due to the fact that, the three previous basis functions have the following properties       0 for 1 for j i j i j i i j l x S x Sinc x i j         (15) In all these three cases Eq. (12) becomes             1 with for , 1, 2,..., i n N n n n n ij j ij ij ij n j x d f x D f x D i j N dx        A (16) The important consequence that derives from relations (14) and (15) is that the matrix  A I is always invertible ( 1   I A ) and thus the ill-conditioning drawback does not occur when the Lagrange polynomials   j l x , Lagrange trigonometric polynomials   j S x or the Sinc function   j Sinc x are chosen. Further details about the weighting coefficients of the previously reported methodologies can be found in [7].