Issue 29

F. Tornabene et alii, Frattura ed Integrità Strutturale, 29 (2014) 251-265; DOI: 10.3221/IGF-ESIS.29.22

  x

n

j dx 

d

    n x

  n

  n A λ

  n











A

i j

N

with

for ,

1, 2,...,

f

(9)

ij

j

i

n

x

i

T

   

   

  d f x d f x   n n

 

n d f x

where   n f

is the vector which contains the derivative of the function   f x



,

, , 

n

n

n

dx

dx

dx

x

x

x

N

1

2

x . In order to perform the derivation, the matrix A should be invertible. This property depends j x  and on the location of the grid points. The unknown parameters vector λ can be  

at all the discrete points i on the basis functions

evaluated from Eq. (6) by simply inverting the matrix A , such as 1   λ A f Subsequently, substituting expression (10) into (9), one obtains       1 n n n    f A A f D f

(10)

(11)

  n D is found as a matrix product between the inverse of the matrix A containing the values of

The differentiation matrix

  n A

in all the grid points and the derivatives of the same functions     n j i x 

 

j i x 

the basis functions

contained in the

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       i j l x S x Sinc x      

0 for 1 for

(15)

j

i

j

i

j

i



i

j



In all these three cases Eq. (12) becomes       n N n ij j d f x D f x   n

  n

  n   ij 

  n



D

i j

N

with

for ,

1, 2,...,

A

(16)

ij

ij

dx



j

1

x

i

 A I is always invertible

The important consequence that derives from relations (14) and (15) is that the matrix ( 1   I A ) and thus the ill-conditioning drawback does not occur when the Lagrange polynomials   j

l x , Lagrange

 

  j S x or the Sinc function

j Sinc x are chosen. Further details about the weighting

trigonometric polynomials

coefficients of the previously reported methodologies can be found in [7].

254