Issue 29

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

Lagrange polynomials (PDQ)

Lagrange trigonometric polynomials (HDQ)

 

S r

 

 

S r

r

j

N

,

0,2 ,  

1,2,...,











L r

 



    l r

j

j

r r

 

r

,          , j

N

,

1,2,...,



    1 j



     1 j

j

j

j

S r

sin

r r L r 

 

 

2

j

N

N

     1 , j





 

   



r r

  

L r

r r L r

r r 



r r

 

N

N

    1 j

 





j

k

k

j

k

S r

S r

sin

,

sin





k

 

 

 

 

k

1, k j k  

1



2

2

k

1, k j k  

1



Jacobi polynomials (Jac)

Legendre polynomials (Leg)



  2 j

j

  1 

j

1



j

d





j



d

   

    , j  



  j  



j





P r  

r

1





J r

r

r

1

1











   2 ! 1 1 j j r 



j

j

j



2 ! j

j

j

dr

j dr

r

 

r

j

N

1,1    

1,2,...,

 



r

j

1,2,..., , N

1,1 ,    

,  

1

 



 

Chebyshev polynomials (I kind) (Cheb I)

Chebyshev polynomials (II kind) (Cheb II)









  r

   sin 1 arccos sin arccos j r  

    cos arccos , j r r

     j T r j

 

1,1 ,        j

N

1,2,...,

U r

r

1,1 ,        j

N

,

1,2,...,







j

j

Chebyshev polynomials (III kind) (Cheb III)

Chebyshev polynomials (IV kind) (Cheb IV)





  r





  r

   

   

   

   

2 1 arccos j 

2 1 arccos j 

sin

cos

2

2

 

    V r

W r

r

1,1 ,        j

N

,

1,2,...,

r

1,1 ,        j

N







,

1,2,...,



  r

  r

j

j

j

j

   

   

   

   

arccos

arccos

sin

cos

2

2

Power or monomial polynomials (Power)

Exponential polynomials (Exp)

  j M r r r  , j

  j E r e 

  1 , j r 

,          , j

N

1,2,...,

 

r

,          , j

N

1,2,...,





j

j

Hermite polynomials (Her)

Laguerre polynomials (Lague)

j

d

1





  2 r 

j

d

 

    1  

, r e r  j r

G r

0,        , j

N

j r

1,2,...,







2

H r

e

, e r

,          , j

N

1,2,...,





j

j

! j e dr  r

j

j

j

j

dr

Bernstein polynomials (Bern)

Fourier polynomials (Fourier)

    

cos         2 j r

   

F r

for even j



 1 !

N

j



 





N j 

 

j

1



B r

r

r

1









1 F r

1,    

1 





    1 !



j

j

j

j

N j

!

j

1   

F r

r

for odd j

sin  

 

j

2

r

j

N



0,1 ,

1,2,...,





   

r

j

N

0,2 ,   

2,3,...,





 

Lobatto polynomials (Lob)

Sinc function (Sinc)





 





1 N r r  

sin



j

 

Sinc r







d

    1 , j 

 



 



j

j

A r

P r r

1,1 ,        j

N

1,2,...,







1 N r r  



j

j

j

dr

r

0,1 ,       j

N

1,2,...,

 and their definition interval used in structural mechanics applications.

Table 1 : List of several basis functions j

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

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