Issue 29

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

Uniform (Unif)

Chebyshev-Gauss-Lobatto (Che-Gau-Lob)

k

1 ,



r r 

k

1       1  

r

k

M r

k

M

,

cos

,

1,2,..., ,

1,1     







k 

1, 2,...,

k 





k

1



1 M k  

1 M r r 

M

M

1



Quadratic (Quad) (only for M odd)

Extended Chebyshev (Extd Cheb) or (Cheb I)



2

k

1 M k 

1





 

2     k  

1,2,...,





r r r 

M

1

2

  T r k , M

M r

,

roots of

1,2,..., ,

1,1     

k 







k

1

   

k

1 M r r 

2

k

k

1 M k 

1

1





  

4       

  

1, M M 

2

1,  

1,...,

k 

 



M

M

1

1

2





Extrema Chebyshev (Extr Cheb) or (Cheb II)

Approximate Legendre (App Leg)

r r 

4 1 k 

1

1

 

cos    

  

r

,

1    

,

k 





k

1

r r r 

  U r k , M

1 M k  

2 M M M   3 8 8

1 M r r 

4 2 

M r

,

roots of

1,2,..., ,

1,1     

k 







k

1

k

1 N r r 

k

M r

1,2,..., ,

1,1     



Legendre-Gauss (Leg-Gau)

Legendre-Gauss-Radau (Leg-Gau-Rad)

r r r 

1 1 M r r  r r  k

 

  r

r

r

P

r P 

,

1,

1,

,

roots of

,

k 

k 



 







k

1

M

k

M

M

1

1



1 M r r 

  P r k  2 , M

r

1, M r

k

1,2,..., , M

r

roots of

2,3,...,

1,1      

1,1     







k

Chebyshev-Gauss (Cheb-Gau)

Legendre-Gauss-Lobatto (Leg-Gau-Lob)

 

 

  

1 M dP r 

r r 

2 3 k 

r r r 

r

1,    r

r

,

1,

cos  

,



k 



k

1

1,     1, r r

,

roots of

,

k 



k

1





M

1 M k  

1

M

k

1

1 M r r 

2 2 M

  

1 M r r 

dr

k

1 M r

2,3,...,

1,1      



k

1, M r

2,3,...,

1,1      



Hermite (Her)

Laguerre (Lague)

r r r 

r r r 

 

 

M H r

M G r

,

roots of

,

,

roots of

,









k 

k 

k

k

1

1

k

k

1 M r r 

1 M r r 

k

M r

k

M r

1,2,..., ,

,

1, 2,..., ,

0,



    



   

 

 

Jacobi (Jac)

Chebyshev-Gauss-Radau (Cheb-Gau-Rad)





 

  

2 1 2 1 k M 

r r r 

r r 

   ,  



M J r

,

roots of

,





k 

k

1

r

,

cos  

,

k 





k

1

k

1 M k  

1 M r r 

1 M r r 

  

k

M r

1, 2,..., ,

1,1     



k

M r

1, 2,..., ,

1,1     



Jacobi-Gauss (Jac-Gau)

Ding et al. [37] distribution

r r r 

    , 2 M   

1,     1, r r

J r

,

roots of

,

k 



k

1

  

 

k

1 2

1





  

 

M

k

1

1 M r r 

k

M

1 2 cos

,  

1,2,...,

k 

 





4 2 1 M



k

1, M r

2,3,...,

1,1      



Chebyshev III (Cheb III)

Chebyshev IV (Cheb IV)

r r r 

r r r 

  V r k , M

  W r k , M

M r

M r

,

roots of

1,2,..., ,

1,1     

,

roots of

1,2,..., ,

1,1     

k 

k 













k

k

1

1

k

k

1 N r r 

1 N r r 

Radau I (Rad I)

Radau II (Rad II)

r r r 

r r r 





  P r P   1 M

  r

  r

 

r

1,   r

P

2 M P r 

,

1,

roots of

,

,

roots of

,

k 



 

 

k 





k

k

1

1

M

k

M

1 M k  

M

1

2

1





1 M r r 

r

r



M

1

k

1, M r

1,2,...,       Table 2 : List of several grid distributions used in structural mechanics applications. 1, M r 1,1 k 

1,2,...,

1,1      



256