next up previous
Next: B.2.2 Rectangle Up: B.2 Two-Dimensional Shape Functions Previous: B.2 Two-Dimensional Shape Functions

B.2.1 Triangle

linear interpolation:

\resizebox{5.0cm}{!}{\includegraphics{/iue/a39/users/radi/diss/fig/shape/lin2D.eps}}

interpolating polynomial function:

u($\displaystyle \xi$,$\displaystyle \eta$) = $\displaystyle \alpha_{1}^{}$ + $\displaystyle \alpha_{2}^{}$$\displaystyle \xi$ + $\displaystyle \alpha_{3}^{}$$\displaystyle \eta$      

shape functions:
N1($\displaystyle \xi$,$\displaystyle \eta$) = 1 - $\displaystyle \xi$ - $\displaystyle \eta$  
N2($\displaystyle \xi$,$\displaystyle \eta$) = $\displaystyle \xi$  
N3($\displaystyle \xi$,$\displaystyle \eta$) = $\displaystyle \eta$  

quadratic interpolation:

\resizebox{5.0cm}{!}{\includegraphics{/iue/a39/users/radi/diss/fig/shape/lin2D_2.eps}}

interpolating polynomial function:

u($\displaystyle \xi$,$\displaystyle \eta$) = $\displaystyle \alpha_{1}^{}$ + $\displaystyle \alpha_{2}^{}$$\displaystyle \xi$ + $\displaystyle \alpha_{3}^{}$$\displaystyle \eta$ + $\displaystyle \alpha_{4}^{}$$\displaystyle \xi^{2}_{}$ + $\displaystyle \alpha_{5}^{}$$\displaystyle \xi$$\displaystyle \eta$ + $\displaystyle \alpha_{6}^{}$$\displaystyle \eta^{2}_{}$      

shape functions:
N1($\displaystyle \xi$,$\displaystyle \eta$) = (1 - $\displaystyle \xi$ - $\displaystyle \eta$)(1 - 2$\displaystyle \xi$ - 2$\displaystyle \eta$)  
N2($\displaystyle \xi$,$\displaystyle \eta$) = $\displaystyle \xi$(2$\displaystyle \xi$ - 1)  
N3($\displaystyle \xi$,$\displaystyle \eta$) = $\displaystyle \eta$(2$\displaystyle \eta$ - 1)  
N4($\displaystyle \xi$,$\displaystyle \eta$) = 4$\displaystyle \xi$(1 - $\displaystyle \xi$ - $\displaystyle \eta$)  
N5($\displaystyle \xi$,$\displaystyle \eta$) = 4$\displaystyle \xi$$\displaystyle \eta$  
N6($\displaystyle \xi$,$\displaystyle \eta$) = 4$\displaystyle \eta$(1 - $\displaystyle \xi$ - $\displaystyle \eta$)  

next up previous
Next: B.2.2 Rectangle Up: B.2 Two-Dimensional Shape Functions Previous: B.2 Two-Dimensional Shape Functions
Mustafa Radi
1998-12-11