Shape function

Shape functions are functions which, when using the finite element method, approximate the real function curve over the element as closely as possible . ${\displaystyle N}$ ${\displaystyle u(x,y)}$

The condition is that the continuity condition is fulfilled . Therefore, although no polynomials of this type can be used, the values ​​in the nodes that are each shared by at least two elements can be used. ${\displaystyle u(x,y)=c_{1}+c_{2}\cdot x+c_{3}\cdot y}$

The function profile sought is approximately determined by interpolating the values ​​in these nodes. In order to express the function progression through the nodes, the shape functions are introduced. These have the property of always being 1 in the current node and 0 in the remaining nodes, so that the function sequence results as:

${\displaystyle u(x,y)=\sum _{i}u_{i}\cdot N_{i}(x,y)}$,

in which

• ${\displaystyle u}$ the value at the node and
• ${\displaystyle i}$ represents the number of the node in the element.

example

The linear shape functions for the unit triangle in the , coordinate system are as follows: ${\displaystyle \xi }$ ${\displaystyle \eta }$

${\displaystyle N(\xi ,\eta )={\begin{pmatrix}1-\xi -\eta \\\xi \\\eta \end{pmatrix}}}$

Inserting the respective coordinates of the three corner points shows the desired functionality: ${\displaystyle (0,0),(1,0),(0,1)}$

${\displaystyle {\begin{aligned}N(0,0)={\begin{pmatrix}1-0-0=1\\0\\0\end{pmatrix}}\Rightarrow u(0,0)&=u_{1}\cdot N_{1}(0,0)+u_{2}\cdot N_{2}(0,0)+u_{3}\cdot N_{3}(0,0)\\&=u_{1}\cdot 1+u_{2}\cdot 0+u_{3}\cdot 0\\&=u_{1}\end{aligned}}}$

${\displaystyle N(1,0)={\begin{pmatrix}0\\1\\0\end{pmatrix}}\Rightarrow u(1,0)=u_{2}}$

${\displaystyle N(0,1)={\begin{pmatrix}0\\0\\1\end{pmatrix}}\Rightarrow u(0,1)=u_{3}}$

swell

• Hans Rudolf Schwarz: Finite Element Method. Teubner, Stuttgart 1991, ISBN 3-519-22349-X