Crack tip elements
============================

Typically, "regular" elements, placed at the crack tip lead to poor convergence and are replaced by "*special*" elements know as crack-tip elements. 

The key behind CTE is to utilize shape functions which are capable of describing the analytical crack tip fields, and more specifically the $1/\sqrt{r}$ singularity. The $\theta$ dependence can be satisfacory described by the elements arrangment. 



```{image} ../CompFM/cte1.png
:alt: Kirsch's plate
:width: 400px
:align: center
```

#### Quarter point (*Barsoum*) elements

QP elements are based on translating nodes from the mid-side of the element to a quarter length distance from the crack tip. 

```{image} ../CompFM/qp1.png
:alt: Kirsch's plate
:width: 400px
:align: center
```

The mapping of the isoparametric coordinates to the local coordinates lead to a singular term compatibale with the analytical stress fields. 

Consider a 1D element for which the displacement field is given by :

$$
u(\xi) = \sum _{a=1}^3 N_a(\xi)u^{(a)}
$$