Quadrilateral Finite Elements must be convex and not self-intersecting. But why?
Solution 1:
It's not a strict requirement, from a mathematical perspective. Instead, it's a semantic requirement: the conditions on convexity (and non-self-intersection, which is just a special-case) are required to guarantee that the mapping from $[0,1]\times[0,1]$ to the quadrilateral is 1:1. Otherwise, one can have multiple points in $uv$ parameter space mapping to the same 'physical' point, and then a sensible interpretation has to be defined for the correct value of whatever function one is modeling at that point.