Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 117

§ 2. Isoparametric Elements

For the treatment of second order elliptic problems on domains with curved bound-
aries, we need to use elements with curved sides if we want to get higher accuracy.
For many problems of fourth order, we even have to do a good job of approximat-
ing the boundary in the C 1 -norm just to get convergence. For this reason, certain
so-called isoparametric families of finite elements were developed. They are a
generalization of the affine families.
For triangulations, isoparametric elements actually play a role only near the
boundary. On the other hand, (simple) isoparametric quadrilaterals are often used
in the interior since this allows us to generate arbitrary quadrilaterals, rather than
just parallelograms.
We restrict our attention to planar domains, and consider families of elements
where every T ∈ T h is generated by a bijective mapping F :

T ref −→ T
(ξ, η) −→ (x, y) = F(ξ,η) = (p(ξ, η), q(ξ, η)).

( 2 . 1 )

This framework includes the affine families when p and q are required to be
linear functions. When p and q are polynomials of higher degree, we get the more
general situation of isoparametric elements. More precisely, the polynomials in
the parametrization are chosen from the same family as the shape functions of
the element (T,,) .

Isoparametric Triangular Elements

The important case where p and q are quadratic polynomials is shown in Fig. 31.
By Remark II.5.4, we know that six points P i ,1 6, can be prescribed. Then

p and q as polynomials of degree 2 are uniquely defined by the coordinates of the
points P 1 ,...,P 6 . In particular, if P 4 , P 5 , and P 6 are nodes at the midpoints of
the edges of the triangle whose vertices are P 1 , P 2 , and P 3 , then obviously we get
a linear mapping.
The introduction of isoparametric elements raises the following questions:
1. Can isoparametric elements be combined with affine ones without losing the
desired additional degrees of freedom?
2. How are the concepts “uniform“ and “shape regular“ to be understood so that
the results for affine families can be carried over to isoparametric ones?
In order to keep the computational costs down, we should use elements with
straight edges in the interior of . This is why elements with only one curved side

≤ i ≤