Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 119

If in addition

diameter(T ) ≤ 2 h and ρ T ≥ h/κ,

then T h is called uniform.

Isoparametric Quadrilateral Elements

Isoparametric quadrilaterals are also of use in the interior since only parallelograms
can be obtained from a square with affine mappings (see Ch. II, §5).

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(affine case)

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(bilinear case)

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Fig. 32. Isoparametric quadrilaterals with bilinear parametrization

[0 , 1] 2 be the unit square. Then
F : p(ξ, η) = a 1 + a 2 ξ + a 3 η + a 4 ξη
q(ξ,η) = b 1 + b 2 ξ + b 3 η + b 4 ξη

Let T ref =

( 2 . 3 )

maps T ref to a general quadrilateral. From the theory of bilinear quadrilateral
elements, we know that the two sets of four parameters are uniquely determined
by the eight coordinates of the four corners of the image of T ref .
In addition, it is clear that when ξ and η are both constant, the parametrization
F is a linear function of the arc length. It follows that the image is a quadrilateral
with straight edges. The vertices are numbered so that the orientation is preserved.
Because of the linearity of the parametrization on the edges, connecting the element
to its neighbors is no problem.

2.2 Remark. T h involving general quadrilaterals with

bilinear parametrizations is shape regular provided there exists a constant κ> 1
such that the following conditions are satisfied:
(i) For every quadrilateral T , the ratio of maximal to minimal edge lengths is
bounded by κ .

A family of partitions