Remarks on the Use of Incompatible Modes
In the analysis of solids, the 4node plane stress/strain and axisymmetric elements
and the 8node brick element are frequently used with incompatible modes,
to improve the bending behavior of the pure displacementbased elements.
The incompatible mode elements can be thought of — and can be formulated — as
loworder enhanced strain elements, in which strains (those
corresponding to the incompatible displacements) are added to the
usual strains derived from the compatible displacements. The elements
are of course formulated to pass the patch tests, see e.g. ref. [1].
The incompatible mode elements are well known to be most powerful when undistorted,
e.g. when rectangular in shape in 2D analyses.
However, it is known that the use of enhanced strain elements can provide difficulties in
large strain (almost incompressible) analysis, see e.g. ref. [2], and the same holds for elements based on
incompatible modes. For such analyses, the displacement/pressure (or u/p) elements available
in ADINA are more reliable and effective.
It seems less well known that the elements based on
incompatible modes can show what may be considered a surprising
behavior in linear analysis, as illustrated in the following example.
Model solved; simplysupported condition; linear analysis
Typical finite element mesh; 4node incompatible mode elements; one element through
thickness and N equal elements along length; in the solutions using 9node elements,
each 4node element is replaced by a 9node element
A simplysupported axisymmetric plate is subjected to pressure loading. A schematic of the model
considered is given above. We also show a typical finite element discretization.
The use of 9node rectangular displacementbased elements and
9/3 elements yields the same convergent result as the number of elements increases, see below.
Even when the number of elements becomes very large, that is, the elements become
extremely thin, the result does not change.
On the other hand, the incompatible mode element results are only acceptable as long as the elements
are not too thin. Indeed, if only results up to N = 100 are considered, the solution has converged. In practice,
surely, no more elements will be used.
However, merely as a convergence study, if we increase the number of
elements further, an increasing displacement at the center of the plate is seen.
The elements near the support cause the displacement to increase unphysically, due to the
incompatibility between the elements. Of course, the increase only occurs when the elements
are very thin and then the increase is also small, but it is good to know about this phenomenon.
Results for simplysupported condition, linear analysis
Repeating the analysis with the fully builtin support condition gives the results in the following figure. The
divergent behavior using the incompatible mode elements is not seen.
Typical finite element mesh with fully builtin support condition
Results for fully builtin support condition, linear analysis
Of course, pointload conditions, as seen in the simplysupported case, need in general special modeling
when the mesh becomes very fine (see Chapter 1 of ref. [1]). However, this study shows that the
displacementbased and u/p elements (while computationally more expensive) are in this case
more reliable than the incompatible mode elements.
References

K. J. Bathe, Finite Element Procedures, Prentice Hall, 1996.

D. Pantuso and K. J. Bathe, “On the Stability of Mixed
Finite Elements in Large Strain Analysis of Incompressible Solids”,
Finite Elements in Analysis and Design, 28, 83104, 1997.
