Remarks on the Use of Incompatible Modes

In the analysis of solids, the 4-node plane stress/strain and axisymmetric elements and the 8-node brick element are frequently used with incompatible modes, to improve the bending behavior of the pure displacement-based elements.

The incompatible mode elements can be thought of — and can be formulated — as low-order 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; simply-supported condition; linear analysis

Typical finite element mesh; 4-node incompatible mode elements; one element through thickness and N equal elements along length; in the solutions using 9-node elements, each 4-node element is replaced by a 9-node element

A simply-supported 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 9-node rectangular displacement-based 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 simply-supported condition, linear analysis

Repeating the analysis with the fully built-in 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 built-in support condition

Results for fully built-in support condition, linear analysis

Of course, point-load conditions, as seen in the simply-supported case, need in general special modeling when the mesh becomes very fine (see Chapter 1 of ref. [1]). However, this study shows that the displacement-based and u/p elements (while computationally more expensive) are in this case more reliable than the incompatible mode elements.

References

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

2. 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, 83-104, 1997.