### Tech Briefs

Surprises in Analyses with Incompatible Modes Elements

The incompatible modes elements have been widely used in practical finite element analysis for many years. In ADINA, the 4-node 2D solid, 8-node 3D solid, and MITC4 shell elements allow the use of incompatible modes as an option, and in fact the incompatible modes option was made the default several years ago.

While it is known that the elements can display unphysical behavior in analyses modeling very large strains, we have now found that meshes of incompatible modes elements can also contain spurious modes in rather simple nonlinear analyses. Indeed, it is only necessary that these elements be geometrically nonlinear under small displacement and small strain conditions and that the elements have large aspect ratio.

The objective in this Brief is to give the results of some illustrative analyses showing this surprising behavior. The theoretical explanation of this behavior and further analysis results are given in the reference.

As a simple example, consider the two-dimensional problem shown in Figure 1. Although a large displacement/small strain formulation is employed, the model is subjected to only numerically small displacements, as shown.

Figure 1  Schematic of two-dimensional problem considered

A frequency analysis is performed after the displacement is applied, and it is found that the first eigenvalue is negative. The corresponding mode shape is shown in Figure 2, along with the modal longitudinal stresses. Clearly this mode is spurious.

Figure 2  Two-dimensional problem: mode shape of first eigenvalue

As a more practical example, consider the analysis of the power screw thread shown in Figures 3 and 4. The material of the power screw is idealized as an elastic-plastic material with bilinear isotropic hardening.  The nut is idealized as rigid, and frictionless contact is assumed between the screw and the nut.

 Figure 3  Schematic of power screw and detail of thread

 Figure 4  Finite element mesh of the power screw thread

A mesh of 8-node incompatible modes elements is used, along with a few 6-node prism elements, and the large displacement/small strain formulation is employed. Notice that the pitch of the thread is included in the mesh. The aspect ratio of the elements in the thread is quite large, with the elements much longer in the tangential direction than in the other directions. However, it would lead to a very inefficient solution if a mesh with good aspect ratio elements were used for this geometry and hence meshes with long slender elements are used.

An axial prescribed displacement is applied to the screw, and the thread of the screw contacts the thread groove of the nut.

When the threads come into contact, a plastic zone forms at the fillet between the screw thread and the screw. On further loading, a spurious mode is triggered. The movie above shows the effective stresses, and the spurious mode is obvious. Notice that the displacements remain small throughout the solution.

This element behavior came as a surprise and we believe that therefore these elements should only be used with great care in nonlinear situations. Since our program development philosophy is that reliability of the analysis results is of primary importance, the incompatible modes option is no longer the default in ADINA from version 9.0.2.

Reference

• T. Sussman and K.J. Bathe, "Spurious modes in geometrically nonlinear small displacement finite elements with incompatible modes", Computers & Structures, 140:14-22, 2014.

Keywords:
Incompatible modes elements, spurious mode, geometrically nonlinear, small strains, small displacements