Remarks on the Shell Elements in ADINA, continued
We presented in the January 30, 2006 News, some remarks on the shell elements available in ADINA. These elements are based on using Mixed Interpolated Tensorial Components and are therefore referred to as MITC shell elements. In this earlier News, we also gave the results obtained with ADINA in a commonly used benchmark problem. These results illustrated the excellent behavior of the elements:
However, there are of course additional commonly used benchmark problems. We present below two such problems and the solutions obtained using the MITC shell elements. In all cases (as in the earlier News referred to above) we have used uniform, non-graded meshes. The figures show that excellent results are obtained, even with rather coarse meshes. Indeed, these problems are not hard to solve and more difficult problems should be tackled in order to fully evaluate the reliability and effectiveness of a shell element, as described in the references given earlier:
Benchmark Problem 1 — Scordelis-Lo Shell
R = 300 in, L = 600 in, φ = 40°, thickness = 3 in
shell weight = 90 lb/ft2
Schematic of Problem
The 1x2 MITC9 element mesh of the region discretized
Benchmark Problem 2 — Hemispherical Pinched Shell
R = 10 m, φ = 18°, thickness = 0.04 m
Schematic of Problem
The 2x2 MITC9 element mesh of the hemispherical pinched shell
(one quarter of shell modeled due to symmetry)
Here are some observations regarding the solutions:
When only a displacement value at a certain point is measured (as in the problems considered above), the solutions can converge from below or above to the "exact" solution. This behavior can be observed for any finite element method depending on the point considered, and underlines the fact that, actually, norms should be used to measure convergence. And also, the norm used must be an appropriate norm, see references given earlier.
Also, when only one point value is measured, a coarser mesh might give a better result than a finer mesh, in particular when both meshes are actually still coarse.
Considering the solutions obtained with the MITC elements, it is remarkable that meshes with only few MITC elements used to model the structures give excellent results! Indeed, using the MITC9 element, in both solutions only about 200 dofs are required to have an error of 1 percent.