As a mechanical engineer, I use Finite Element Analysis (FEA) software at work to calculate the way metal parts deform, their stresses, and the frequencies at which they vibrate. I recently found Elmer, a free FEA program. To test how accurate Elmer is, I calculated the lowest vibration frequency of a cantilevered beam using Elmer, and compared it to the hand calculation for the frequency. I investigated the impact of mesh fineness, and element type on the accuracy of the answer, and showed that second order tetrahedral elements should be used.
Beam Geometry and Material:
thickness (into the page) = .01 m
height = .03 m
Material: Aluminum (generic)
Young’s Modulus: 70 *10^9
Poisson’s Ratio: .35
Density: 2700 kg/m^3
Modeling: FreeCAD (dimensions must be in m)
- Export geometry in .brep file
- Export mesh as .msh file
FEA Analysis: Elmer
- Linear elastic eigenvalue analysis, set matl + constrain left end of beam to x=y=z=0
The shape of the beam when it is vibrating at this frequency is:
I varied the fineness of my mesh between 1-.1 values of ‘Element Size Factor’ where .1 is the finest. The finest and coarsest meshes are below.
Element Size Factor of 1.0 resulted in 220 tetrahedra elements
Element Size Factor of 0.1 resulted in a mesh of 57,748 tetrahedra elements:
I used 1st order and 2nd order tetrahedral elements (triangular pyramid). The calculated frequencies are:
|Element Size Factor||Tets_1st_order (Hz)||Tets_2nd_order (Hz)|
This results in the following error values vs element size factor (fineness):
|Element Size Factor||Tets_1st_order||Tets_2nd_order|
The big take-away here is that for any modal analysis, one should be using second order tets in Elmer, and that should get you within 2% of the answer predicted by hand calculations.
If you’d like to run my analysis yourself, or examine the results, download my zipped file below. I included a folder with the 2nd order tetrahedron results meshed with an element size factor of 0.4. I’ve also included the Freecad geometry file, brep geometry file, and the gmsh .msh mesh file. tets_2nd_size04_cant_beam.zip