Errors Arising in Finite Element Analysis |
The Finite Element Method involves approximating a structure (assuming a structural analysis is being carried out) and there are several potential sources of error or difficulties. The main ones are briefly discussed below.
1. The model of the structure may contain a number of simplifications. Often this involves omitting small details. This is only satisfactory if the stresses in the areas where these details have been omitted are low. It must be remembered that sharp radii can greatly increase the stress. Normally it is best to start with a very simple representation of the actual component, analyse it and see if it is behaving as expected. If it is then more detail can be added in stages, repeating the analysis every time further detail has been added. By doing this it may be possible to gain an appreciation of the amount of detail that needs to be included.
Internal corners with zero radius will give rise locally to infinite stress. Modern software will normally detect this
potential problem and generate large numbers of small elements in the area. This may well significantly increase memory requirements
and running time. It is much better to put in a finite internal corner fillet radius close to the actual value (it is
obviously not possible to manufacture internal corners with zero radius). It should be noted that automatic meshing routines
will need to generate a lot of elements in the vicinity of small radius fillets as normally one element side should not exceed
45^{o} of curve and maximum length to width (aspect) ratios are usually between 5 and 10. These limitations to not
apply to 'P' type elements available in some software.
This problem does not arise when simplifying modeling with zero radius external fillets.
Many packages as default use meshing software that generates tetrahedral elements. In the case of components containing thin sections this can lead to the generation of surprisingly large numbers of elements. In such cases it may be more efficient to model the component as a thinwall structure and specify rectangular plate or shell elements.
2. Element Order. When using most FEA packages ('H' method), the analyst selects the element order, but with some packages, ie PTC Mechanica 'Structure' ('P' method) the selection of the element order may be left partially to the software. Where the user is specifying the element order, it is important that an appropriate combination of mesh density and element order be chosen, as otherwise results will be of low accuracy. This means that great care is needed in areas of rapidly changing stresses, eg notches etc.
3. Loads and Boundary Conditions. Although the calculation of the load(s) and the choice of correct boundary conditions may seem straight forward, it is often somewhat more difficult than initially thought. This is particularly so for problems involving torsional loading. Applying a force to the end of a wrench applies a direct force in addition to the moment!
A significant for almost all mechanical engineering applications, is that the loads will be fluctuating and allowances will normally have to be made. For example when someone is climbing a step ladder, the force they apply to the rungs will greatly exceed their body weight. For many products (including different types of ladders) design codes, often in the form of British Standards exist, which spell out the type of testing a product should withstand. Obviously a design to meet the standard needs to be analysed for the specified loads. For critical components testing may be needed.
Wind and wave loadings are two types where wide fluctuations occur and some specialist knowledge and understanding of statistics are needed. During the past 30 years the Automotive industry has spent a lot of time and money gathering actual data from test vehicles operating in a wide variety of conditions.
A problem when applying boundary conditions that are nominally 'fixed' is that for FEA this means 'infinitely' stiff whereas in engineering 'fixed' boundaries cannot be infinitely stiff.
4. Numerical. Computed values, such as stresses and strains, are evaluated at 'Gauss' points, which are inside element boundaries. Values at other positions are interpolated or extrapolated. If this is done across a boundary between two elements, then it should be reasonably accurate, but extrapolating to the edge of an element on the edge of a structure, where the stresses will probably be at the highest and of most interest, can lead to significant errors in rapidly changing stress fields if the mesh density or the element order is too low.
Contour plotting routines also use interpolation, a listed maximum value may be different to the maximum on a contour plot.
Irregular element geometry is a source of error. Because of the transformations carried out in most packages, the further the element geometry departs from regular geometry (eg rectangular elements depart from square) the greater will be the error. Most packages have sophisticated checks and give quite detailed warnings, but for 'P' type FEA, where very considerably departure from a regular shape is acceptable, you may be faced with the option of having to relax element tolerances to allow the automatic meshing program ('Auto Gem' in Mechanica Structure) to complete or to reduce the number of elements so the job will fit available resources.
5. General Warning. Modern pre-processors make it reasonably easy to pick up errors in geometry, however they do nothing to pick up incorrectly keyed in loads.
David J Grieve, 28th September 2012.