Modelling errors are introduced via simplifying assumptions made in the formulation of the mathematical (usually simplified) representation.

Discretization errors occur a continuous mathematical model is discretised into a FE model.

Solution errors arise from the process of the numerical solution of the FE equations.It should also be noted that errors can be made by users incorrectly interpreting the results. To minimize the danger of this happening,

**always work in SI units**.The Finite Element Method involves approximating a structure (assuming a structural analysis is being carried out) and there are several potential sources of error.

The following are the main sources of error:

**1. The model of the structure may contain a number of simplifications.**Often this involves omitting small details (this is referred to as 'defeaturing'). 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 also to be included.

**Stress Singularity**

All components have finite radii at corners, however for small radii a common simplification is to ignore the radius and make the corner 'sharp'. This may not matter for an external corner, however a sharp re-entrant corner results in a stress singularity - refining the FEA mesh will give increasing (without limit!) stress values as the element size in reduced. The stress results are meaningless (the displacement results may be acceptable) and a reasonable approximation of the radius must be used in the model. One way of reducing this problem is to model the component with a material which can model plastic deformation, however strain at the sharp re-entrant corner will remain infinite.

If stresses are not of interest, for example if modal frequencies are being computed, then the inclusion of a sharp re-entrant corner will not affect the results and the simplification will help to simplify the model.

**Point Constraint Singularities**

Simplifying spot welds, bolts or rivets as point constraints will cause serious errors as the maximum displacements tend to infinity as the mesh is refined.

A point constraint can be used if the load on a component is balanced and the analyst wants to restrict rigid body motions.

**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 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.

**Element size**

FEA analysis is carried out on an assembly of discrete elements, not on a continuous structure. The smaller the element size (the finer the mesh), the smaller the discretisation error, but computation time increases.

**Connecting different types of element**

Where two element edges are coincident, the function describing the displacements of the two edges must be the same. This can create difficulties when thin sections of components are connected to thick sections, e.g: cooling fins on a cylinder. The body of the cylinder may well be meshed with solid tetrahedra but for eficiency the fins will probably need to be meshed with thin plates. Appropriate linking will have to be done to ensure the model is a reasonable representation the component.

**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 issue 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. For example even if a beam is welded to a large, solid piece of metal, the large piece of metal will still deform slightly when the beam is loaded.

**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.

- Always check that the software is operating on the values you thought you typed in!
- Always do a rough 'back of the envelope' check on the results.
- Always check the deformed shape to see if it is roughly what you expected.