Discretization Errors

Discretization errors are to errors dependant on the FEM method. Discretization errors can be controlled by using reliable and tested FEM techniques. To do this we need to identify how the results we expect from the analysis are dependent on the choice of discretization. The aim is not to find the most exact results but find the results which are not expressively dependent on the choice of discretization.

Convergence

Convergence is the process where the discretization errors are minimised by meticulously making changes to the choice of discretization and monitoring the changes on the anticipated results.

h-Convergence (h signifies size of element)

h-convergence is achieved by methodically refining the mesh size of the discretized FE model.

Mesh refining imply refining a coarse mesh to a fine mesh size. This means increasing the degree of freedom in the FE model and approximating a more accurate solution.

p-Convergence (p signifies order of element)

p-convergence is achieved when the element order that discretize the FE model is upgraded to a higher order. This means that a higher order of the shape function is used to solve the same mesh methodically. Obviously changing the element order increases the degree of freedom on the FE model as higher order shape function corresponding to the higher order element which approximates a more accurate solution.

Convergence Variations (In both h & p)

Both h & p convergence is done uniformly across the model. However it is also possible to perform convergence methods in a non-uniform but a strategic way where only the areas of stress concentration on the FE model will have mesh refinement or change to higher element order because we know that the stresses in these areas are of more importance.

E.g. Mesh refinement for area of hole on a plate. Areas around the hole are the locations for stress concentration.

This method involve lower number of iteration for an exact approximation of reliable results.

Oddities

What if the solution fail to converge??

  • Stress Singularities

Presence of sharp re-entrant corners on the FE model show that the results does no converge to a finite value, rather it keeps diverging. The expected result can diverge to an infinite value. E.g. The stress value on a re-entrant corner node will tend to increase on iterations after convergence.

The divergence of result show error in mathematical model error i.e. Sharp re-entrant corner.

Solution: Use a better mathematical model, remove sharp re-entrant corners (Stress singularities) with fillets. This is similar to the real parts even if they appear sharp. Although the area around the fillets will still show higher stresses but it will converge on iterations.

Another solution will be use on material models which can represent plasticity. Which will limit the stress value to the plastic limit of the material. Although the strain at the re-entrant corner node will still remain large.

FE models with sharp re-entrants corners can be used for displacement analysis, they do not pose displacement singularities. They can be also used to analysis stresses far from the re-entrant corner areas. However for corner areas, fillet must be modelled to evaluate FEA results.

  • Displacement Singularities

These are the effect of point constraints on the mathematical models. E.g. fixing a single node, single point support for a 2D analysis.

The FEA results shows that the displacement values for such analysis always diverge.

Point support is similar to the sharp re-entrant node and cannot exist in a real structure and is to be avoided in a FE model.

Stress convergence for FE model with displacement singularities also diverge.

Singularities are a not a consequence of the FEM, but are imposed but the mathematical model. We can only identify the singularities by seeing the convergence data.

The presence of singularities does not mean the results are entirely incorrect as long as we know their presence and the implications. E.g. FE Model with sharp re-entrant corner can be utilized to study displacement and vibration analysis. However they are unreliable to analyse stress values near the re-entrant corner.