# The Importance of Mesh Convergence

## Part 1

This two-part article describes what might be one of the most overlooked issues that affect accuracy, namely; mesh convergence. This refers to the smallness of the elements required in a model to ensure that the results of an analysis are not affected by changing the size of the mesh. We have encountered occasions where mesh size is just accepted as a historical legacy that cannot be changed, and no knowledge of its affect on accuracy is available. This is bad practice.

This article is directly applicable to static stress analysis. Whilst the issue of mesh size is important in all analyses, there are other issues that affect the selection of an appropriate element size in more advanced analyses. It relates to the majority ‘h element’ programs; ‘p element’ programs (e.g. Pro Mechanica) converge on a result in the solution process and, to a large extent, are not dependant on element size.

## How to do a Convergence Study

### The Convergence Curve

The formal method of establishing mesh convergence requires a curve of a critical result parameter (typically some kind of stress) in a specific location, to be plotted against some measure of mesh density. At least three convergence runs will be required to plot a curve which can then be used to indicate when convergence is achieved or, how far away the most refined mesh is from full convergence. However, if two runs of different mesh density give the same result, convergence must already be achieved and no convergence curve is necessary.

Figure 1 – a 4 point convergence curve

### Local Mesh Refinement

In theory, for each successive level of mesh refinement in the convergence study, all elements in the model should be split in all directions. While the latter requirement is important, it is not necessary to carry this out on the whole model: St Venant’s Principle implies that local stresses in one region of a structure do not affect the stresses elsewhere. From a physical standpoint then, we should be able to test convergence of a model by refining the mesh only in the regions of interest, and retain the unrefined (and probably unconverged) mesh elsewhere. We should also have transition regions, from coarse to fine meshes, suitably distant from the region of interest (at least 3 elements away for linear elements).

### Boundary Geometry – a Related Effect

A common influence on stress results when using linear (straight sided) elements to represent a curved surface or edge is that the geometry of the boundary will be better represented, as the mesh is refined. This is a modelling or geometry effect, different to mesh convergence, which is numerical. It is worth being aware of the distinction between these two affects.

## Meshing Strategy

The idea of using only local mesh refinement for a convergence study can be extended. If a model is required to produce accurate stresses only at certain regions of interest, the role of all elements away from these regions is one of only representing geometry and transmitting load. This demands a much lower level of mesh refinement than for accurate stress prediction. Thus, these elements can be considerably larger, subject to the constraints of permitting both reasonable quality transitions and geometry representation.

Using larger elements away from regions of interest in a model is common practice but a more subtle point is, providing they don’t misrepresent the geometry and suitable mesh transitions can be carried out; these elements can be considerably larger than those in regions of interest, without jeopardising accuracy. Contrast this as a meshing strategy against that of filling an entire model with small, high-quality elements, to improve ‘overall’ accuracy. This latter approach is inefficient and unlikely to improve accuracy in static stress analysis with an implicit code.

## Summary

• Every department should have some basic data on mesh convergence for their models.

• A number of runs of a model with increasing levels of mesh refinement in the areas of interest can be used to demonstrate mesh convergence.

• Element sizes distant from a region do not significantly affect the results in that region, providing they do not grossly misrepresent the distant geometry.