** Written by Janet Wolf **
Finite Element Analysis (FEA) software has advanced tremendously in ease-of-use and robustness to the point that meshing complex geometry is often as easy as pushing a button. But getting a model to mesh and getting good results can be two different things, and the user has to ask themselves: How do I know if the mesh is actually any good? And how many elements do I really need? The mesh density (the amount and distribution of elements) can have profound effects on the results! This article discusses what to consider when setting mesh sizing and how to check the resulting mesh.
Mesh Density Considerations
FEA is the process of dividing geometry into smaller pieces (elements), applying loads and boundary conditions to those elements, and then solving the matrix equations assembled from the mesh. Theoretically, the more elements used in the model, the closer the results get to the actual behavior (as modeled). However, we can’t analyze an infinite number of elements, so FEA becomes a balance between accuracy and efficiency when creating a model. Meshing is sometimes referred to as an “art form” and can be very subjective, but asking the following questions can give some direction:
What type of analysis are you performing?
Advanced analyses tend to have more rigorous mesh requirements than linear analyses. Some examples: sliding contact problems may need a finer mesh to capture the changing status behavior accurately; advanced materials such as plasticity, hyperelasticity, etc., often require finer meshing to capture large strain gradients; and large deformation analyses need a finer mesh to accommodate large changes in shape during the analysis.
Where is the area of interest?
Instead of refining the entire mesh, focus elements on the area of interest if it is known ahead of time– using a more targeted approach to meshing can improve solution time. Coarse elements are usually adequate for force transfer, and can be used in areas where information on stresses are not needed, and then elements can be invested where they are needed. If the area of interest is not known ahead of time, a coarse mesh can be analyzed to identify areas that need refinement in subsequent analyses.
How do small features affect the results?
Small holes, fillets, bosses, cutouts, and lettering can all create meshing headaches as the software tries to model the detail with adequate elements, creating a very fine, localized mesh. Rather than mesh everything, it’s best to assess whether the feature is necessary for the analysis: is it in a critical area? Will it affect the load path? Fillets can improve stresses by removing sharp corners that cause hot spots, so including them at areas of interest can be important. It is up to the user to decide what features to include, weighing accuracy against solution time.
What kind of results do you need?
Accurate displacement results don’t require as fine of meshes as stress results for linear analyses– if the analysis is being performed to ensure the design doesn’t displace too much, a coarser mesh can be used than if you are checking the stresses against an allowable value.
What kind of elements are you using?
Some element types are more stiff in bending and require more elements through the thickness of a part in order to properly capture bending behavior. This problem is often referred to as “locking”, and as a rule of thumb: elements with mid-side nodes (quadratic elements) are less prone to locking than elements with just corner nodes (known as linear elements or draft elements). Even when using mid-side elements, it is a good idea to use more than one element through the thickness if bending is significant.
Also, some element shapes are more sensitive to distortion. For example, a hexahedral (brick) element can be elongated but still provide good results. A tetrahedral element, however, will develop small angles when lengthened in one direction, which can provide less accurate results.
Checking Element Quality
After a mesh has been generated, the user should evaluate the mesh by looking at the elements’ quality. Element quality is very subjective– a mesh that gives good results for one set of boundary conditions and loads might give terrible results for another. However, it is still a useful tool to get a general idea of how good a mesh is. There are many mesh quality metrics used by analysts, some of which are implemented as checks in FEA software. Each metric assesses a different characteristic of the elements, and can affect results in different ways. A few common quality metrics are listed below:
Aspect Ratio is the ratio of the longest side to the shortest side of an element. An ideal element has an aspect ratio of 1, but as mentioned earlier, some elements (hexahedral) are less sensitive to large aspect ratios.
Skewness is a measure of how close to ideal an element face is; values range from 0 (ideal) to 1 (degenerate). It is calculated by comparing the actual element size, to the size of an ideal element that would fit in the same size circle as the original element.
Jacobian Ratio This ratio is based on the determinant of the Jacobian Matrix, which is used in FEA to convert element matrices from being based on their theoretical shape (e.g., perfect square, triangle) to being based on their actual shape. An ideal element has a Jacobian Ratio of 1, and the further this value is from 1 the further the element is from ideal.
There are additional mesh metrics, some of which are software specific. Which mesh metric should be used? That is up to the user, although it might be dictated by project requirements and/or FEA software capabilities.
Reviewing results are a good way to assess a mesh, and there are other tools that can be used to check the mesh after solution. A few methods are listed below:
Unaveraged Results The stress values are calculated at the nodes for each element they’re connected to. The values calculated at a node per element will differ, which creates an uneven stress pattern around a node. FEA software programs often average the stresses at the nodes to present a smooth stress contour, but this can hide discontinuities in stress. A coarse mesh will have higher differences in stresses than a fine mesh, so plotting the unaveraged stress can help users identify regions that are good candidates for mesh refinement. Looking at these values also helps you to assess how much you may be underestimating stress by using the averaged values.
Stress Error Another way to assess the effect of the averaged vs unaveraged stress is the Stress Error or Structural Stress Error. This plot shows the energy due to the mismatch of stresses at the nodes. The actual number should be treated qualitatively, not quantitatively: there is no “good” or “bad” value of this number. It is used to compare results for similar models with similar loading, and is good to help pinpoint areas that need refinement.
Convergence Study Theoretically, the more elements used in a model, the closer the results get to the actual values. If the user refines the mesh and solves the revised model, the results will improve. This can be done repeatedly, but after a number of iterations the improvement in the results will start to decline; this is referred to as Mesh Convergence. It is up to the user to decide what percent difference in results between meshing iterations can be considered “converged”. This method can be very time consuming and compute intensive– the user may find that their hardware limits the ultimate mesh size. In these cases, it is best to use a targeted meshing approach and focus on areas of interest.
FEA is a very powerful tool that can give great insight into designs and reduce engineering uncertainty. However, the tool is limited by its inputs, with the mesh being a critical part of that. Asking the questions above and critically assessing the results can give the user better confidence in the mesh and improve the quality of the results.