When performing Finite Element Analysis (FEA), there are many factors that can influence the accuracy of the simulation, the most critical of which may include mesh size and element quality. Mesh size, as you might guess, refers to the geometric dimensions of the element: length, width, and height. Element quality, though, is a bit more complicated: several measurements, including warpage, aspect ratio, Jacobian, skew, and minimum/maximum angle must be considered when discussing element quality.
Before we examine the different types of element quality measures, it is first important to understand why element quality matters. Simply put, better quality mesh yields more accurate results! But what exactly do we mean when we say “better quality”? As a general rule, better quality means the element’s shape more closely resembles a square for quad elements and an equilateral triangle for tria elements. As elements deviate from these “perfect” shapes, the shape functions (which govern the interpolation between integration points) begin to break down and yield less reliable answers.
Article 4: Element Quality Check Criteria
Although this list is certainly not comprehensive, the most common (and, generally, most important) element quality measures include the following (in no particular order):
Aspect Ratio is a measure of an element’s longest length divided by its shortest length and is essentially an indicator of slender elements. Although there are no distinct rules for quality, an aspect ratio below 5 is generally considered to be good quality; however, values of up to 10 may be considered acceptable based on the specific use case.
Jacobian, deemed to be one of the most important element quality criteria, is a bit more complex than the other quality measures. Used to measure the deviation of a given element from an ideally shaped element, the Jacobian value is a complex measure calculated by mapping an ideal element in parametric coordinates onto the actual element in the global coordinates; the amount that the ideal element needs to “stretch” to match the actual element is the basis for the Jacobian calculation. A perfectly shaped element will have a Jacobian value of 1.0; however, any value greater than 0.6 is generally considered acceptable (depending on your application and required level of accuracy, of course).
Warpage, which measures how “bent” an element is (the amount by which an element deviates from perfectly planar), is a quality measure that only pertains to quad-based elements (since a 3-noded element is inherently planar). Warpage values below 10 are typically acceptable, with a warpage measure of 0 indicating a perfectly planar element.
Skewness is a measure of angular deviation with respect to the “perfect” element shapes discussed earlier (square and equilateral triangle). Although it can be difficult to achieve, especially in areas of complex geometry, an ideal skew value is below 0.50.
Element Quality Example
Now, let’s take a look at a quick example of how mesh quality can influence the accuracy of predictions by performing a simple validation exercise using a cantilevered beam. For this example, let us imagine a beam with a length of 100 mm and a solid box section that is 10 mm by 10 mm.
By performing a simple hand calculation and comparing the results with FEA predictions performed using various meshes, we can observe the impact that element quality has on accuracy. For the case mentioned above, the handbook calculation for deflection is 1.92 mm. As we can see in the contour plots below, the deflection prediction using high quality mesh is within 1% of the theoretical value whereas the model with poor element quality has an error of slightly over 5%.
Finding Bad Elements
Luckily, as users of Abaqus CAE, we have a neat tool that can help us understand the type of problems we might have in our mesh before we ever even run the model. It can also let us know which elements will throw ‘warnings’ by the solver and if any of them will throw ‘errors’, meaning the solution will not even pass pre.exe.
In the mesh module of CAE, there is an icon that looks like this:
When you click it, a dialogue box will appear and allow checks of Shape, Size and Analysis (warning and errors). Initially these will be populated with Abaqus defaults, but you can use this tool to assess the quality of your elements against your own requirements, too. The nice thing here is that the elements that fail the checks will light up in your model, making it extremely easy to diagnose and fix problems dynamically while you mesh. When we apply the check to our ‘bad aspect ratio’ model, you can see that all of the elements light up yellow, meaning that they will pass pre.exe, but Abaqus is going to flag them as potentially problematic for the analysis.
As we’ve seen today, element quality can have a notable influence on the results of your simulation. In fact, poor mesh quality can lead to erroneous solutions altogether, especially when you consider compounding interpolation errors that can occur in large models subjected to complex loading. For this reason, it is advisable that analysts pay particular attention when meshing areas of interest (e.g., potential high-stress locations) during the initial model build. Stay tuned for future installments of our Element Selection blog series for more tips and tricks that can be used when determining the elemental makeup of your Finite Element Analysis (FEA) model!
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