Not long ago, I wrote a blog titled “How Do I Know If My Mesh is Good Enough?”. This post dealt with the issue of mesh density and how it affects solution accuracy in a finite element analysis. It also mentioned that element type and shape also affect accuracy. This article focuses on element shapes: specifically, what’s meant by poor element shapes and how can they affect solution accuracy.
Many finite element analysts are too preocuppied with other preprocessing considerations to consider element shapes unless their meshing tool checks shape quality automatically. This can be a mistake, however, as poor element shapes will often cause convergence problems in nonlinear analyses and can produce inaccurate results, especially when they’re located regions of a model where critical results are essential.
A number of element shape metrics can be used to evaluate shape quality. These include Jacobian ratio, aspect ratio, corner angle, midside node position, parallel deviation, warping factor, and composite quality measures. In all cases, they measure the ability of an element to map data between element (numerical) space and real, physical space, which is a critical step in FEA. Some of these metrics are shown in Figure 1, above. The ideal element shape is one that has near 90 degree angles for quadrilaterals and hex elements, and near 60 degree corners for triangles and tetradral elements, while also having low elongation (aspect ratio). The element surfaces should also be relatively flat, as measured by the warping factor.
The ANSYS software is a good example of a finite element code that includes automatic element shape checking. By default, it checks both Jacobian ratio and a composite quality measure. These two measures together are quite comprehensive because they account for the placement of midside nodes, skewness, warping, curvature, aspect ratio, and ultimately the element’s ability to map data. Figure 2 shows the Jacobian ratios for different 2D element shapes. As you can see, large curvatures due to midside nodes deviating from a straight edge and large corner angles result in large the Jacobian ratios. The element quality number used by ANSYS as a composite measure of shape quality is based on the ratio of the volume of an element to a sum of its edge length. In 2D, a value of 1 indicates a perfect cube, square, or triangle while zero indicates an element that has zero or negative area. A simple example is shown in Figure 3 for a single 2D quad element.
Figure 2: Jacobian Ratios for Different 2D Element Shapes

Figure 3: Element Quality for a PoorlyShaped Quadrilateral

All of this element shape information is great, but how much do poorlyshaped elements really affect critical results? To illustrate this, we can look at the example for the same 2D plane stress model used in the earlier study of mesh density. In this model (Figure 4), a 2D bracket was constrained at its top end and subjected to a shear load at the edge on the lower right. The maximum Von Mises stress converged to a value of 14,754 psi with a very high mesh density and high quality element shapes in the critical region, as shown in Figure 4. Figure 5 shows the mesh used in the current study. This mesh is coarser than the converged mesh used in the earlier study but fine enough to predict stresses within 1.5% of the conveged stress with wellshaped elements. Meshes with a range of element quality were evaluated. The best and worstshaped of these evaluated meshes in the critical region are shown in the upper right of Figure 5. A comparison of the maximum nodal averaged and unaveraged stress results are listed in Table 1 for these different meshes. Notice that the maximum stress solution deviates from the converged value by 0.75% for a Jacobian ratio of 3.75, 2.4% for a ratio of 29.4, and 8.1% for a ratio of 34. This last ratio corresponds to the mesh with highly skewed elements shown in the upper right of Figure 5. Table 1 also shows that the difference between the unaveraged stress and the converged stress is considerably greater for the mesh with significantly skewed elements.
While this doesn’t represent a comprehensive test of the effects of poor element shapes on results accuracy, it provides a solid example of its effects. It‘s important to note, however, that the influence of element shape on results is not easy to predict, especially because so many other factors affect results accuracy. These factors include the mesh density, element type selection, response type (linear or nonlinear), and the accuracy of inputs such as loads, constraints, and material properties. So the overall error can be considerably different from the error caused by element shapes alone.
Finite element codes such as ANSYS will automatically check the Jacobian ratio and composite element quality, and will warn users if the values exceed specified limits. However, it’s good practice to check these error measures yourself, especially the Jacobian ratio, which provides a direct indication of the ability of a mesh to map data from element space to real space. As illustrated by the bracket example, poorlyshaped elements in regions of your model where results accuracy is critical can cause significant inaccuracies which can ultimately influence important design decisions. We are very interested to know of any experience you have with poorlyshaped elements affecting your results. Have they caused real problems, how did you find them, and what did you do to fix them?
Figure 4: HighlyQuality Refined Mesh with Converged Stress Results

Figure 5: 2D Bracket Model Geometry and Mesh

Table 1: Maximum Stresses For Meshes With Good to PoorlyShaped Elements
