Element Quality Frivolity: Are My Elements Bad?

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In FEA, elements are like dogs in that they are all good boys. However, the FEA textbooks would have you believe that that bad quality elements are bad and they should feel bad. They may say so without elaborating, the reality being that element quality requires a good grasp of the shape functions at hand. Since I don’t have a good grasp of shape functions (and my dog ate that chapter of my textbook) I set about the easier method of investigating how element quality impacts some simple linear static solves. The results have made me less critical of “poor” quality elements – but don’t tell the elements I called them “poor”, they are stressed as is.

Figure 1: Do you fix your element quality?

 

The first place I hunted for element quality examples was the OptiStruct documentation. OS-0720 is a cantilever beam study which shows how various element breeds handle In-plane Extension, In-plane Bending, Transverse Bending, and Twist. The beams in this study are shown in Figure 2 and have either rectangular, parallelogram, or trapezoidal elements with one element through the height of the beam. Figure 3 shows deflection results normalized to the theoretical value. This study shows how CQUAD4 elements can struggle to resolve in In-plane Bending, particularly if they have trapezoidal distortion. My “all elements are good boys” claim is strained here, particularly when the bad elements come in packs.

Figure 2: OS-0720 CQUAD model

Figure 3: OS-0720 results showing the fallibility of poor quality CQUAD4

 

But what if most of the mesh was composed of quality elements and there was a single bad element – a wolf among sheep – at a point of interest? How much would the results be thrown off? I made a large plate mesh with a small hole in the middle. My model is shown in Figure 4 where symmetry boundary conditions are applied and a bad element is made at the top of the hole, where we would expect peak stress. Density around the hole was varied to study the effects of a coarse vs fine mesh and these meshes are shown in Figure 5.

Figure 4: Plate with a hole boundary conditions

Figure 5: Plate with a hole mesh refinement with good elements

Poor quality elements were inserted by pawing nodes on the element at the top of the hole until the Jacobian was less than 0.5. This made elements with either a very obtuse angle at one corner or a comparatively smaller angle at the other corner, as shown in Figure 6.

Figure 6: Plate with a hole model types of element distortion

Stress results were referenced to the theoretical stress of 30 – this value stems from a nominal stress of 10 and stress concentration factor of 3 for an infinite thin plate with a circular hole. Results for coarse vs fine mesh refinement are shown in Figure 7.

Figure 7: Plate with a hole model corner stress results for coarse and fine mesh

I barked at how close the stress values of poor quality elements were to the good quality baseline. As shown in Figure 8, the peak corner stress in poor quality elements was sometimes at the wrong corner. Sometimes an inward element had higher stress, but not significantly higher.

Figure 8: Plate with a hole model corner stress results for poor quality elements

A summary of the stress difference between poor and good quality elements at the top of the hole is shown in Figure 9 and Figure 10 for corner stress and centroid stress respectively.

Figure 9: First order corner stress good elements vs bad elements for various mesh refinement

Figure 10: First order centroid stress good elements vs bad elements for various mesh refinement

The same results are shown in Figure 11 and Figure 12 for second order elements.

Figure 11: Second order corner stress good elements vs bad elements for various mesh refinement

Figure 12: : Second order centroid stress good elements vs bad elements

Both studies indicate that poor quality elements will change the stress results at their location, but perhaps not to huge extent. All elements are good okay boys again?

Next, I pup-ared some plate models with simple boundary conditions, shown in Figure 13, that should all have uniform stress distribution. Figure 14 shows these CQUAD4 elements with error message generating vertex angles, skew angles, and aspect ratios for a variety of mesh refinements.

Figure 13: Uniform stress model boundary conditions

Figure 14: Uniform stress model mesh

For CQUAD4 elements, the element defects had no impact on stress or displacement results. The exception is bending, where mesh refinement affects displacement. Figure 15, Figure 16, Figure 17, and Figure 18 show stress and displacement, where any stress difference between elements is in the numerical noise.

Figure 15: Uniform stress model XTension

Figure 16: Uniform stress model YTension

Figure 17: Uniform stress model XYShear

Figure 18: Uniform stress model YBending

The same studies were run with CQUAD8 elements, shown in Figure 19. The poor quality elements here did have significantly lower stress values than their litter mates.

Figure 19: Uniform stress model CQUAD8

So poor quality elements are capable of resolving uniform stress distribution, but what if there’s many poor quality elements throughout the mesh? I concluded my investigation with a cantilever beam model similar to the beginning; this time poor quality elements are buried among the good quality elements and there are multiple elements through the beam’s height. In Figure 20, the poor quality beam is above and the good quality beam is below.

Figure 20: Poor quality beam boundary conditions

For both CQUAD4 and CQUAD8 elements, overall displacement results for the poor quality beam is very close to the good quality beam, as shown in Figure 21.

Figure 21: Poor quality beam displacement (1.132 expected)

Similarly, Figure 22 shows overall stress is very similar between the beams. However, stress close to a poor quality element (Point is different from the stress at the same location in the lower beam. Measurements for these figures is shown in Table 1. Generally, the CQUAD8 beam is less sensitive to poor element quality.

Figure 22: Poor quality beam corner stress (Expected Stress at Point A = 18.75 Expected Stress at Point B = 16.56, using stress for a cantilever beam in bending sigma = yFL/I  )

Table 1: Poor Quality  Beam Displacement and Stress

These studies indicate that the presence of poor quality elements might not invalidate models. Keep in mind these results are for linear static shell elements and that quality may have a bigger impact for different simulations or different result types, particularly if the elements have parallelogram or trapezoidal distortion. As for myself, I will certainly be less dogmatic with element quality requirements.