In my last entry, I wrote about how to obtain accurate stress results by ensuring that there were at least two elements across the thickness in areas of interest. However to obtain accurate results, the quality of the mesh itself must be taken into account.
In order to talk about mesh quality, there are a couple important metrics that need to be taken into consideration.
The aspect ratio of an ideal tetrahedral element is 1.0. This is a ratio of the longest edge to the shortest normal dropped from a vertex to the opposite face, normalized with respect to the shortest normal dropped from a vertex to the opposite face of a perfect tetrahedral element. A general rule of thumb is to not have more than 10% of the elements with an aspect ratio higher than 10. Extremely large values >> 40 should be closely examined to determine where they exist and whether the stress results in those areas are of interest or not.
Figure 1: This is an example of a first order tetrahedral solid element with an aspect ratio of 1.0. The aspect ratio is the ratio of the longest edge to the shortest normal dropped from a vertex to the opposite surface normalized with respect to a perfect tetrahedral element. In this case, the shortest normal has a value of 1.0 because it is normalized with respect to itself.
Figure 2: On the left is a picture of a tetrahedral element with an aspect ratio of 1.0. The element on the left has a much larger aspect ratio.
In SolidWorks Simulation, you can create a mesh plot showing the values of the aspect ratio within the mesh after it has been generated.
Figure 3: Right click on the Mesh icon in the Simulation tree> “ Create Mesh Plot”
Figure 4: You have the choice to create a mesh plot, an aspect ratio plot, or a jacobian plot.
Figure 5: We see that the maximum aspect ratio in this mesh is 3.79, well within acceptable limits
The second metric used to determine mesh quality is the Jacobian Ratio. This method is only available for second order (High quality) mesh elements. Parabolic (second order) elements are able to map curvilinear geometry more accurately than the first order linear elements. The midside nodes are placed on the actual geometry of the model, and in extremely sharp or curved boundaries, the edges can cross over each other. This can result in a negative jacobian ratio which will cause the solver to fail.
Figure 6: The Jacobian is a measure of the curvature of the edge and distortion at the mid-side node.
Figure 7: This is an example of a 2-D representation of an element with a Negative Jacobian ratio. The curvature of the geometry that the element was trying to map was too great for the size of the element, causing the edge to collapse in on itself creating a negative jacobain ratio. This will cause the solver to fail.
As with the first order elements and the aspect ratio, the jacobian ratio of a perfect tetrahedral element with linear edges is 1.0. The jacobian ratio of an element increases as the curvature of the edges increase and are calculated at the selected number of Gaussian Points for each tetrahedral element. In general, elements with a jacobian ratio less than 40 are acceptable.
You can also create a Mesh Check Plot similar to the Aspect ratio check in SolidWorks.
Figure 8: The Jacobian check plot shows that the only areas where there are elements with a non-unity Jacobian value are areas where there is actual curvature of the geometry. Here the maximum value of the Jacobian is 1.095, and there is no need to further refine the mesh.