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Element Aspect Ratio

December 14, 2010

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.

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9 Comments
  1. George permalink

    I have a simulation where elements in my area of interest have an aspect ratio of about 10-15. The jacobian ratio in that area is good, about 1. Which of the two is more important for an evaluation of the quality of the mesh? Will I get results with precision ? Generally, what is the acceptable aspect ratio for a simulation in SW?

    • Hey George,
      I believe that an element aspect ratio of less than 10 is generally acceptable. in regard to which one is more important, I am not sure which of the two is more important, but less than 40 for jacobian, but not negative, and less than 10 should be good bounds for good results. also I think that in areas where the element aspect ratio or jacobian are high or above the bounds will have poorer results.

      – Andrew

  2. George permalink

    Thanks a lot for the answer
    George

  3. George permalink

    Hi again Andrew. My question might seem silly to you. Suppose I have a part and I want to add a fixture. I want to restrain its movement along ONE AXIS only. The green arrow that appears in SW shows the direction towards the body CAN or CAN’T move? I’ m asking because every time I m trying to add a restrain along one direction only (along x, or along y, or along z, or along the radial direction for cylinders), SW gives me the option “reverse direction” and when I click on it the direction of the green arrow changes. Pretty confusing. I’d highly appreciate your help.

    • Hey George,

      The green arrow indicates the direction that is being fixed (can’t move). If the green arrow is reversing direction when you click on “reverse direction” then it might be that you have specified a displacement. I am guessing that you used the advanced fixtures option and are using the “use reference geometry” option? If you specify a 0 displacement then it is fixed in the direction of the green arrows, or is displaced equivalently.

      Hope this helps,

      -Andrew

  4. George permalink

    Yes, i used advanced fixtures and reference geometry. Suppose i want to define a very small displacement, let’s say 0,001 mm towards the +x axis (to the right). The green arrow should also point towards +x or towards the opposite direction? Thanks in advance. It’s a pity you stopped working on this blog. It’s really helpful.

    • Hey George, the direction of the arrow will indicate the direction of displacement, so in this case you would want the arrow to point towards the +x direction when you want to define a .001mm displacement in the +x direction or if you want the reverse, in the -x direction. Yeah I stopped working on this blog after I changed jobs and went back to school for a Master’s degree, but if you have more questions feel free to ask, I’ll try to get to answer them if I can.

      If you have a local Solidworks Reseller/VAR they may offer classes in Solidworks Simulation. I find one of the best parts about that is it comes with a manual which has helpful examples. Unfortunately Solidworks does not sell these manuals unless you take a class, but there are also other books you can purchase from third party authors as well.

      -Andrew

      • George permalink

        Thanks a lot for the help and the info. I have to use SW FEM calculations for my job but unfortunately I didn’t have any FEM lessons during my university years, so I ‘ve bought a book covering basic FEM theory and SW FEM and I m trying to figure everything out by myself. it is surprising that I didn’t find this basic answer anywhere. Good luck with your Master’s degree. Cheers

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