Friday, August 10, 2012

The Jacobian: Finite Element Mesh Quality

This is as technical as engineering comes. Bring up a Jacobian to a bunch of finite element engineers and hopefully they will all know what you are talking about. This is so technical that it it typically only covered in senior level college classes or graduate school classes. Although, if you do cover it in college you will probably do the actual matrix equations, even though in real life (the business world) a computer does it in fractions of a second.

Now more specifically the Jacobian, which is short for the Jacobian Matrix Determinate, is really the best measure of finite element mesh quality. It is one number which defines how good or bad an element is. The Jacobian is a measure of the normals of the element faces relative to each other. Unfortunately, Hypermesh does not show the element face normals on solid elements, but it is basically an arrow on each face pointing out perpendicular to the face. The range of a Jacobian is from 1, a perfect cube, to something lower, -1 or even lower. The smallest Jacobian I have seen was -1.45. When the element face normals start to cross, that is they are not perpendicular to each other, your element quality gets worse.

For several examples I created the image below. In all of the elements, except for the red one, I simply translated one node (vertex) to a new location and kept the other seven in the original cube positions. You can see that as the node moves farther away from the cube position the element quality gets worse.

Sample Jacobians (J): Orange Cube J = 1.0; Blue J = .942 (z is .9 for one point); Purple J = .883 (z and y are .9 for one point); Pink J = .398 (z and y are .5 for one point); Green J = -.409 (z and y are -.1 for one point); Tan J = -.130 (z and y are .1 for one point); Red J = 1.0 (z is 3 for all four end points); Light Blue J = .072 (x y and z are .5 for one point)

How bad is bad? Abaqus will not run a job with a Jacobian below 0, at least not for me. Ansys on the other hand has less strict mesh quality requirements. Often times Abaqus will not run a solid element Jacobian below 0.2 and a shell element Jacobian below 0.3. And yes, I have had one element with a negative Jacobian prevent an Abaqus job with over 100,000 elements from running. If you can get all of the Jacobians in your model above .5 you can typically say you have a good quality mesh.

If you liked this click on something below or leave a comment, please. I can do more like this, but only if there is demand.

24 comments:

  1. Hi, thx for this article, I always wonder what was this quality criterion because element with the same jacobian could have different shapes.

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    1. But you can compare other mesh quality parameters

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  2. Thanks for the article. It is simple,clear and to the point.

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  3. Wow it seems finite element analysis is a bit over my head at the moment, I'll have to start a bit simpler.

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  4. it is really good explanation, simple to understand, great work...

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  5. very simple explanation, perfect information, thank you!

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  6. what happens if jocobain fails in analysis results

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  7. But how to see this jacobian value in abaqus??

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  8. can you tell at what point jacobian was found, as jacobians vary at every point in a geometry.I jus couldn understand at what point those values in figures were found

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    1. Vishnu, I don't believe anyone answered your question in regards to what point the Jacobian is calculated. Typically most software packages do it at the integration points, and for elements with multiple integration points this means the Jacobian evaluation ends up with multiple values. Then the min and max of the values are used to provide a ratio, which is why the value of the Jacobian typically is between -1 and 1. It is important to understand exactly what the software package you are using does, so it is best to ask the application engineers for details on this. I hope this helps :)

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  9. The image that you have created to explain the element distortion is awesome. You have done a great job, I hope you'll write more articles like this.

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  10. Great article. Simple and succint.

    Thanks, here from Brasil.

    ;)

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  11. Hi.... i work with Hypermesh... i got max jacobian above 0.6.. but Hypermesh default value is more than 0.7.... my mesh is qualify or not?

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