# How to determine the Jacobian Ratio for triangle element?

I am trying to implement an algorithm to find the Jacobian ratio for each triangle in mesh as a part of mesh quality check.

Let's say that I have vertices of the triangle: $$P_1(x_1, y_1, z_1)$$, $$P_2(x_2, y_2, z_2)$$, $$P_3(x_3, y_3, z_3)$$. Having this data how to determine the Jacobian ratio of the triangle?

• What is the Jacobian ratio? – nicoguaro Mar 6 at 13:24
• it is a measure of the deviation of a given element from an ideally shaped element – Aravindh SK Mar 6 at 13:37
• Do you have a reference where it is defined? – nicoguaro Mar 6 at 13:37
• I read on random website. – Aravindh SK Mar 6 at 13:39
• I believe that there are 3 things that define the mesh quality: 1) Aspect Ratio: which is the ratio between the largest and the smallest edge of the element. The closer to 1, the better. 2) Internal angles: they should be close to the angles of an element with no distortion. 3) Jacobian determinant: must be positive. – Gustavo Costa Apr 7 at 19:04

## 2 Answers

It seems that the Jacobian ratio is defined as the ratio between the maximum and minimum Jacobian determinant in an element [1, 2]. And, that a value between 0.33333 and 1 is good-enough [3].

Nevertheless, for linear elements, the Jacobian is constant and thus the same over each element. As mentioned by @GustavoCosta, 3 descriptors commonly used for element quality check are aspect Ratio, internal angles, and Jacobian determinant. But there are many more as mentioned in a previous answer and references there. You might also want to check reference 4.

### References

1. Kwok, W., & Chen, Z. (2000, October). A Simple and Effective Mesh Quality Metric for Hexahedral and Wedge Elements. In IMR (pp. 325-333).

2. Bi, Z. (2017). Finite Element Analysis Applications: A Systematic and Practical Approach. Academic Press.

3. Bucki, M., Lobos, C., Payan, Y., & Hitschfeld, N. (2011). Jacobian-based repair method for finite element meshes after registration. Engineering with Computers, 27(3), 285-297.

4. Shewchuk, J. R. (2002). What is a good linear finite element? interpolation, conditioning, anisotropy, and quality measures (preprint). University of California at Berkeley, 73, 137.

I suppose you have calculated the determinant of the Jacobian, and you may calculate the Jacobian ratio by the ratio of the maximum and the minimum of the determinant value. Please refer to this link if you have difficulties to calculate the Jacobian.

• Can you add a reference where they define this measure? – nicoguaro Mar 7 at 16:23
• You may refer to the "Basic Finite Element Method as Applied to Injury Biomechanics" books, they give examples to calculate the Jacobian ratio in Gaussian quadrature nodes. The definition of the Jacobian Ratio can be found in "Finite Element Analysis Applications: A Systematic and Practical Approach". – Lele Mabur Mar 8 at 3:24
• @Lele Mabur, I have already read but I didn't understand properly how they have defined J matrix. – Aravindh SK Mar 8 at 9:38
• @AravindhSK sorry for late response, I suggest you look an open source software for FE mesh generator, e.g. gmsh. They have build in function API to call in some programming language. Maybe the documentation will help you a bit. Thanks – Lele Mabur May 20 at 5:54
• @AravindhSK take a look at gitlab.onelab.info/gmsh/gmsh/-/blob/master/Mesh/… if you familiar with C++ – Lele Mabur May 20 at 6:11