Computational Science Stack Exchange is a question and answer site for scientists using computers to solve scientific problems. It only takes a minute to sign up.
Sign up to join this community
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?
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.
Kwok, W., & Chen, Z. (2000, October). A Simple and Effective Mesh Quality Metric for Hexahedral and Wedge Elements. In IMR (pp. 325-333).
Bi, Z. (2017). Finite Element Analysis Applications: A Systematic and Practical Approach. Academic Press.
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.
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.
