# Is mesh orthogonality important for FEM?

While studying mesh quality metrics in literature and software documentation, I've seen discussions about mesh orthogonality in Finite Volume Method (FVM) contexts, but not for Finite Element Method (FEM). For FEM, the metrics generally discussed and evaluated are skewness, smoothness and aspect ratio, for example.

This made me wonder: is mesh orthogonality also important for FEM?

EDIT: I will clarify my question by presenting the definition of orthogonality that I'm referring to. For two quadrilateral elements such as the orthogonality angle $$\theta$$ is the angle between the vector that connects the centroids of the elements $$\textbf{d}$$ and the vector normal to the surface connecting the elements $$\textbf{n}$$.

## 2 Answers

Yes. The constants that appear in the interpolation estimates upon which finite element error estimates are based contain minimum and maximum angles of triangles/tetrahedra (or similar geometric measures for quadrilaterals/hexahedra). These constants are smallest whenever you have equilateral triangles or square quadrilaterals.

• Thank you for answering the question. You are saying that equilateral elements are better for orthogonality. As far as I know, equilateral elements are also better for element skewness. So, is evaluating orthogonality between elements equivalent to evaluating element skewness? – Eduardo May 5 at 18:36
• I don't know the term "element orthogonality". How do you define it? In the end, you want all angles of all elements to be as equal as possible. That is achieved by using equilateral triangles. – Wolfgang Bangerth May 5 at 19:17
• I updated my answer by presenting the definition of orthogonality angle that I'm referring to. – Eduardo May 5 at 21:01
• @Eduardo It;s hard to define "skewness" for anything except a straight-sided quadrilateral, and for automatic mesh generation simplex-shaped elements (triangles or tetrahedra - and not necessarily with straight sides) are easier to work with than quadrilaterals. – alephzero May 6 at 1:53
• @Eduardo I see. The statement was that you get the smallest errors for elements with the most uniform angles. For triangles that leads to equilateral triangles. For quadrilaterals, that would be rectangles -- but a deeper analysis shows that you really want squares. – Wolfgang Bangerth May 6 at 13:06

It depends on the problem being solved and the element formulation.

The orthogonality criterion you state may not be as important as the shape of each element.

For example in structural mechanics, with an irregular (e.g. automatically generated and refined) mesh and isoparametric elements, an important criterion for accuracy is the largest angle between sides meeting at a node.

A "very thin" triangle with angles of 89, 89 and 2 degrees is pretty harmless (but not very efficient since it doesn't occupy much area), but a "reasonable looking" triangle with angles 120, 30, and 30 is significantly worse than a triangle which is close to equilateral (or rather equiangular).

• Thank you for bringing the dependence of the problem to the discussion. Do you have references of mesh quality in the context of structural mechanics? – Eduardo May 6 at 12:00