# Why are quad elements better than tria elements in fem/ cae analysis?

Why are quad elements better than tria elements in fem/ cae analysis?

### The triangular element

The triangular element  is called as a constant strain triangle(CST) element. Value of the stress tensor in the whole element will be constant .  This may not be true in the real situation. So this approximation may create a lot of error.

That is the reason, we are not prefering triangular elements. But we will be forced to use this if the geometry is very complecated due to the difficulty in meshing with quad elements.

The CST nature of this element is coming from the nature of the shape function .

Sreenath.A.M
Asst. Professor
Mechanical Engineering Department
National Institute of Technology
Calicut,India

Since your question makes unqualified reference to quads and triangles, permit me to offer you an answer that, too, is (almost) equally unqualified: They aren't! Quads aren't better than triangles; triangles are better than quads!!

To see how, feel free to invite me to conduct a course emphasizing fundamentals of FEM, at your organization :) [More seriously, Tata Autocomp and I were seriously talking of doing this a few years ago.]

OK, to give you some idea of how I approach such issues in my course:

The real considerations here is: the computational cost (say "complexity") of the resulting algebraic system and that of any further processing, for a given degree of accuracy. The computational cost may be estimated by looking at, say, the total number of DOFs generated in building the FE model, as also the cost involved in computing the secondary (or gradient) fields, including performing their interpolations at "other" points (if and as necessary).

If both FE models using both quads and triangles (of suitable orders) are roughly the same in terms of their implied computational costs, then, by themselvs, triangles can have an edge in terms of better representing curved boundaries.

However, for certain problems involving rectangular domains and certain kinds of loading conditions and constraints, quads might have a slight edge in terms of accuracy for the primary variable (displacements). The difference isn't much. And, if you think of the discontinuities in the gradient field, off-hand, I think, even that does not matter much, if at all---indeed, a reversal might occur in the sense that triangles actually could perform better than the quads.

In either case (whether quads perform better or triangles), the difference is not much: off-hand, often, 7.5%, 5%, or even less. The usual "accuracy" requirement (i.e. deviation from the continuum case) can easily have a tolerance of a similar magnitude---or even more!

Finally, realize that computers have advanced real fast. Many of the golden observation of yesterday's don't any longer make any sense. You can always find, in yesteryears' literature, an isolated result in favor of quads vs. another one in favor of triangles. However, also think of the biggest algebraic system (matrices) that could
be completely loaded in-core, and solved fast, using the yesteryear's machines vs.
today's.

Considering everything (range of problems, advances in computing, etc.), as of today, the debate of quads-vs-triangles (or bricks/hex-vs-tets) may be taken to be more or less completely meaningless.

But then, again, just to remind you in case you have by now forgotten because of my long (or "detailed," "elaborate," etc. sort of) reply: You should know that I am always looking for commercial opportunities for conducting my FEM course. I have even upgraded/revised my slides and the notes I circulate as a part of this course, now! Feel free to recommend me to the decision makers in your organization (and call me over if you are the decision-maker), by email: aj175tp AT yahoo DOt co DoT in .

--Ajit

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