Why not Rectangular Mesh for Arbitrary Shape Domain?

[mdn]

Hi all,
It is well know that, rectangular or quadrilateral element gives better result over triangular meshes in finite element method.
I know, Rectangular meshes has more discretization error, but my question is that, if i refine mesh up to 4 to 6 pixel width of screen, where i am wrong?

 

[Simon Bridge]

Depends what you are trying to achieve and how you do the refinement.

What do you mean: "gives better result"? ... surely that depends on the desired result?

 

[AlephZero]

It is well know that, rectangular or quadrilateral element gives better result over triangular meshes in finite element method.

I know, Rectangular meshes has more discretization error

So you are saying they are both better and worse, at the same time??

Second-order isoparametric triangular elements (with midside nodes) are as good as second-order quadrilaterals for most purposes. The big advantage of triangular and tetrahedral elements is that automatic mesh generation and mesh refinement is easy, using Delaunay triangularization for example. With quadrilaterals, you can end up with a few nodes with a different nodal valency to the rest, which can cause local errors in the solution. Or, you end up with a mesh that is all quadrilaterals except for a few triangles.

 

[mdn]

No, i mean even though Rectangular Meshes has more discretization error(Mapping of domain by elements), but it can overcome by refining very fine mesh.
actually this question come to my mind, because i made automatic rectangular mesh generation program in Flash,
and getting better result by using this rectangular mesh in solver. Generally what i observed many research worker use the triangular mesh, instead if i use rectangular mesh for arbitrary shape, what should have been problem in future?.
thanks.

 

[mdn]

Depends what you are trying to achieve and how you do the refinement.

What do you mean: "gives better result"? ... surely that depends on the desired result?

The result i getting from solver by using this rectangular mesh.

 

[Simon Bridge]

The result i getting from solver by using this rectangular mesh.

What about it?

 

[mdn]

What about it?

Respected Sir, thank you for your reply.
I will explain the stuff what i did and what is problem.
I developed 2D FEM Solver in Java and Mesh generation program using rectangular elements in Flash.
I solved Helmholtz equation for arbitrary shape wave guide, and obtained result compared with others papers as well as analytical data and found to be good agreement with it.
I have attached two sample meshes and field plot which output from solver, generated in my program.
my questions is very simple, should i use such Rectangular element mesh for any arbitrary domain? Is there any problem to use such rectangular element mesh for arbitrary shape domain? because i want to avoid master element (used for bilinear mapping ) and Jacobian.

 

[AlephZero]

Maybe it doesn't matter for your problem, but approximating the boundary as a "staircase" of horizontal and vertical lines would be a horrible idea in many FE applications.

If the boundary conditions involve derivatives normal to the boundary, even approximating a curved boundary as a polygon with straight sided elements is an order of magnitude worse than using curved sided elements, because the direction of the normals can be very important.

If you need about 80 elements across the diameter of your circle to get "good" results, something is probably wrong. About 8 elements across the diameter should be enough, if you modeled the boundary correctly.

 

[mdn]

That's why i used word "pixel width" in first question. I can refine this mesh again to minimize "staircase" space.
again there is no problem of number of elements, this is structured element it has following features
1. easy to increase the order of approximation at we wish. (more DOF)
2. Direct surface integration
3. no master element
4. and last, easy to assemble
that's why i made little compromise in accuracy (Approximation! We can't expect exact answer).
any way, i got your message. thanks once again.

 

[AlephZero]

I can refine this mesh again to minimize "staircase" space.

The length of the boundary of your mesh will never converge to the correct value. For a circle of radius $r$, your mesh always has a boundary of length $8r$, not $2\pi r$.

If your boundary conditions involve an integral around the boundary, your meshes will not converge to the correct solution as you increase the number of elements, or the order of the elements.

 

[mdn]

The length of the boundary of your mesh will never converge to the correct value. For a circle of radius $r$, your mesh always has a boundary of length $8r$, not $2\pi r$.

Does it means all straight edges of element, those are touched to physical boundary has sum of length of edges is equal to = 8r?
Because touched edges of elements on physical boundary is i considered main boundary, and what ever boundary integral operation, have made on touched edge only.

 

[Simon Bridge]

Parallel to AlephZero;
...objects expressed in terms of pixels should refer to the rendering on the screen rather than to the underlying mathematics. On-screen everything is restricted to a rectangular grid due to the geometry of the screen.

I am wondering if you have properly separated the computation (helmholtz solutions for a specified waveguide) from the on-screen representation of it.

 

[mdn]

Respected sir,
could you give me comment on paper, just i prepared for arbitrary shape waveguides using rectangular element.
Shall i attach paper?

 

[Simon Bridge]

I'll leave that to AlephZero;
If this is a research paper, you should use citations to discuss your choice of paradigm (rectangular vs triangular) in a short section near the start or as an appendix.

Aside: you don't have to call people "respected sir" all the time.
English language can be quite informal among technical types - it is what is in your head that counts and too much formality can get in the way of cooperating to solve problems.

 

[mdn]

Parallel to AlephZero;
...objects expressed in terms of pixels should refer to the rendering on the screen rather than to the underlying mathematics. On-screen everything is restricted to a rectangular grid due to the geometry of the screen.

I am wondering if you have properly separated the computation (helmholtz solutions for a specified waveguide) from the on-screen representation of it.

Would you give me comment on attached example?

 

[AlephZero]

There is nothing wrong with using quadrilateral elements if you want, but you should be making a mesh that properly matches the boundary of the geometry, for example like this:

 

[mdn]

thanks, you indeed AlephZero!
Now i being coding on quadrilateral element as you described.

 

[NumericalFEA]

thanks, you indeed AlephZero!
Now i being coding on quadrilateral element as you described.

Quadrilateral meshing programs that would deliver high-quality meshes (low number of ill-conditioned elements) are hard to come by, but there are some very good meshers (mostly commercial).