Why not Rectangular Mesh for Arbitrary Shape Domain?

In summary, the conversation discusses the use of rectangular or quadrilateral elements in finite element method, which are known to provide better results compared to triangular meshes. However, there is a trade-off as rectangular meshes also have more discretization errors. The use of triangular and tetrahedral elements is often preferred due to their ease of automatic mesh generation and refinement. The conversation also touches on the issues of approximating boundaries as a "staircase" of horizontal and vertical lines and the importance of considering the direction of normals when using curved sided elements. In addition, the conversation addresses the use of rectangular elements for arbitrary domain and the potential problems that may arise from it.
  • #1
mdn
49
0
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?
 
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  • #2
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?
 
  • #3
mdn said:
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?? :confused:

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.
 
  • #4
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.
 
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  • #5
Simon Bridge said:
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.
 
  • #6
The result i getting from solver by using this rectangular mesh.
What about it?
 
  • #7
Simon Bridge said:
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.
 

Attachments

  • Circular Mesh.png
    Circular Mesh.png
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  • Double Ridge Mesh.png
    Double Ridge Mesh.png
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  • Circular TE21.png
    Circular TE21.png
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  • #8
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.
 
  • #9
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.
 
  • #10
mdn said:
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.
 
  • #11
AlephZero said:
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.
 
  • #12
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.
 
  • #13
Respected sir,
could you give me comment on paper, just i prepared for arbitrary shape waveguides using rectangular element.
Shall i attach paper?
 
  • #14
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.
 
  • #15
Simon Bridge said:
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?
 

Attachments

  • Right Angle Triangle Waveguide.docx
    183.6 KB · Views: 366
  • #16
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:
circle3.256.gif
 
  • #17
thanks, you indeed AlephZero!
Now i being coding on quadrilateral element as you described.
 
  • #18
mdn said:
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).
 

1. Why is a rectangular mesh not suitable for an arbitrary shape domain?

A rectangular mesh is not suitable for an arbitrary shape domain because it will result in poor representation of the geometry. The mesh will not conform to the shape of the domain, leading to distorted elements and inaccurate solutions. Additionally, a rectangular mesh is not efficient in capturing the complex features and boundaries of arbitrary shapes.

2. Can a rectangular mesh be used for an arbitrary shape domain if it is refined enough?

While refining a rectangular mesh can improve the accuracy of solutions, it cannot completely overcome the issues of poor representation and inefficiency. The mesh will still have a fixed structure that cannot conform to the shape of the domain, resulting in inaccurate solutions and increased computational cost.

3. What are the advantages of using a non-rectangular mesh for an arbitrary shape domain?

A non-rectangular mesh, such as a triangular or unstructured mesh, can conform to the shape of the arbitrary domain and accurately capture its features and boundaries. This results in more accurate solutions and a more efficient use of computational resources. Non-rectangular meshes also allow for more flexibility in mesh size and refinement, leading to better overall results.

4. Are there any specific applications where a rectangular mesh can be used for an arbitrary shape domain?

In certain applications where the shape of the domain is relatively simple and uniform, a rectangular mesh may be sufficient. For example, in some structural analysis problems, a rectangular mesh can be used for domains with straight edges and uniform thickness. However, for most cases, a non-rectangular mesh will provide better results.

5. What are some common challenges in using a non-rectangular mesh for an arbitrary shape domain?

The main challenge in using a non-rectangular mesh is the increased complexity in mesh generation and manipulation. Non-rectangular meshes require specialized software and algorithms, and may also require more computational resources. Additionally, the quality of the mesh may be affected by the shape and curvature of the domain, so careful mesh refinement and quality checks are important.

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