FEM Method in 2D with Triangular Elements

In summary, the conversation discusses the FEM method in 2D with rectangular elements and the difficulty in learning triangular elements due to their complexity and dependence on global coordinates. The request for a simple numerical example for a temperature problem is made, and it is noted that the assembly process for triangular elements can be confusing, especially when dealing with different orientations and sizes. Resources and tips are shared for understanding and implementing the assembly process, as well as the importance of element orientation and shape.
  • #1
bob012345
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I have worked through the basics of the FEM method in 2D with rectangular elements and now am trying to learn triangular elements but need a simple enough numerical example.
I have worked through the basics of the FEM method in 2D with rectangular elements and now am trying to learn triangular elements but need a simple enough numerical example. It seems triangular elements are far more complicated and also depend on the global coordinates thus each local stiffness matrix is different. At least that's my current understanding. If anyone can point me to a simple enough 2D problem worked out with triangular elements for a temperature problem I would be grateful. Simple is defined by a handful of elements and basically just involving temperature, not things like stress and strain. Also, it's mainly the assembly process I am struggling with especially when triangular elements are in different orientations or different sizes. Thanks.
 
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  • #2
Did you really mean to post this in the DiffEq forum, or did you mean to post in ME or another forum? I can move it if you meant to post in a different forum.
 
  • #3
berkeman said:
Did you really mean to post this in the DiffEq forum, or did you mean to post in ME or another forum? I can move it if you meant to post in a different forum.
Sorry, I thought since FEM is used to solve differential equations that this is the right place and also I started some threads in the past here on FEM but anywhere you think is best is fine. Thanks.
 
  • #4
No need to be sorry. I just wanted to be sure to place this thread start in the best forum. We'll see how things shake out by tomorrow. :smile:
 
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  • #5
Maybe this helps:
https://www.unm.edu/~bgreen/ME360/2D Triangular Elements.pdf

But I just want to note that in general also for quadrilateral elements the stiffness matrix depends on the orientation and shape of the element and are thus different for each element. Only if you have a grid of perfectly equal quads which sides are aligned with the coordinate system this part might be omitted.
 
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  • #6
Arjan82 said:
Maybe this helps:
https://www.unm.edu/~bgreen/ME360/2D Triangular Elements.pdf

But I just want to note that in general also for quadrilateral elements the stiffness matrix depends on the orientation and shape of the element and are thus different for each element. Only if you have a grid of perfectly equal quads which sides are aligned with the coordinate system this part might be omitted.
Thanks. I actually have seen that before and it helps a little but is still confusing because it solves for stress and strain not temperature which just adds a layer of unfamiliar material. I can do this for one element but the assembly process where triangular elements are in different orientations confounds me still. If I compute each local stiffness matrix separately based on its global coordinates, would that account for each elements orientation? I can only think of computing a simple case of a few elements and see if things make sense.

Also, triangular elements don't have to be equilateral do they? I assume they can be any kind of triangle
 
  • #7
bob012345 said:
Thanks. I actually have seen that before and it helps a little but is still confusing because it solves for stress and strain not temperature which just adds a layer of unfamiliar material. I can do this for one element but the assembly process where triangular elements are in different orientations confounds me still. If I compute each local stiffness matrix separately based on its global coordinates, would that account for each elements orientation? I can only think of computing a simple case of a few elements and see if things make sense.

The local stiffness matrix is in global coordinates. So this indeed accounts for its orientation. You then need to number all degrees of freedom. With this you can define the global stiffness matrix. If a degree of freedom of a certain node is shared between two or more elements you just add all the values of their respective local stiffness matrices.

bob012345 said:
Also, triangular elements don't have to be equilateral do they? I assume they can be any kind of triangle

True, they do not have to be equilateral, as long as you don't go overboard with it. If the element is too severely deformed you get numerical issues.
 
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  • #8
Suppose I have a simple two element problem like the reference you sent me with nodes labeled counterclockwise and the element number inside;
geogebra-export (15).png


Then suppose the local stiffness matrix for element 1 is

$$k_1 = \begin{bmatrix}
a_{1,1} & a_{1,2} & a_{1,3}\\
a_{2,1} & a_{2,2} & a_{2,3}\\
a_{3,1} & a_{3,2} & a_{3,3}
\end{bmatrix}$$

And for element 2 is;

$$k_2 = \begin{bmatrix}
b_{1,1} & b_{1,2} & b_{1,3}\\
b_{2,1} & b_{2,2} & b_{2,3}\\
b_{3,1} & b_{3,2} & b_{3,3}
\end{bmatrix}$$

I believe the local stiffness matrix for element 1 will go into the global stiffness matrix like this;

$$K_1 = \begin{bmatrix}
a_{1,1} & a_{1,2} & a_{1,3}&0\\
a_{2,1} & a_{2,2} & a_{2,3}&0\\
a_{3,1} & a_{3,2} & a_{3,3} &0\\
0 & 0 & 0 & 0
\end{bmatrix}$$

but I am confused as to how to add the second one.
 
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  • #10
bob012345 said:
Suppose I have a simple two element problem like the reference you sent me with nodes labeled counterclockwise and the element number inside;
View attachment 305603

Then suppose the local stiffness matrix for element 1 is

$$k_1 = \begin{bmatrix}
a_{1,1} & a_{1,2} & a_{1,3}\\
a_{2,1} & a_{2,2} & a_{2,3}\\
a_{3,1} & a_{3,2} & a_{3,3}
\end{bmatrix}$$

And for element 2 is;

$$k_2 = \begin{bmatrix}
b_{1,1} & b_{1,2} & b_{1,3}\\
b_{2,1} & b_{2,2} & b_{2,3}\\
b_{3,1} & b_{3,2} & b_{3,3}
\end{bmatrix}$$

I believe the local stiffness matrix for element 1 will go into the global stiffness matrix like this;

$$K_1 = \begin{bmatrix}
a_{1,1} & a_{1,2} & a_{1,3}&0\\
a_{2,1} & a_{2,2} & a_{2,3}&0\\
a_{3,1} & a_{3,2} & a_{3,3} &0\\
0 & 0 & 0 & 0
\end{bmatrix}$$

but I am confused as to how to add the second one.
Ok, I think the contribution of element #2 to the global stiffness matrix will go like this;

$$K_2 = \begin{bmatrix}
b_{3,3} & 0 & b_{3,1}&b_{3,2}\\
0 & 0 & 0 & 0\\
b_{1,3} &0& b_{1,1} & b_{1,2}\\
b_{2,3} &0& b_{2,1} & b_{2,2}

\end{bmatrix}$$I used an ordering like this since the counterclockwise sense of nodes for element #2 goes as {3,4,1};

341
b_{1,1}b_{1,2}b_{1,3}3
b_{2,1}b_{2,2}b_{2,3}4
b_{3,1}b_{3,2}b_{3,3}1

I'm fairly sure this is correct.
 
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  • #12
If you want examples involving heat transfer, check "A First Course in Finite Elements" by J. Fish and T. Belytschko and "A First Course in the Finite Element Method" by D.L. Logan. Other books with examples of 2D element calculations focus on solid mechanics problems. Including my favorite collection of 2D FEA examples - "Finite Element Analysis" by U. Borker.
 
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  • #13
For code examples, you can have a look at the online database of John Burkhardt:
https://people.sc.fsu.edu/~jburkardt/
Just go to your favorite language and scroll down to the examples fem*
 
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What is the FEM Method in 2D with Triangular Elements?

The Finite Element Method (FEM) is a numerical technique used to solve partial differential equations in engineering and science. In 2D problems, triangular elements are used to discretize the domain and the FEM method is applied to each element to obtain a solution.

How does the FEM Method in 2D with Triangular Elements work?

The FEM Method in 2D with Triangular Elements works by dividing the domain into smaller triangular elements. Each element is then represented by a set of equations based on the governing partial differential equation. These equations are then solved simultaneously to obtain an approximate solution for the entire domain.

What are the advantages of using the FEM Method in 2D with Triangular Elements?

The FEM Method in 2D with Triangular Elements offers several advantages, such as its ability to handle complex geometries and boundary conditions, its flexibility in choosing element sizes and shapes, and its accuracy in approximating solutions for a wide range of problems.

What are the limitations of the FEM Method in 2D with Triangular Elements?

While the FEM Method in 2D with Triangular Elements has many advantages, it also has some limitations. It can be computationally expensive for large-scale problems, and the accuracy of the solution depends on the quality of the mesh and the choice of element size and shape.

What are some applications of the FEM Method in 2D with Triangular Elements?

The FEM Method in 2D with Triangular Elements has a wide range of applications in engineering and science, including structural analysis, heat transfer, fluid dynamics, and electromagnetics. It is also commonly used in finite element analysis software for design and optimization purposes.

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