Deriving the FEA formulation using triangular elements

In summary, the conversation discusses the difficulties of understanding the 2D finite element formulation and the resources available for learning it. The speaker also mentions their background in engineering and their attempts to learn the material through online resources and classes. They express their desire for someone to help guide them in understanding the formulation and solving the PDE in variational form. The conversation also suggests using a book with worked out examples and learning how to transform triangles and rectangles to the unit triangle or rectangle. It is also recommended to write a solver in a programming language to better understand the concepts. The conversation ends with a clarification that the integral is in 2D and the number of nodes increases the size of the matrix.
  • #1
maistral
240
17
Hi, it's been a while since I last posted. Anyway, so I went through the trouble of enrolling in two finite element analyses classes and yet, they still don't teach how the 2D formulation has been made. I'll list the things that 'I know' already to get some things clear.

  1. I know how to derive the 1D finite element formulation using the Galerkin linear weighting functions (in the end I studied this myself as the classes I took did not teach this).
  2. I know that you can mechanically solve the 1D formulation using the generic force = stiffness matrix × displacement method (this is what I initially knew before taking the undergrad and graduate classes, and yet this is the one they teach here. Yep, waste of money).
  3. I also know the shape functions for 2D (the triangle thing, where T(x,y) = Ti(xi,yi)Ni(x,y) + Tj(xj,yj)Nj(x,y) + Tk(xk,yk)Nk(x,y) then the weighting functions are Ni(x,y), Nj(x,y), and Nk(x,y), which is integrated to the differential equations as in 1D).
So I'm trying to figure out how did everything get derived since I don't want to blindly follow formulations without any information of how they arrived... there. So I took some resources from the internet and upon checking them, my nose instantly bled :DD. They were talking about integrations along volumes and there was even an integral with a small circle in the middle! Apparently some deep knowledge in mathematics courses are needed in order to understand these, and most of them involve integrating an entire vector which just made me more dizzy and confused.

I actually wanted to avoid those things as I don't know how they would be relevant to me as I'm not really a mathematics major/graduate, I'm an engineering graduate. I am truly sorry for asking help here if it's too much to ask as I really wanted to know where did these things come from, or at least guide me until I achieve the 'unsimplified' but 'clear' solution I wanted - what I'm hoping for is someone be able to guide me with the pertinent theorems in order to arrive with the default, unsimplified formulation similar to the 1D case; something like:

Ti × (insert lengthy integral) + Tj × (insert lengthy integral) + Tk × (insert lengthy integral) = 0.

Anyway, I started off by using the Laplace equation. This is what I have now:

111.png


Anyway, these are the questions.

  1. I know I am supposed to multiply the entire PDE with the weighting function. Which weighting function do I use? I remember in the 1D case you have to integrate the differential equations twice, one for the ith case and one for the jth case. Am I correct to assume that I integrate this differential equation thrice, seeing there are three vertices in the triangular element? So that would mean I integrate the PDE; one for the ith, one for the jth case, and one for the kth case?
  2. I am assuming that you have to use double integration along the x-axis and the y-axis. My problem is about the limits. Which limits do I use? If the bottom limit of the integrands are xi and yi, what would be the upper limit?
I hope, and pray very dearly, that someone helps me. I even intend to spend some money just to learn this (I already did and they just taught me stuff that I know already which is extremely disheartening). Please, help me. Thank you.
 
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  • #2
Here's a book that is low on math and high on worked out examples that you can follow by hand:
https://www.amazon.com/dp/0070087148/?tag=pfamazon01-20
You might also want to study this in case you haven't done so already:
https://www.amazon.com/dp/0070552215/?tag=pfamazon01-20

You probably know chapters 1 and 2, study chapters 3 and 6 for 2D stuff. If you want to know why it works, you have to know about the variational form though.
The first thing to learn is how to transform any triangle and rectangle to the unit triangle or rectangle. Then learn how to integrate a function on any triangle by transforming it to the unit triangle.
Then realize that solving the pde in variational form just means discretizing the domain into a lot of triangles, transforming the problem from the unit triangle (because all these neat integration methods only work from -1..+1) to the actual triangle, and construct the global mass matrix and solve the system Ax=b

It helps to write an actual solver in e.g. matlab, scilab, python or whatever you have available that has some basic matrix-vector stuff.

1. no, the integral is in 2D. The number of nodes does increase the size of the matrix. Try to see this by just integrating a function in 2D on some rectangle (with known solution so you can check the outcome).
2. After discretization, the integration becomes the sum of the integration over the unit triangle/rectangle.

Hope this helps.
 
  • #3
bigfooted said:
Here's a book that is low on math and high on worked out examples that you can follow by hand:
https://www.amazon.com/dp/0070087148/?tag=pfamazon01-20
You might also want to study this in case you haven't done so already:
https://www.amazon.com/dp/0070552215/?tag=pfamazon01-20

You probably know chapters 1 and 2, study chapters 3 and 6 for 2D stuff. If you want to know why it works, you have to know about the variational form though.
The first thing to learn is how to transform any triangle and rectangle to the unit triangle or rectangle. Then learn how to integrate a function on any triangle by transforming it to the unit triangle.
Then realize that solving the pde in variational form just means discretizing the domain into a lot of triangles, transforming the problem from the unit triangle (because all these neat integration methods only work from -1..+1) to the actual triangle, and construct the global mass matrix and solve the system Ax=b

It helps to write an actual solver in e.g. matlab, scilab, python or whatever you have available that has some basic matrix-vector stuff.

1. no, the integral is in 2D. The number of nodes does increase the size of the matrix. Try to see this by just integrating a function in 2D on some rectangle (with known solution so you can check the outcome).
2. After discretization, the integration becomes the sum of the integration over the unit triangle/rectangle.

Hope this helps.

Hi! Thanks for replying. I'll try and read the resources you pointed out. Thank you very much!
 

1. What is the purpose of deriving the FEA formulation using triangular elements?

The purpose of deriving the FEA formulation using triangular elements is to develop a mathematical model that can accurately simulate the behavior of a structure under different loading conditions. Triangular elements are commonly used in FEA because they can provide a good balance between accuracy and computational efficiency.

2. How are triangular elements used in FEA?

Triangular elements are used in FEA by dividing a complex structure into smaller, simpler triangles. These triangles are then connected to each other at their nodes, creating a mesh. The equations of equilibrium and compatibility are then applied to each individual triangle, allowing for the overall behavior of the structure to be determined.

3. What are the advantages of using triangular elements in FEA?

There are several advantages of using triangular elements in FEA. These include their ability to accurately model curved boundaries, their ability to handle complex geometries, and their computational efficiency. Additionally, triangular elements are less prone to numerical errors compared to other types of elements, making them a popular choice in FEA.

4. What are the limitations of using triangular elements in FEA?

While triangular elements have many advantages, they also have some limitations. One major limitation is their inability to accurately model stress concentrations at sharp corners or edges. This can lead to inaccurate results in certain cases. Additionally, triangular elements may not be suitable for all types of structures and may require more elements to accurately model certain geometries.

5. How is the FEA formulation derived using triangular elements?

The FEA formulation using triangular elements is derived by applying the principles of equilibrium and compatibility to each individual triangle in the mesh. This involves setting up a system of equations for each node, which can then be solved using numerical methods. The final FEA formulation is a set of equations that can be used to calculate the displacements, strains, and stresses at each node in the mesh.

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