Deriving the FEA formulation using triangular elements

[maistral]

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) = T_i(x_i,y_i)N_i(x,y) + T_j(x_j,y_j)N_j(x,y) + T_k(x_k,y_k)N_k(x,y) then the weighting functions are N_i(x,y), N_j(x,y), and N_k(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 . 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:

T_i × (insert lengthy integral) + T_j × (insert lengthy integral) + T_k × (insert lengthy integral) = 0.

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

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 i^th case and one for the j^th 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 i^th, one for the j^th case, and one for the k^th 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 x_i and y_i, 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.

 

[bigfooted]

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.

 

[maistral]

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!