FEM heat transfer problem

Reload · April 14, 2012, 16:29

Hello everyone!

I'm trying to make a program that uses finite element method (FEM) to transient heat transfer. I'm using Matlab and already have a meshing algorithm. I'm trying to understand how to go further. The best for me would be to understand a simple example. So I've made one – a simple cube.

I've put everything in a *.pdf file to make everything clear by proper text format. I marked all my (5) questions in red. They are also numbered so you can write answers to "question no #". Everything else is written in pdf. Link to download:
http://wyslijto.pl/plik/g66y5hcqtt - click in FEM [v0.1].pdf

In shortcut: I don't understand solving a FEM heat transient problem. I try to make it in Matlab. I made a simple cube example, it's made from 6 tetrahedra. I'm looking for someone to help me with my problems by solving this example or showing me a text algorithm how to do this - step by step. I will be really grateful for any help.

Best regards, Reload.

Rami · April 16, 2012, 06:45

Reload,

It's simple, and I'll just sketch the FEM procedure for the case in hand (conduction, transient solution).

Let us write the equation (using tensor notation) as:
$\rho C_{p}\partial T/\partial t=-q_{i,i}+G$
with
$q_{i}=-kT_{,i}$

Assume T distribution within an element is
$T=\sum_{I}N{_I}T{_I}$
with

and

being the nodal shape functions and temperatures, and the sum is on the element nodes.
Now, apply Galerkin's method, i.e., integrate both hands multiplied by

over the element volume.

The LHS will be
$\partial T_{J}/\partial t\int_{V}\rho C_{p}N_{I}N_{J}dV$

The RHS, after using intergation by parts and the Gauss theorem, yields
$-\int_{\partial V}N_{J}q_{n}dA-T_{J}\int_{V}N_{I,i}kN_{J,i}dV+\int_{V}N_{J}GdV$
The first term on the RHS is the boundary normal flux, integrated over the element surface.

Now sum over all the elements and assemble to get the global matrix equation
$M_{IJ}\dot{T}_{J}=BC \: term -K_{IJ}T_{J}+F_I$

You have to specify appropriate initial conditions, the shape functions (apparently you used linear functions) and a time scheme (simple differencing will do). You also have to specify the boundary conditions. You may wish to choose G distribution (e.g., use the same shape function as for T), and - voila! That's the FEM in a nutshell.

I hope I have clarified your doubts,
Rami

April 14, 2012, 16:29	FEM heat transfer problem	#1
Reload New Member Rafal Join Date: Apr 2012 Posts: 1 Rep Power: 0	Hello everyone! I'm trying to make a program that uses finite element method (FEM) to transient heat transfer. I'm using Matlab and already have a meshing algorithm. I'm trying to understand how to go further. The best for me would be to understand a simple example. So I've made one – a simple cube. I've put everything in a .pdf file to make everything clear by proper text format. I marked all my (5) questions in red. They are also numbered so you can write answers to "question no* #". Everything else is written in pdf. Link to download: http://wyslijto.pl/plik/g66y5hcqtt - click in FEM [v0.1].pdf In shortcut: I don't understand solving a FEM heat transient problem. I try to make it in Matlab. I made a simple cube example, it's made from 6 tetrahedra. I'm looking for someone to help me with my problems by solving this example or showing me a text algorithm how to do this - step by step. I will be really grateful for any help. Best regards, Reload.

April 16, 2012, 06:45		#2
Rami Senior Member Rami Ben-Zvi Join Date: Mar 2009 Posts: 155 Rep Power: 17	Reload, It's simple, and I'll just sketch the FEM procedure for the case in hand (conduction, transient solution). Let us write the equation (using tensor notation) as: $\rho C_{p}\partial T/\partial t=-q_{i,i}+G$ with $q_{i}=-kT_{,i}$ Assume T distribution within an element is $T=\sum_{I}N{_I}T{_I}$ with and being the nodal shape functions and temperatures, and the sum is on the element nodes. Now, apply Galerkin's method, i.e., integrate both hands multiplied by over the element volume. The LHS will be $\partial T_{J}/\partial t\int_{V}\rho C_{p}N_{I}N_{J}dV$ The RHS, after using intergation by parts and the Gauss theorem, yields $-\int_{\partial V}N_{J}q_{n}dA-T_{J}\int_{V}N_{I,i}kN_{J,i}dV+\int_{V}N_{J}GdV$ The first term on the RHS is the boundary normal flux, integrated over the element surface. Now sum over all the elements and assemble to get the global matrix equation $M_{IJ}\dot{T}_{J}=BC \: term -K_{IJ}T_{J}+F_I$ You have to specify appropriate initial conditions, the shape functions (apparently you used linear functions) and a time scheme (simple differencing will do). You also have to specify the boundary conditions. You may wish to choose G distribution (e.g., use the same shape function as for T), and - voila! That's the FEM in a nutshell. I hope I have clarified your doubts, Rami Last edited by Rami; April 17, 2012 at 05:35.

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