Look at this doc it can help you. Good luck.

http://bookzz.org/book/445240/b2543f

Thank you so much for the link and support. I actually needed this problem for heat transfer in solids which I should have mentioned it. Anywyas this book looks great. Thanks a lot. I'm not able to find the analytical solution to the problems so that I can use it to generate using FEM and compare the solutions.

Sunag R A.

Hi Sunag,

If your only goal is to test your FE implementation, you can build "by hand" analytical solutions to your problem (that will be physically irrelevant) as follow:

1- Choose a simple domain (e.g. a square in 2D)

2- Decide what type of initial and boundary conditions you want to apply (e.g. periodic / Dirichlet /Neumann / Robin condition)

3- Pick any (regular) function that satisfies these boundary conditions (and is "complex enough" so that the comparison has sense) e.g. u(x,y,t) = x(x-1)y(y-1)exp(-t) for the square [0,1]^2 with full homogeneous Dirichlet condition.

4- Apply the differential operator into consideration to this function to determine the "source term" which would give you such solution.

5- Run FE on this domain, for these inital and boundary conditions and source term, and compare the solution to the function you picked.

You can find examples of this procedure in the Fenics book https://fenicsproject.org/documentation/ (and many others, but I like this one, and it's free :-) )

Hi Remi,

   With respect to above mail, I want to do FE Implementation for the known analytical solution(basic one as you said square or rectangle) from any resources. I'm trying hard to find the resources towards this. So, I need only the analytical solution from the resource, so that I can take it further with the FE method.

  Apart from that, I checked the link.But it is for Fluids and I searched for the analytical solutions in the book. I am not able to find one. Could you please help me out with respect to this?

Thanks and Regards,

Sunag R A.

Hi Sunag,

I am a little confused by your answer above, I thought I answered to question 1. and 2. in your first post, but I should have missed something. Perhaps could you reformulate your question ? Adding a short description of the kind problem you'd like to solve (i.e. related physics, partial differential equation and boundary conditions) may help.

The book I linked deals not only with fluids (perhaps you meant that your problem il for fluids, apologizes in this case) but embeds many exemples from different physical backgrounds.

Looking at section 1.1.1, which deals with the simplest model problem (the Poisson equation), you'll see that they apply the method I described above (in section 1.1.3, "Implementation"), i.e. first choose a solution, then build the problem from it.

Hi Remi,

  The question is as follows:

1. Define a problem (ie domain, heat source, conductivity and Boundary conditions) such that the analytical solution is known. The most relevant aspect is that you explain the strategy that you followed to define such a problem/

Hint : Given a set of data ( heat source conductivity and boundary conditions) it is usually very difficult to find the analytical solution. Is it easier to derive the required data for a given solution?

2. This is the question . I tried researching the solution in terms of fins, Iron box, and few other practical heat transfer. But, I'm not able to find one simple problem with an analytical solution. It should be a simple rectangular component with 2D heat transfer.

 Hope you understood. If I had any mistake in Understanding in your last message. Please correct me.

Thanks and Regards,

Sunag R A.

Well, it seems that my first answer above was exactly an answer to the "Hint" which is given, don't you think ? :-)

That's very true. I'll try following the steps as said and I'll check the fenics link which you sent throughly.Also, I'll try my best to get the problem. Can I contact you in between if I get any problem? Can u share your mail ID if possible? 

Thanks and Regards,

Sunag R A.