Solving 2D Heat Equation w/ FEM & Galerkin Method

[bob012345]

TL;DR Summary

Solving a 2D temperature problem but having an issue with the proper setup of the Galerkin method. All my terms are zero.

I want to solve the 2D heat equation

$$\frac{∂^2 {T}}{ ∂x^2} + \frac{∂^2 {T}}{ ∂y^2} = 0$$

The only boundary conditions is I will specify the edge temperatures but there are no heat sources.

So I create an average temperature function $\tilde{T}$ and weighting functions $S_i$ over a rectangular element with four nodes with i=1,2,3,4.

Using numbers instead of letters for clarity I wish to solve first for the weighting functions assuming a form for my average temperature function;
$$\tilde{T} = b_1 + b_2x +b_3y +b_4xy$$

I solve for the coefficients to get my weighting functions and my average temperature function;

$$S_1 = (1 - \frac{x}{l}) (1 - \frac{y}{w})$$
$$S_2 = \frac{x}{l}(1 - \frac{y}{w})$$
$$S_3 = \frac{xy}{lw}$$
$$S_4 = \frac{y}{w}(1 - \frac{x}{l})$$

Then;
$$\tilde{T} = T^e = S_1T_1 + S_2T_2 + S_3T_3 + S_4T_4$$

Following the Galerkin method I minimize the Residual

$$\iint_A S_i R \,dA = \iint_A S_i (\frac{∂^2 \tilde{T}}{ ∂x^2} + \frac{∂^2 \tilde{T}}{ ∂y^2} )\,dA = 0$$

Now I use integration by parts to eliminate the second order derivative. This is just for the x derivative part of one of the four equations;

$$\iint_A S_i \frac{∂^2 \tilde{T}}{ ∂x^2}\,dA = \int_Y \Big[ S_i \frac{∂ \tilde{T}}{ ∂x} \Big |_{x_1}^{x_2} - \int_X \frac{∂S_i}{∂x} \frac{∂ \tilde{T}}{ ∂x} \,dX \Big] dY$$ where the index goes to 4.

My problem is when I integrate this to get my node equations all the terms go to zero. I mean the two integrals exactly cancel for each node variable. My question is this, did I set up this correctly with the integration by parts over a double integral? I used the integration of parts for the x integral then when manipulating the x derivative terms. Thanks.

EDIT: I think I found the issue. It is how to evaluate the first term after integration by parts. I was substituting the full expression for $\frac{∂ \tilde{T}}{ ∂x}$ back in when it should be evaluated as the slope at the limits of integration. I found it by revisiting my 1D reference problem. Thanks.

 

[wrobel]

Summary: Solving a 2D temperature problem but having an issue with the proper setup of the Galerkin method. All my terms are zero.

I want to solve the 2D heat equation

∂2T∂x2+∂2T∂y2=0

it is not the heat equation it is the Laplace

 

[bob012345]

it is not the heat equation it is the Laplace

There are no heat sources. Only boundary edge temperature constrains.

 

[wrobel]

I understand. Nevertheless it is the Laplace equation.

 

[bob012345]

I understand. Nevertheless it is the Laplace equation.

I agree but can you tell me how I should include the boundary conditions? I am a bit fuzzy on that. Should there be some form of source term? Thanks.

 

[wrobel]

To solve the Laplace equation $\Delta u=0$ in a rectangular region $$0\le x\le a,\quad 0\le y\le b$$ with known boundary conditions
$$u(x,0)=\alpha(x),\quad u(0,y)=\beta(y),\quad u(a,y)=\gamma(y),\quad u(x,b)=\psi(x)$$
you can find a solution as a sum of solutions of the following two problems
$$\Delta u_1=0,\quad u_1(x,0)=0=u_1(x,b)=0,\quad u_1(0,y)=\beta(y),\quad u_1(a,y)=\gamma(y)$$
and
$$\Delta u_2=0,\quad u_2(x,0)=\alpha(x),\quad u_2(x,b)=\psi(x),\quad u_2(0,y)=0= u_2(a,y)$$
Further
$$u_1=\sum_{k=0}^\infty v_k(x)\sin(\pi k y/b),\quad u_2=\sum_{k=0}^\infty w_k(y)\sin(\pi k x/a)$$

 

[wrobel]

oh, I have just noticed the word "FEM" that is not my field sorry

 

[bob012345]

I resolved how to do the calculations for this part;

$$ \int_X \frac{∂S_i}{∂x} \frac{∂ \tilde{T}}{ ∂x} \,dX dY$$

But am having conceptual issues with this part around the 2D element;
$$ \int_Y S_i \frac{∂ \tilde{T}}{ ∂x} \Big |_{x_1}^{x_2}$$

Any suggestions would be appreciated. Thanks.