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Homework
=Problem 6: Newtons Law of Colling=

Given
The partial Differential Equation is

1.Develop Weak form similar to (3)-(6) page 34-3.
The weak form will be as below,

Lets calculate the derivative of the following factor, Therefore, Replacing in the integral we obtain, Applying the Gauss Theorem on the first term we obtain, We can recognize that the Heat Flux is Given by, Eventually We have,

Organizing We get, Simplifying we get, For the Case when Conductivity if the Identity Matrix, we have following equations, Also We can Write as,

2) Develop Semidiscrete Equations (ODE's) similar to (3) page 23-3. Give Detailed Description for all the quantities.
Performing the Discretization on the Dominium we have,

Where N is the vector of the Interpolation (Lagrange Polynomial) and $$\displaystyle u_{e},w_{e}$$ are the Nodal Values of u & w Respectively. The Derivative With respect to $$\displaystyle x_{1}$$, $$\displaystyle x_{2}$$ are Replacing on the Semi Discrete Weak form we obtain, From the Last Equation we can Obtain the Conductivity Matrix, the Heat source vector and the Capacitance matrix. The Conductivity Matrix is Given by

Further the Capacitance Matrix is Given by,

And the Heat Source is

3.Solve G2DM2.0/D1 Using 2D LIBF (similar to HW 6.6) till $$10^{-6}$$ accuracy at center. Plot Solution in 3D w/Contour.
To solve G2DM2.0/D1 Following Routine was developed. The following figure shows the result for 16 points.

To solve the problem using LIBF we will devide the x-axis and Y-axis in 2 parts initially. In that way the Approximated solution by LIBF is given by:

We Have developed a COMSOL code to plot this routine. The COMSOL code is as follows

Appendix
The Plots for the Routine Developed along with the intended 3-D contour is as below.





Comment:

We can see from the above figures that there also exists a negative value, which is the same with our results from HW6.10.

Comment: The Contour is plotted for $$\displaystyle n_{el}=4$$. The above MATLAB code is developed for larger values of $$n_{el}$$.
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