ME3291: The governing equation for the temperature distribution with time on a 2D square plate measuring 1 unit by1 unit is: Numerical Methods In Engineering Assignment, NUS, Singapore

• The governing equation for the temperature distribution with time on a 2D square plate measuring 1 unit by1 unit is

∂T/∂t = ∂²T/∂x² + ∂²T/∂y²

subjected to the Dirichlet boundary conditions for T  provided in Fig.1.  You are to obtain the following:

(a) The temperature contour plot on the square plate with time, say at t=0.01, 0.1, and at steady state. (You can provide contours at other times too to depict the convergence of the results at a steady state.) Take the initial condition at t=0 as T=0.0 for the whole domain.
(b) Separately, program and compute for the Laplace Equation

∂²T/∂x² + ∂²T/∂y²  = 0

and obtain the solution for comparison to the steady-state solution in (a).

For the above, you have to show clearly how you treat the Dirichlet boundary conditions and provide a listing of your program, and other pertinent workings. The various contour plots can be carried out using the Techp1ot or any other suitable software.

• The same governing equation for the temperature distribution with time on a 2D square plate measuring  1  unit by 1 unit is given as

∂T/∂t = ∂²T/∂x² + ∂²T/∂y²

In this case, the boundary conditions are given as the Dirichlet type for 3 sides of the plate and reflected as follows

0  ≤  x  ≤  1.0, y = 0,            T = 0.0
0  ≤  x  ≤  1.0, y = 1.0,         T = 1.0
0  ≤  y  ≤  1.0, x = 0,            T = 0.0

and the Neumann boundary condition for

0  ≤  y  ≤  1.0, x = 1.0,

is given as

∂T/∂x = 0.0.

Obtain the temperature contour plot on the square plate with time, say at=0.01, 0.1, and at steady state.