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ME3291: Numerical Methods In Engineering Assignment, NUS, Singapore The governing equation for the temperature distribution with time on a 2D square plate measuring
 University National University of Singapore (NUS) Subject ME3291: Numerical Methods In Engineering
Posted on: 10th Aug 2023

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

(1) 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, 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.

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(2) 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 (Fig. 2),

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.

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