Consider the transient heat conduction equation where θ is the non-dimensional temperature, Ω is the square domain of side 2, and Γ is its boundary with the boundary conditions shown in Fig. 6.7.3....

Consider the transient heat conduction equation

where θ is the non-dimensional temperature, Ω is the square domain of side 2, and Γ is its boundary with the boundary conditions shown in Fig. 6.7.3. Determine the temperature field inside the domain as a function of position and time when kˆ = k0(1+βθ) with k0 = 1.0 and β = 0.4×10−3 . Assume zero initial condition θ(x, y, 0) = 0, exploit the biaxial symmetry, and use a uniform mesh of 4 × 4 nine-node rectangular elements in a quadrant, the Crank–Nicolson scheme (i.e. α = 0.5) with ∆t = 0.05, and error tolerance of  = 10−3 for the direct iteration method. Plot the temperature θ(x0, 0, t) as a function of time for x0 = 0.0, 0.5, and 0.75. The total number of time steps are such that the temperature reaches a steady state.