Thermal field 2D, Dirichlet

Example: a plate with a heat source

Consider a two-dimensional example with an internal heat source. Take a square plate Ω\Omega​=[0,1]×[0,1]{} = [0, 1] \times [0, 1] with thermal diffusivity a2a^2​=0.004{} = 0.004. The edges of the plate are held at zero temperature (Dirichlet condition), and four point heat sources are placed near the corners, f(x,y)f(x, y)​=∑k=14Pk⋅δ(x−xk, y−yk){} = \sum_{k=1}^{4} P_k \cdot \delta(x - x_k,\, y - y_k). Initially the plate is heated non-uniformly. Over time, the initial heat escapes through the cold edges while the sources sustain hot regions, and the solution settles to a steady profile. The problem statement reads

∂T(x,y,t)∂t\displaystyle \frac{\displaystyle \partial T(x, y, t)}{\displaystyle \partial t} =a2⋅(∂2T(x,y,t)∂x2\displaystyle {} = a^2 \cdot \biggl( \frac{\displaystyle \partial^2 T(x, y, t)}{\displaystyle \partial x^2}​+∂2T(x,y,t)∂y2)\displaystyle {} + \frac{\displaystyle \partial^2 T(x, y, t)}{\displaystyle \partial y^2} \biggr)​+∑k=14Pk⋅δ(x−xk, y−yk),\displaystyle {} + \sum_{k=1}^{4} P_k \cdot \delta(x - x_k,\, y - y_k),T(x,y,0)\displaystyle T(x, y, 0) =T0(x,y),\displaystyle {} = T_0(x, y),T∣∂Ω\displaystyle T|_{\partial \Omega} =0.\displaystyle {} = 0.
(7.2)
Fig. 7.2. The 2D Dirichlet problem with an internal heat source. The colour shows the temperature T(x,y,t)T(x, y, t) at each point of the plate; the scale is shown on the right.