Thermal field 3D, Dirichlet

Example: heating a ball with sources

Consider a three-dimensional transient example. Take a ball Ω\Omega ={ r:∣r∣= \{\, \mathbf{r} : |\mathbf{r}| ≤R }\le R \,\} with thermal diffusivity a2a^2 =0.3= 0.3. The surface of the ball is held at zero temperature (Dirichlet condition), and six point heat sources are placed symmetrically inside, f(r)f(\mathbf{r}) =∑k=16Pk δ(r−rk)= \sum_{k=1}^{6} P_k \, \delta(\mathbf{r} - \mathbf{r}_k) (at the vertices of an octahedron). Initially the ball is cold; over time the sources heat it up while heat escapes through the cold surface, and the solution settles to a steady profile. The problem statement reads

∂T∂t\displaystyle \dfrac{\partial T}{\partial t} =a2(∂2T∂x2+∂2T∂y2+∂2T∂z2)\displaystyle = a^2 \left( \dfrac{\partial^2 T}{\partial x^2} + \dfrac{\partial^2 T}{\partial y^2} + \dfrac{\partial^2 T}{\partial z^2} \right) +∑k=16Pk δ(r−rk),\displaystyle + \sum_{k=1}^{6} P_k \, \delta(\mathbf{r} - \mathbf{r}_k),r\displaystyle \mathbf{r} ∈Ω,\displaystyle \in \Omega,[2mm]T(r,0)\displaystyle [2mm] T(\mathbf{r}, 0) =0,\displaystyle = 0,[1mm]T∣∂Ω\displaystyle [1mm] T|_{\partial \Omega} =0.\displaystyle = 0.
(7.6)
Fig. 7.5. A 3D transient Dirichlet problem: heating of a ball by six internal sources. The colour shows the temperature T(r,t)T(\mathbf{r}, t).