Using the swarm intelligence algorithms in solution of the two-dimensional inverse Stefan problem (Q524765)

The authors consider an inverse Stefan problem posed in a rectangular domain \(\left[ 0,b\right] \times \left[ 0,d\right] \) containing a liquid phase (\(k=1 \)) and a solid (\(k=2\)) phase, based on the heat equation \(c_{k}\rho _{k} \frac{\partial }{\partial t}T_{k}(x,y,t)=\lambda _{k}\nabla ^{2}T_{k}(x,y,t)\), \(k=1,2\), with homogeneous Neumann boundary conditions on the pieces \( \left\{ x=0\right\} \) and \(\left\{ y=0\right\} \) of the boundary and with the Robin-type boundary condition \(\lambda _{k}\frac{\partial }{\partial n} T_{k}(x,y,t)=\alpha (x,y,t)(T_{k}(x,y,t)-T_{\infty })\) on the two other pieces, where \(T_{\infty }\) is the ambient temperature. On the interface between the two phases, Stefan boundary conditions are imposed and the solution starts from an initial temperature \(T_{0}\). Given a finite number of sensors located at \((x_{i},y_{i})\) inside the rectangle and a finite number of measurements at times \(t_{j}\), one obtains given values \(U_{ij}\) of the temperature, which lead to the functional \(J(\alpha )=\sum_{i,j}(T_{ij}-U_{ij})^{2}\) which has to be minimized with respect to the heat transfer coefficient \(\alpha \), where \(T_{ij}\) is the computed temperature at \((x_{i},y_{i},t_{j})\). The direct Stefan problem is solved using a finite difference method. The first main part of the paper presents the main features of the ant colony and artificial bee colony algorithms which will be used for the resolution of the optimization problem. The authors here do not introduce any regularizing term in the cost functional \(J \). The second main part of the paper presents results of the numerical simulations and the authors finally apply their algorithms to an example dealing with the solidification of an aluminium probe.