Numerical analysis and simulation of a bio-thermal model for the human foot

This paper analyses the numerical convergence and the approximation of the finite element method applied to a biothermal nonlinear model for the bare foot. As far as we know, there is not a previous finite element analysis of this well-known bioheat equation. Thus, this work can be seen as a first step to study the coupling with energy and mass transfer models (water, vapor and gas) at the textiles surrounding the foot. The model is posed as a steady partial differential equation for the temperature field, and a non-linear boundary condition on the external boundary, where heat losses due to convection, radiation and evaporation are considered. The existence and the uniqueness of the solution is proved for the weak formulation and also for its finite element approximation by using arguments of monotone operators. Then, numerical convergence and an a priori error estimates result are obtained. Some numerical simulations are presented to show the accuracy of the numerical method and the behavior of the solution, being qualitatively acceptable and, in some cases, validated against experimentation, being quantitatively correct too. Interesting conclusions are followed from the analysis of the model parameters as well as from the comparison of 2D and 3D solutions.