Determination of the heat transfer coefficients in transient heat conduction

The determination of the space- or time-dependent heat transfer coefficient which links the boundary temperature to the heat flux through a third-kind Robin boundary condition in transient heat conduction is investigated. The reconstruction uses average surface temperature measurements. In both cases of the space- or time-dependent unknown heat transfer coefficient the inverse problems are nonlinear and ill posed. Least-squares penalized variational formulations are proposed and new formulae for the gradients are derived. Numerical results obtained using the nonlinear conjugate gradient method combined with a boundary element direct solver are presented and discussed.