A neat flux-based weak formulation for thermal problems which develops Biot’s variational principle

We propose a weak form of the transient heat equations for solid bodies, as a time-dependent spatial variation of the heat displacement vector field, whose time derivative is the heat flux. This develops the variational principle originally proposed by Biot, inasmuch Fourier’s law is embedded as a holonomic constraint, while energy conservation results from the variation (the vice-versa from Biot). This is a neat formulation because only the heat displacement appears in the variational equations, whereas Biot’s form also involved the unknown temperature field: Fourier’s law is used only a posteriori to recover the temperature. Since the heat displacement is generally more regular than the temperature field, it represents a natural variable in problems with material inhomogeneities, uneven radiation, thermal shocks. The three-dimensional analytical set-up is presented in comparison with Biot’s, for boundary conditions accounting for radiation and convection. A mechanical analogy with the equilibrium of an elastic bar with viscous constraints is suggested for the one-dimensional case. The variational equations are implemented in a finite element code. Numerical experiments on benchmark problems, involving high temperature gradients, confirm the efficiency of the proposed approach in many structural problems.