Extended framework of Hamilton's principle in heat diffusion

•Variational formulation for heat diffusion including 2nd sound effects is provided.•It recovers all the governing PDEs with proper use of initial/boundary conditions.•It provides sound basis to develop various unified space-time FEMs.•For its validity, the simplest unified space-time FEM is developed with examples.

Based upon the extended framework of Hamilton's principle, a variational formulation in heat diffusion, as well as second sound phenomena is developed. This formulation is compatible with the initial and boundary conditions of a well-posed heat problem, and it correctly accounts for the governing partial differential equations, as its Euler-Lagrange equations. Thus, this new formulation provides a sound base to develop various unified space-time finite element methods. In order to validate the finite element representation over both space and time in the context of this formulation, two-dimensional lower-order space-time finite element methods are also developed with numerical investigations on representative examples.