In this paper the development of a physically consistent phase-field theory of solidification shrinkage is presented. The coarse-grained hydrodynamic equations are derived directly from the N -body Hamiltonian equations in the framework of statistical physics, while the constitutive relations are developed in the framework of the standard phase-field theory, by following the variational formalism and the principles of non-equilibrium thermodynamics. To enhance the numerical practicality of...