Metal foams have both structural and functional properties, offer advantages in different fields such as automotive and aircraft etc., and thus gain more and more technical interest. In this study, a computational model is presented which allows the numerical simulation of the bubble expansion and solidification during metal foaming processes. Mathematical modelling of the different relevant physical properties and effects during the metal foam expansion and solidification is discussed in this paper. This model considers the broad variety of the complex boundary conditions, the simple and multiphase fluid flow with complex geometries, the stabilization of the foams, and the mutual interactions between the bubbles. The Navier-Stokes equation is applied to elucidate the fluid flows in the liquid and in the mushy zone. And the diffusion equation for the hydrogen transport, the distribution functions of the temperature, the density and kinetic equations are also be proposed and discussed.