In the framework of the foam process modelling, this paper presents a numerical strategy for the direct 3D simulation of the expansion of gas bubbles into a molten polymer. This expansion is due to a gas overpressure. The polymer is assumed to be incompressible and to behave as a pseudo-plastic fluid. Each bubble is governed by a simple ideal gas law. The velocity and the pressure fields, defined in the liquid by a Stokes system, are subsequently extended to each bubble in a way of not perturbing the interface velocity. Hence, a global velocity-pressure-mixed system is solved over the whole computational domain, thanks to a discretization based on an unstructured first-order finite element. Since dealing with an Eulerian approach, an interface capturing method is used to follow the bubble evolution. For each bubble, a pure advection equation is solved by using a space-time discontinuous-Galerkin method, coupled with an r-adaptation technique. Finally, the numerical strategy is achieved by considering a global mesh expansion motion, which conserves the amount of liquid into the computational domain during the expansion. The expansion of one bubble is firstly considered, and the simulations are compared with an analytical model. The formation of a cellular structure is then investigated by considering the expansion of 64 bubbles in 2D and the expansion of 400 bubbles in 3D.