Finite element simulation of the filling of thin moulds

In this work we present a nite element formulation to simulate the lling of thin moulds. The model has as starting point a 3-D approach using either div-stable elements, such as the Q2=P1 element (tri-quadratic velocities and piecewise linear, discontinuous pressures) or stabilized nite element formulations. The tracking of the free surface is based on the Volume-Of-Fluid (VOF) method. The velocity pro le is assumed to be parabolic in the direction normal to the mid-plane, so that one element along the width of the mould is enough to reproduce this pro le if this element is quadratic. The velocity is prescribed to zero on the upper and lower surfaces and the normal to the mid-plane is also prescribed to zero. In the case of div-stable elements, the pressure pro le is prescribed to be constant along the width of the mould. This can be achieved by using as interpolation degrees of freedom the pressure values at the element centroid and its derivatives in the directions tangent and normal to the mid-plane, and prescribing the latter to zero. No modi cations are needed when stabilized formulations are employed. To advance in time the function used in the VOF technique, we use a constant velocity across the width of the mould, which is taken as the projection on the tangent plane of each element of the nodal velocities. This is needed in order to have mass conservation