The flow rate
corresponds to the flow rate passing through the side surface formed by the range from the mid-point of the two constituent sides of the quadrilateral element, including node α.

Degree of fill calculation

The finite element formulation composed by superimposing Eq. (17) on the entire problem domain requires the boundary condition of either the pressure or the flow rate. In general, in the operation of a TSE, the extrusion amount is often intentionally controlled using a weight-controlled-feeder, and it is smaller than the full capacity of the extruder; thus, it is called the “starved state.”
Setting the boundary conditions of pressure and flow together on the same boundary results in excessive constraint, making it impossible to obtain a solution. This difficulty is solved by adding an unknown quantity called the degree of fill. However, even when the degree of fill is added, the finite element equation does not allow the setting of both the pressure and the flow rate boundary condition on the same boundary surface.
The basic idea of a one-dimensional calculation algorithm of the FAN method is expanded to two dimensions as a quantification method of the unfilled element. The pressure boundary condition at the inflow surface (feeding port) is set to atmospheric pressure, and the flow rate is set. A prescribed (backpressure) value obtains the pressure boundary condition at the outflow surface (head). A finite element calculation is performed, and the pressure of each element is determined. At this time, there is no guarantee that the calculated pressure value of the outflow surface will agree with the prescribed value.
As incompressible conditions are assumed, the pressure dependence of the material properties are ignored, and the equations of continuity and motion are invariant to the addition and subtraction of constant pressure. The flow behavior does not change if the pressure gradient is unchanged. The pressure distribution is recalculated using the down-wind pressure update scheme in