The 3D Lagrangian Integral Method. Henrik Koblitz Rasmussen.

The numerical simulation of 3D time-dependent viscoelastic flow is of interest in connection with plastic moulding operations. In a large range of polymer processing methods, the most important issue is to find the temporal development of the free surface (or interface) of the plastic. This are processes such as thermo-forming, gas-assisted injection moulding and all kind of simultaneous multi-component polymer processing operations. Though, in all polymer processing operations free surfaces (or interfaces) are present and the dynamic of these surfaces are of interest. In the "3D Lagrangian Integral Method" to simulate viscoelastic flow, the governing equations are solved for the particle positions (Lagrangian kinematics). Therefore, the transient motion of surfaces can be followed in a particularly simple fashion even in 3D viscoelastic flow. The "3D Lagrangian Integral Method" is described in H.K. Rasmussen, JNNFM, Vol. 84, p. 217-232 (1999), H.K. Rasmussen, JNNFM, Vol. 92, p. 227-243 (2000) and H.K. Rasmussen, JNNFM, Vol. 106, p. 107-120 (2002), for respectively the Upper Convected Maxwell, the Rivlin-Sawyers (includes the K-BKZ) and a generalized Molecular Stress Function (MSF) constitutive model. All codes converge with second order, both with respect to discretization of the space and the time dimension. Note that the second order convergence in time in Rasmussen (2002) is obtained by integrating the first order differential equation describing the Molecular Stress Function with a second order Runge-Kutta integration method. In any development of a numerical method for viscoelastic flow it is important to focus on the constitutive equation associated to the method. For instance the K-BKZ model is not adequate to describe both shear and extensional flow using the same constitutive equation. This can be resolved using some kind of Molecular Stress Function (MSF) model allowing the use of dissipative convective constraint release in the constitutive equation (see M.H. Wagner, P. Rubio and H. Bastian, J. Rheol. Vol. 45, p. 1387-1412 (2001) ). The implemented constitutive equation in Rasmussen (2002) is inspired by Wagner et al. (2001). However the constitutive equation does not include the second of the two second order Rivlin-Ericksen tensor, only the first of the Rivlin-Ericksen tensor. Hence only the constitutive equations from Wagner et al. (2001), with the parameter a2=0 are included in the generalized Molecular Stress Function (MSF) constitutive model from Rasmussen (2002). Though using the elongational and shear measurements, from Wagner et. al. (2001) and A. Bach, H. K. Rasmussen and O.Hassager, J. Rheol. Vol. 47, p. 429-441 (2003), performed on a linear and a branched polymer melt, it is possible to construct a constitutive equation based on the first (second order) Rivlin-Ericksen tensor only, which can describe both the shear and the elongational data accurately.