For 2D and 3D viscoelastic flow, the standard differential viscoelastic models can be used for both isothermal and non-isothermal flows. For shell models, a multi-mode integral viscoelastic model for blow molding or thermoforming applications can also be used. This method makes use of a membrane element. The fluid constitutive equations are written as follows:

(17–11)

(17–12)

(17–13)

where is an additional viscous component.

The numerics for viscoelastic blow molding are based on an extension of the shell element for fluids (described in Shell Elements for 3D Models). The algorithm is of the "explicit-implicit" type. At each time step, viscoelastic extra stresses are evaluated explicitly by integrating Equation 17–12 in time over the current time step.

These stresses are then added to the right-hand side of the momentum equation, where they act as body forces (denoted by in Equation 17–14, below). Then velocity, thickness, and positions (collectively denoted by the unknown vector ) are evaluated using the following equation:

(17–14)

Equation 17–14 is specific to the Euler implicit time integration scheme (the default and most recommended scheme for blow molding), and is a mass matrix that is affected mostly by inertia terms. In order to calculate the velocity, thickness, and position at time from the values at time , the algorithm requires a stiffness matrix . The optional viscosity is not sufficient to guarantee good stability of the algorithm, so a "numerical viscosity" called is introduced. The value of affects both and .

In Equation 17–14, the matrix contains contributions of both and , whereas is affected only by . Ansys Polydata automatically evaluates as the sum of all individual mode viscosities divided by 30. This empirical value generally produces good and stable results. When time steps are sufficiently small, this numerical viscosity (which appears with the subscript on the left-hand side of Equation 17–14, and with the subscript on the right-hand side) has no influence on the time accuracy of the results. This algorithm is similar to the method described in Integral Viscoelastic Models for integral viscoelastic fluids.

Note that the viscosity that appears in the constitutive model affects the rheological response of the material, whereas does not in the limit of small time steps.

The procedure for modeling viscoelastic blow molding or thermoforming in Ansys Polyflow is the same as for other 3D problems using the shell element. See Inputs for Shell Contact Detection for details.