A fully-coupled implementation of the contravariant deformation tensor formulation for viscoelastic flows

We present a fully-coupled numerical scheme for computing steady and time-dependent viscoelastic flows. The scheme relies on the contravariant deformation tensor formulation and uses a Newton–Raphson iteration to solve the non-linear system of equations. The contravariant reformulation allows for the computation and implementation of the analytical Jacobian relatively easily, especially compared to other reformulations such as the log-conformation. The contravariant deformation tensor rotates in steady state shearing flows, which is solved here by “resetting” it as a pre-processing step in the numerical scheme, rather than a post-processing step. We use the finite element method with standard stabilization techniques (SUPG and DEVSS-G) for the spatial discretization. The numerical scheme is tested in three viscoelastic flow problems which are studied in terms of stability and accuracy: planar Couette flow, 2D flow around a cylinder and 3D flow around a sphere. For all problems, quadratic convergence is observed in both the difference between iterations and the residuals during the Newton–Raphson procedure. Moreover, we observe that the residuals are several orders smaller than the difference between iterations. A distinct advantage of the numerical scheme presented here, is that it significantly relaxes the requirement on the time-step size in time-dependent problems, as compared to explicit or semi-implicit methods. Moreover, steady states can be efficiently computed if the initial guess in the Newton–Raphson iteration is close enough to the solution.