Adaptive refinement of a viscoelastic flow problem with the explicitly elliptic momentum equation

Abstract

The explicitly elliptic momentum equation (EEME) of King et al. is applied with adaptive mesh refinement and a mixed Galerkin finite-element method for two model problems: planar and axisymmetric sudden contraction flow of a Maxwell fluid. Meshes for these problems were of an unstructured triangular nature, and were refined based on the mean of the seven residual equations of interest. Elements with high residuals were bisected, forming two new elements, and the solution was recomputed on the new, more refined mesh. Depending on Weissenberg number this process can or cannot be continued. Results with this method were mixed. Using the EEME formulation with quadratic interpolation for the velocities and stresses and linear interpolation for pressure (QQL) we obtained a significant improvement over QQL interpolation with the traditional Cauchy form of the momentum equation (CME). The quality of the solution was better for the EEME—QQL and higher Weissenberg number limits were found. However, quadratic velocity, linear stress and pressure interpolation (QLL) gave results for which the quality was similar using EEME and CME. The economy of mesh refinement is an important benefit of this technique since good solutions have been obtained using a highly refined mesh only near the singularity. If we covered our mesh with elements the size of our smallest element, we would need over 200 000 elements, rather than the 473 actually used. The most important point about adaptive finite-element methods is that we add degrees of freedom only in regions where the solution tells us to.