Stabilized finite element methods for a fully-implicit logarithmic reformulation of the Oldroyd-B constitutive law

Abstract

Logarithmic conformation reformulations for viscoelastic constitutive laws have alleviated the high Weissenberg number problem, and the exploration of highly elastic flows became possible. However, stabilized formulations for logarithmic conformation reformulations in the context of finite element methods have not yet been widely studied. We present stabilized formulations for the logarithmic reformulation of the Oldroyd-B model by Saramito (2014) based on the Variational Multiscale framework and Galerkin/Least-Squares method. The reformulation allows the use of Newton’s method due to its fully-implicit nature for solving the steady-state problem directly. The proposed stabilization methods cure instabilities of convection and compatibility while preserving a three-field problem. The spatial accuracy of the formulations is assessed with the four-roll periodic box, revealing comparable accuracy between the methods. The formulations are validated with benchmark flows past a cylinder and through a 4:1 contraction. We found an excellent agreement to benchmark results in the literature. The algebraic sub-grid scale method is highly robust, indicated by the high limiting Weissenberg numbers in the benchmark flows.