An adaptive discrete Newton method for a regularization-free Bingham model

Abstract

Developing a numerical and algorithmic tool which correctly identifies unyielded regions in the yield stress fluid flow is a challenging task. Two approaches are commonly used to handle the singular behaviour at the yield surface, i.e. the Augmented Lagrangian approach and the regularization approach, respectively. Generally in the regularization approach for the resulting nonlinear and linear problems, solvers do not perform efficiently when the regularization parameter gets very small. In this work, we use a formulation introducing a new auxiliary stress [1]. The three field formulation of yield stress fluids corresponds to a regularization-free Bingham formulation. The resulting set of equations arising from the three field formulation is treated efficiently and accurately by a monolithic finite element method. The velocity and pressure are discretized by the higher order stable FEM pair Q_2⁄(P_1^disc ) and the auxiliary stress is discretized by the Q_2 element.

Furthermore, this problem is highly nonlinear and presents a big challenge to any nonlinear solver. We developed a new adaptive discrete Newton's method, which evaluates the Jacobian with the directional divided difference approach [2]. The step size in this process is an important key: We relate this size to the rate of the actual nonlinear reduction for achieving a robust adaptive Newton's method. The resulting linear subproblems are solved using a geometrical multigrid solver. We analyse the solvability of the problem along with the adaptive Newton method for Bingham fluids by doing numerical studies for different prototypical configurations, i.e. "Viscoplastic fluid flow in a channel" [2], "Lid Driven Cavity", "Flow around cylinder", and "Bingham flow in a square reservoir", respectively.

References

[1] A. Aposporidis, E. Haber, M. A. Olshanskii, A. Veneziani. A Mixed Formulation of the Bingham Fluid Flow Problem: Analysis and Numerical Solution, Comput. Methods Appl. Mech. Engrg. 1 (2011), 2434–2446.

[2] A. Fatima, S. Turek, A. Ouazzi, M. A. Afaq. An Adaptive Discrete Newton Method for Regularization-Free Bingham Model, 6th ECCOMAS Young Investigators Conference 7th-9th July 2021, Valencia, Spain. doi: 10.4995/YIC2021.2021.12389.