Robust adaptive discrete Newton method for regularization-free Bingham model
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Date
2023-03
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Abstract
Developing a numerical and algorithmic tool which accurately detects unyielded
regions in yield stress fluid flow is a difficult endeavor. To address these issues, two common approaches are used to handle singular behaviour at the yield surface, i.e. the augmented Lagrangian approach and the regularization approach. Generally, solvers do not operate effectively when the regularization parameter is very small in the regularization approach. In this work, we use a formulation involving a new auxiliary stress tensor, wherein the three-field formulation is equivalent to a regularization-free Bingham formulation.
Additionally, a monolithic finite element method is employed to solve the set of equations resulting from the three-field formulation accurately and effciently, where the velocity, pressure fields are discretized by the higherorder stable FEM pair Q2=Pdisc1 and the auxiliary stress is discretized by the Q2 element.
Furthermore, this article presents a novel adaptive discrete Newton method for solving highly nonlinear problems, which exploits the divided difference approach for evaluating the Jacobian. The step size of the solver is dynamically adjusted according to the rate of nonlinear reduction, enabling a robust
and efficient approach. Numerical studies of several prototypical Bingham fluid configurations ("viscoplastic
fluid flow in a channel", "lid driven cavity" and "rotational Bingham flow in a square reservoir") are used to analyse the performance of this method.
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Keywords
viscoplastic fluid, regularization-free, discrete Newto method, FEM, divided difference, Bingham fluid