An adaptive discrete Newton method for regularization-free Bingham model

[EN] Developing a numerical and algorithmic tool which correctly identifies unyielded regions in 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, solvers do not perform efficiently when the regularization parameter gets very small. In this work, we use a formulation introducing a new auxiliary stress. The three field formulation of the yield stress fluid corresponds to a regularization-free Bingham formulation. The resulting set of equations arising from the three field formulation is solved efficiently and accurately by a monolithic finite element method. The velocity and pressure are discretized by the higher order stable FEM pair Q2/Pdisc 1 and the auxiliary stress is discretized by the Q2 element. Furthermore, this problem is highly nonlinear and presents a big challenge to any nonlinear solver. Therefore, we developed a new adaptive discrete Newton method, which evaluates the Jacobian with the divided difference approach. We relate the step length to the rate of the actual nonlinear reduction for achieving a robust adaptive Newton method. We analyse the solvability of the problem along with the adaptive Newton method for Bingham fluids by doing numerical studies for a prototypical configuration ”viscoplastic fluid flow in a channel”.We would like to thank the Deutsche Forschungsgemeinschaft (DFG) for their financial support under the DFG Priority Program SPP 1962. The authors also acknowledge the support by LS3 and LiDO3 team at ITMC, TU Dortmund UniversityFatima, A.; Turek, S.; Ouazzi, A.; Afaq, MA. (2022). An adaptive discrete Newton method for regularization-free Bingham model. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 180-189. https://doi.org/10.4995/YIC2021.2021.12389OCS18018

An Adaptive Discrete Newton Method for Regularization-Free Bingham Model

Developing a numerical and algorithmic tool which correctly identifies unyielded regions in 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, solvers do not perform efficiently when the regularization parameter gets very small. In this work, we use a formulation introducing a new auxiliary stress. The three field formulation of the yield stress fluid corresponds to a regularization-free Bingham formulation. The resulting set of equations arising from the three field formulation is solved 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. Therefore, we developed a new adaptive discrete Newton method, which evaluates the Jacobian with the divided difference approach. We relate the step length to the rate of the actual nonlinear reduction for achieving a robust adaptive Newton method. We analyse the solvability of the problem along with the adaptive Newton method for Bingham fluids by doing numerical studies for a prototypical configuration ”viscoplastic fluid flow in a channel”

Robust Monolithic Multigrid FEM Solver for Three Field Formulation of Incompressible Flow Problems

Numerical simulation of three field formulations of incompressible flow problems is of interest for many industrial applications, for instance macroscopic modeling of Bing-ham, viscoelastic and multiphase flows, which usually consists in supplementing the mass and momentum equations with a differential constitutive equation for the stress field. The variational formulation rising from such continuum mechanics problems leads to a three field formulation with saddle point structure. The solvability of the problem requires different compatibility conditions (LBB conditions) [1] to be satisfied. Moreover, these constraints over the choice of the spaces may conflict/challenge the robustness and the efficiency of the solver. For illustrating the main points, we will consider the three field formulation of the Navier-Stokes problem in terms of velocity, stress, and pressure. Clearly, the weak form imposes the compatibility constraints over the choice of velocity, stress, and pressure spaces. So far, the velocity-pressure combi-nation took much more attention from the numerical analysis and computational fluid dynamic community, which leads to some best interpolation choices for both accuracy and efficiency, as for instance the combination Q2/P1disc. To maintain the computational advantages of the Navier-Stokes solver in two field formulations, it may be more suitable to have a Q2 interpolation for the stress as well, which is not stable in the absence of pure viscous term [2]. We proceed by adding an edge oriented stabilization to overcome such situation. Furthermore, we show the robustness and the efficiency of the resulting discretization in comparison with the Navier-Stokes solver both in two field as well as in three field formulation in the presence of pure viscous term. Moreover, the benefit of adding the edge oriented finite element stabilization (EOFEM) [3, 4] in the absence of the pure viscous term is tested. The nonlinearity is treated with a Newton-type solver [5] with divided difference evaluation of the Jacobian matrices [6, 7]. The resulting linearized system inside of the outer Newton solver is a typical saddle point problem which is solved using a geometrical multigrid method with Vanka-like smoother [8, 9]. The method is implemented into the FeatFlow [10] software package for the numerical simulation. The stability and robustness of the method is numerically investigated for ”flow around cylinder” benchmark [7, 11]

Monolithic Newton-Multigrid Solver for Multiphase Flow Problems with Surface Tension

We have developed a monolithic Newton-multigrid solver for multiphase flow problems which solves velocity, pressure and interface position simultaneously. The main idea of our work is based on the formulations discussed in [1], where it points out the feasibility of a fully implicit monolithic solver for multiphase flow problems via two formulations, a curvature free level set approach and a curvature free cut-off material function approach. Both formulations are fully implicit and have the advantages of requiring less regularity, since neither normals nor curvature are explicitly calculated, and no capillary time restriction has to be respected. Furthermore, standard Navier-Stokes solvers might be used, which do not have to take into account inhomogeneous force terms. The reinitialization issue is integrated within the formulations. The nonlinearity is treated with a Newton-type solver with divided difference evaluation of the Jacobian matrices. The resulting linearized system inside of the outer Newton solver is a typical saddle point problem which is solved using a geometrical multigrid method with Vanka-like smoother using higher order stable Q_2/P_1^disc FEM for velocity and pressure and Q_2 for all other variables. The method is implemented into an existing software package for the numerical simulation of multiphase flows (FeatFlow). The robustness and accuracy of this solver is tested for two different test cases, static bubble and oscillating bubble, respectively

Monolithic Newton-Multigrid Solver for Multiphase Flow Problems with Surface Tension

[EN] We have developed a monolithic Newton-multigrid solver for multiphase flow problems which solves velocity, pressure and interface position simultaneously. The main idea of our work is based on the formulations discussed in [1], where it points out the feasibility of a fully implicit monolithic solver for multiphase flow problems via two formulations, a curvature-free level set approach and a curvature-free cutoff material function approach. Both formulations are fully implicit and have the advantages of requiring less regularity, since neither normals nor curvature are explicitly calculated, and no capillary time restriction. Furthermore, standard Navier-Stokes solvers might be used, which do not have to take into account inhomogeneous force terms. The reinitialization issue is integrated with a nonlinear terms within the formulations.The nonlinearity is treated with a Newton-type solver with divided difference evaluation of the Jacobian matrices. The resulting linearized system inside of the outer Newton solver is a typical saddle point problem which is solved using the geometrical multigrid with Vanka-like smoother using higher order stable FEM pair $Q_2/P^{\text{disc}}_1$ for velocity and pressure and $Q_2$ for all other variables. The method is implemented into an existing software packages for the numerical simulation of multiphase flows (FeatFlow). The robustness and accuracy of this solver is tested for two different test cases, i.e. static bubble and oscillating bubble, respectively [2].Muhammad Aaqib Afaq would like to thank Erasmus Mundus INTACT project, funded by the European Union as part of the Erasmus Mundus programme and the National University of Sciences and Technology (NUST) for their financial support. The authors also acknowledge the support by LS3 and LiDO3 team at ITMC, TU Dortmund University.Afaq, MA.; Turek, S.; Ouazzi, A.; Fatima, A. (2022). Monolithic Newton-Multigrid Solver for Multiphase Flow Problems with Surface Tension. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 190-199. https://doi.org/10.4995/YIC2021.2021.12390OCS19019

Robust adaptive discrete Newton method for regularization-free Bingham model

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