Finite Element Methods for p-Navier Stokes

Funding agency	Linz Institute of Technology (LIT)
Project number	LIT-JKU-2017-04-SEE-004
Principal investigator	Ioannis Toulopoulos
Research scientist	Svetoslav Nakov
Duration	2019-02-01 – 2021-01-31

This website documents the work in the project LIT-JKU-2017-04-SEE-004 of the Linz Institute of Technology concerning the numerical simulation of incompressible Non-Newtonian flows modeled by power-law Navier-Stokes systems.

The method

• This project concerns the development of novel, efficient and accurate finite element methods, for discretizing simplified viscoplastic models
  \begin{align} \mathbf{u}_t -\mathrm{div}\big( (\delta +|\nabla\mathbf{u}|)^{p-2} \nabla \mathbf{u}\big) = & \mathbf{f}, \qquad\text{in $\Omega \times (0,T)$}, \qquad (1.1) \end{align}
• and more general non-Newtonian models, where the stresses have a $p$-$$power nonlinear dependence on the shear rate ($p$-{INS})$:$
  \begin{align} \mathbf{u}_t -\mathrm{div} \mathbf{S}(\mathbf{Du})+(\nabla \mathbf{u})\mathbf{u}+ \nabla P = & \mathbf{f}, \qquad\text{in $\Omega \times (0,T)$}, \qquad (1.2a)\\
  \nabla\cdot\mathbf{u} = & 0, \qquad\text{in $\Omega\times (0,T)$}, \qquad (1.2b) \end{align}
  the unknowns are the velocity $\mathbf{u}$ and the pressure $P$, while $\mathbf{Du}=\frac{1}{2}(\nabla\mathbf{u}+(\nabla\mathbf{u})^\top)$ denotes the symmetric velocity gradient, $|\mathbf{Du}|$ is the shear rate, $\mathbf{S}(\mathbf{Du}) = \frac{1}{Re}(\delta+|\mathbf{Du}|)^{p-2}\mathbf{Du}$ is the extra stress tensor with $\delta > 0$ to be a regularization parameter related to the behaviour of the fluid, and $\mathbf{f}$ is the external body force. For $p=2$, system (1.2) describes Newtonian fluid motions, for $p<2$, the fluid exhibits shear-thinning behavior (viscosity decreasing by increasing the shear rate measure), and for $p>2$, shear-thickening behaviour.
• The discretization and numerical treatment of these models is of great importance since they are met in many industrial processes, e.g., chemical engineering, food technology, glaceology, biofluid mechanics, etc.

Figure 1: Examples of non-Newtonian models in engineering

• Although the numerical solution of these type of problems has been a subject of research investigation in the last decades, the  $p$-nonlinear form of the extra stress tensor creates further difficulties. The devising of efficient numerical methods for these models is still a challenging problem. The classical steps for the numerical treatment of these fluid flow problems is first a space discretization, then a time discretization using Runge-Kutta methods and lastly an application of nonlinear iteration solvers. This usually brings restrictions between the time step $dt$ and the mesh size $dx$.
• In this project, we investigate space-time finite element methods that help on to discretizing the problem in space and in time simultaneously and in a unified way. This means that the fully discrete scheme is developed entirely by following the finite element discretization concept.
  The time variable is considered as an additional spatial variable, lets say $t:=x_{d+1}$ and in that way, the time derivative term in (Description 1.2) will play the role of an advection term in $x_{d+1}$ direction. Furthermore, we propose a method for speeding up the convergence properties of the nonlinear-iterative solver.
• The construction of a efficient space-time finite element methods combined with efficient iterative techniques for solving the $p$-{INS} of interest, as well the scientific implementation of the corresponding algorithms on a computer platform, are the main objectives of this project.

National and International Collaborators

• AC2T research GmbH, at Austrian Excellence Center for Tribology, Austria
• Institute of Applied Mathematics, Leibniz University, Hannover, Germany
• Johann Radon Institute for Computational and Applied Mathematics (RICAM), an institute of the Austrian Academy of Sciences, JKU Linz, Austria
• RISC Software GmbH, Softwarepark Hagenberg, Austria
• Research center of Inria Sophia Antipolis, France

A postgraduate fellowship from the AC2T research GmbH, Austrian Excellence Center for Tribology, was awarded to the Institute of Computational Mathematics, Johannes Kepler University, Linz. Thesis title: "Applications of variational inequalities in contact problems and Finite Element discretizations", winter and summer semester 2020.

Related publications

• I. Toulopoulos
• A Unified Time Discontinuous Galerkin Space-Time Finite Element Scheme for Non-Newtonian Power Law Models
• December 2022
• doi
• I. Toulopoulos
• Numerical Solutions of Quasilinear Parabolic Problems by a Continuous Space-Time Finite Element Scheme
• September 2022
• doi
• I. Toulopoulos
• Unified Stable Space-Time Finite Element Scheme for Non-Newtonian Power Law Models
• Submitted, 2021
• NuMa-Report 2021-06
• I. Toulopoulos
• A Continuous Space-Time Finite Element Scheme for Quasilinear Parabolic Problems
• Submitted, 2021
• NuMa-Report 2021-01
• J. A. Iglesias and S. Nakov
• Weak Formulations of the Nonlinear Poisson-Boltzmann Equation in Biomolecular Electrostatics
• Journal of Mathematical Analysis and Applications, vol. 511 (1), Article 126065, 2022
• arXiv
• S. Nakov and I. Toulopoulos
• Convergence Estimates of Finite Elements for a Class of Quasilinear Elliptic Problems
• Computers & Mathematics with Applications, vol. 104, pp. 87-112, 2021
• NuMa-Report 2020-03
• J. Kraus, S. Nakov and S. Repin
• Reliable Computer Simulation Methods for Electrostatic Biomolecular Models Based on the Poisson–Boltzmann Equation
• Computational Methods in Applied Mathematics, vol. 20, no. 4, pp. 643-676, 2020
•  
• I. Toulopoulos
• Viscoplastic Models and Finite Element Schemes for the Hot Rolling Metal Process
• 2020
•  
• I. Toulopoulos
• A Model and Numerical Investigation for Rolling Metal Process Using Continuous Finite Element Discretizations
• 2019
• doi
• S. Nakov and I. Toulopoulos
• Finite Element Methods for Simplified Viscoplastic Models
• 2019
•  
• S. Nakov, E. Sobakinskaya, T. Renger and J. Kraus
• Adaptive Goal Oriented Solver for the Linearized Poisson-Boltzmann Equation
• 2019
• doi

Theses

• Stefan Tyoler
• The p-Laplace problem in an isogeometric setting
• Master thesis, Johannes Kepler University Linz, 2021
• epub
• Svetoslav Nakov
• The Poison Boltzmann Equation: Analysis A Posteriori Error Estimates and applications
• PhD thesis, Johannes Kepler University Linz, 2019
• epub