Physics-Compatible Finite Element Methods for Scalar and Tensorial Advection Problems

289 Pages·2019·7.586 MB·English

by  Christoph Lohmann

Christoph Lohmann
Physics-Compatible
Finite Element
Methods for Scalar
and Tensorial
Advection Problems
Physics-Compatible Finite Element
Methods for Scalar and Tensorial
Advection Problems
Christoph Lohmann
Physics-Compatible
Finite Element
Methods for Scalar
and Tensorial
Advection Problems
Christoph Lohmann
Dortmund, Germany
Dissertation Technische Universität Dortmund, Fakultät für Mathematik, 2019
Erstgutachter: Prof. Dr. Dmitri Kuzmin
Zweitgutachter: Prof. Dr. Matthias Möller
Tag der Disputation: 08. Mai 2019
ISBN 978-3-658-27736-9 ISBN 978-3-658-27737-6 (eBook)
https://doi.org/10.1007/978-3-658-27737-6
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Acknowledgments
Firstandforemost,Iwouldliketoexpressmysinceregratitudetomyadvisor
Prof. Dr. Dmitri Kuzmin for his guidance and continuous support during
my PhD studies. He gave me the opportunity to work on this exciting
topic and to attend many scientific conferences. His time invested in various
fruitful discussions is highly appreciated. I also would like to thank him for
proofreading this thesis and for providing me with valuable feedback.
Furthermore, I would like to take this opportunity to thank Prof. Dr.
Stefan Turek for significantly contributing to my academic education and
for already hiring me as an undergraduate student to work at the “Institute
of Applied Mathematics (LS III)”. In this context, I like to express my deep
gratitudetoallformerandcurrentmembersofLSIII.Itwasagreatpleasure
to be part of your group and to make new friendships! Thank you very
much for all your social and moral support during the past years! Special
thanks also go to the whole FEAT team for always helping me out in case
of software problems.
In addition, I acknowledge the financial support provided by the German
Research Association (DFG) without which this work would have been
impossible. At this point, a special thanks goes to Omid Ahmadi who
worked with me on the DFG project that this thesis is mainly concerned
with.
I am very grateful to Dr. John N. Shadid for making my exciting stay at
Sandia National Laboratories possible. It was a fruitful collaboration and a
warm atmosphere with you and Dr. Sibusiso Mabuza!
I also thank Prof. Dr. Matthias Möller and Prof. Dr. Matthias Röger for
kindly agreeing to act as reviewers and examiners in the defense of this
thesis.
Finally, I am profoundly grateful to my family and friends for their
invaluable support during my PhD studies.
Thank you so much!
Contents
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Equations of fluid dynamics 15
2.1 Scalar conservation laws . . . . . . . . . . . . . . . . . . . . 16
2.1.1 Physics . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.2 Well-posedness . . . . . . . . . . . . . . . . . . . . . 19
2.1.3 Maximum principles . . . . . . . . . . . . . . . . . . 28
2.2 Incompressible Navier-Stokes equations . . . . . . . . . . . 33
2.2.1 Physics . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Discretization 35
3.1 Finite elements . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1 Triangulation . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2 Piecewise-polynomial subspace . . . . . . . . . . . . 37
3.1.3 Degrees of freedom and basis functions . . . . . . . . 39
3.2 Steady problem . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 Galerkin method . . . . . . . . . . . . . . . . . . . . 40
3.2.2 Error analysis . . . . . . . . . . . . . . . . . . . . . . 41
3.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . 42
3.3 Unsteady problem . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.1 Galerkin method . . . . . . . . . . . . . . . . . . . . 44
3.3.2 Time integrator . . . . . . . . . . . . . . . . . . . . . 45
3.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . 46
4 Limiting for scalars 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Terms and definitions . . . . . . . . . . . . . . . . . 55
4.2.2 Steady problem . . . . . . . . . . . . . . . . . . . . . 57
4.2.3 Unsteady problem . . . . . . . . . . . . . . . . . . . 63
VIII Contents
4.3 Low order method . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.1 Steady problem . . . . . . . . . . . . . . . . . . . . . 73
4.3.2 Unsteady problem . . . . . . . . . . . . . . . . . . . 84
4.4 Fractional step approach . . . . . . . . . . . . . . . . . . . . 88
4.4.1 Steady problem . . . . . . . . . . . . . . . . . . . . . 89
4.4.2 Unsteady problem . . . . . . . . . . . . . . . . . . . 100
4.5 Monolithic approach . . . . . . . . . . . . . . . . . . . . . . 103
4.5.1 Steady problem . . . . . . . . . . . . . . . . . . . . . 103
4.5.2 Unsteady problem . . . . . . . . . . . . . . . . . . . 127
4.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . 136
4.6.1 Steady problem . . . . . . . . . . . . . . . . . . . . . 136
4.6.2 Unsteady problems . . . . . . . . . . . . . . . . . . . 139
5 Limiting for tensors 151
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . 152
5.1.2 Properties of interest . . . . . . . . . . . . . . . . . . 152
5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.3 Low order method . . . . . . . . . . . . . . . . . . . . . . . 161
5.3.1 Steady problem . . . . . . . . . . . . . . . . . . . . . 161
5.3.2 Unsteady problem . . . . . . . . . . . . . . . . . . . 165
5.4 Fractional step approach . . . . . . . . . . . . . . . . . . . . 168
5.4.1 Semidefinite limiting strategies . . . . . . . . . . . . 173
5.5 Monolithic approach . . . . . . . . . . . . . . . . . . . . . . 177
5.5.1 Steady problem . . . . . . . . . . . . . . . . . . . . . 178
5.5.2 Unsteady problem . . . . . . . . . . . . . . . . . . . 189
5.6 Numerical examples . . . . . . . . . . . . . . . . . . . . . . 196
5.6.1 Steady problem . . . . . . . . . . . . . . . . . . . . . 197
5.6.2 Unsteady problem . . . . . . . . . . . . . . . . . . . 202
6 Simulation of fiber suspensions 211
6.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.2 Fluid dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 214
6.2.1 Discretization . . . . . . . . . . . . . . . . . . . . . . 215
6.2.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . 218
6.2.3 Iterative solver . . . . . . . . . . . . . . . . . . . . . 219
6.3 Orientation dynamics . . . . . . . . . . . . . . . . . . . . . 221
6.3.1 Orientation discretization and splitting. . . . . . . . 221
6.3.2 Closures . . . . . . . . . . . . . . . . . . . . . . . . . 224
6.3.3 Temporal discretization . . . . . . . . . . . . . . . . 239
Contents IX
6.3.4 Selected three dimensional closures . . . . . . . . . . 247
6.4 Numerical examples . . . . . . . . . . . . . . . . . . . . . . 251
7 Conclusions 263
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Abstract
This thesis presents the author’s contributions to the development and anal-
ysis of bound-preserving numerical methods for hyperbolic equations. A
special focus is placed on physics-compatible approaches to simulation of
fiber suspension flows. In this application, a second order tensor field is
frequently introduced to approximate the orientation distribution of rigid
slender particles in a Newtonian carrier fluid. To simulate its evolution in a
physically admissible and stable manner, the unit trace property must be
preserved and eigenvalues must stay nonnegative. The methodology devel-
oped in this work leads to numerical approximations which are guaranteed
to satisfy these requirements under suitable time step restrictions.
Before considering the evolution of tensor quantities, the state of the
art in the field of algebraic flux correction (AFC) for continuous finite
element discretizations of scalar hyperbolic equations is reviewed. Existing
algorithms and the underlying theory are supplemented with new results.
Stabilization techniques based on the AFC approach modify the entries of
finite element matrices so as to make the scheme bound-preserving. Low
order approximations with desired properties can easily be constructed by
adding discrete diffusion operators. In high order nonlinear extensions,
the accuracy of the solution is improved by removing redundant diffusion
in smooth regions. Two ways to perform such corrections are described
in this thesis: (i) fractional-step algorithms which add bound-preserving
antidiffusive fluxes to a low order predictor and (ii) monolithic approaches
which incorporate nonlinear corrections into the residual of the low order
method. Both kinds of AFC schemes are backed by proofs of generalized
discrete maximum principles (DMPs), which imply preservation of global
bounds in particular. Transient transport problems are discretized in time
by the θ-scheme. Upper bounds for admissible time steps are derived
from sufficient conditions for the validity of DMPs. For the monolithic
AFC discretization of the steady state advection equation, results on the
existence of a (unique) solution and corresponding a priori error estimates
are presented.
A major highlight of this work is the extension of AFC tools to the nu-
merical treatment of symmetric tensor quantities. The proposed algorithms
XII Abstract
constrain the eigenvalue range of evolving tensor fields by imposing local
discretemaximumprinciplesonthemaximalandminimaleigenvalues. Using
this design principle and corresponding generalizations of the theoretical
framework, robust property-preserving tensor limiters are introduced fol-
lowing the analysis and design of their scalar counterparts. The proofs of
generalized DMP properties employ spectral decompositions and positive
semidefinite programming tools.
In the last part of this work, a frequently used model of fluid-fiber flows
is considered. The involved nonhomogeneous transport equation for the
second order orientation tensor is discretized using an operator splitting
approach. Under suitable assumptions and time step restrictions, preser-
vation of physical properties is shown for the forward and backward Euler
time discretizations of the tensorial ODE that governs the local orientation
dynamics. Finally, a numerical method for solving the fully coupled problem
is developed on the basis of customized solvers for individual subproblems.
Numerical results for the fiber suspension flow through an axisymmetric
contraction illustrate the potential of the proposed solution strategy.