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Electronic Theses, Treatises and Dissertations The Graduate School
2010
Level Set and Conservative Level SetMethods on Dynamic Quadrilateral GridsSvetlana Simakhina
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THE FLORIDA STATE UNIVERSITY
COLLEGE OF ARTS AND SCIENCES
LEVEL SET AND CONSERVATIVE LEVEL SET METHODS
ON DYNAMIC QUADRILATERAL GRIDS
By
SVETLANA SIMAKHINA
A Dissertation submitted to theDepartment of Mathematicsin partial fulfillment of the
requirements for the degree ofDoctor of Philosophy
Degree Awarded:Summer Semester, 2010
The members of the committee approve the Dissertation of Svetlana Simakhina defended on
June 10th, 2010.
Mark SussmanProfessor Directing Dissertation
Michael RoperUniversity Representative
David KoprivaCommittee Member
Brian EwaldCommittee Member
Janet PetersonCommittee Member
The Graduate School has verified and approved the above-named committee members.
ii
I dedicate the dissertation to my middle and high school teacher of mathematics, LyudmilaMulyakaeva, PhD, from Kokchetav (Kazakhstan), who patiently taught all her students toenjoy the beauty of mathematics and simplicity of logic in everything we do in our lives.
iii
ACKNOWLEDGEMENTS
I would like to thank my adviser, Professor Mark Sussman, for his tremendous help in
my grad research work. His help was not only in dedicating his time to share new ideas
and valuable discussions with me, but also in financial support over the past three years,
which was a part of his NSF grant. It was a big honor to be recognized with a Research
Assistantship as a part of his big research.
I would like also to express my gratitude to the committee members. I appreciate their
time, suggestions and comments they made about this research.
I would like to acknowledge Department of Mathematics for education and for giving
me opportunity to work as a Teaching Assistant over my first two years at FSU. It was a
valuable experience.
I express my appreciations to department faculty and stuff for their work that made it
possible for me to complete my study and this research.
Special thanks go to my friends and relatives in Tallahassee, Russia, Kazakhstan and all
over the world, whose constant support and friendship I am lucky to have. I am especially
glad to mention our friend Natalie Rubik for her selfless tremendous help to my family and
for her love to us and our kids. It’s my greatest pleasure to thank my husband, Yevgeny
Goncharov, the smartest and kindest person I have ever met, for his unwavering belief,
support and love. Thank you to my daughters, Polina and Olesya, for being the most
beautiful part of my life.
iv
TABLE OF CONTENTS
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Computing two phase flow: definition of the problem . . . . . . . . . . . 31.2 Focus of research: interface capturing on a dynamic, stretched, curvilinear
grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2. DYNAMIC STRETCHED CURVILINEAR GRIDS . . . . . . . . . . . . . . . 132.1 Variational grid generation . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Stretched quadrilateral grid . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Rectilinear grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Monitor functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3. INTERFACE CAPTURING WITH CLS METHOD . . . . . . . . . . . . . . 263.1 CLS algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2 CLS advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 CLS reinitialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 CLS remapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.5 CLS reinitialization revisited . . . . . . . . . . . . . . . . . . . . . . . . . 303.6 LTE of CLS method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4. LS WITH GLOBAL VOLUME FIX ON A DYNAMIC, STRETCHED GRID 364.1 Algorithm for LS with global volume fix on a dynamic, stretched, grid . 374.2 Reinitialization of the LS function with improved preservation of the zero
level set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3 Global volume fix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4 Numerical results: LS with global volume fix . . . . . . . . . . . . . . . . 47
5. TUNNEL QUASI-CUBIC INTERPOLATION ALGORITHM . . . . . . . . . 655.1 Tunnel quasi-cubic spline on an orthogonal 2-d grid . . . . . . . . . . . . 665.2 Tunnel quasi-cubic local interpolation on a general quadrilateral grid . . 68
v
6. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.1 CONCLUSIONS - grid generation . . . . . . . . . . . . . . . . . . . . . . 706.2 CONCLUSIONS - Eulerian front tracking . . . . . . . . . . . . . . . . . 71
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
vi
LIST OF TABLES
4.1 Rate of convergence for Zalesak’s problem . . . . . . . . . . . . . . . . . . . 54
4.2 Computational time and error, Zalesak’s problem 50x50. . . . . . . . . . . . 55
4.3 Computational time and error, Zalesak’s problem 100x100. . . . . . . . . . . 55
4.4 Computational time and error, Zalesak’s problem 200x200. . . . . . . . . . . 55
4.5 Rate of convergence for deforming velocity test problem . . . . . . . . . . . . 60
4.6 Computational time and error, a deforming velocity field, grid 50x50. . . . . 62
4.7 Computational time and error, a deforming velocity field, grid 100x100. . . . 62
vii
LIST OF FIGURES
1.1 LS and CLS functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Triangular non-structured body-fitted grid. . . . . . . . . . . . . . . . . . . . 14
2.2 Quadrilateral grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Discretization of nonlinear Laplace equations. . . . . . . . . . . . . . . . . . 17
2.4 A stretched curvilinear quadrilateral grid. . . . . . . . . . . . . . . . . . . . 19
2.5 Discretization of grid generating nonlinear ODE. . . . . . . . . . . . . . . . . 20
2.6 2-d rectilinear grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 Semi-Lagrangian finite volume scheme illustration. . . . . . . . . . . . . . . . 28
3.2 CLS reinitialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 CLS local truncation error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1 LS remapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Semi-Lagrangian scheme for LS advection. . . . . . . . . . . . . . . . . . . . 40
4.3 LS contours without redistancing and volume fix. . . . . . . . . . . . . . . . 41
4.4 LS contours with redistancing and volume fix. . . . . . . . . . . . . . . . . . 41
4.5 Volume change inside interface in a deforming velocity field. . . . . . . . . . 42
4.6 Global volume fix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.7 Zalesak’s test on dynamic rectilinear 50x50. . . . . . . . . . . . . . . . . . . 48
4.8 Zalesak’s test on dynamic rectilinear 100x100. . . . . . . . . . . . . . . . . . 49
4.9 Zalesak’s test on dynamic rectilinear 200x200. . . . . . . . . . . . . . . . . . 50
4.10 Zalesak’s test on dynamic quadrilateral 50x50 grid. . . . . . . . . . . . . . . 51
viii
4.11 Zalesak’s test on dynamic quadrilateral 100x100 grid. . . . . . . . . . . . . . 52
4.12 Zalesak’s test on dynamic quadrilateral 200x200 grid. . . . . . . . . . . . . . 53
4.13 Computational time and numerical error, Zalesak’s problem. . . . . . . . . . 54
4.14 Zalesak’s problem, grid 100x100. . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.15 Zalesak’s problem, grid 200x200. . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.16 Deforming divergence free velocity field. . . . . . . . . . . . . . . . . . . . . 58
4.17 An interface in a deforming velocity field on a uniform grid. . . . . . . . . . 59
4.18 Volume variation on a uniform grid. . . . . . . . . . . . . . . . . . . . . . . . 60
4.19 An interface in a deforming velocity field on a dynamic rectilinear grid. . . . 61
4.20 Volume variation on a dynamic rectilinear grid. . . . . . . . . . . . . . . . . 62
4.21 An interface in a deforming velocity field on a dynamic quadrilateral grid. . . 63
4.22 Volume variation on a dynamic quadrilateral grid. . . . . . . . . . . . . . . . 64
4.23 Computational time and error, smooth interface in deforming field. . . . . . 64
5.1 Quasi-cubic interpolation on a rectilinear grid . . . . . . . . . . . . . . . . . 66
5.2 Tunnel quasi-cubic interpolation . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.1 Zalesak’s problem (rotating counter clockwise) using LS/global mass fixalgorithm. Dynamic block structured AMR grid with two levels of adaptivityis used. Problem computed in a polar coordinate system. Effective fine gridresolution: 200 cells in the r direction and 400 cells in the θ direction. . . . 71
6.2 Computation of an axisymmetric gas bubble rising in liquid. using LS/globalmass fix algorithm. Frames go from left to right, top to bottom. Dynamicblock structured AMR grid with one level of adaptivity is used. Effective finegrid resolution is 64x128. Periodic boundary conditions at the top and bottomof the computational domain. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.3 Comparison of the computed rise speed for an axisymmetric cap bubble,between LS/global mass fix and CLSVOF methods. Dynamic block structuredAMR grid with one level of adaptivity is used. Effective fine grid resolutionis 64x128. Periodic boundary conditions at the top and bottom of thecomputational domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
ix
6.4 Comparison of the computed volume for an axisymmetric cap bubble, betweenLS/global mass fix and CLSVOF methods. Dynamic block structured AMRgrid with one level of adaptivity is used. Effective fine grid resolutionis 64x128. Periodic boundary conditions at the top and bottom of thecomputational domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
x
ABSTRACT
The work in this thesis is motivated by the application of spray combustion. If one
develops algorithms to simulate spray generation, for example the primary break-up of
a liquid jet in a gas cross-flow, then a body-fitted or Lagrangian methods would require
“surgery” in order to continue a simulation beyond the point at which a droplet is torn into
multiple droplets. The liquid volume must also be conserved in simulating spray generation.
In this thesis, an Eulerian front tracking method with conserved fluid volume is developed
to represent and update an interface between two fluids. It’s a level set (LS) method with
global volume fix, and the underlying grid is a structured, dynamic, curvilinear grid.
We compared our newly developed method to the coupled level set and volume of fluid
method (CLSVOF) for two strategic test problems. The first problem, the rotation of
a notched disk, tests for robustness. The second problem (proposed in this thesis), the
deformation of a circular interface in an incompressible, deforming, velocity field, tests for
order of accuracy.
We found that for the notched disk problem, the CLSVOF method is superior to the
new combined level set method/curvilinear grid method. For a given number of grid points,
the CLSVOF method always outperforms the combined level set/curvilinear grid method.
On the other hand, for the deformation of a circular interface problem, the combined level
set/curvilinear grid method gives better accuracy than the CLSVOF method, for a given
number of grid points. Unfortunately the new method is more expensive because a new
mesh must be generated periodically.
We note that the volume error of the new level set/curvilinear grid algorithm is
comparable to that of the CLSVOF method for all test cases tried.
xi
We prove that the conservative level set (CLS) method has O(1) local truncation error
in an advection scheme. The following developments of the conservative level set (CLS)
method are presented in the thesis: new CLS function remapping algorithm and new CLS
reinitialization algorithm. The new developments allow one to implement the CLS method
on a dynamic quadrilateral grid but don’t remedy the order of the method.
A new algorithm for quasi-cubic interpolation is presented. Quasi-cubic interpolation has
been used for local polynomial interpolation on an orthogonal mesh before, but never on a
general, non-orthogonal curvilinear mesh. The new (tunnel quasi-cubic) algorithm enables
one to find a global piece-wise polynomial interpolation of degree three on an orthogonal
mesh, and to find a local polynomial interpolation of degree three on a curvilinear mesh.
xii
CHAPTER 1
INTRODUCTION
To compute solutions to problems of multiphase flow with applications to atomization and
spray [1, 2], the numerical method must be both robust and accurate. A critical design issue
for numerical algorithms is to track the interface separating gas and liquid accurately.
The primary objective of this thesis is to explore Eulerian interface tracking algorithms
for dynamic, stretched, curvilinear grids. Previously, adaptive mesh refinement (AMR)
[3, 4] has been used to simulate two-phase flow in which droplets and bubbles are generated.
The dynamic, stretched, curvilinear grid algorithms developed in this thesis share the same
property as AMR methods in that the interface moves through the grid instead of the grid
being fit to the interface. A distinction in this research, in contrast to AMR interface tracking
methods, is that resolution is gained without increasing the number of grid points. Therefore,
storage costs, and perhaps CPU time too, might be reduced using stretched curvilinear grids.
Simulating two phase flow is difficult because the material interface separating gas and
liquid can develop corners and cusps. For many applications the material interface can
break up into multiple droplets or bubbles. Droplets or bubbles that are formed can then
merge. Expressed in mathematical terms, an interface that is complex is not everywhere
smooth, since the normal to the interface is not everywhere continuous. If one were to use
conventional high order methods without limiting, to represent the interface, then Gibb’s
phenomenon will appear at corners. The Gibb’s phenomenon can pollute the accuracy on
the smooth parts of the interface. On the other hand, low order methods exhibit too much
numerical diffusion.
A direct way to represent and update the gas-liquid interface is to explicitly define the
interface as a set of connected markers [5] (xint(t), yint(t))n, n = 1 : N . Assuming that the
interface velocity is given, the markers can be integrated in time in a Lagrangian way using
1
the equations:
dxint/dt = uint, dyint/dt = vint.
An approach that uses connected markers [5], or body fitted grid approach [6] is not
robust for simulating complicated interfaces in atomization and spray problems. Just at the
moment a droplet forms, or two bubbles merge, a parameterized interface becomes multi-
valued. Furthermore, no amount of regridding can remove “small elements” during the
process of mergers or break-ups. Instead, artificial surgery must be applied in order to carry
a computation beyond break-up or merging if one is using a body-fitted or front-tracking
approach [7, 8].
Level set methods [9, 10], in contrast to a methods that uses connected markers, can
easily treat interfaces that have corners or treat interfaces in which bubbles or drops can
merge or break-up. Furthermore, level set methods are easily generalizable to curvilinear
grids [11]. A serious drawback of level set methods is that they do not preserve the volume
of enclosed incompressible flow regions. To rectify the volume conservation issue, researchers
have attempted the following strategies:
• coupled level set method and volume of fluid method [12],
• the conservative level set method [13, 14],
• coupled level set method and particle methods [15, 16],
• compute the level set solution on a grid that has higher resolution than the grid used
to represent the underlying velocity field [17, 18],
• apply a global mass constraint [19, 20].
Methods that add more resolution to represent the interface [15, 16, 17, 18, 4], still do
not rectify the volume conservation issue. These methods only reduce the rate with which
mass is lost or gained. For example, the simulation of a cap bubble rising in a liquid will
never achieve a steady state if its mass is steadily lost regardless of how small the rate of
mass loss is. On the other hand, the coupled level set and volume of fluid method [21],
[12], conservative level set methods [13], or level set methods coupled with a global mass
2
constraint [19, 20] will enable one to compute steady motion of bubbles (see Figures 6.2, 6.3,
and 6.4).
There are auxiliary drawbacks to these last three approaches. For the CLSVOF method,
the geometric operations necessary to preserve volume during the dynamic curvilinear grid
remapping step (and simultaneously preserve overall second order accuracy) are prohibitively
complicated. We refer the reader to [22] for an example of a conservative remapping
algorithm. A conservative remapping step implemented for CLS methods is comparatively
easier. But we prove in this thesis that CLS methods are not second order accurate. Also
CLS methods cannot preserve the advection of a straight line interface on a uniform static
grid, let alone on a dynamic curvilinear grid. A level set method together with a global mass
constraint has the draw back that mass can be redistributed from one separate fluid volume
to another.
We now pose the following modus operandi: if one can develop a global mass constraint
that does not have the side effect of redistributing volume between geometrically unconnected
fluid volumes, then the best method for dynamic curvilinear grids would be a level set
method together with a global mass constraint. The method might have the accuracy that
a conservative level set method lacks, and at the same time avoid complicated conservative
remapping algorithms.
So, in this thesis, we develop a new Eulerian front tracking level set method for dynamic,
stretched, curvilinear grids. Our new method implements a global mass constraint to preserve
volume. Besides developing the new interface tracking method, we introduce new techniques
to quantify the effectiveness of existing front tracking approaches.
1.1 Computing two phase flow: definition of theproblem
Simulating incompressible immiscible two phase flow has applications in many areas such as
microfluidics, weather prediction, oil production, spray combustion and atomization in spray
[23].
In the bulk liquid regions, and the bulk gas regions, the governing equations for immiscible
two phase flow are:
ρLDuL
Dt= −∇pL + 2µL∇ · DL + ρLg, ∇ · uL = 0, x ∈ liquid,
3
ρgDug
Dt= −∇pg + 2µg∇ · Dg + ρgg, ∇ · ug = 0, x ∈ gas,
D =∇u + (∇u)T
2.
The interfacial jump conditions on the boundary Γ between liquid and gas, are:
(2µLDL − 2µgDg) · n = (pL − pg + σκ) n and uL = ug, x ∈ Γ,
uL = ug, x ∈ Γ,
dx
dt= u(x(t), t), x ∈ Γ,
κ = ∇ · n, x ∈ Γ.
ρL, µL, uL, pL correspond to the liquid density, viscosity, velocity, and pressure, respectively.
ρg, µg, ug, pg correspond to the gas density, viscosity, velocity, and pressure, respectively.
D is the rate of deformation tensor. g corresponds to the gravitational force directed
in the negative vertical direction. σ is the surface tension coefficient. κ is the interfacial
curvature.
Regardless of whether one implements a level set method for representing the air-water
interface [9], phase field method [24], volume-of-fluid method [12], conservative level set
method [13], or front tracking method [5], one can rewrite the above system of equations for
two-phase flow as follows [19] (here we use the level set representation in order to write out
the equations):
ρDu
Dt= ∇ · (−pI + 2µD) + ρgz − σκ∇H, (1.1)
∇ · u = 0,
Dφ
Dt= 0, (1.2)
κ(φ) = ∇ · ∇φ|∇φ| ,
4
H(φ) =
1 φ ≥ 0,0 φ < 0,
ρ = ρLH(φ) + ρG(1−H(φ)),
µ = µLH(φ) + µG(1−H(φ)).
For a given physical domain Ω = ΩL ∪ Ωg with two fluids (for notation simplicity lets
take ΩL as liquid, Ωg as gas) such that ΩL ∩ Ωg = ∅, the density and viscosity are
ρ =
ρL in liquid ΩL,
ρg in gas Ωg,
µ =
µL in liquid ΩL,
µg in gas Ωg.
The level-set equation, (1.2), is an auxiliary equation whose solution φ represents the
location of the gas-liquid interface. φ is continuously defined in a physical domain in such
a way that, φ(x, t) > 0 if x ∈ ΩL (in liquid) and φ(x, t) < 0 if x ∈ Ωg (in gas). The
zero level φ(x, t) = 0 of this function implicitly represents the interface position between
these two fluids. The material interface velocity is assumed to be a contact discontinuity in
which the liquid velocity and gas velocity match at the interface. The level set function φ
is constant (and equal to zero) on the material interface for all time t. So, if the interface is
parameterized by (x(t), y(t)) and the velocity of the interface is governed by the velocity (u, v)
at the liquid/gas interface (which is continuous, since the interface is a contact discontinuity),
then one hasdφ(x(t), y(t), t)
dt=∂φ
∂x
dx
dt+∂φ
∂y
dy
dt+∂φ
∂t= 0, (1.3)
since dx/dt = u and dy/dt = v along the interface, and u, v are continuous around the
interface, one has, u∂φ
∂x+ v
∂φ
∂y+∂φ
∂t= 0, i.e. Dφ/Dt = 0. If φ is governed by Dφ/Dt = 0,
then the interface will be represented implicitly by the zero level set of φ for all time t ≥ 0.
We remark, that including an auxiliary variable, such as the level set function φ, is
important in order to maintain accuracy.
If one does not add an auxiliary variable, then the material interface would have to
be implicitly “captured” by solving the following two advection equations for density and
viscosity:
Dρ
Dt= 0,
Dµ
Dt= 0. (1.4)
5
Since ρ and µ are discontinuous across the interface, the numerical discretization of these
equations would directly lead to either excessive ringing (if a high order method is used) or
numerical dissipation. Numerical dissipation is manifested by the smearing of the density
profile so that the interface thickness continues to grow the longer one integrates (1.4).
Because of these numerical artifacts of front “capturing” schemes, the volume enclosed by
the (ρL + ρg)/2 contour of ρ is not preserved.
1.2 Focus of research: interface capturing on adynamic, stretched, curvilinear grid
This research is a part of a larger research project in order to understand and simulate
spray combustion. We focus our attention here on methods and models which allow one
to accurately and efficiently numerically simulate spray generation. Special care has to be
taken to simulate the primary break-up phase in the break-up of a liquid with a gaseous
cross-flow. For this particular application one can not use body-fitted grids or front tracking
Lagrangian methods since these methods are applicable only in the preliminary evolution of
a liquid jet prior to break-up.
In this research, we focus on Eulerian front tracking approaches (see e.g. [20, 13, 12]).
Because the spray generation problem possesses multiple spatial scales, we shall explore
Eulerian front tracking methods with dynamic, stretched, curvilinear grids that increase
resolution at the gas-liquid interface. So far, Eulerian interface tracking methods for
incompressible flow have been developed for adaptive mesh refinement (AMR) [18], [25]
where the number of grid points is not fixed. We are aware of only one work [26] (2009)
in which an Eulerian interface tracking method is developed in the context of dynamic,
stretched, curvilinear grids. In that work, the authors used the phase field method which
suffers from the same accuracy issues as the conservative level set method (see Chapter 3).
In this thesis, we explore a front tracking method for dynamic, stretched curvilinear grids,
which overcomes the accuracy problems of the phase field method [26] (or the conservative
level set method), and conservs volume.
1.2.1 Interface capturing
Volume preservation is important when developing an Eulerian interface tracking method.
In the context of a dynamic, stretched, curvilinear grid, a high order remapping step for a
6
volume-of-fluid method, or a coupled level set and volume of fluid method (CLSVOF) [12]
would be prohibitively complicated to implement. Therefore, we develop herein a level set
method together with a “global mass constraint” in order to accurately track complicated
interfaces on a dynamic, stretched, structured, curvilinear grid.
A level set (LS) [27] method tracks the interface position using an implicit continuous
function, φ, defined in such a way that φ > 0 in the liquid and φ < 0 in the gas; the zero level
contour of this function represents the interface between liquid and gas. The LS method has
the following attractive property: the interface representation with an implicit continuous
function φ defined in the whole domain simplifies the numerical calculation of a solution
of two-phase flow because there is no need to develop complicated interface reconstruction
algorithms as with, for example, the coupled level set and volume-of-fluid method [12]. The
sign of the level set function determines whether a computational cell is a liquid cell or a
gas cell, and then appropriate physical properties (e.g. density and viscosity) are assigned
[21]. Usually, but not necessarily, the level set function is maintained as the signed distance
to the gas-liquid interface (see Fig.1.1(a)). For example, if an interface is given by xΓ ∈ Γ
then a signed distance LS function is
φ(x) =
infxΓ∈Γ|x− xΓ| if x ∈ Ω1,
− infxΓ∈Γ|x− xΓ| if x ∈ Ω2.
(1.5)
The location of an interface, whose motion is governed by the velocity field U, is
integrated in time by solving the following advection equation in the whole computational
domain,
φt + U · φ = 0. (1.6)
Exact solutions of (1.6) correctly reflects the position of an interface moving with speed
U, as shown in (1.3), but the LS function can develop large gradients in a direction normal to
the zero level set. The numerical solution of the level set equation will therefore progressively
become less accurate as the LS function develops larger and larger gradients. Therefore, in
many numerical implementations of the LS method, the LS function is maintained as a
distance function by periodically reinitializing the level set function [9], [18], [17], [28]. The
level set reinitialization step replaces the original level set function with a new level set
function that shares the same zero contour, but is a distance function. Numerically, one can
7
reinitialize the level set function by solving the following equation to steady state.
φτ + sign(φ)(| φ| − 1) = 0. (1.7)
Steady state solutions to the above equation gives a distance function with | φ| = 1 [9].
If ∇ ·U = 0, then exact solutions of (1.6) will preserve the volume enclosed by the zero
levelset of φ. Unfortunately, the numerical solution of (1.6) does not necessarily preserve the
volume enclosed by the zero levelset of φ. Observe that since ∇ ·U = 0, one can write (1.6)
in conservation form,
φt +∇ · (Uφ) = 0. (1.8)
Even if one discretizes (1.8) in conservation form, thereby preserving the discrete integral of
φ, this does not imply that the volume enclosed by the zero level set of φ is preserved. There
is a trade-off between conservation and accuracy. One could take φ to represent density,
in which case one preserves volume but loses accuracy, or one could take φ to represent
a smooth distance function, in which one does not necessarily preserve volume, but the
resulting method is accurate.
The following are approaches that address the problem of volume conservation: coupled
level set-volume of fluid method (CLSVOF) [12], particle level set (PLS) [15], marker level
set method (MLS) [16], LS with volume fix [19], [29]. In this thesis we shall address the
problems of accuracy and volume conservation on a dynamic curvilinear grid, which has not
been explored previously.
Recently an idea of using a conservative LS function for two phase flow was proposed
[13]. A conservative level set function (with zero thickness) is one inside one fluid and zero
inside another fluid:
ψ(x, t) =
1 if x ∈ Ω1,
0 if x ∈ Ω2.
To improve convergence and stability of the numerical solution of the conservative level
set equation, a thin hyperbolic tangent interfacial layer is introduced to smear ψ and make
it continuous.
ψ(x, t) =1
1 + ed(x,t)/ǫ=
1
2
(
1− tanh
(
d(x, t)
2ǫ
))
, (1.9)
where d(x, t) is a signed distance to an interface, d(x, t) < 0 in Ω1 and d(x, t) > 0 in Ω2.
The parameter ǫ is suggested in [13] to be of the order of the grid size in order to make
the width of the interfacial layer to be several cells.
8
(a) Signed distance LS function (b) CLS function with ǫ = 0.02, ψ(x, y) =1
1 + ed(x,y)
ǫ
Figure 1.1: LS and CLS functions.
The 0.5 level of the smeared out CLS function implicitly represents the interface position
(Fig.1.1(b)). Conservative advection which at the same time stabilizes the profile across the
interface is given by
ψt + · (Uψ) =1
µ∇ · (−ψ(1− ψ)n+ ǫ(∇ψ · n)n),
where n =∇ψ|∇ψ| is a normal to the interface.
(1.10)
Here µ is an empirical constant. An integral of the function ψ over a physical domain will be
conserved if a conservative numerical scheme is used to solve the equation (1.10) [13]. One
can see after applying the discrete divergence theorem to (1.10), that the discrete integral of
ψ is preserved as long as U · nwall = 0 and ∆ψ · nwall = 0.
As will become apparent later on in this thesis there is a trade-off between accuracy and
volume preservation based on the thickness ǫ. If ǫ is too small (e.g. on the same order as
the grid size), then there is a reduction in the order of accuracy associated with discretizing
(1.10). On the other hand, if ǫ is too large, then preserving the integral of ψ does not carry
over to preserving the volume enclosed by the 0.5 level of ψ.
Olsson et al. [13] used a conservative finite element method and split the stabilized CLS
advection equation (1.10) into two steps. First, they advect the CLS interface by solving
ψt + · (Uψ) = 0 (1.11)
9
for t = tn +t with initial condition ψn ≈ ψ(tn). Then, given the resulting ψ∗, normals to
the interface are calculated
n =∇ψ∗
|∇ψ∗| . (1.12)
Then the hyperbolic tangent profile (1.9) across the interface is restored by solving
ψτ +∇ · (ψ(1− ψ)n) = ǫ∇ · ((∇ψ · n)n) (1.13)
to a steady state solution.
The conservative level set method has the advantage over the standard level set method
in that it admits improved volume conserving solutions while at the same time preserving
the simplicity of implementation, e.g. no complicated geometrical operations needed as with
the VOF method. On the other hand, stiffness of the CLS function ψ introduces greater
numerical errors than the errors encountered when numerically integrating the original LS
function φ. We investigate CLS numerical error in Chapter 3.
1.2.2 Dynamic, stretched, curvilinear grids
One way to have smaller numerical error in interface position, and to save computational
time, is to use a dynamic, stretched, curvilinear mesh [30, 31, 32], which concentrates
resolution near the zero level set of the level set function φ (or, e.g., the 1/2 contour of
the conservative level set function ψ) without increasing the total number of grid points.
A moving mesh method has been implemented for solving two-phase flows in which the
interface is represented and updated using the phase field approach [26]. We are not aware
of a dynamic curvilinear mesh implementation using the level set method. We remark that
the phase field approach [26] is similar to the conservative level set method; the interfacial
thickness parameter ǫ for a phase field approach is on the order of α√h in contrast to the
interfacial thickness used for the conservative level set method: αh. The resulting order of
accuracy for phase field methods, given this interfacial thickness, is still first order accurate
using the same arguments that we used to analyze the order of accuracy of the conservative
level set method (see Chapter 3).
The level set method has been implemented on a static curvilinear mesh for simulating
two-phase flows [11]. Also, the level set method has been implemented on dynamic structured
and unstructured grids generated using adaptive mesh refinement (AMR) for simulating two-
phase flows [33, 3, 4, 34, 35]. In contrast to dynamic, stretched, curvilinear grid generation
10
in which the total number of grid points are fixed (the target of our research), the total
number of grid points is not fixed for AMR implementations.
We remark that a hybrid block-AMR on a static curvilinear grid was implemented for
magnetohydrodynamics applications in [36] and for compressible multi-fluid flow applications
by [37].
1.2.3 Overview of level set front tracking algorithm on dynamic,
stretched curvilinear grids; Overview of new developments.
In the following algorithm we give an outline of the Eulerian, level set front tracking
algorithm (for a passively advected interface, with solenoidal interface velocity u) on dynamic
curvilinear grids, that has been developed for this thesis.
1. Given the time step ∆t, level set function φn, velocity u, and grid node positions xn.
2. Compute new grid node positions based on the location and curvature of the zero levelset
of φn: xn+1. Define dx = xn+1 − xn.
3. Remap φn(xn+1) = Iφn(xn + dx).
4. upredict = u(xn+1).
5. ucorrect = Iupredict(xn+1 −
t2
upredict).
6. Advect the LS function: φ∗
n+1 = Iφn(xn+1 − ucorrectt).
7. Reinitialize LS: φτ + S(φ)(1 − |∇φ|) = 0, τ → ∞, to the steady state φ∗∗
n+1, taking φ∗
n+1
as an initial condition.
8. Correct volume φn+1 = φ∗∗
n+1− (V0−Vn+1)/Pn+1, where Pn+1 and Vn+1 are the length and
the volume of the zero level of φ∗∗
n+1.
What is unique about this research is that we explore level set front tracking on dynamic,
stretched, curvilinear grids. In particular, we take into consideration (1) order of accuracy,
(2) volume preservation characteristics, and (3) robustness for our newly developed front
tracking algorithm, in the context of dynamic, stretched, curvilinear grids.
These are the highlights of our newly developed level set algorithm that will be discussed
and analyzed in the ensuing chapters of this thesis:
11
• An elliptic PDE grid generation algorithm has been developed that increases resolution
in the vicinity of the zero level set of φ and further increases grid resolution in regions
of large curvature of the zero level set of φ. The grid generation algorithm and the
introduction of unique monitor functions are described in Chapter 2.
• In order to avoid CFL stability constraints associated with skewed elements generated
by our stretched grid algorithm, and in order to preserve an overall second order
accuracy, a semi-Lagrangian approach is used for level set advection. I is a newly
developed, efficient, tunnel quasi-cubic interpolation method, for curvilinear grids,
described in Chapter 5.
• The remap step in our new algorithm uses the same interpolation algorithm as for the
semi-Lagrangian advection step. Therefore, the remap algorithm preserves an overall
second order of accuracy.
• Our new algorithm is trivially “straight line preserving”; a property that does not
hold for either phase field methods or conservative level set methods. We analyze the
accuracy of conservative level set methods in Chapter 3 which motivates why we have
implemented a level set method together with a global mass constraint instead.
• Our new algorithm is trivially volume preserving.
• We have tested our new algorithm on the problem of a rotating notched disk (Zalesak’s
problem). We note that this test problem involves a non-deforming (distance function
preserving) velocity field.
• A new benchmark problem is devised based on a deforming velocity field (non distance
function preserving) that enables one to faithfully test the order of accuracy of the
reinitialization algorithm.
• More details of our level set algorithm are presented in Chapter 4.
• A summary of this research, and discussions of future work are presented in Chapter 6.
12
CHAPTER 2
DYNAMIC STRETCHED CURVILINEAR GRIDS
There are two main choices of grid geometry used for 2-dimensional numerical problems
in fluid dynamics: triangular and quadrilateral. The choice of the grid geometry is
application dependent. For example, a triangular grid (Fig.2.1) is convenient if it’s
necessary to have a body fitted grid with a geometry that does not undergo topological
transformations; triangular grids allow for easy cell adaptation. In contrast to a triangular
(often unstructured) grid, a structured quadrilateral grid lends itself well to multigrid
acceleration and trivial domain decomposition for implementation on parallel computers. If
a grid is not required to be body-fitted, or if the geometry is too complex, then a structured
quadrilateral grid is a viable option.
There are two main approaches for how to generate a quadrilateral grid. One approach is
based on adaptive mesh refinement (AMR); the number of grid points is not fixed when using
AMR implementations. The other approach is based on generating an optimal, stretched,
dynamic, curvilinear mesh; keeping the number of grid points fixed.
Within the AMR classification, there have been developed block structured adaptive
mesh refinement (AMR) methods (Fig.2.2(a)) [36] and quadtree or oct-tree based methods
[38]. Often, in AMR implementations using quadrilateral elements, a uniform orthogonal
grid is introduced in a domain as the coarsest grid layer. Next a refinement layer is generated
by dividing some cells into four equal cells (8 equal cells in 3D). This process is repeated
until the desired resolution is reached in the region of interest. In time dependent problems
adaptivity can be obtained by dynamically adding and removing grid layers.
Another way to generate a dynamic grid is to keep the number of grid nodes fixed and
to redistribute these nodes to achieve optimal grid configuration trying to minimize or at
least to bound error estimation of a numerical solution (Fig.2.2(b)). A non-degenerate grid
13
Figure 2.1: Triangular non-structured body-fitted grid.(http://www.cs.cornell.edu/home/vavasis/qmg2.0/example.html)
(a) Block-structured adaptive mesh refine-ment.
(b) Stretched structured quadrilateral grid.
Figure 2.2: Quadrilateral grids.
mapping x(ξ, ν), y(ξ, ν) is derived by attempting to minimizing a functional that is based on
an a-priori error estimate of the level set equation. This functional can be chosen to generate
14
orthogonal or quasi-orthogonal grids, to control node spacing, grid skewness, and isotropy.
A detailed overview of numerical grid generation in a general domain is given in [39].
In this research we focus on controlling node spacing by solving an elliptic equation in
order to generate a stretched curvilinear grid in a rectangular domain. We use dynamic
structured (number of grid points is fixed) quadrilateral non-body-fitted grids in the context
of a moving interface. The essence of this problem suggests the use of a dynamic grid
refined near an interface between two fluids where material properties of flow like density
and viscosity are discontinuous.
2.1 Variational grid generation
Lets consider a 2-d physical domain ~X = (x, y) ∈ Ω ∈ ℜ2 and a corresponding computational
domain ~ξ = (ξ, η) ∈ Ξ = [0, 1]2. Assume we know some error estimation of a numerical
solution in the domain (x, y) ∈ Ω ∈ ℜ2 where we solve numerically a two phase flow problem.
To find a grid which uniformly distributes the error is equivalent to minimizing the error
in L1 norm [40]. We assign a monitor function σ(x, y) > 0 based on the numerical error
estimation and solve the following optimization problem for ~X(~ξ)
∫
Ξ
(
∇ ~X(~ξ))T
M( ~X)∇ ~X(~ξ)d~ξ → min, (2.1)
where M(x, y) is a diagonal 2× 2 matrix with a given monitor function on its diagonal
M(x, y) =
[
σ(x, y) 00 σ(x, y)
]
.
A solution of the unconstrained optimization problem is given by corresponding Euler-
Lagrange equation
∇ · (M(−→X )∇−→X (
−→ξ )) = 0.
A solution−→X (−→ξ ) of this nonlinear Laplace equation minimizes the functional (2.1). If σ(x, y)
is a continuous smooth nonnegative function the solution of the given minimization problem
provides a one-to-one mapping from (ξ, η) to (x, y) [41]. The following system of nonlinear
equations
· (M(x, y) x(ξ, η)) = 0,
· (M(x, y) y(ξ, η)) = 0(2.2)
is to be solved numerically to generate a stretched 2-d grid.
15
To get a simple intuition about these equations (2.2) lets consider a 1-dimensional
equation for x(ξ) with a monitor function depending on x only. Then we have
(σ(x)xξ)ξ = 0⇒ σ(x)xξ = C.
Bigger error estimate σ requires smaller grid size xξ. Constant C is needed to scale the
generated grid to fit it into a necessary interval in x.
2.2 Stretched quadrilateral grid
Discretization of the first equation for x in the nonlinear system (2.2) is illustrated on Figure
2.3. (xi,j, yi,j), i = 1 : N, j = 1 : M are coordinates of grid nodes. As an example we take
discretization around (2, 2) node (shown as a bold blue point). The monitor function σ(x, y)
is evaluated on the cell faces σ3/2,2, σ5/2,2, σ2,3/2, σ2,5/2, these points are indicated with red
square markers on solid grid lines on Figure 2.3. Central difference discretization of the first
equation for x in (2.2) at the node (2, 2) gives
(x3,2 − x2,2)σ5/2,2 − (x2,2 − x1,2)σ3/2,2 + (x2,3 − x2,2)σ2,5/2 − (x2,2 − x2,1)σ2,3/2 = 0,
−x2,2(σ3/2,2 + σ5/2,2 + σ2,3/2 + σ2,5/2) + x1,2σ3/2,2 + x3,2σ5/2,2 + x2,1σ2,3/2 + x2,3σ2,5/2 = 0.
Boundary conditions assign grid lines to be orthogonal to domain boundaries. Central
difference discretization of the nonlinear Laplace equations (2.2) together with appropriate
Neumann and Dirichlet boundary conditions gives the following nonlinear algebraic system
to solve for xi,j and yi,j:
xi,jΣi,j − xi−1,jσi−1/2,j − xi+1,jσi+1/2,j − xi,j−1σi,j−1/2 − xi,j+1σi,j+1/2 = 0,
yi,jΣi,j − yi−1,jσi−1/2,j − yi+1,jσi+1/2,j − yi,j−1σi,j−1/2 − yi,j+1σi,j+1/2 = 0,
for i = 2 : N − 1, j = 2 : M − 1,
(2.3)
16
Figure 2.3: Stretched quadrilateral grid generation: discretization of nonlinear Laplaceequations (2.2).
where Σi,j = σi−1/2,j + σi+1/2,j + σi,j−1/2 + σi,j+1/2. Boundary conditions are
BC
(xi,1 − xi,2)σi,3/2 = 0, i = 1 : N,
(xi,M − xi,M−1)σi,M−1/2 = 0, i = 1 : N,
x1,j = xleft,
xN,j = xright,
(y1,j − y2,j)σ3/2,j =, j = 1 : M,
(yN,j − yN−1,j)σN−1/2,j =, j = 1 : M,
yi,1 = ylow,
yi,M = yup.
(2.4)
Since the function σ(x, y) in general depends on x and y, this system of equations (2.3) is
nonlinear. The matrix in the algebraic system (2.3) is 5-diagonal symmetric positive definite.
This comes from the fact that σ(x, y) > 0, on each row (and column) the diagonal element is
equal to the sum of off-diagonal elements and we have Dirichlet boundary conditions on the
left and right side for x and on the top and bottom for y. All eigenvalues for Laplace type
discretized equations of this sort are strictly positive, consequently the matrix is positive
definite. Hence the system has a unique solution.
To solve the nonlinear system of algebraic equations (2.3), (2.4) we used fixed-point outer
iterations and SSOR-PCG method as inner iterations. We point out that this could not be
the fastest method to obtain the solution. One can use any nonlinear CG method, BFGS,
17
Newton-SOR or SOR-Newton, CG-Newton, Newton-CG and so on. One can use Jacobian
free iterative methods [42] for solving nonlinear systems of equations which is more general
in application (doesn’t require a positive definite matrix). Since we have a nice 5-diagonal
symmetric positive definite matrix on each inner iteration, any linear algebraic system solver
can be used to solve Ax = b, Ay = c in inner iterations. We decided to use fixed-point/SSOR-
PCG, because it’s simple in implementation and fast enough for the purpose of this research.
Here is a fixed point/SSOR preconditioned CG algorithm we used. In shorthand
notations we have A(x, y)x = b and A(x, y)y = c, where unknown vectors x and y of
size 1 × NM represent coordinates of naturally ordered grid nodes (Fig.2.3) (xi,j, yi,j),
i = 1 : N, j = 1 : M .
Algorithm
Given an initial guess (x0i,j, y
0i,j) of grid nodes and A0 = A(x0, y0), set k = 0.
1. Solve for xk+1, yk+1 two linear algebraic systems Akxk+1 = a, Akyk+1 = b with a
symmetric positive definite matrix Ak (SSOR precondition linear CG method).
2. Update Ak+1 on a new grid (tunnel quasi-cubic interpolation of σ(x, y)).
3. If a residual (|Ak+1xk+1− b|2 + |Ak+1yk+1− c|2) > tolerance and k < kmax set k = k+1
and go to step 1.
On Figure 2.3 a new position on the node (2, 2) is shown after step 1 as an intersection
of new grid (dashed blue) lines. To update matrix A on a new grid we interpolate a monitor
function σ at the points marked with red squares on new (dashed) grid lines . A bilinear,
bicubic or quasi-cubic interpolation on a quadrilateral grid can be used at this step. We used
a tunnel quasi-cubic interpolation proposed in this research and explained in Chapter 5.
2.3 Rectilinear grid
One way to accelerate the grid generation process is to assume that grid stretching occurs
in x and y directions separately.
To generate a 2-d rectilinear (orthogonal) structured grid the system of equations (2.2)
is simplified after setting grid lines orthogonality conditions xη = 0, yξ = 0, and taking
K(x) = maxyσ(x, y), L(y) = max
xσ(x, y) as monitor functions. We are to solve two nonlinear
ODEs
18
(a) 40 × 40 grid. A=10, B=0, k=0.025 inthe monitor function (2.9).
(b) 30× 30 grid. A=5, B=0.5, k=0.025 inthe monitor function (2.9).
Figure 2.4: A stretched curvilinear quadrilateral grid.
dK(x)dx(ξ)
dξ2= 0, (2.5a)
dL(y)dy(η)
dη2= 0. (2.5b)
Discretization of the nonlinear ODE (2.5a) for generating x-grid lines (vertical black) is
illustrated on Figure 2.5. Central difference discretization of (2.5a) around the x-grid line
number 2 (see Fig. 2.5) gives
(x3 − x2)K5/2 − (x2 − x1)K3/2 = 0,
−x2(K5/2 +K3/2) + x3K5/2 + x1K3/2 = 0.
Here a monitor function K3/2 = K(x3/2) = K(x1 + x2
2), K5/2 = K(x5/2) = K(
x2 + x3
2) is
evaluated on vertical dashed lines numbered 3/2 and 5/2 on the Fig.2.5.
Central difference discretization of the nonlinear ordinary differential equations (2.5)
together with appropriate Dirichlet boundary conditions gives the following two nonlinear
algebraic systems with symmetric positive definite 3-diagonal matrices to solve for xi and
yj.
xi(Ki−1/2 +Ki+1/2)− xi−1Ki−1/2 − xi+1Ki+1/2 = 0, for , i = 2 : N − 1,
x1 = xleft,
xN = xright,
(2.6)
19
Figure 2.5: Rectilinear grid generation: discretization of a nonlinear differential equation(2.5a).
yj(Lj−1/2 + Lj+1/2)− yj−1Lj−1/2 − yj+1Lj+1/2 = 0, for , j = 2 : M − 1,
y1 = ylow,
yM = yup.
(2.7)
Here (xi, yj), i = 1 : N, j = 1 : M are coordinates of grid nodes. The monitor functions K(x)
and L(y) are evaluated between grid nodes and in general depend on x and y accordingly,
which makes these two algebraic systems nonlinear. In shorthand, we are to solve two
independent nonlinear algebraic systems A(x)x = a and B(y)y = b, where matrices A(x)
and B(y) are 3-diagonal symmetric positive definite (sinceK(x) and L(y) are strictly positive
functions). These systems can be solved by nonlinear CG methods, any Newton method.
We solve them by fixed point/Thomas algorithm as follows.
Algorithm
Given an initial guess x0i of grid nodes x-coordinates, A0 = A(x0), set k = 0.
1. Solve for xk+1 a linear algebraic systems Akxk+1 = a with exact Thomas solver.
20
(a) Circular interface (b) Elliptic interface
Figure 2.6: 2-d rectilinear grid.
2. Interpolate a monitor function K(xk) on a new x-grid K(xk+1) (update a coefficient
matrix Ak+1).
3. If a residual (|Ak+1xk+1 − a|2) > tolerance and k < kmax set k = k + 1 and go to step
1.
Step 2 in this algorithm is done as follows. Given a monitor function Kki+1/2, i = 1, N −1
on an old k-th grid (black lines), we find a monitor function Kk+1s+1/2, s = 1, N − 1 on blue
dashed lines associated with a new (k + 1)-st grid (see Fig.2.5),
Kk+1s+1/2 =
Kki+3/2(x
ki+1/2 − xk+1
s+1/2) +Kki+1/2(x
k+1s+1/2 − xk
i+3/2)
xki+1/2 − xk
i+3/2
, if xk+1s+1/2 ∈ (xk
i+1/2, xki+3/2],
Kk3/2, if xk+1
s+1/2 ≤ xk3/2,
KkN−1/2, if xk+1
s+1/2 > xkN−1/2.
An analogous algorithm is used to find y grid lines.
2.4 Monitor functions
A monitor function is a crucial part in the generation of dynamic, stretched, curvilinear
grid generation. It gives some measure of local solution activity and allows grid generation
to concentrate grid nodes in these regions aiming to equidistribute (or at least to bound)
numerical error.
21
One of popular choices of a monitor function is σ(x, y) =√
1 + A| f(x, y)|2, where f is
a solution of a given problem, user defined parameter A controls concentration of generated
grid nodes adapted in regions of high solution variation. As was shown in [43] this choice
of a monitor function corresponds to equidistribution of error of a piece-wise linear solution
interpolation. In [44] vorticity of an incompressible viscous flow is taken in place of f ,
parameter A = 0.5. Often velocity U is considered to serve as a singularity measure of
a solution of Navier-Stokes equations. Another choice of a solution variation is weighted
average of rate of change and divergence of velocity, i.e. a monitor function is in the form
σ(x, y) = 1 +A| ·U(x, y)|+B|U(x, y)|.In the context of interface capturing in a two-phase flow, a grid has to be refined near the
interface because in two-phase flow one will have discontinuous physical properties across
the interface between the two fluids. In [45] the following simple adaptive Cartesian grid
refinement criteria was used: ”split any cell whose edge length exceeds its distance to the
interface”. Only near an interface this condition is satisfied, where distance to the interface
is small and cell edge length can exceed the distance to the interface, then we split the cell.
A level set method of the second order convergence was reported in the paper.
We propose new monitor functions for moving interface problems. On the figure 2.4 two
different monitor functions σ(x, y) are used to generate a stretched grid. A monitor function
σ(x, y) = 1 +Ae−d(x,y)2
4k , (2.8)
generates a normal distribution of grid cells depending on distance d(x, y) to the interface
(see figure 2.4(a)). A monitor function
σ(x, y) = 1 +
(
A+B|d(x, y)|e− d(x,y)2
4k
max |d(x, y)|
)
e−d(x,y)2
4k (2.9)
in addition to a normal distance distribution puts smaller grid cells where an interface has
a more radical curvature (see figure 2.4(b)). In the functions (2.8), (2.9) user prescribed
constants A and B control the relative size of cells. If we evaluate σ(x, y) on the interface
then distance d = 0, exponent is equal to one. The monitor function (2.8) is equal to A+ 1.
If we are far away from the interface then d is big, exponent goes to zero, the monitor
function goes to 1. So we get A+ 1 times finer grid near the interface then further from the
interface. User prescribed parameter k sets a width of a band near an interface where we
22
want to refine a grid. Outside this band the monitor functions approximately equal to one,
a grid is almost uniform. The monitor function (2.9) in addition to setting smaller cells near
the interface, picks up information about interface curvature (the term with B) and after
normalizing it uses this information for additional grid refinement where the interface has a
bigger curvature.
For orthogonal rectilinear grid generation according to (2.5) we need monitor functions
depending on one variable. For x grid lines the monitor function we suggest is
K(x) = maxyσ(x, y). (2.10)
For y grid lines the monitor function is
L(y) = maxx
σ(x, y). (2.11)
On the Figure 2.6 we show examples of two rectilinear grids generated based on (2.5) grid
generation equation with (2.10) and (2.11) monitor functions.
The two choices of a monitor function (2.8) and (2.9) are purely heuristic. To have
more justified grid adaptation one can use the following reasonings to derive a new monitor
function.
Assume we are interested in grid adaptation near a moving interface. We take the base
of the monitor function in the form
σ(x, y) = 1 + (A+ E(x, y))e−d(x,y)2
4k . (2.12)
It generates grid cells on an interface A+1 times finer than grid cells farther from an interface.
In addition, if we want to optimize a numerical error along an interface we consider an
additional refining coefficient E(x, y) which represents a local truncation error of a numerical
scheme solving interface advection.
For example, lets consider the LS advection equation (1.6) discretized with the explicit
two-stage Runge-Kutta method in time and the finite difference second order upwind gradient
in space. For simplicity lets consider a uniform orthogonal grid, a circular interface centered
23
at (0, 0) and a constant velocity field U = (u, v). Then we have
φn+2/3i,j = φn
i,j −2
3t(
uDx(φni,j) + vDy(φ
ni,j))
,
φn+1i,j = φn
i,j −ut
4
[
3Dx(φn+2/3i,j ) + Dx(φ
ni,j)]
− vt4
[
3Dy(φn+2/3i,j ) + Dy(φ
ni,j)]
,
where Dx(φi) = (3φi − 4φi−1 + φi−2)/2x, if u > 0,
Dx(φi) = −(3φi − 4φi+1 + φi+2)/2x, if u < 0.
(2.13)
If a LS function is a distance function then a local truncation error of this second order
method is
LTE(x, y) ∼ t3
6φttt +tx2uφxxx
(
1− 2tvy
)
+ty2vφyyy
(
1− 2tux
)
= −(u3φxxx + v3φyyy + 3uv2φxyy + 3u2vφxxy)
+tx2uφxxx
(
1− 2tvy
)
+ty2vφyyy
(
1− 2tux
)
= − t32(x2 + y2)5/2
(uy − vx)2(ux+ vy)
−3tx2uxy2
(x2 + y2)5/2
(
1− 2tvy
)
− 3ty2vyx2
(x2 + y2)5/2
(
1− 2tux
)
=
= −1
2t3(φ)2|U−U · ∇φ|2(U · ∇φ)
−3t(φ)2
[
x2uφxφ2y
(
1− 2tyv
)
+y2vφyφ2x
(
1− 2txu
)]
,
(2.14)
where ∇φ = (φx, φy) is a gradient, φ = φxx + φyy = κ = 1/(x2 + y2) is a curvature of level
contours.
If an interface is a three times differentiable closed contour and LS is a signed distance
function, then locally at any point (x, y) contours of this LS function can be seen as a segment
of a circle with a radius R(x, y) = 1/φ(x, y). Taking into account that θ is an angle between
velocity U and a normal n = φ to the interface we have u cos θ + v sin θ = U · n = U · φand u sin θ − v cos θ = U · s = U− (U · φ) φ, where s⊥n. We can use the same formula
as a local truncation error of the numerical scheme for a generally defined interface
LTE(x, y) ∼ −1
2t3(φ)2|U−U · ∇φ|2(U · ∇φ)
−3t(φ)2
[
x2uφxφ2y
(
1− 2tyv
)
+y2vφyφ2x
(
1− 2txu
)]
.(2.15)
This way we obtain the coefficient E(x, y) for the monitor function (2.12), i.e.
σ(x, y) = 1 + (A+ E(x, y))e−d(x,y)2
4k , (2.16)
24
where
E(x, y) = B|LTE(x, y)|e− d(x,y)2
4k
max |LTE(x, y)| . (2.17)
Note, that a LS function fails to have ∇φ and φ at some points. This fact should not
influence the proposed numerical grid generation because if an interface contour is three
times differentiable (an assumption made to derive a second order numerical scheme) then
there exists a neighborhood near the interface where ∇φ and φ are well defined. Since
Gaussian function e−d2
4k across the interface goes to zero exponentially fast farther from the
interface, we need ∇φ and φ only in a narrow band of width about 4k near the interface.
25
CHAPTER 3
INTERFACE CAPTURING WITH CLS METHOD
The CLS method was implemented on an adaptive unstructured triangular grid in [13].
We are not aware of LS or CLS methods implemented on a dynamic, stretched, structured
quadrilateral grid. The CLS method is very promising, but it can be shown that the CLS
method does not preserve straight lines, and we show in Section 3.6 that the LTE of the
advection equation in this method is of the order O(1). Despite all the weak points of the
CLS method, we tried to improve it in order to achieve second order convergence. It’s still a
topic of investigation. In this Chapter some improvements of the CLS method are presented
which can be beneficial for further CLS developments. We propose a new conservative CLS
remapping algorithm and new CLS reinitialization equation. These two developments allow
one to implement the CLS method on a dynamic, stretched, quadrilateral grid, albeit, not
with the same accuracy as a level set implementation, and not yet straight line preserving.
3.1 CLS algorithm
Here is an algorithm of interface advection using a CLS method on a dynamic structured
quadrilateral grid.
Lets assume we are given a computational domain Ω with a divergence free velocity field
U = U(x, t),x ∈ Ω. The velocity field could be described by two phase flow Navier-Stokes
equations, or it could be given as a function of space variables and time. The initial interface
position is given by 0.5 level of the CLS function ψ0(x) = 11+ed0(x)/ǫ , where d0(x) is a signed
distance to the interface at the initial time, positive inside gas and negative inside liquid.
Here is an algorithm of interface advection using a CLS method on a dynamic curvilinear
grid.
Algorithm
26
1. Generate a grid adapted near current interface position.
2. Remap ψn(x) onto a new grid in a conservative way.
3. Find ψ∗
n+1(x) solving the equation (1.11) with a conservative numerical scheme.
4. Find normals (1.12). Taking ψ∗
n+1(x) as an initial condition solve (1.13) to steady
state. It gives ψn+1(x).
5. Increment time and go to step 1.
The rest of this Chapter is dedicated to description of a semi-Lagrangian advection scheme
to solve the CLS advection equation (1.11), CLS reinitialization proposed by Olsson [13],
new CLS remapping procedure and newly proposed CLS reinitialization equation which is
theoretically more substantiated. Local truncation error of an advection step is derived.
3.2 CLS advection
We suggest the following conservative second order semi-Lagrangian finite volume scheme
for discretization of the CLS advection equation (1.11)
ψn+1i,j = ψn
i,j −tVi,j
∑
L(Un+1/2 · n)ψn+1/2
, (3.1)
where ψni,j is an averaged value (see Fig.3.1) of the function over the cell of volume Vi,j,
∑
is
taken over the (i, j)-th cell faces with corresponding outward normals n and lengths L, the
velocity Un+1/2 and the function ψn+1/2 are evaluated on the cell faces. The midpoint rule
in time is done by utilizing cubic or quasi-cubic upwind interpolation along characteristics
ψn+1/2(x) = ψn(x−Unt2
).
See an efficient quasi-cubic tunnel interpolation algorithm on a general quadrilateral grid
proposed in this research in Chapter 5. If a velocity field is time dependent then a predictor-
corrector method is used to find Un+1/2 = 12
(
Un + Un+1predict
)
.
3.3 CLS reinitialization
In [13] the following CLS profile reinitialization equation was proposed,
ψτ +∇ · (ψ(1− ψ)n) = ǫ∇ · ((∇ψ · n)n), (3.2)
27
Figure 3.1: Illustration of a semi-Lagrangian finite volume scheme (3.1). ψni,j is cell averaged,
velocity Un+1/2 and ψn+1/2 are on cell faces. Outward normals to cell faces are n.
where normals n to the interface are calculated ones at the beginning and kept fixed during
the reinitialization process
n =
( ∇ψ|∇ψ|
)
τ=0
. (3.3)
The equation (3.2) is to be solved to a steady state solution and can be solved with any
explicit conservative scheme. The authors in [13] solve the equation semi-implicitly in
28
the context of the finite element method. We discretize (3.3) with a centered difference
approximation and find n on cell faces ones at τ = 0. Then we solve (3.2) with forward
Euler method in time and second order finite volume in space as follows.
ψm+1i,j = ψm
i,j −t1
V i,j
∑
L [ψ(1− ψ)− ǫ(∇ψ · n)] (n · n), (3.4)
where∑
is taken over all cell faces with corresponding outward normals n and lengths L.
The value ψmi,j of the CLS function is cell centered averaged over the (i, j)-th cell with volume
Vi,j.
3.4 CLS remapping
Conservative remapping could be done by piecewise polynomial function reconstruction at
each cell followed by grid intersection weighted conservative interpolation. If grids are close
to each other with the same connectivity then the swept-region method can be used [46], [22].
We propose another method which takes advantage of nice properties of the CLS function,
namely, the known shape of the profile of the CLS function across the interface (1.9).
Assume we have two grids (x, y)old and (x, y)new. The CLS function ψold is given at the
old cell centers. We want to find ψnew such that∫
Ω
ψolddxdy =∫
Ω
ψnewdxdy and retain the
second order of a numerical solution.
Algorithm
1. Calculate normals to the interface nold =ψold
| ψold| on the old grid.
2. Find nnew on a new grid using a second order 2-d interpolation (see efficient quasi-cubic
tunnel interpolation algorithm proposed in this research in Chapter 5).
3. Remap a CLS function onto a new grid using a fast inaccurate but conservative scheme.
For example, we equally distribute function mass φoldi,j V
oldi,j among all new recipient cells
which share area with the old donor cell. Here V denotes cell area. After this we have
mass ψ∗
i,jVnewi,j at the new grid cells. Division by V new
i,j gives intermediate CLS values
ψ∗
i,j at the new cell centers.
4. Take the intermediate ψ∗ as an initial condition and solve the CLS reinitialization
equation
ψτ +∇ · (ψ(1− ψ)nnew) = ǫ∇ · ((∇ψ · nnew)nnew)
29
using a conservative numerical scheme and normals nnew found in step 2. A steady
state solution gives ψnew.
Note that an integral of ψ over the physical domain is conserved because each step of this
algorithm is conservative. There is no need in finding cell faces intersections. Moreover, if two
grids are sufficiently close to each other, then step 3 can be simplified by performing simple
conservative injection φoldV old → φ∗V new. Step 4 will correct this rough approximation
because the most important part in this algorithm which defines the error of the algorithm is
normals calculation. For example, if cubic or quasi-cubic interpolation is used to interpolate
normals and a second order in space numerical scheme is used to find a steady state solution
or the CLS reinitialization equation then overall the proposed remapping algorithm is of the
second order.
Unfortunately, any remapping procedure can introduce spurious oscillations. In the case
of CLS remapping this oscillations can be very noticeable being a consequence of stiffness of
the CLS function. This fact drew our attention to the reinitialization procedure of the CLS
function proposed by Olsson et al. [13] and described in the subsection 3.3 in this paper.
We revisit CLS reinitialization procedure to explore possibility to smooth the profile and
diminish spurious oscillations associated with numerical advection and remapping of a stiff
function.
3.5 CLS reinitialization revisited
In [13] to reconstruct an exponential profile across the interface the following equation is
solved to a steady state solution,
ψτ +∇ · (ψ(1− ψ)n∗) = ǫ∇ · ((∇ψ · n∗)n∗). (3.5)
Here the normal n∗ = (∇ψ/|∇ψ|)τ=0 to the interface is calculated ones and is kept fixed
during the reinitialization process.
Since the CLS function is very steep across the interface, advection or remapping of the
function can introduce essential oscillations in ψ and in ψ as well. Before calculating n∗ we
have to smoothψ, otherwise n∗ can be point out in opposite directions in neighboring cells.
In [14] to diminish influence of spurious oscillations in ψ and ∇ψ, a distance LS function φ
was used to calculate normals to the interface n∗ =
( ∇φ|∇φ|
)
τ=0
in the CLS reinitialization
30
equation (1.13), where φ was reconstructed by taking inverse of the CLS function ψ on the
interface followed by fast marching redistancing method. The equations (1.11) and (1.13)
were discretized using a finite difference scheme. Good stability and volume conservation
properties were reported in the paper.
It’s much easier from a numerical point of view, to calculate normals to the interface from
a distance LS function which is linear across the interface as it’s done in [14], than from a
steep CLS function. So we reconstruct the signed distance function φ from the CLS function
ψ and then calculate normals n∗ =
( ∇φ|∇φ|
)
τ=0
from a distance LS function and use this
normals in the reinitialization (and remapping) process of the CLS function ψ. We must
underline the fact that a numerical steady state solution of the equation (3.5) in ℜd, d > 1 is
not unique for a given field of normal directions and the result depends on the initial guess
of ψ at the beginning of reinitialization and on the way we solve the equation (3.5). In fact,
this equation mimics behavior of an exponential profile in 1-d case ignoring the fact that we
deal with a surface in 2-d case and a higher order surface in 3-d.
To show that a numerical solution of (3.5) is not unique for a given field of normal
directions, lets consider a simple model of a line interface in a 2-d physical domain. We
know that the exponential profile of ψ can be very disturbed after advection or remapping.
Even a line interface in a context of a CLS function is advected numerically inexactly as
opposed to exact advection of a line with a LS function. Lets assume that line interface
position is disturbed by small perturbations (see Fig.3.2(a)). But assume that the field of
normal directions is known exactly, i.e. it’s constant in 2-d domain. After applying the CLS
reinitialization equation (3.5) we get a disturbed profile again (see Fig.3.2(b)). Changing the
initial perturbation we get other steady state solutions. The position of the interface will be
still disturbed because the equation (3.5) reconstructs an exponential profile in the normal
to the interface direction only and doesn’t “see“ what happens in the tangential direction.
But we know that it’s essential in 2 and 3-d case to restore a constant in the tangential
direction. This is the difference between a 1-dimensional CLS function and 2, 3-dimensional
ones.
Here is a new idea how to restore an exponential profile in the normal direction and get
constant contours in the tangential direction to the interface. We solve
ψτ +∇ · (ψ(1− ψ)n∗) = ǫ∇ · ((∇ψ · n∗)n∗ +∇ · (∇ψ · s∗)s∗) for τ →∞, (3.6)
31
(a) Disturbed CLS function for a line interface. (b) Reinitialized CLS function using original reini-tialization equation (3.5).
(c) Reinitialized CLS function using newly pro-posed reinitialization equation (3.6).
Figure 3.2: CLS reinitialization.
where n∗ =
( ∇φ|∇φ|
)
τ=0
is a normal to the interface and s∗ is a vector tangential to the
interface such that s∗ · n∗ = 0. This new reinitialization procedure is tested and proven
more stable and efficient for the CLS method. It finds numerically a unique CLS function
with a given constant function integral and it distributes the volume along the given normal
directions. The last term with s∗ corrects the profile in the tangential direction and ensures
uniqueness of the numerical solution. On the Figure 3.2(c) the CLS function is much
smoother than on Figure 3.2(b) although the same number of iterations were used in the
reinitialization process of the initially disturbed CLS profile shown on Figure 3.2(a).
32
3.6 LTE of CLS method
A local truncation error of the numerical scheme (3.1) is
LTE(ψ(x, y)) ∼ Cu
12(2 + C2u2)x3ψxxx. (3.7)
First of all, we see why a line interface is not advected exactly with a second order numerical
scheme as opposed to exact straight line advection with a signed distance LS function. It
happens because the third derivative of the CLS function is not equal to zero on the interface.
To see this lets consider a line interface placed at x0 and moving with constant speed, then
we have
ψxxx =1
ǫ3(
6ψ2 − 6ψ + 1) (
ψ2 − ψ)
, (3.8)
where ψ(x, y) = 1
1+expx−x0
ǫ
. If we draw ψxxxx3 as a function of distance to the line
interface (Fig.3.3) we see that the maximum error in any advection scheme of the second
order will be exactly on the line interface, at the point of inflection of ψ(x, y). Moreover,
LTE(ψǫ(x/2, y)) = LTE(ψǫ/2(x, y)) and ψǫ(x/2, y) = ψǫ/2(x, y). It means that error in the
0.5 contour, which is interface position, is of the order O(ǫ) = O(x).Now lets consider a simple case of a uniform orthogonal grid, a circular interface of
radius R0 centered at (0, 0) translated with a constant speed in the direction of x-axis, i.e.
U = (u, 0), u > 0 and t = Cx. The third derivative in x of the CLS function is equal to
ψxxx =(1− 4f + f 2)fx3
(1 + f)4ǫ3R3+
3f(f − 1)xy2
ǫ2(1 + f 2)R4+
3fxy2
ǫ(1 + f)2R5
=(1− 4f + f 2)f
(1 + f)4ǫ3d3
x +3f(f − 1)
ǫ2(1 + f 2)dxd
2yd+
3f
ǫ(1 + f)2dxd
2y(d)2,
(3.9)
where f(x, y) =1
1 + ed(x,y)
ǫ
, R =√
x2 + y2, dx = x/R, dy = y/R, d = 1/R with a signed
distance function d(x, y) =√
x2 + y2 − R0 positive inside one fluid and negative inside
another.
Since we can view a (three times differentiable) closed interface in (x, y) plane and a
signed distance function near such interface locally as a segment of a circle with a radius
R = 1/κ = 1/d, we can consider the LTE formula (3.7) with (3.9) as LTE of a generally
defined interface moving to the right. All error estimations and conclusions can be generalized
for advection of any sufficiently smooth interface.
33
Figure 3.3: ψxxxx3 as a function of a signed distance to the interface. ǫ = x and astraight line interface is considered. Maximum LTE= C0ψxxxx3 (3.7) is on the interface.
What do we see from the obtained error estimation? We can compare errors (2.14) and
(3.7) of the LS and CLS solutions after advecting the interface one time step. The LTE
(2.14) is simplified in the case of a circular interface moving to the right with constant speed
U = (u, 0) and t = λx
LTE(φ(x, y)) ∼ −(
1
2λ2u2 + 3
)
x3(φ)2uλφxφ2y. (3.10)
It’s interesting to note that LS solution is exact at θ = 0, θ = π and θ = ±π/2 whereas
CLS solution is exact at θ = ±π/2, where θ is an angle between velocity and a normal to
the interface. And finally, the most important observation is that LTE of the CLS function
is of the order O
(x3
ǫ3,x3
ǫ2R,x3
ǫR2
)
. To get good volume conservation and to capture an
interface sufficiently close we take ǫ = O(x) and x = O(R) [13]. This choice of ǫ gives
34
O(1) order of the LTE of CLS and O(x) order in interface position. It could be shown
that it happens not only in this particular choice of a numerical scheme. For any numerical
scheme and for the given choice of constants LTE convergence will not be observed with
the CLS method. We can hope for an order of LTE o(1) if we assign ǫ ∼ xk, k < 1 as
authors suggest [13]. In fact, LTE = O(x) of a CLS solution can be obtained if we put
ǫ ∼ x2/3, x ∼ R3/2. We are faced with a trade off, better conservation leads to bad
interface position and vise versa. In [13] and [47] the second order convergence in interface
position in numerical experiments is reported and in [13] nonconvergent cases are mentioned
with the choice of ǫ = O(x). The theoretical derivations above show that if we choose
the CLS method with ǫ = O(x) for nice conservation then there is a little hope to get
convergence rate higher than one with any numerical scheme.
The main goal of this research is to implement an interface capturing algorithm on a
dynamically, stretched, curvilinear, grid with appropriate accuracy properties. At the very
least, an Eulerian front tracking method should be “straight line preserving.” We have not
been able to figure out how to improve the CLS method in order to satisfy the “straight
line preserving” property. Therefore, we have abandoned the CLS method in favor of a LS
method together with the implementation of a global mass constraint.
35
CHAPTER 4
LS WITH GLOBAL VOLUME FIX ON A
DYNAMIC, STRETCHED GRID
Based on our analysis of the error associated with the CLS method (see Chapter 3), we have
decided to approach interface tracking on a dynamic curvilinear grid using the LS method.
In order to overcome the volume preservation errors associated with the LS method, we have
decided to implement a global mass constraint. The decision to implement a global mass
constraint is based on the following reasons:
• methods that reduce mass error by adding more grid resolution near the interface (e.g.
particle level set methods [15] or marker level set methods [16]) only effect the rate of
mass loss; these methods do not help if it is important to preserve mass over a long
period of time.
• The coupled level set and volume of fluid method [12] possesses excellent mass preser-
vation properties (over long periods of time) but this method would be prohibitively
difficult to implement on a dynamic, stretched, curvilinear grid.
• Concurrently, a method for consistently enforcing a global mass constraint in the
presence of many droplets or bubbles that merge or break-up is being developed.
In [19], [29], [20] simple global volume fix procedures were applied to the original LS
method in order to keep the volume of a bubble conserved.
These volume fix methods are the most attractive due to their unbeatable simplicity.
We have implemented a LS method together with a simple global volume fix on a dynamic,
stretched, quadrilateral grid. One can rightly object that a global volume fix redistributes
mass incorrectly uniformly “blowing up” a bubble to restore its initial volume. On the other
36
hand, if a level set advection scheme is accurate enough, then the mass to be redistributed at
each time step will be very small, and will not effect the overall dynamics. The goal of this
research is to implement a volume conserving, accurate, simple interface tracking method
on a dynamic, stretched, quadrilateral grid. Accuracy in capturing an interface especially in
the regions where an interface has large curvature, a potentially dangerous region for mass
loss, will be achieved by reducing cell size in these regions. The monitor functions that we
proposed in Chapter 2 are tuned to produce higher resolution grids in interfacial regions
that possess large curvature. In addition to exact volume conservation, our approach, LS
together with global volume fix, exactly advects a straight line front. Our approach has
the “straight line preserving” property regardless of whether we compute on uniform static
grids, or dynamic, stretched curvilinear grids. Our algorithm for advection and remapping
uses cubic interpolation which is exact for linear and quadratic functions. We could have
used linear interpolation too, albeit, linear interpolation would reduce the performance of
our algorithm (we have found that linear interpolation results in an overly diffusive method).
We remark that since a signed distance LS function is linear and is advected and remapped
exactly, a volume constraint is unnecessary for a moving straight line interface. An overall
second order discretization of the LS method is chosen based on the observation that a LS
function is continuous but can have a discontinuous first derivative. Higher order schemes will
be reduced to lower order anyway at regions in which the level set function has discontinuous
derivatives.
4.1 Algorithm for LS with global volume fix on adynamic, stretched, grid
We suggest the following algorithm. We are given an initial grid x0, a cell centered signed
distance LS function φ0(x0) and an initial volume V0 inside φ0 < 0 at time t = 0. A cell
centered divergence free velocity field U is given, for example, by Navier-Stokes equations
solution. Set n = 0, Tgrid = 0.
1. If Tgrid < 10 then set xn+1 = xn and go to step 4.
2. Generate a new adaptive grid xn+1, dx = xn+1 − xn, set Tgrid = 0.
3. Remap φn(xn+1) = Iφn(xn + dx) (I is the tunnel quasi-cubic interpolation proposed
in Chapter 5).
37
4. Find Upredict = U(xn+1) (via a prescribed velocity field formula or Navier-Stokes
solution).
5. Find Ucorrect = IUpredict(xn+1−t2
Upredict) (I is the tunnel quasi-cubic interpolation
proposed in Chapter 5).
6. Advect the LS function: φ∗
n+1 = Iφn(xn+1 −Ucorrectt), (I is the tunnel quasi-cubic
interpolation proposed in Chapter 5).
7. Reinitialize LS: φτ +S(φ)(1−|∇φ|) = 0, τ →∞, to the steady state φ∗∗
n+1, taking φ∗
n+1
as an initial condition (see Section 4.2).
8. Correct volume φn+1 = φ∗∗
n+1 − (V0 − Vn+1)/Pn+1, where Pn+1 and Vn+1 are the length
and the volume of the zero level of φ∗∗
n+1 (see Section 4.3).
9. Increment n and Tgrid and go to step 1.
On a stationary grid we omit steps 1-3. On a dynamic grid we generate a new grid on
every 10-th time step. Algorithms for generating structured rectilinear and quadrilateral
grids that have high grid resolution near the interface are explained in the Chapter 2.
On a dynamic, stretched, grid we have tried to combine steps 3-6 in the form φn+1(xn+1) =
Iφn(xn−Unt+dx). But this combined remapping/advection step reduces the overall order
of convergence of the above algorithm on a dynamic, stretched, grid to first order.
Remapping of a cell-centered LS function in step 3 is done by utilizing a quasi-cubic
interpolation on a curvilinear grid. On Figure 4.1 solid black grid lines correspond to the
n-th old grid, dashed blue grid lines show the newly generated (n+ 1)-st grid. For example,
the new LS function value φn+1 at the cell center indicated with a blue square marker is
found with a quasi-cubic interpolation (see Chapter 5) on a 12-point stencil indicated with
red dots. In the boundary cells bi-linear interpolation is used which is done on a 4-point
stencil.
In steps 5-6 we move an interface by solving an advection equation with a space variable
velocity field φt + U(x) · φ = 0 in the formdφ
dt= 0 using the following semi-Lagrangian
scheme. The characteristics equation for this problem isdx
dt= U(x). We discretize the
characteristics equation asx1 − x0
t = U( x1/2) ⇒ x0 = x1 −tU( x1/2), where x1 = xn+1
38
Figure 4.1: Cell-centered LS function remapping. Old grid lines are solid black. New gridlines are dashed blue. An interpolated point is at the new cell center marked with a bluesquare. A quasi-cubic interpolating stencil for this point is shown with red dots.
in the algorithm above. To find a mid-point velocity U( x1/2) we use a predicted velocity
U( x1) = Upredict to find x1/2 = x1 − t2
U( x1) first (see Fig. 4.2). Then we ”correct” the
velocity using a quasi-cubic interpolation U(x1/2) = Ucorrect = IUpredict(x1−t2
Upredict) (see
Chapter 5 for an interpolation algorithm). An approximate solution of the advection equation
then becomes φ(x1, tn+1) ≈ φ(x0, tn) ≈ φn(x1−tU( x1/2)) = φn(xn+1−tUcorrect). What
we do on each time step is we find an upwind value of a LS function in the direction of a
mid-point velocity (see Figure 4.2).
To demonstrate the importance of the redistancing step 7 and the volume correction
step 8, we run the algorithm above without these steps. As an example we consider a
smooth circular interface in a deforming divergence free velocity field (this new test problem
is explained in details in Section 4.4). On Figure 4.3 we see that with time a distance LS
39
Figure 4.2: A semi-Lagrangian scheme solving the LS advection equationdφ
dt= 0.
φn+1(x1) = φn(x0) = φn(x1 −tUcorrect), where Ucorrect = U(x1 − t
2Upredict).
function is deformed which makes big errors in the advection step 6. As the result a part
of an interface is pinched of (see a blue contour on Figure 4.3(b)). Level curves on this
pictures are getting close to each other, which indicates that gradient of the LS function
is big. Compare the same test problem but with redistancing and volume fix applied on
each time step, see Figure 4.4. The interface is smooth throughout the trip, LS gradient is
uniform.
Figure 4.5 shows volume change inside the zero level with and without redistancing and
volume fix applied in this problem. Volume variation without redistancing and volume fix
(Fig. 4.5(a)) for this problem exceeds 7%. With redistancing applied to this problem,
volume variation is about 4% (Fig. 4.5(b)). Whereas redistancing and global volume fix
reduce volume variation to less then 0.1% (Fig. 4.5(c)).
4.2 Reinitialization of the LS function with improvedpreservation of the zero level set.
Advecting a LS function in a deforming velocity field can lead to big gradients in the level set
function φ, which can reduce the quality of the solution as shown in the previous section. The
40
(a) Time T = 2π/3. (b) Time T = 4π/3. (c) Time T = 2π.
Figure 4.3: LS contours without redistancing and volume fix. Initially a circular interfaceR = 0.35 centered at (0, 0.45) is moving and deforming in a divergence free velocity field(4.3). Blue - zero level, black - contours of the LS function, green dots - grid’s cell centers.
(a) Time T = 2π/3. (b) Time T = 4π/3. (c) Time T = 2π.
Figure 4.4: LS contours with redistancing and volume fix. Initially a circular interfaceR = 0.35 centered at (0, 0.45) is moving and deforming in a divergence free velocity field(4.3). Blue - zero level, black - contours of the LS function, green dots - grid’s cell centers.
reinitialization process aims to restore the gradients of the LS function to be |∇φ| ≈ 1 while
keeping the zero level of this function unchanged. In the paper [9], where a LS method was
used for two-phase flow, a LS function was taken in the form of a signed distance function.
The LS reinitialization equation (1.7) φτ + S(φ)(| φ| − 1) = 0 was solved using a second
order ENO scheme with upwinding towards the interface position. Such a discretization
of (1.7) will perturb the zero level set, especially in the vicinity of segments with large
curvature. To keep the interface position unchanged during LS reinitialization one needs to
41
(a) Without redistancing and vol-ume fix.
(b) With redistancing only. (c) With redistancing and volumefix.
Figure 4.5: Volume change inside interface in a deforming velocity field.
restrict upwinding by taking values only from one side of the interface. One sided, finite
difference, discretizations that replace nodes on the opposite side of the interface with nodes
on the interface will have stability problems when the zero level set is near a grid node.
In [28] an interface position preserving stable first order in space LS reinitialization routine
was successfully implemented and tested. They reduced the reinitialization time step in the
vicinity of the zero level set in order to avoid stability problems.
An alternate way of interpreting Russo and Smereka’s work[28] is that they performed
one-sided differencing and at the same time they smeared the sign function. One can
choose a smoothed out sign function S(φ) = 2Hǫ(φ) − 1 instead of a sharp discontinuous
S(φ) = 2H(φ) − 1 sign function in the LS reinitialization equation (1.7). Here H denotes
the Heaviside step function, and Hǫ is a smoothed out Heaviside function. This way
2Hǫ(φ)− 1, which is nearly zero near the interface, slows down changes of φ in the equation
φτ +(2Hǫ(φ)− 1)(|φ|− 1) = 0 while maintaining a normal speed of the LS reinitialization
process away from the interface (where |2Hǫ(φ)− 1| ≈ 1).
In our implementation of the LS reinitialization algorithm, we have chosen a smoothed
sign function of the form,
S(φ0) = 2Hǫ(φ0)− 1 =
2
1 + e−φ0/ǫ− 1,
where the smoothing coefficient, ǫ, based on our experience, is problem dependent.
We note that for Zalesak’s notched disk benchmark problem, the velocity field is a non-
deforming velocity field. For this benchmark problem, LS reinitialization is not required
42
since the distance property is not perturbed in the presence of a solid-body motion velocity
field. LS reinitialization can even lower the overall accuracy of a level set algorithm for
test problems in which the velocity is non-deforming. For Zalesak’s problem, we can
choose the reinitialization sign function to be smoothed with ǫ = 3 max“critical” cells
(xi,j,yi,j).
Alternatively, for Zalesak’s problem, we can choose the reinitialization sign function to be
zero in a “critical cell.” We call two cells ”critical“ if the LS function changes its sign between
these two cells. We tried to set S(φ) = 0 at ”critical” cells in order to eliminate interface
motion while solving the LS reinitialization equation. This method gives very good results
for problems in which the velocity field is non-deforming, but lowers the overall accuracy
of the algorithm to first order for benchmark problems involving a deforming velocity field.
To keep the interface in the position it was before reinitialization, while maintaining second
order accuracy in the presence of a deforming velocity field, we have tested the following
algorithm:
• Calculate φreinit as a result of standard LS reinitialization.
• Take a weighted average of φreinit and φ0:
wi,j =∣
∣2Hǫ(φ0)− 1
∣
∣min
( |φ0i,j − φreinit
i,j |δi,j
, 1
)
,
φ = wφreinit + (1− w)φ0,
where δi,j is the longest distance to the neighboring cell center.
A different situation, from that observed for Zalasek’s problem (non-deforming velocity
field), occurs when we solve a problem with a smooth interface in a complex divergence
free deforming velocity field. In the absence of LS reinitialization, a distance LS function
fails to hold | φ| = 1 and the gradient can have large values which lowers the overall
accuracy of the LS method. In the second set of experiments, where we consider a smooth
interface advected in a deforming divergence free velocity field, we choose ǫ = 0.02 in the
smoothed out sign function. This choice of ǫ gives a perfect overall second order convergence
of the algorithm. We remark, that a drawback of this reinitialization approach is that we
have found ǫ to be problem dependent. Results are sensitive to ǫ.
To find the steady state solution φreinit of the LS reinitialization equation
φτ + S(φ0)(| φ| − 1) = 0,
43
the first order forward Euler scheme is used for time discretization
φm+1 = φm −τS(φ0)(|Dφm| − 1),m = 0, 1, 2...,
and the second order upwind finite difference discretization of the gradient
D = (Dx, Dy).
On an orthogonal grid an upwind gradient is
Dx(φ) =
D+x (φ), if D+
x (φ)φ < 0 and (D+x (φ) +D−
x (φ))φ < 0,
D−
x (φ), if D−
x (φ)φ > 0 and (D+x (φ) +D−
x (φ))φ > 0,
0, otherwise,
Dy(φ) =
D+y (φ), if D+
y (φ)φ < 0 and (D+y (φ) +D−
y (φ))φ < 0,
D−
y (φ), if D−
y (φ)φ > 0 and (D+y (φ) +D−
y (φ))φ > 0,
0, otherwise.
One sided weighted finite difference operators D+ and D− on a nonuniform grid are
D+x φi = −φi, (xi+2 + xi+1 − 2xi)
(xi+1 − xi)(xi+2 − xi)+
φi+1(xi+2 − xi)
(xi+1 − xi)(xi+2 − xi+1)− φi+2(xi+1 − xi)
(xi+2 − xi+1)(xi+2 − xi),
D−
x φi =φi(2xi − xi−2 − xi−1)
(xi−1 − xi)(xi−2 − xi)− φi−1(xi − xi−2)
(xi−1 − xi,j)(xi−2 − xi−1)+
φi−2(xi − xi−1)
(xi−2 − xi−1)(xi−2 − xi).
These operators are simplified in the case of a uniform grid
D+x φi = −(3φi − 4φi+1 + φi+2)/2x,
D−
x (φi) = (3φi − 4φi−1 + φi−2)/2x.
Similar operators are used to find derivatives in the y, ζ or χ directions.
On a curvilinear quadrilateral grid (ζ, χ) a gradient of φ is equal to
φ = (φx, φy) = (ζxφζ + χxφχ, ζyφζ + χyφχ) =
(
yχφζ − yζφχ
J,−xχφζ + xζφχ
J
)
,
where J is a Jacobian
J = xζyχ − xχyζ .
Covariant basis vectors ~a1 = (xζ , yζ) and ~a2 = (xχ, yχ) are discretized by second order
weighted central differences, for example
xζ(xi,j, yi,j) ≈xi,j31 − xi−1,j32
2131
+xi+1,j21 − xi,j31
3231
,
44
where
21 = ζi,j − ζi−1,j =√
(xi,j − xi−1,j)2 + (yi,j − yi−1,j)2,
32 = ζi+1,j − ζi,j =√
(xi+1,j − xi,j)2 + (yi+1,j − yi,j)2,
31 = 32 +21.
Upwind formulas are used to find curvilinear derivatives of φ, i.e.
φζ ≈
D+ζ (φ), if D+
ζ (φ)φ < 0 and (D+ζ (φ) +D−
x (φ))φ < 0,
D−
ζ (φ), if D−
ζ (φ)φ > 0 and (D+ζ (φ) +D−
ζ (φ))φ > 0,
0, otherwise,
φχ ≈
D+χ (φ), if D+
χ (φ)φ < 0 and (D+χ (φ) +D−
χ (φ))φ < 0,
D−
χ (φ), if D−
χ (φ)φ > 0 and (D+χ (φ) +D−
y (φ))φ > 0,
0, otherwise.
4.3 Global volume fix
In this section we explain the global volume fix procedure which is based on ideas from [19]
and [29].
We are given a cell-centered LS function on a 2-d structured grid. The zero level contour
implicitly represents the interface position between two fluids. For simplicity we assume
that there is only one simply connected interface contour in the physical domain. There is
a way to generalize this approach to multiple interfaces merging and dividing, we leave this
generalization as a further research topic. Let us assume we know the initial “true” volume
V0 inside a closed interface contour where φ(x) < 0.
The underlying velocity field, U, is divergence free which implies that the volume enclosed
by any closed contour of the level set function is preserved assuming that one can solve the
level set equation (1.2) exactly. Unfortunately, there is no guarantee that discretizations of
(1.2), even if discretized in conservation form, have the same volume preserving property.
To correct the volume we change the LS function value by a simple shift up or down
φ(x)← φ(x) + (V − V0)/P.
Volume V is calculated as
V =∑
k
H(φk)Vk,
45
where Vk is the k-th cell volume, H(φk) is smoothed Heaviside function
H(φk) =
1, if φk 6 −min(k-th cell faces)/2,
0, if φk > min(k-th cell faces)/2,
0.5(1− 2φk/min(k-th cell faces)), otherwise.
In the Figure 4.6(a), the above formula for volume inside the red interface gives a sum of
partial volume of light blue cells plus full volume of dark blue cells.
Perimeter P of the interface is a sum of all cell faces lengths across which a LS function
changes its sign, i.e.
P = 0,
for j = 1 : M − 1 and i = 1 : N − 1
P = P +√
(xi,j+1 − xi,j)2 + (yi,j+1 − yi,j)2, if φi,jφi+1,j ≤ 0,
P = P +√
(xi+1,j − xi,j)2 + (yi+1,j − yi,j)2, if φi,jφi,j+1 ≤ 0.
In the Figure 4.6(b) the stair step perimeter approximation of the red interface is shown in
blue.
(a) Volume calculation. (b) Perimeter calculation.
Figure 4.6: Global volume fix.
46
4.4 Numerical results: LS with global volume fix
Zalesak’s notched disk.
Zalesak’s notched disk is moving in a prescribed circular counter clockwise velocity field [48]
u(x, y) = − 2π
6.28(y − 0.5),
v(x, y) =2π
6.28(x− 0.5).
(4.1)
Since the velocity field is purely rotational, it should not deform an interface, nor change its
volume. This problem is a good diagnostic for Eulerian interface tracking algorithms since
it measures how well an algorithm behaves in the vicinity of discontinuous first derivatives.
After one full rotation at time T = 6.28 the notched disk is back to the upward position,
where a numerical error at ”critical“ cells is measured using L1 and L∞ norms. The LS
function is stored at cell centers. Two adjacent cells are called ”critical“ if the signed distance
LS function φ changes its sign between these two cells, or equal to zero in one of these cells.
Given K ”critical“ cells of volumes Vk and the exact solution φexc we define the numerical
error measure as
L1(error) =
∑
k=1,K
|φ(k)− φexc(k)|Vk
∑
k=1,K
Vk
,
L∞(error) = maxk=1,K
|φ(k)− φexc(k)|.
(4.2)
In all experiments below we use the numerical algorithm described in Section 4.1 with
t = min(x/|u|,y/|v|)/8 for advection and τ = min(x/|u|,y/|v|)/16 for 3-step
LS reinitialization process.
On Figures 4.7, 4.8, 4.9 Zalesak’s disk test problem is solved on dynamic adaptive
rectilinear grids of the size 50× 50, 100× 100 and 200× 200 correspondingly.
The grids are generated and updated on every 10-th timestep as described in Section
2.3. User defined parameters in rectilinear grid generation are A = 4, B = 1, k = 0.005
in the monitor functions (2.10-2.11). The number of iterations in the fixed-point/Thomas
algorithm is restricted by 500 and tolerance is set to be 10−16.
On Figures 4.10, 4.11, 4.12 Zalesak’s test problem is solved on dynamic adaptive
quadrilateral grids of the size 50× 50, 100 × 100 and 200× 200 correspondingly. The grids
47
(a) Interface moving counter clockwise. (b) Grid and solution.
(c) Close up grid and level curves. Bold blue curveis the zero level, exact solution is in red.
(d) Volume variation about V = 0.058225 insidezero level against time.
Figure 4.7: Zalesak’s test problem solved with LS/global volume fix method on a dynamicrectilinear 50x50 grid.
are generated and updated on every 10-th timestep as described in Section 2.2. User defined
parameters in quadrilateral grid generation are A = 4, B = 1, k = 0.005 in the monitor
48
(a) Interface moving counter clockwise. (b) Grid and solution.
(c) Close up grid and level curves. Bold blue curveis the zero level, exact solution is in red.
(d) Volume variation about V = 0.058225 insidezero level against time.
Figure 4.8: Zalesak’s test problem solved with LS/global volume fix method on a dynamicrectilinear 100x100 grid.
function (2.9). In a grid updating routine which is based on fixed point-SSOR-PCG method,
the number of inner iterations is restricted by 20, outer iterations by 40, and tolerance is set
to 10−16.
Computational time and numerical errors of the LS/global volume fix algorithm for this
problem on grids 50x50, 100x100, 200x200 are given in the tables 4.2, 4.3, 4.4. Numerical
49
(a) Interface moving counter clockwise. (b) Grid and solution.
(c) Close up grid and level curves. Bold blue curveis the zero level, exact solution is in red.
(d) Volume variation about V = 0.058225 insidezero level against time.
Figure 4.9: Zalesak’s test problem solved with LS/global volume fix method on a dynamicrectilinear 200x200 grid.
error is calculated at ”critical” cells where LS function changes sign. A table illustrating
the rate of convergence of Zalesak’s problem is shown in table 4.1. Logarithmic plots of
time and L1, L∞ norms of numerical error are shown on the Figures 4.13(a) and 4.13(b).
Computational time on all three types of grids exhibits O(N2) growth on N × N number
of grid cells. Numerical error is decreasing as O(N−1.5) in average for all experiments on all
types of grids, which indicates that the implemented algorithm is between first and second
order convergent.
50
(a) Interface moving counter clockwise. (b) Grid and solution.
(c) Close up grid and level curves. Bold blue curveis the zero level, exact solution is in red.
(d) Volume variation about V = 0.058225 insidezero level against time.
Figure 4.10: Zalesak’s test solved with LS/global volume fix method on a dynamicquadrilateral 50x50 grid.
Three conclusions can be drawn from these results. First, the dynamic, stretched,
curvilinear grids that were automatically generated for Zalesak’s problem do not lower order
of convergence of the algorithm. Second, dynamic, stretched, curvilinear grids increase the
accuracy in the interface position. Third, the global volume constraint does not lower the
order of convergence.
51
(a) Interface moving counter clockwise. (b) Grid and solution.
(c) Close up grid and level curves. Bold blue curveis the zero level, exact solution is in red.
(d) Volume variation in time around V = 0.058225inside interface.
Figure 4.11: Zalesak’s test solved with LS/global volume fix method on a dynamicquadrilateral 100x100 grid.
Final interface position on different grids is plotted on one picture for comparison.
100x100 grid is on Figure 4.14 and 200x200 is on Figure 4.15. Red contour is the exact
interface position, black - uniform grid, blue - nonuniform orthogonal, green - curvilinear
quadrilateral. On the same pictures a solution by coupled level set - volume of fluid method
on a uniform grid (this result belongs to Mark Sussman (FSU)) is drawn with a black dotted
52
(a) Interface moving counter clockwise. (b) Grid and solution.
(c) Close up grid and level curves. Bold blue curveis the zero level, exact solution is in red.
(d) Volume variation in time around V = 0.058225inside interface.
Figure 4.12: Zalesak’s test solved with LS/global volume fix method on a dynamicquadrilateral 200x200 grid.
line. From these picture we see that the suggested LS/global fixed volume algorithm is
comparable in accuracy with CLSVOF method if it is implemented on a dynamic, stretched,
curvilinear grid. The advantage of our method is its simple implementation on dynamic
curvilinear grids as opposed to the difficult and time consuming geometric considerations in
the CLSVOF method.
53
(a) Logarithmic plot of computational time. (b) Logarithmic plot of L1 (solid) and L∞ (dashed)norms of numerical error.
Figure 4.13: Computational time and numerical error for Zalesak’s problem. Solved withLS/global volume fix algorithm (Chapter 4). Red - uniform stationary grid, blue - dynamicnonuniform orthogonal, green - dynamic quadrilateral grid.
Table 4.1: Rate of convergence for Zalesak’s problem. L1 error
Grid size Uniform Rectilinear Quadrilateral50x50 0.015 0.005901 0.004532
100x100 0.007589 0.001525 0.001443200x200 0.00199 0.0006749 0.0005233
Circular interface in a divergence free velocity field in a rectangular
domain.
Where as the previous set of experiments tested capturing a nonsmooth interface in a pure
rotational non-deforming (distance preserving) velocity field, the next set of experiments are
designed to demonstrate accuracy for a smooth interface capturing problem in a deforming
divergence free velocity field
u(x, y) = γy(a2 − x2),
v(x, y) = γx(y2 − b2).(4.3)
See Figure 4.16. This velocity field is new and to the best of our knowledge have not been used
before. The velocity field has nice properties: it’s divergence free in a rectangular domain
54
Table 4.2: Computational time and error of LS with global volume fix on a 50x50 grid,Zalesak’s problem. Numerical error is calculated at ”critical“ cells near an interface.
Grid type Time (sec) L1 error L∞ errorUniform 14.445 0.015 0.1225
Rectilinear 41.991 0.005901 0.01463Quadrilateral 277.097 0.004532 0.01531
Table 4.3: Computational time and error of LS with global volume fix on a 100x100 grid,Zalesak’s problem. Numerical error is calculated at ”critical“ cells near an interface.
Grid type Time (sec) L1 error L∞ errorUniform 115.547 0.007589 0.01557
Rectilinear 359.91 0.001525 0.008871Quadrilateral 2152.975 0.001443 0.007311
Table 4.4: Computational time and error of LS with global volume fix on a 200x200 grid,Zalesak’s problem. Numerical error is calculated at ”critical“ cells near an interface.
Grid type Time (sec) L1 error L∞ errorUniform 939.299 0.00199 0.008074
Rectilinear 2941.728 0.0006749 0.003007Quadrilateral 17641.727 0.0005233 0.004275
(x, y) ∈ [−a, a] × [−b, b] with wall boundary conditions and simulates bubble deformation
like in a two-phase flow while rotating a bubble with γ angular speed.
In the experiments below initially a circle of radius R = 0.35 is centered at (0, 0.45).
Under the velocity field with γ = 2π, a = b = 1 the interface is rotating in clockwise direction
with angular velocity γ = 2π deforming in a drop like shape. We stop clockwise rotation at
time T = 2π and reverse the velocity field u(x, y) = −γy(a2− x2), v(x, y) = −γx(y2− b2) to
return the interface into its initial position, where L1 and L∞ norm of a numerical error is
calculated at ”critical“ cells.
In the Figure 4.17 experiments are done on uniform stationary grids 50x50 and 100x100.
55
Figure 4.14: Zalesak’s problem, 100x100 grid. Red contour - exact solution. LS/globalvolume fix algorithm: black solid - on a stationary uniform grid, blue - on a dynamicnonuniform orthogonal, green - on a dynamic curvilinear grid. Black dotted contour - coupledlevel set/volume of fluid method on a uniform stationary grid.
In the Figures 4.17(a) and 4.17(b) initial position of the interface is shown in red. Three
green contours show captured interface position at time T = 2π/3, 4π/3 and 2π. After one
full rotation clockwise a velocity field is reversed and an interface is returned to its initial
position. Blue contours show the interface on its way back rotating in counterclockwise
direction. The exact solution at the final moment coincides with the red contour. In the
Figures 4.17(c) and 4.17(d) cell centers of the grids are shown with green dots, the blue
56
Figure 4.15: Zalesak’s problem, 200x200 grid. Red contour - exact solution. LS/globalvolume fix algorithm: black solid - on a stationary uniform grid, blue - on a dynamicnonuniform orthogonal, green - on a dynamic curvilinear grid. Black dotted contour - coupledlevel set/volume of fluid method on a uniform stationary grid.
contour is the interface at time T = 2π and level curves after LS reinitialization are shown
in black. The last pare of pictures 4.18(a) and 4.18(b) shows volume variation inside the
zero level contour.
The second experiment (Fig.4.19) is done on a dynamic, stretched, rectilinear grid. The
grid is updated every 10-th time step as described in Section 2.3. A monitor function is taken
as in (2.10-2.11) with user defined parameters of the adaptive grid A = 4, B = 1, k = 0.005.
The number of iterations in the fixed-point/Thomas algorithm is restricted by 500 and
57
Figure 4.16: A divergence free velocity field (4.3) in a rectangular domain.
tolerance is set to be 10−16.
The third experiment (Fig.4.21) is done on a dynamic, stretched, quadrilateral grid. The
grid is updated every 10-th time step as described in Section 2.2. A monitor function is
taken as in (2.9) with user defined parameters of the adaptive grid A = 4, B = 1, k = 0.005.
The stretched quadrilateral grid at time T = 2π is demonstrated on Figure 4.21. In a grid
updating routine which is based on fixed point-SSOR-PCG method, the number of inner
conjugate gradient iterations is restricted by 20, outer fixed point iterations - by 40, and
tolerance is set to 10−16.
Computational time and L1, L∞ norms of numerical error at cells adjacent to the
interface are given in the tables 4.6 and 4.7. A table illustrating the rate of convergence
of our method is shown in Table 4.5. Logarithmic plots of time and L1, L∞ norms of
numerical error are shown on the Figures 4.23(a) and 4.23(b). Computational time on all
three types of grids exhibits O(N2) growth on N ×N number of grid cells. Numerical error
58
(a) (b)
(c) (d)
Figure 4.17: A circular interface in the velocity field (4.3)on uniform 50x50 (left) and100x100 (right) grids. Upper: interface going clockwise (green) and backwards (blue), exactsolution in red. Lower: grid - green dots, interface at time T = 2π - blue, level curves -black.
is decreasing as O(N−2) in average for all experiments on all types of grids, which indicates
that the implemented algorithm (LS/global volume fix on dynamic stretched quadrilateral
grids) for this problem has second order convergence. We note that the order of accuracy
measured for the Zalesak’s benchmark problem is larger than first order but not second
59
(a) (b)
Figure 4.18: Volume variation in time around V = 0.3812 inside interface. A circularinterface in the velocity field (4.3)on uniform 50x50 (left) and 100x100 (right) grids.
order. This is expected since the exact solution for the LS function for Zalesak’s problem
has a discontinuous first derivative in the vicinity of a nonsmooth interface.
Table 4.5: Rate of convergence for deforming velocity test problem. L1 error
Grid size Uniform Rectilinear Quadrilateral50x50 0.005954 0.003097 0.001954
100x100 0.0009414 0.0008001 0.0005429
60
(a) (b)
(c) (d)
Figure 4.19: A circular interface in the velocity field (4.3)on dynamic adaptive rectilinear50x50 (left) and 100x100 (right) grids. Upper: interface going clockwise (green) andbackwards (blue), exact solution in red. Lower: grid - green dots, interface at time T = 2π- blue, level curves - black.
61
(a) (b)
Figure 4.20: Volume variation in time around V = 0.3812 inside interface. A circularinterface in the velocity field (4.3)on dynamic adaptive rectilinear 50x50 (left) and 100x100(right) grids.
Table 4.6: Computational time and numerical error of LS with global volume fix, a circularinterface in a deforming velocity field on 50x50 grid. Numerical error is calculated at”critical“ cells near an interface.
Grid type Time (sec) L1 error L∞ errorUniform 28.169 0.005954 0.01625
Rectilinear 89.306 0.003097 0.01475Quadrilateral 543.358 0.001954 0.009827
Table 4.7: Computational time and error of LS with global volume fix, a circular interface ina deforming velocity field on 100x100 grid. Numerical error is calculated at ”critical“ cellsnear an interface.
Grid type Time (sec) L1 error L∞ errorUniform 224.214 0.0009414 0.004658
Rectilinear 674.1301 0.0008001 0.004729Quadrilateral 4256.5498 0.00054287 0.003252
62
(a) (b)
(c) (d)
Figure 4.21: A circular interface in the velocity field (4.3)on dynamic adaptive quadrilateral50x50 (left) and 100x100 (right) grids. Upper: interface going clockwise (green) andbackwards (blue), exact solution in red. Lower: grid - green dots, interface at time T = 2π- blue, level curves - black.
63
(a) (b)
Figure 4.22: Volume variation in time around V = 0.3812 inside interface.A circular interfacein the velocity field (4.3)on dynamic adaptive quadrilateral 50x50 (left) and 100x100 (right)grids.
(a) Logarithmic plot of computational time. (b) Logarithmic plot of L1 (solid) and L∞ (dashed)norms of numerical error.
Figure 4.23: Computational time and numerical error. A smooth interface in a deformingvelocity field. Solved with LS/global volume fix algorithm (chapter 4). Red - on a stationaryuniform grid, blue - on a dynamic nonuniform orthogonal, green - on a dynamic quadrilateralgrid.
64
CHAPTER 5
TUNNEL QUASI-CUBIC INTERPOLATION
ALGORITHM
In two phase flow problems an interpolation is used in interface reconstruction, data
remapping and in semi-Lagrangian numerical schemes. Bicubic spline interpolation is a
piecewise polynomial approximation P3(x, y) = 1, x, x2, x3 × 1, y, y2, y3 of a function
f(x, y) of two variables. This interpolation referred as direct cubic interpolation was derived
as local interpolation on a structured curvilinear 2-d grid in [49]. Quasi-cubic interpolation
is a polynomial approximation which can be represented by P3\x3y3, x3y2, x2y3, x2y2, i.e.
terms of higher order than 3 are dropped. Both interpolations, bicubic and quasi-cubic,
are exact for polynomials of degree 3 on orthogonal grids. In contrast to bicubic spline
interpolation where a 16-point stencil is used, in quasi-cubic interpolation a stencil consists
of 12 nodes. This simplification of a cubic interpolation is used in the weather prediction
community in semi-Lagrangian numerical schemes as local interpolation on a rectilinear
structured grid.
A quasi-cubic interpolation is done as follows. We use a linear interpolation on the upper
two and lower two points of a 12-point stencil to evaluate values at the upper and lower
intermediate points (points (x, y)1 and (x, y)4 on the Fig.5.1). We find a cubic interpolation
at points (x, y)2 and (x, y)3 on the Fig.5.1. At last we interpolate cubicly a value at the point
of interest (x, y)∗ along the line going trough four intermediate points. This interpolation
requires 3 cubic and 2 linear 1-d interpolations on a rectilinear grid as opposed to 5 cubic
interpolations in a bicubic local interpolation. In addition to efficiency, corner points are
not used in a quasi-cubic interpolation. This eliminates the need for boundary conditions at
corner points of a physical domain.
In both bicubic and quasi-cubic local interpolations coordinates of the intermediate points
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Figure 5.1: Quasi-cubic interpolation on a rectilinear grid
(x, y)1, (x, y)2, (x, y)3, (x, y)4 have to be found. If on a rectilinear grid it’s straightforward
to do, on a general quadrilateral grid additional operations and considerations have to be
done to find these points. In the tunnel quasi-cubic algorithm which we propose here there
is no need in the upper and lower intermediate points (labeled as points (x, y)1, (x, y)4 on
the Fig.5.1).
5.1 Tunnel quasi-cubic spline on an orthogonal 2-dgrid
To explain a new algorithm let us start with a rectilinear grid. Assume we are given
12 values of a function f at the nodes labeled on the Fig.5.2(a). We want to find
a polynomial interpolant P = 1, x, x2, x3 × 1, y, y2, y3\x3y3, x3y2, x2y3, x2y2 in the
center cell of the given stencil. The main idea behind this method is to decompose the
interpolating surface into a sum of three components. A bilinear component fits values
of an interpolated function f at the nodes (2:3,2:3). Two other tunnel looking surfaces
fit partial derivatives fx and fy at the nodes (2:3,2:3). The first step is to find a unique
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(a) Orthogonal rectilinear grid (b) Horizontal tunnel
(c) Curvilinear quadrilateral grid (d) z = x3+y3. Red dots are interpolationpoints on a curvilinear grid (blue dots),N=32.
Figure 5.2: Tunnel quasi-cubic interpolation
bilinear surface P lin(x, y) = 1, x, × 1, y satisfying P lin(2:3,2:3)=f(2:3,2:3). Next, we
subtract this surface from all 12 interpolated values f = f − P lin of the given stencil. At
this stage we should get zeros at the nodes (2:3,2:3), i.e. f(2:3,2:3)=0. We evaluate partial
derivatives fx and fy at the nodes (2:3,2:3) using a central difference formula if a grid is
uniform and a weighted central difference formula if a grid is nonuniform. Now we find a
surface P hor = 1, x×1, y, y2, y3 satisfying P hor(2:3,2:3)=0 and P hory (2:3,2:3)=fy(2:3,2:3).
Note that this surface looks like a horizontal tunnel shown on Fig.5.2(b) with cubicals
in the vertical direction and lines in the horizontal direction. Then we find a similar
tunnel surface oriented vertically P ver = 1, y × 1, x, x2, x3 satisfying Pver(2:3,2:3)=0
and P verx (2:3,2:3)=fx(2:3,2:3). Finally, adding up all three components we get a quasi-cubic
spline P = P lin + P hor + P ver interpolating the function f at the central cell of the given
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stencil.
Representation of an interpolating surface as a composition of three subsurfaces enables
us to find efficiently a quasi-cubic spline defined in the cell, while a traditional quasi-cubic
algorithm was derived for local interpolation at a point only. It worths to mention that this
procedure can be easily generalized for a 3-dimensional spline interpolation.
5.2 Tunnel quasi-cubic local interpolation on ageneral quadrilateral grid
Now let’s consider a 2-d curvilinear quadrilateral grid. Given 12 values of a function f at the
nodes shown on the Fig.5.2(c), we want to interpolate the function at the star point (x∗, y∗).
Algorithm
1. Find a bilinear component of an interpolating surface P lin = 1, x × 1, y satisfying
P lin(2 : 3, 2 : 3) = f(2 : 3, 2 : 3). Interpolate the bilinear surface at the star point
P lin(x∗, y∗). Subtract the bilinear surface from the function at all 12 interpolating
nodes, i.e. f = f − P lin.
2. Find an intersection (xint, yint) of two ”vertical” sides of the central cell up to prescribed
tolerance. Then draw a line going through (xint, yint) and (x∗, y∗). If these two sides are
parallel then draw a line going through (x∗, y∗) and parallel to the ”vertical” cell faces.
Intersection of this line with the two other sides of the central cell gives intermediate
interpolating points (xlo, ylo) and (xhi, yhi).
3. Fit two cubicals P ver = 1, ζ, ζ2, ζ3 along horizontal sides of the central cell satisfying
P ver(2 : 3, 2 : 3) = 0 and P ver(1, 2 : 3) = f(1, 2 : 3), P ver(4, 2 : 3) = f(4, 2 : 3).
That would be two curves of a vertical tunnel surface. Evaluate these cubical curves
at the “lo“ and “hi“ points. Linearly interpolate P ver(x∗, y∗) using P ver(xlo, ylo) and
P ver(xhi, yhi).
4. Calculate curvilinear partial derivatives fχ along ”vertical” sides of the cell at the
nodes (2:3,2:3) using a wighted central difference formula. Linearly interpolate these
derivatives at the points “lo“ and “hi” fχ(xlo, ylo) and fχ(xhi, yhi). Fit a cubical
P hor = 1, χ, χ2, χ3 satisfying P hor(xlo, ylo) = 0, P hor(xhi, yhi) = 0, P horχ (xlo, ylo) =
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fχ(xlo, ylo), Phorχ (xhi, yhi) = fχ(xhi, yhi). This cubical curve is a part of a horizontal
tunnel surface. Evaluate this cubical curve at the star point P hor(x∗, y∗).
5. Sum up three components and get a quasi-cubic interpolation at the star point
P (x∗, y∗) = P lin(x∗, y∗) + P ver(x∗, y∗) + P hor(x∗, y∗).
A quasi-cubic interpolation is exact for cubic polynomials on an orthogonal rectilinear grid
and converges with the second order on a smoothly generated curvilinear quadrilateral grid.
In our case we generate grids with twice differentiable mapping (see Chapter 2). A numerical
experiment with a quasi-cubic interpolation on a curvilinear grid illustrated on the Fig.5.2(d)
confirms second order convergence. Discrete L2-norm of the error at the interpolating points
is calculated and compared on different grids for different surfaces. Curvilinear grids were
generated as described in Chapter 2 for N = 16, 32, 64, 128. The error is going to zero
as O(N2) on the curvilinear quadrilateral grids. The experiments show that the proposed
algorithm of a quasi-cubic interpolation is exact for cubic surfaces on orthogonal grids and
exact for quadric surfaces on curvilinear grids.
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CHAPTER 6
CONCLUSION
In this thesis, an Eulerian, level set, front tracking method has been developed in the context
of dynamic, stretched, curvilinear grids. A global mass constraint has been developed which
overcomes the volume preservation problems associated with level set methods.
6.1 CONCLUSIONS - grid generation
An elliptic grid generation algorithm has been implemented, together with new monitor
functions which are customized for interface tracking problems. Our proposed monitor
functions are designed to move interfaces with more equally distributed (hence minimized)
numerical error near an interface. A monitor function (2.9), in addition to concentrating
grid cells near an interface, places smaller grid cells where an interface has bigger curvature.
These regions with large curvature are potentially vulnerable to volume loss and bigger
numerical errors. Monitor functions for generating stretched, orthogonal rectilinear grids
(in contrast to non-orthogonal curvilinear grids) with the above properties are derived as
well (2.10),(2.11). A more theoretically substantiated monitor function based on the local
truncation error (LTE) of a numerical scheme advecting an interface (2.12),(2.17) is derived.
All these monitor functions require user defined grid refinement parameters which control
relative grid size refinement as well as width of a refinement region near an interface.
In future research, the dependence of the numerical solution error with respect to the user
defined monitor function parameters needs to be quantified in more detail. Also in the future,
whereas we have developed a nonlinear elliptic solver based on the fixed point method (outer
loop) and the PCG method (inner loop), one can accelerate the grid generation algorithm
further by implementing multigrid techniques.
Also in future research, it is proposed to combine block structured AMR techniques
70
Figure 6.1: Zalesak’s problem (rotating counter clockwise) using LS/global mass fix algo-rithm. Dynamic block structured AMR grid with two levels of adaptivity is used. Problemcomputed in a polar coordinate system. Effective fine grid resolution: 200 cells in the rdirection and 400 cells in the θ direction.
with stretched grid techniques in order to further reduce the error and computational time
associated with a given number of grid points. Please see Figure 6.1 which show results in
which we use block structured adaptive mesh refinement together with polar coordinates in
order to compute Zalesak’s test problem.
6.2 CONCLUSIONS - Eulerian front tracking
Throughout the research of this dissertation, it is discovered that there is a trade-off between
volume preservation and accuracy of a front tracking method; the trade-off is based on the
numerical representation of the interface.
An interface tracking method that represents the interface by a smooth distance function
(LS method) will exhibit high accuracy, but poor volume preservation. On the other hand,
an interface tracking method that represents the interface by a profile function with thickness
proportional to the grid size (CLS method), will preserve volume very accurately, but will
exhibit a low order of accuracy in the shape of the tracked interface. We have shown that
71
the CLS method has an O(1) local truncation error when the thickness parameter is of the
order of mesh size. We have developed improvements to the CLS reinitialization algorithm,
albeit the overall CLS method is still not “straight line preserving.”
So, a level set method (LS) together with a global mass fix has been developed for
this thesis. Our developed method preserves volume and is shown to exhibit second order
accuracy. The LS/global volume fix method on a dynamic, stretched, quadrilateral grid is
tested on a number of interface capturing problems and shows comparable in numerical error
results with the CLSVOF method. The LS/global volume fix method is relatively easy to
implement on a dynamic curvilinear grid as opposed to difficult geometric considerations
in VOF or CLSVOF methods. The LS/global volume fix method has the second order
convergence on a smooth interface on all three types of grids tested in this research:
stationary uniform, dynamic nonuniform rectilinear and dynamic nonuniform quadrilateral
grids. Zalesak’s notched disk problem solved with this method gives between first and second
order convergence.
We remark that our LS/global volume fix algorithm is based on a semi-Lagrangian
approach (therefore avoiding CFL constraints) for the advection and remapping operations.
The advection and remapping operations are highly dependent on the interpolation method
used. For this research, a new tunnel quasi-cubic interpolation algorithm has been developed
for stretched, curvilinear, grids. A quasi-cubic interpolation method is known to be more
efficient than bicubic interpolation and has been used on an orthogonal grid before. A new
algorithm proposed in this research enables us to derive a local quasi-cubic interpolation
algorithm on a general curvilinear grid and to derive a global quasi-cubic spline interpolation
on a nonuniform orthogonal grid. Our new interpolation scheme is of the second order if
the underlying generated grid has a twice differentiable mapping function. As a note for
further development, we consider generalization of the tunnel quasi-cubic algorithm from
2-d to 3-d. Also for future research, we plan to base the level set reinitialization algorithm
on a semi-Lagrangian approach, thereby eliminating the dependence on smoothing the sign
function as discussed in section 4.2. The new tunnel interpolation scheme would then be
integral for all three level set operations: advection, remapping, and reinitialization. Figures
6.2, 6.3, 6.4, illustrate preliminary results in which cubic interpolation is used for advection
and reinitialization, on a dynamic block structured AMR grid (as opposed to a dynamic,
stretched, curvilinear grid) in order to compute the rise of a gas bubble in liquid. The
72
bubble velocity and mass are preserved better using the new method, than with CLSVOF.
(Navier-Stokes equations solver to obtain this numerical results was graciously given by Mark
Sussman (FSU).)
In order to quantify the accuracy of newly developed algorithms, a new benchmark
problem has been devised for testing the accuracy of algorithms for a deforming interface in
the context of an incompressible flow field. Our new benchmark test complements existing
benchmark tests [50]. Existing benchmark tests for passive advection of an interface either
prescribe a velocity which does not deform the interface (e.g. Zalesak’s problem) or generate
singularities in the interface which are only amenable to the investigation of the robustness
of an interface advection algorithm; not amenable to investigating the order of accuracy in
simulating a deforming interface.
73
Figure 6.2: Computation of an axisymmetric gas bubble rising in liquid. using LS/globalmass fix algorithm. Frames go from left to right, top to bottom. Dynamic block structuredAMR grid with one level of adaptivity is used. Effective fine grid resolution is 64x128.Periodic boundary conditions at the top and bottom of the computational domain.
74
Figure 6.3: Comparison of the computed rise speed for an axisymmetric cap bubble, betweenLS/global mass fix and CLSVOF methods. Dynamic block structured AMR grid with onelevel of adaptivity is used. Effective fine grid resolution is 64x128. Periodic boundaryconditions at the top and bottom of the computational domain.
75
Figure 6.4: Comparison of the computed volume for an axisymmetric cap bubble, betweenLS/global mass fix and CLSVOF methods. Dynamic block structured AMR grid with onelevel of adaptivity is used. Effective fine grid resolution is 64x128. Periodic boundaryconditions at the top and bottom of the computational domain.
76
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80
BIOGRAPHICAL SKETCH
Svetlana Simakhina
Svetlana Simakhina was born in USSR in 1975. In 1998 she graduated from Tomsk State
University, Tomsk, Russia with a university diploma which is equivalent to USA’s Bachelor
degree in Mechanics. The same year she was admitted to a graduate school at Surgut State
University, Surgut, Russia. She got married and moved to USA in 2001. In the winter of
2002 she was admitted to a graduate school at the University of Illinois at Chicago, USA,
and in the summer of 2003 she completed her Master’s degree in Applied Mathematics and
Computer Science. She enrolled in the doctoral program at FSU in the fall of 2005.
81
