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Computer Aided Geometric Design

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Contents lists available at

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Computer Aided Geometric Design

www.elsevier.com/locate/cagd

Topology analysis of time-dependent multi-fluid data using the Reeb

graph

✩

Fang Chen

a

,

∗

, Harald Obermaier

b

, Hans Hagen

a

, Bernd Hamann

b

, Julien Tierny

c

,

Valerio Pascucci

c

a

Department of Computer Science, University of Kaiserslautern, Kaiserslautern 67663, Germany

b

Institute
for Data Analysis and Visualization, Department of Computer Science,
University of California, Davis, One Shields Avenue, Davis, CA 9561

6, USA

c

72 South, Central Campus Drive, Salt Lake City, UT 84112, USA

article info

abstract

Article history:

Available online xxxx

Keywords:

Multi-phase fluid

Level set

Topology method

Point-based multi-fluid simulation

Liquid–liquid extraction is a typical multi-fluid problem in chemical engineering where

two types of immiscible fluids are mixed together. Mixing of two-phase fluids results in

a time-varying fluid density distribution, quantitatively indicating the presence of liquid

phases. For engineers who design extraction devices, it is crucial to understand the density

distribution of each fluid, particularly flow regions that have a high concentration of the

dispersed phase. The propagation of regions of high density can be studied by examining

the topology of isosurfaces of the density data. We present a topology-based approach to

track the splitting and merging events of these regions using the Reeb graphs. Time is used

as the third dimension in addition to two-dimensional (2D) point-based simulation data.

Due to low time resolution of the input data set, a physics-based interpolation scheme is

required in order to improve the accuracy of the proposed topology tracking method. The

model used for interpolation produces a smooth time-dependent density field by applying

Lagrangian-based advection to the given simulated point cloud data, conforming to the

physical laws of flow evolution. Using the Reeb graph, the spatial and temporal locations

of bifurcation and merging events can be readily identified supporting in-depth analysis of

the extraction process.

©

2012 Elsevier B.V. All rights reserved.

1. Introduction

In chemical engineering, liquid–liquid extraction is a widely used method, where compounds of one liquid are separated

by mixing it with a finely dispersed solvent (

Drumm et al., 2008

). Numerical simulations model this process as a multi-fluid

flow with two liquid phases. These multi-fluid simulations can be used to improve the design parameters of extraction

devices in order to optimize the extraction process.

The desired output of these simulations is the predicted evolution of the density distribution for each of the phases,

which indicates how well the two liquids are mixed. The density of a phase in a given space denotes the fraction of

the space occupied by the fluid. In liquid–liquid extraction, a finely dispersed solvent results in low density values for the

second phase in large regions of the data set. Examples for Computational Fluid Dynamics (CFD) codes capable of simulating

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This
research was supported by the International Research and
Training Group at the University of Kaiserslautern (IRTG) and
Deutsche

Forschungsgemeinschaft (DFG, German research foundation).

*

Corresponding author.

E-mail addresses:

chen@cs.uni-kl.de

(F. Chen),

harald.obermaier@itwm.fhg.de

(H. Obermaier),

hagen@cs.uni-kl.de

(H. Hagen),

hamann@cs.ucdavis.edu

(B. Hamann),

jtierny@sci.utah.edu

(J. Tierny),

pascucci@sci.utah.edu

(V. Pascucci).

0167-8396/$ – see front matter

©

2012 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.cagd.2012.03.019

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F. Chen et al. / Computer Aided Geometric Design

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two-phase flows are Fluent (commercial), OpenFoam (open source) and finite point set method (FPM) (

Kuhnert and Tiwari,

2003

). The latter is used throughout this work and produces scattered point sets, carrying velocity and density information.

Typically, there are three types of multi-phase fluid models tackling different flow regimes:

volume of fluid

(VOF) modeling

which focuses on tracking the interface of the two fluids for slug and surface flow; Eulerian multi-phase modeling which

deals with heat and momentum transfer between the phases; and a discrete phase modeling of the mixture. The last type

of modeling is widely used for simulating bubbly flow and slurry flow, such as fluid mixture in a bubble column reactor

for liquid–liquid extraction. However, traditional approaches of capturing fluid material boundaries cannot handle the case

where multiple regions within a cell are occupied by different fluids, as is the case in finely dispersed liquids. Thus a higher

level visualization technique is required for the understanding of these flow density distributions.

We present a topology-based approach for studying the volume fraction field given by an arbitrarily distributed numer-

ically computed point set. The simulation data we are looking at is a two-dimensional, time-varying fluid field discretized

by particles with associated density and velocity values. The low time resolution of this reference data set complicates

the tracking of material boundaries, as a non-physically-based interpolation scheme can result in wrong topology. For this

matter, we make use of a physically-based interpolation scheme that improves correspondences between time steps of the

simulation and allows for more robust feature extraction.

The major goal of this paper is to develop plausible and practical interpolation schemes for point-based multi-fluid

density data, and to characterize fluid interface behavior with Reeb graphs. To identify interesting time intervals where

regions that are densely occupied by a certain phase of fluid split or merge, we first define these regions by a certain

level

set

of the density field. Using a physically-based interpolation scheme, we compute a time-continuous density field at a

re-sample grid points. Finally we carry out a topological analysis of the extracted time-varying level sets using the Reeb

graph. The major contributions of this work are as follows:

•

It proposes an interpolation scheme for point-based time-dependent density data sets which have no connectivity

information. The proposed interpolation scheme is capable of handling sparse data with large time intervals, preserving

both the physical properties as well as topology of the flow.

•

It introduces a framework to extract and analyze fluid interface topology. The framework is practical for the analysis

point-based multi-fluid data sets. It offers novel views and tools for domain experts for further analysis and estimation

of solvent efficiency.

The paper is organized as follows: In Section

1.1

, we introduce related work about particle-based fluid simulation and

topology-based feature tracking techniques. In Section

2

, suitable interpolation schemes are applied separately for time

and spatial direction in order to obtain a topological-clean extraction of the level sets. Section

3

contains the example and

topology analysis of our result. The final section highlights some areas of possible improvements as well as future work

concerning our method.

1.1. Related work

A

level set

of a scalar field is given by the set of points with identical scalar value. Our idea of studying the level sets

of the density field was inspired by existing research which focuses on material boundary and fluid interface tracking,

including the

front tracking

(

FT

)

method

(

Unverdi and Tryggvason, 1992

),

level set method

(LSM) (

Osher and Sethian, 1988

),

and

volume of fluid method

(

VOF

)(

Hirt and Nichols, 1981

).

The FT method (

Unverdi and Tryggvason, 1992; Terashima and Tryggvason, 2009; Gloth et al., 2003

)advectsthemarked

interface from an initial configuration and keeps the topology of the interface constant during the simulation. Therefore,

this method is limited to topological changes in multi-phase fluids, such as merging or breaking of droplets.

The LSM was introduced by Osher (

1988

) in 1988. The material boundary or interface is defined as the zero set (

Osher

and Fedkiw, 2001

,

2003

;

Sethian, 1985

) of the given scalar field. Sethian (

2003

) and Lakehal (

2002

) applied the concept to

fluid simulation. In 2002, Enright et al. (

2002

) combined Lagrangian marker particles with LSM to obtain and maintain a

smooth geometrical description of the fluid interface. However, it has been pointed out by Müller (

2009

) and Garimella et

al. (

2005

) that material volume is not well-preserved in the level set method, which is a main drawback of this approach.

The VOF method (

Hirt and Nichols, 1981

) is one of the best established interface volume tracking methods currently

in use (

Marek et al., 2008; Sussman and Puckett, 2000

). It employs the idea of using mass conservation for the volume of

each fluid. Apart from VOF, other interface tracking algorithms include

simple line interface

(SLIC) (

Noh and Woodward, 1976

)

and piecewise linear interface construction (PLIC) (

Rider and Kothe, 1998

). In the SLIC method, the interface is taken to be

perpendicular or parallel to coordinate axes directions, while in PLIC, the interface is given by a piecewise linear function

with arbitrary orientation.

However, the interface tracking algorithms mentioned above do not fully apply to liquid–liquid extraction simulation

as one phase of fluids is fully dispersed and no continuous interface exists between the two fluids. Therefore, a material

interface is not traceable in the case of slurry flow. Instead, topology-based techniques for the analysis of level sets are

more suitable and useful in this context because of their ability to capture the absolute and relative behavior of small-scale

features like dispersed bubbles directly (

Tierny et al., 2009

).