Transportation-based Visualization of Energy Conversion

Oliver Fernandes, Steffen Frey and Thomas Ertl

VISUS, University of Stuttgart, Allmandring 19, Stuttgart, Germany

Keywords:

Energy Transport, Glyph-based Visualization, Glyph Placement, Volume Rendering, Energy Flux, Flow

Dynamics.

Abstract:

We present a novel technique to visualize the transport of and conversion between internal and kinetic energy

in compressible ﬂow data. While the distribution of energy can be directly derived from ﬂow state variables

(e.g., velocity, pressure and temperature) for each time step individually, there is no information regarding

the involved transportation and conversion processes. To visualize these, we model the energy transportation

problem as a graph that can be solved by a minimum cost ﬂow algorithm, inherently respecting energy con-

servation. In doing this, we explicitly consider various simulation parameters like boundary conditions and

energy transport mechanisms. Based on the resulting ﬂux, we then derive a local measure for the conversion

between energy forms using the distribution of internal and kinetic energy. To examine this data, we em-

ploy different visual mapping techniques that are speciﬁcally targeted towards different research questions.

In particular, we introduce glyphs for visualizing local energy transport, which we place adaptively based on

conversion rates to mitigate issues due to clutter and occlusion. We ﬁnally evaluate our approach by means of

data sets from different simulation codes and feedback by a domain scientist.

1 INTRODUCTION

With the ever growing computational power avail-

able, ﬂow simulations performed by scientists and en-

gineers become more realistic and detailed. There-

fore, the analysis of occurring ﬂow phenomena be-

comes increasingly difﬁcult, and available tools may

not be adequate to the task. New visualization tech-

niques are necessary to be able to explore the dif-

ferent aspects of phenomena occurring in the data.

In particular, the consideration of conserved quanti-

ties proves to be a valuable asset in both mathemat-

ics and physics, and is ultimately fundamental for our

understanding of nature. Used as a basis for model-

ing ﬂow dynamics, the conservation laws of physics

are employed to establish a set of partial differential

equations to describe the dynamics of a ﬂuid in mo-

tion. Basically, this is done by ﬁrst determining the

composition of a conserved quantity, and identifying

its constituents. The different ways the quantity can

change over time is then examined by modeling ap-

propriate physical interactions. The quantity needs to

be analyzed in a temporal context, as only examining

a given state of the simulation, even when considering

all deﬁning variables, may not reveal all dynamics in-

volved. For example, the visualization of the velocity

does not sufﬁce to fully understand the dynamics (i.e.

transport) of momentum for the simple case of an in-

compressible steady laminar ﬂow simulation. In this

particular case, the solution is fairly simple: by taking

into account the deviatoric stress tensor, the true mo-

mentum ﬂux ﬁeld can be derived for each component,

and analyzed via standard vector ﬁeld visualization.

However, this is not the case in general. Usually,

all dynamics calculated by solving the partial differ-

ential equations cannot be extracted directly from the

state of the ﬂuid. In a more interesting example of a

compressible ﬂow simulation, one cannot directly de-

rive, given the initial and ﬁnal state of the ﬂuid, how

an energy change has occurred. The ﬁnal state can,

in theory, be reached by various combinations of ex-

changing heat or having work done on its volume.

In this paper, we propose an approach to visualize

this process for arbitrary ﬂow data by examining the

exchange of total energy between control volumes.

The resulting energy ﬂux is in turn used to compare

the composition of energy types in the initial state

with the ﬁnal state. Based on this comparison, a visual

representation is generated showing the spatial distri-

bution for the conversion between different forms of

energy.

In Sec. 2 we give an overview on related work. Af-

ter providing a quick outline of our overall approach

and describing the underlying fundamentals regard-

Fernandes O., Frey S. and Ertl T.

Transportation-based Visualization of Energy Conversion.

DOI: 10.5220/0006098200520063

In Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications (VISIGRAPP 2017), pages 52-63

ISBN: 978-989-758-228-8

ing energy densities occurring in ﬂow (Sec. 3), we

then discuss and evaluate the individual components

of our approach, which we consider to be the main

contributions of our paper:

• We introduce our approach to estimate energy

conversion for visualization by reformulating and

solving it as a transportation problem (Sec. 4).

• We then discuss several mapping approaches for

the visualized quantity to generate meaningful

renderings from the obtained energy data (Sec. 5).

• We evaluate the utility of our approach with dif-

ferent data sets and an expert study with a domain

scientist (Sec. 6).

We ﬁnally conclude our work in Sec. 7.

2 RELATED WORK

Since the visualization of energy and energy transport

can lead to signiﬁcant insights of the explored data

set, a few approaches to the problem have been sug-

gested. However the intricate nature of the problem,

both theoretically and technically, proves it to be a

difﬁcult task to produce a concise and comprehensive

visualization of all the information contained.

Many visualization techniques to explore ﬂow

data have been developed over the years. Relevant for

the approach discussed in this paper are several prior

works, relating to various steps of our approach.

Already in 1905, visualizing the participation of

energy in an experiment or a simulation, often so-

called Sankey diagrams were (Sankey, 1905) (and still

are) used, to give an abstracted overview of energy ex-

change. Of course, any further information regarding

the spatial distribution is lost.

Many geometry-based techniques have been de-

rived processing the underlying ﬁelds of the data

to generate geometry representing its characteristics.

Among these are the well-known streamlines and

similar integration techniques resulting in geome-

try, which resemble the velocity vector ﬁeld of the

ﬂow. For an overview of geometric visualizations, see

(McLoughlin et al., 2010).

Often a mixture of texture-based and geometry-

based approach already reveals much information

about the ﬂow, for example done with LIC on isosur-

faces in (Laramee et al., 2004). An extension of these

are stream tubes, which are densely seeded stream-

lines on a closed curve, therefore creating a tube-like

structure (Schroeder et al., 1991). Various improve-

ments have been made to enhance the usage of stream

tubes, for example (McLoughlin et al., 2013). An

interesting property of these is that there is no ﬂow

across the tube mantle (per deﬁnition). In this con-

text, Meyers and Meneveau (Meyers and Meneveau,

2012) suggested so-called energy transport tubes to

visualize the transport of kinetic energy in statistically

steady ﬂows. Again, deﬁning an energy tube mantle

as the two-dimensional manifold where energy ﬂux

does not have a normal component (i.e. total energy

stays the same summed up across the cross-section

of such a tube), they give a good visualization of the

propagation of kinetic energy. However, they do not

help in exploring phenomena beyond transport of an

energy form. Also, arbitrary sources and sinks can-

not be included as easily. They also do not consider

the different forms of energy present in a ﬂow simula-

tion, and how the composition of the total energy can

reveal additional information about the data.

With our approach aggregating energy transport

information to yield a scalar value for the conversion,

direct visualization techniques need to be considered.

Very common are techniques like volume render-

ing which are used for scalar ﬁelds of ﬂow data such

as density. Still being an active part of research, vol-

ume rendering has many applications in scientiﬁc vi-

sualization, e.g. (Kroes et al., 2011). Also more

lighting-based techniques can help to present the data

more intuitively, as for example shading cues, em-

ployed in (Boring and Pang, 1996) for vector ﬁeld

ﬁelds, such as the velocity. A good overview of direct

visualization for higher order data, and possible visu-

alization approaches is given in (Sadlo et al., 2011).

Glyphs provide a very good means of visualizing

multivariate and higher order data. Our approach re-

sults in multivariate data represented by a dense graph

of the data set with both nodes and edges having rele-

vant scalar values applied (node count on the order of

cell count). A concise yet intuitive way of interpret-

ing the data is needed. However, for multivariate data,

there is no straightforward standard technique to visu-

alize involved quantities directly. An overview of us-

ing glyphs to represent multivariate data can be found

in (Borgo et al., 2013) and (Ropinski et al., 2011).

Custom glyphs incorporating all dimensions and vari-

ables of the data can be designed, as is done for exam-

ple in vector uncertainty visualizations (Wittenbrink

et al., 1996), where the classic “hedgehog” vector

ﬁeld visualization is enhanced with additional infor-

mation on the uncertainty. A novel approach of us-

ing the glyph in a multi-scale fashion, where different

zoom levels of the same dense glyph layout revealed

different information about the data set, was pre-

sented in (Hlawatsch et al., 2011). This also encom-

passes customizing glyphs to ﬁt the underlying data,

as shown in (Kraus and Ertl, 2001) or (Tong et al.,

2016). Glyphs also work well for ﬂow data when con-

Transportation-based Visualization of Energy Conversion

sidering the tensors (Jacobian, Hessian) of ﬂow quan-

tity derivatives, and are a popular tool for visualiz-

ing higher order quantities (Schultz and Kindlmann,

2010). An example of combining direct and glyph

visualization in an integrated approach is shown in

(

Ufﬁnger et al., 2011).

Our technique determines the ﬂow of energy be-

tween subsequent time steps to visualize the conver-

sion processes. Conceptually, this is related to the

earth mover’s distance (EMD). Basically, it deter-

mines the minimum cost of turning one distribution

into the other. First, it constructs a complete graph,

taking each cell of the data as a node (i.e., every pair

of nodes is connected by an edge). In our approach,

we use a minimum-cost ﬂow solver to compute the

distance (e.g., (Goldberg and Tarjan, 1990)). In more

detail, we employ a widely used technique in this cat-

egory: the cost-scaling push-relabel algorithm (Gold-

berg and Tarjan, 1986). It maintains a preﬂow and

gradually converts it into a maximum ﬂow by mov-

ing ﬂow locally between neighboring vertices using

push operations under the guidance of an admissible

network maintained by relabel operations. This algo-

rithm will ﬁnd an optimal solution (i.e., one with min-

imal associated cost). In this paper, we use Google’s

implementation of this algorithm

. Its complexity is

O(n

· m · log(n · c)), where n denotes the number of

nodes in the graph, m denotes the number of edges,

and c is the value of the largest induced edge cost in

the graph. Due to this comparably high complexity,

we exploit certain properties, particularly to signif-

icantly reduce the number of edges to make its di-

rect application to cells of a data set computation-

ally feasible. The EMD is particularly popular in the

computer vision community where distances are typ-

ically computed between color histograms, e.g., for

image retrieval (Rubner et al., 2000). To interpolate

between commonly used distributions in computer

graphics (like BRDFs, environment maps, stipple pat-

terns, etc.), Bonneel et al. (Bonneel et al., 2011) de-

compose distributions into radial basis functions, and

then apply partial transport that independently consid-

ers different frequency bands.

3 MOTIVATION AND

FUNDAMENTALS

In this section, we give a quick overview of the prob-

lem determining the transport of total energy for visu-

alization within a given domain, and outline our ap-

proach to solve this problem. We then give a short

https://developers.google.com/optimization

introduction to the relevant fundamentals of energy in

ﬂuid dynamics. The section concludes with a brief

summary of the minimum cost ﬂow algorithm.

3.1 Motivation for Visualizing Energy

Transport

Any process can be divided into variables describing

the state, e.g. velocity, pressure etc., and variables de-

scribing the dynamics as for example various ﬂuxes.

The energy transport we aim to visualize in this pa-

per, can be classiﬁed as pertaining to the dynamics of

any ﬂow. However, these dynamics involved propa-

gating the system between states cannot be easily ex-

tracted in an arbitrary ﬂow, but enable useful insights,

as for example, predictions for the evolution of the

system. In real life phenomena, dynamical data of-

ten cannot be captured, and is unavailable for analysis

(e.g. momentum exchange in global weather air ﬂow

data). Using computational ﬂuid dynamics, the evo-

lution equations (Navier-Stokes equations, see sec-

tion 3.2) modeling the ﬂow are known. Even so, there

are various approaches to discretize and solve these

equations, possibly leading to the resulting dynamics

being difﬁcult to extract from the calculation. Even

if possible, doing this highly depends on the type of

simulation and solver (e.g., implicit or explicit), as

well as the respective implementation. In all cases,

retrieving the state variables is fairly easy.

Our approach estimates the energy dynamics of

any arbitrary system, for which the state variables

are known at different time steps (Sec. 4). One of

the two prerequisites necessary for the employed al-

gorithm is to know how the amount of energy in a

given energy form is related to the state variables of

the system. The other prerequisite is information re-

garding all additional external sources and sinks (or

an approximation thereof), to satisfy the energy con-

servation law. The composition of the total energy,

i.e., the types of energy involved, may be chosen arbi-

trarily for our calculation. This is an important prop-

erty of our approach in that it is widely applicable to

data from many kinds of sources. In particular, using

only an initial and ﬁnal state, knowledge of the ac-

tual evolution equations, which, as explained above,

might not always be available, are not necessary to

compute the energy dynamics. However, any addi-

tional knowledge of the domain, or knowledge of the

transport mechanisms, can be included to improve the

accuracy of the estimated energy ﬂuxes.

To render the myriad of energy ﬂux information

yielded by our visualization, we discuss several map-

pings for the data in Sec. 5). These include a vol-

ume rendering for exploration and overview, as well

IVAPP 2017 - International Conference on Information Visualization Theory and Applications

as a glyph-based approach. The glyph approach is

further enhanced by automatically placing the glyphs

for minimal occlusion, while retaining the most im-

portant information.

3.2 Energy in Fluid Dynamics

Several forms of energy are present in a compress-

ible ﬂow data set. First, there is kinetic energy, for

any mass moving at a non-zero velocity. Other kinds

of energy include internal energy, a thermodynami-

cal quantity dependent on the pressure and tempera-

ture. While we focus on these two types of energy,

various other forms of energy can effortlessly be con-

sidered and implemented in a straight forward fash-

ion. In the following, we give a brief overview of the

partial differential equations governing compressible

ﬂow. We also give a quick summary of the energy

types involved and their calculation.

Being independent of underlying governing equa-

tions, we only need to consider the forms of energy

present in a given data set, regardless how the data

was actually produced. For the data sets in this pa-

per, this speciﬁcally means considering the internal

and kinetic energy forms. For all known phenom-

ena in compressible ﬂow, the Navier-Stokes equations

(NSE, 1) prove a very accurate and general model

(Hasert, 2014; Landau and Lifshitz, 1987).

Derived from basic conservation equations for

mass, momentum and energy (yielding (1a), (1b) and

(1c) respectively), the compressible NSE take the fol-

lowing form:

∂

ρ + ∇· (ρu) = 0 (1a)

∂

(ρu) + ∇ · (ρu ⊗ u + pI) = ∇ · σ

(1b)

∂

(ρε) + ∇ · (ρu(ε + p)) = ∇ · (σ

· u) − ∇ · q − B

(1c)

p = ρRT, (1d)

where ρ is the mass density, u the velocity, p the pres-

sure and T the temperature. ε denotes the speciﬁc

energy per unit mass (i.e. the energy density), and

the deviatoric stress tensor. q is the the heat ﬂux.

The ﬁnal equation closing the set is the equation of

state for an ideal gas (1d), using the gas constant R.

B is the sum of conservative external ﬁelds. However,

for numerically solving this set of partial differential

equation, various terms need to be approximated and

modeled accordingly. In some cases, the exact mod-

eling may not be available, as for measured data. The

energy conservation is expressed in terms of the spe-

ciﬁc energy per mass unit ε:

ε = e +

|u|

. (2)

e = c

T is the internal energy. Multiplication with ρ

deﬁnes an energy density e

tot

. In the following, the

term energy will be used synonymous to energy den-

sity for brevity.

Using the NSE as a basis, the types of energy occur-

ring in compressible ﬂow can be derived. A given

energy distribution can be split into several distinct

parts. For one, there is the internal energy given by

int

= ρc

T with the heat capacity c

at constant vol-

ume. The heat capacity c

can be expressed with a

material parameter, the adiabatic index constant γ, as

= R/(γ−1), ﬁnally yielding an internal and kinetic

energy of

int

ρT

γ − 1

(3)

kin

ρ|u|

(4)

Other forms could also be taken into considera-

tion, for example potential energy in external ﬁelds B,

given with e

pot

= ρgz.

The NSE as expressed in (1), do not encompass

all possible ﬂow physical phenomena. Various exten-

sion (e.g. Magnetohydrodynamics, which also con-

sider the Maxwell equations (Goedbloed and Poedts,

2004)) can be found, possibly adding further forms

of energy to the composition of the total energy. The

data sets considered in this paper focus on the inter-

nal and kinetic energy, and assume no external ﬁelds.

In any case, the conserved quantity is the total en-

ergy e

tot

, given by the sum of all the occurring energy

types. For the cases considered in this paper, this is

simply given by e

tot

= e

kin

+ e

int

To fully describe the ﬂow problem, or more

speciﬁcally the energy conservation problem, the sys-

tem needs to be closed, or the interaction with the

surrounding environment must be considered. This

is usually done by restricting measurements or sim-

ulation to a certain domain, for which the boundary

conditions are known or can be modeled sufﬁciently.

3.3 Minimum Cost Flow Problem

On the basis that energy only gets exchanged in a

close neighborhood of a cell, we formulate our en-

ergy exchange with a minimum cost ﬂow problem. In

this subsection, we give a brief summary of this algo-

rithm and its terminology.

A minimum cost ﬂow problem examines the trans-

port of a scalar quantity through a network, deﬁned by

a graph G. Two sets of nodes are given: a source set

S and a destination set D. All nodes s,d in S,D are

assigned an integral scalar value s

called supply

and demand respectively. The supply and demand for

each node in S,D must be chosen so that

∑

= −

∑

. (5)

Transportation-based Visualization of Energy Conversion

Equation 5 expresses that the quantity in question is

preserved. A graph connecting the two sets is estab-

lished by creating a set E of directed edges between

source nodes and destination nodes. Only connec-

tions from S to D are allowed, connections back or

within S or D are disallowed. S,D and E make up the

graph G which deﬁnes the input for a minimum cost

ﬂow algorithm.

The problem now consists of moving supply from

S to D, that is, assigning a set of so-called ’ﬂow’ val-

ues f

for each edge e, so that:

∀s ∈ S : −

∑

f ∈I

f = s

, I = { f

| e starts at s}

∀d ∈ D :

∑

f ∈I

f = d

, I = { f

| e ends in d}

(6)

In simple terms, this means that all the supply needs to

be transported to cover all the demand, and can only

be moved along an edge deﬁned in E. No source node

can distribute more than it has supply, and no destina-

tion node can receive more than it has demand.

This condition alone potentially still allows a myr-

iad of solutions. To further limit the possibilities, a

more speciﬁc one can be selected by introducing a

secondary condition. This requires assigning a cost c

to every edge e in E, which can be arbitrarily adjusted.

With the assignment of c

, the selected solution must

additionally minimize the so-called total cost c

∑

e∈E

(7)

The cost value c

can be lowered for certain edges to

favor these edges being chosen to move the supply

quantity.

4 EXTRACTION OF ENERGY

TRANSPORT

To enable a comprehensive visualization of the en-

ergy dynamics, we ﬁrst need to extract the appropriate

quantities. The second part is to map these quantities

to a renderable representation, explained in Sec. 5.

Initially, the value for the two relevant energy den-

sities from the state variables, employing equations

3 and 4, is calculated. After determining the energy

contained in each cell for two consecutive time steps,

we calculate an approximation of the ﬂux for the to-

tal energy e

tot

between the two time steps. Since the

quantity is conserved, the total change of the energy

in the domain between time steps can be determined

by taking the initial total energy and considering all

additions and subtractions at the domain boundary

(Reynolds transport theorem (Leal, 2007)).

Obviously energy changes can only be transported

by a mechanism (i.e. energy itself does not have

a ’speed’), and therefore the limiting speed of this

mechanism also limits the distance energy can be

moved. For a sufﬁciently small time step, this trans-

port can be assumed to only occur within a small

neighborhood for each cell. This extraction step

“splits off” the energy transport information and pre-

pares it to be mapped to an appropriate visual repre-

sentation. In the following paragraphs, we reformu-

late our problem of determining the energy ﬂow as

a transportation problem. The outcome can then be

directly processed by a standard minimum cost ﬂow

solver to eventually calculate the energy ﬂux.

For the rest of this discussion, the term energy

refers to the total energy density. Matching the en-

ergy ﬂux problem to be solvable by a minimum cost

ﬂow algorithm involves the following steps:

1. Deﬁne all cells of the input grid in the ﬁrst time

step as nodes in S and associate the correct energy

representing the supply s

. Analogously, deﬁne

the cells in the second time step as nodes in D

with demand d

2. Model the correct boundary conditions for the

nodes in both S and D.

3. Convert the ﬂoating point values to integers (re-

quired by the solver), and normalize the energy

to fulﬁll equation (5) in a distribution-preserving

fashion.

4. Determine an edge set E connecting the two time

steps and solve the deﬁned minimum cost ﬂow

problem. In our setup, we employ the google or-

tools solver to evaluate the minimum cost ﬂow

problem.

In the following paragraphs, we explain these steps in

more detail.

1. Preparing the Energy Data. As an example for

given input data without compromising generality, we

consider the ﬂuid state to be deﬁned by velocity u,

pressure p and temperature T in the following subsec-

tions. The step 1 can be implemented in a straightfor-

ward manner. With the velocity u, the pressure p and

the temperature T provided for each cell by the sim-

ulation, the mass density ρ can be calculated using

equation (1d). Using the mass density, the energies

can be determined with equations (3) and (4), leading

to an energy e

tot

(see section 3.2). With the equation

of state, similar calculations can be done for any set

of state variables.

2. Handling the Boundary Conditions. Modeling

the correct boundary conditions for step 2 is more

complex. In particular, the source and destination

time steps cannot be treated symmetrically: Any

IVAPP 2017 - International Conference on Information Visualization Theory and Applications

amount of quantity entering the domain has to be

considered in the source step, while quantity leaving

the domain only applies to the destination time step.

In order to keep the energy conservation assumption

valid, the exchange of energy over the boundaries

must be fully accounted for. This will also ensure

condition (5) is automatically fulﬁlled.

To correctly map the boundary effects, we need to

identify domain boundaries where energy exchange

is possible. For brevity, we only discuss a few impor-

tant exemplary boundary conditions occurring in our

data sets. A correct modeling for missing (or even

completely different) conditions can be easily derived

based on these ideas.

For each of the state variables u, p and T a bound-

ary condition can be deﬁned as explained in sec-

tion 3.2. Periodic boundaries need to be constructed

with the effect that an addition of energy is modeled

in the source time step S, while the exact same re-

moval is implemented in the destination D for the pe-

riodic neighbor. Dirichlet boundary conditions allow

for both adding and removing energy from the do-

main, depending on the prescribed value and the ac-

tual value of the quantity in the adjacent control vol-

ume. For example an inﬂow boundary condition (u,

p and T prescribed) can be modeled by adding a cell

(called boundary node) to the source set S for all grid

cells at a location fulﬁlling the condition (i.e. touch-

ing the inﬂow boundary), and supplying the state vari-

able according to the prescribed values.

Neumann boundary conditions also may change

the energy contained in the domain. As a prominent

example, consider the outﬂow boundary condition,

e.g. where derivatives of u, and T are zero, while

the value of p is ﬁxed. The mass ﬂux itself can trans-

port energy out of the domain, and the difference in p

may exchange energy too. The removed energy can

be calculated taking the amount of mass crossing the

boundary (scalar product of mass ﬂux and boundary

normal), while the pressure is set to the ﬁxed value of

the simulation parameter. Now the associated energy

can be calculated and, just as for the inﬂow condition,

a set of additional cells (i.e. boundary nodes) need to

be added into the destination set D, to be used with the

calculated energy. For different combinations of Peri-

odic, Dirichlet and Neumann conditions for the state

variables, the calculation of energy exchange must be

accounted for individually. Some boundary condi-

tions do not allow for an exchange in the ﬁrst place,

and therefore do not require a special treatment.

Fig. 1(b) shows a schematic of a cross section of

our beam in cross ﬂow data set. The gray boxes indi-

cate areas, to which energy exchange is possible, and

needs to be handled by boundary nodes. Note that

only the connectivity to the data set and energy den-

sity of these boundary nodes is needed, the geometry

is irrelevant.

3. Normalization and Quantization. After prop-

erly setting up the energy distributions in the previ-

ous steps, the rescaling and quantizing step 3 is per-

formed, to fulﬁll equation 5 and to turn the ﬂoating

point energy to an integral supply/demand value for

the minimum cost ﬂow algorithm. This involves sim-

ply multiplying all energy values by a ﬁxed value and

rounding down the result to the closest integer.

This quantization, and numerical precision issues, or

even insufﬁciently accurate modeling or estimation of

the boundary condition, can cause the precondition

(5) to fail. To remedy the problem, a normalization

is done to adjust the higher total domain energy to

the lower one. Simply scaling all values by a con-

stant factor (the ratio of the total domain energies) is

not possible, since the energy values are already quan-

tized. Instead, a unit of supply/demand from a node

in the set with the higher total domain energy is re-

moved in a random fashion, until both time steps con-

tain the same accumulated total energy. To keep the

overall distribution approximately the same, cells get

selected for removal proportional to their energy con-

tent.

4. Building and Solving the Graph. Now that the

nodes S,D, and associated supply and demand s

of the graph have been established, the edge set E can

be constructed in step 4. Following the assumption

at the beginning of this section, that energy can only

propagate to cells within a certain neighborhood, the

edges can be constructed accordingly. For each node

in the source S, a neighborhood set is selected in the

destination set D based on spatial proximity. Fig. 1(a)

shows the most simple subgraph for a single desti-

nation cell (blue). This cell can potentially receive

an energy ﬂow from the source cells (framed in red)

located in the neighborhood. The ﬁgure only shows

immediate neighbors in two dimensions for simplic-

ity, and may actually be an arbitrary unstructured grid

in three dimensions with connection to further neigh-

bors in addition to the nearest. The neighborhood

to which edges are constructed, is determined by a

face-neighbor relationship, and recursive collection of

nearby cells, until a certain maximum distance from

the original cell is achieved.

This distance is determined by the maximum

speed at which a mechanism can transport energy,

multiplied by a constant factor, to account for errors

introduced in the quantization/normalization step.

This can happen when the algorithm is forced to move

a unit of energy further than it could have propagated

physically, and can be controlled by the edge cost c

Transportation-based Visualization of Energy Conversion

Boundary

(a) Simpliﬁed minimum

cost ﬂow subgraph for a

single cell (blue) and con-

nected source cells (red).

Inﬂow Outﬂow

Outﬂow

(b) Boundary conditions

example, here for the

beam in cross ﬂow data

set.

Figure 1: Establishing the minimum cost ﬂow network

graph, and how boundaries are considered.

(see below). Note that a cell’s source node has an

edge to the same cell’s destination node. A boundary

node, depicted in Fig. 1(a) with a dashed red line, is

only connected to it’s adjacent “spawning” node.

All edges e in the set E get assigned a cost c

based

on Euclidian distance between the cell centers. There-

fore, the actual geometry of boundary nodes is arbi-

trary, since the boundary node only has a single edge,

connecting it to the source/destination cell in the data

domain to which it injects, or from which it removes

energy, respectively, and must fulﬁll equation (6). We

promote the use of shorter edges, based on the fact

that the physical energy transport is continuous, and

the assumption that the time interval between time

steps is very small. We do this by deﬁning the edge

cost c

using the squared Euclidian norm on the vec-

tor connecting the cell centers s,d: c

= ks − dk

With construction of the edge set E the energy

transport has been fully mapped to a minimum cost

ﬂow problem. To solve the problem, we employ a

generic minimum cost ﬂow algorithm with the prop-

erty of ﬁnding an optimal solution, in the sense that

the total cost c

(see equation (7)) for this solution is

a global minimum for all possible solutions.

By executing the minimum cost ﬂow algorithm on

the graph G a ﬂow value f

for each edge in E is de-

termined. These ﬂow values can be interpreted as the

energy needed to be transported from starting (source)

nodes to ending (destination) nodes, to achieve the en-

ergy distribution in the second time step starting from

the ﬁrst. In addition, the total cost c

for the gener-

ated solution can be interpreted as a coarse indicator

for the accuracy of the approximation.

5 VISUAL MAPPING OF ENERGY

TRANSPORT QUANTITIES

Obviously, the representation provided by the min-

imum cost ﬂow algorithm cannot be displayed in a

straightforward manner, since the graph G is usually

very dense. Even when considering the fact that the

outﬂow values from source nodes contain the same in-

formation as the inﬂow values from destination nodes

(only sorted differently) there is still too much infor-

mation per cell. For each destination cell for example,

there can be several (incident) edges, of which each

in turn carries two relevant scalar values, the energy

distribution s

(e.g. encoded in the ratio e

int

tot

) in

the source cell, and the energy ﬂow amount f

. Addi-

tionally, the destination cells energy ratio d

has to be

considered to extract actual conversion information.

We show several approaches which help to explore

the energy conversion under different aspects. These

mappings may also be combined to complement each

other.

5.1 Energy Conversion

First, we deﬁne a scalar ﬁeld giving an overview of

the energy conversion. This ﬁeld incorporates the

transported amount, as well as the ratios and their spa-

tial distribution over the entire domain. The propaga-

tion direction of energy is lost, however. To calculate

an expressive scalar value C for the conversion, given

the input as explained in the introduction part of this

section, we use a point-based intermediate data repre-

sentation mapping each point to a single scalar.

For our intermediate representation, we construct

a point p

for each of the incident edges in the graph

G. The geometry of p

is simply the cell center of

the destination node. To deﬁne a scalar value C

max

for

each point, we calculate s

and d

for the source and

destination cells. We then take the absolute value of

the difference between s

and d

and weight it with

the amount of energy f

transported over the edge:

max

= ks

− d

k · f

(8)

This yields a scalar for the conversion ratio be-

tween consecutive times in each cell, regardless of

cell geometry or absolute values, while still encoding

all important information.

5.2 Volume Rendering

With Volume Rendering being a good tool to show

spatial distributions, we use it to generate overview

approaches of the energy transport.

IVAPP 2017 - International Conference on Information Visualization Theory and Applications

Total Energy Flow Amount. The simplest visualiza-

tion of the energy transport data is summing up the in-

ﬂow of energy f

over all incident edges of the graph

for each cell. Displaying this scalar in a volume ren-

dering can give a quick overview of where energy was

transported between cells, and the amount. However,

any directional information, as well as energy compo-

sition information is lost (e.g. Fig. 3(a)).

Direct Conversion Rendering With the Conversion

quantity deﬁned above (Sec. 5.1) being a scalar, a di-

rect volume render can be performed.

Both representations can then be rendered as a com-

mon scalar ﬁeld as a volume rending. The resulting

scalar ﬁeld is then rendered by common volume ren-

dering techniques (e.g. Fig. 3(d)).

5.3 Glyph-based Mappings

To preserve the information of local energy transport

on the scale of a cell, we employ glyphs for the mul-

tivariate data.

Transport Glyph. This visualization has the advan-

tage of keeping all the information calculated intact,

including directions of transport, as well as energy ra-

tios. The simplest way of displaying all the graph’s

information is directly showing a subgraph for each

node in a set of selected destination cells.

For every edge from a source cell to the destination

cell, a glyph is constructed. For the energy remain-

ing in the cell, a sphere is drawn, otherwise a cone

is shown, pointing towards the destination cell. The

radius of the sphere, or the cone base respectively, en-

codes the amount of inﬂow f

from the source cell,

while the color shows the conversion ratio. The vi-

sualization enables a detailed examination of energy

ﬂow and conversion for a smaller region. The glyphs

may be placed randomly or manually.

Importance-Weighted Glyph Placement. To avoid

the disadvantages Transport Glyph (occlusion) and

Direct Conversion Rendering (aggregated informa-

tion) have, both variants can be joined into a com-

bined visualization. Based of the conversion ratio

(Sec. 5.1) as an importance weight, cells are selected

as candidates for displaying glyphs. Using a user-

deﬁned threshold for the conversion ratio C

max

, a cell

is added to a set of glyph candidates. Since display-

ing a glyph for all candidates would still lead to mas-

sive occlusion, candidates are then randomly picked

to represent a small neighborhood. The size of this

neighborhood can also be interactively chosen by the

user, offering an overview vs. detail trade-off. This

allows for several valid placement distributions. To

select one, we run the placing algorithm several times,

and keep the distribution with the most glyphs, as this

(a) Pressure ﬁeld (b) Temperature ﬁeld

Figure 2: Volume renderings of the input ﬁelds. The images

show the state variables for the beam in cross ﬂow data set

from the destination time step for comparison to our ap-

proach shown in ﬁgure 3.

is the one offering the most information. Note that

the glyph distribution can be controlled by any scalar

value as weighting, and may be used in conjunction

with completely different feature detection methods.

6 RESULTS

We employ several ﬂow simulation data sets to dis-

cuss the results achieved with our approach and

demonstrate its utility. These data sets were pro-

duced by different computational ﬂuid solvers. For

the beam in cross ﬂow data set, most simulation pa-

rameters were known, particularly an exact account-

ing for all the boundary conditions of all variables. In

summary, the data consists of an unstructured, time-

varying grid containing 14469 cells. The topology is

constant, and 7 boundary patches are deﬁned. The

state of the ﬂuid is recorded in the velocity, pressure

and temperature. The boundary conditions are set to

appropriate Dirichlet values for the inlet, “slip” for the

side walls, “outﬂow” for top and back walls, and “no-

slip” for the ﬂoor. See below for exact values. For the

5jets data set (provided by K.L. Ma, UC Davis), only

the state variables were available. The data consists

of a regular grid, converted to the corresponding un-

structured mesh consisting of 262144 cells. The state

variables provided are internal energy, density and ve-

locity. We choose the time interval between time steps

to be sufﬁciently small for our assumptions to apply.

The boundary condition had to be reconstructed by

considering the data close to the boundary, and es-

timating sensible boundary condition types and val-

Transportation-based Visualization of Energy Conversion

(a) Total Energy Flow (b) Sparse Visualization (c) Zoomed in on structure (d) Conversion

Figure 3: All representations of the energy transport result generated by the minimum cost ﬂow algorithm. (a) The Total

Energy Flow clearly shows the decrease of energy along the x-axis. (b),(c) The Transport Glyph gives a detailed look of

a region of interest (the beam) and the surrounding energy ﬂow including the composition of the energy. (d) Finally, the

Conversion shows the energy exchange between kinetic and internal energy for the entire domain.

ues. For the nozzle data set, all necessary parame-

ters were known and implemented accordingly. De-

tails can be extracted from the publicly available data

set (see Sec. 4). The calculations for the minimum

cost ﬂow algorithm and the subsequent visualization

were performed on an Intel i7-2600K without multi-

threading. In our implementation, we used Google’s

implementation of a minimum cost ﬂow solver to de-

termine the energy ﬂux from our constructed graph.

In the following, we give a short summary of the

data sets, and present our examination results for our

visualization and the implemented mappings in sec-

tions 6.1, 6.2 and 6.3 respectively. We then discuss its

computational performance (section 6.5), and ﬁnally

present our expert study (section 6.4).

6.1 Beam in Cross Flow

For this dataset, many of the deﬁning parameters for

the energy transport problem are known, and can be

directly implemented. The foam-extend-3.1

pack-

age for ﬂow simulations was used to generate this

simulation data set. It is a fork of the well-known

OpenFOAM package

. The constants employed are

= 100,T

= 100,R = 1, γ = 1.4. The images in

Fig. 2 show volume renderings of the pressure, den-

sity and velocity magnitude ﬁelds, respectively. We

produced a result for all the representations explained

in Sec. 5, which can be seen in Fig. 3.The image in

Fig. 2(a) shows the Total Energy Flow Amount moved

into a destination cell. One can clearly see the di-

minishing energy transport from the inlet (on the left)

towards the outlet (on the right). This immediately

implies that energy, being only injected on the inlet

boundary condition, dissipates from the domain, at

approximately constant rate. This suggests that the

energy leaves the domain through the top boundary

http://www.extend-project.de/

http://openfoam.org/

(being an outﬂow), and the lower boundary (being a

“no-slip” wall). However, this visualization type does

not show a detailed transport of energy, nor any en-

ergy composition information for the region of inter-

est in the data set, the beam structure in the center.

For this, the Transport Glyph can be employed.

The images 3(b) and 3(c) show glyphs depicting

the individual subgraphs for manually selected cells

around the structure. Since in this simple case the

region of interest is known, a manual selection is pos-

sible. The large sphere in the center implies that for

the lower middle part of the structure, not too much

energy is moved, and the small cone pointing into the

sphere shows the small amount of energy entering the

cell. The other cones and spheres can be interpreted

in a similar fashion, and it can be easily seen that the

energy ﬂow is similar to the mass ﬂow. However, con-

sidering Fig. 2(c), differs signiﬁcantly, as the energy

has other modes of transportation than convection.

The visualization gives a detailed view of the trans-

port for individual cells, and is applicable when the

region of interest is small. Finally, conversion per

cell is rendered in the last ﬁgure showing the Energy

Conversion. Employing the volume rendering tech-

nique allows for a complete overview of the whole

domain.

6.2 5jets

For the 5jets data set, the producing parameters are

not known. With our technique, only knowledge of

the boundary conditions is necessary. Information

about the boundary condition can be derived by as-

suming a given boundary condition, and testing if pa-

rameters can be found that match the values on cell

boundaries. After empirically determining the bound-

ary conditions to be outﬂow on all sides and the top,

and a slip condition for the bottom, as well as 5 cir-

cular patches being the inlets for the jets, we modeled

our setup accordingly. Since the inlet velocity was

IVAPP 2017 - International Conference on Information Visualization Theory and Applications

(d) Density (e) Internal energy (f) Momentum

Figure 4: Volume renderings of the input ﬁelds using com-

mon visualization techniques. The images show the state

variables for the 5jets (top row) and the nozzle (lower row)

data set from the destination time step for comparison to our

approach shown in Fig. 5 and Fig. 6.

found to vary over time, we also employed further ad-

justments to the empirically determined inlet velocity.

With our approach still being very stable against these

inaccuracies, these assumptions allow for an analysis.

In Fig. 4 we show how the destination time step’s state

variable ﬁelds look like in an internal energy, density

and velocity magnitude volume rendering.

Fig. 5 shows the application of the Importance-

Weighted Glyph Placement. A low-opacity volume

rendering of the density for the destination time step

has been added for context. The utility of the (in-

teractively adjustable) conversion threshold parame-

ter C

max

becomes apparent when used to explore the

conversion distribution as a whole, and ﬁnd areas with

a fairly high (or low) conversion (top row, Fig. 5(a)-

Fig. 5(c)). Zooming in on a high conversion area iden-

tiﬁed this way, energy conversion can be examined

per cell. The glyph on the left of Fig. 5(d) receives

a high amount of energy from the cells in the area

below (→ large diameter on cones), only the fraction

transferred from the cell directly below is heavily con-

verted between energy forms (→ saturated red). The

glyph on the right has little conversion and a low in-

ﬂow of total energy from other cells. On the other

hand, the energy remaining in the cell (→ encoded in

sphere), is subject to a fair amount of conversion.

6.3 Nozzle

The nozzle data set, created with OpenLB

, is a good

example of a more turbulent simulation. Parameters

deﬁning the simulation can be extracted from the ex-

amples bundled with the solver. In Fig. 4 we show the

destination time step’s state variable ﬁelds, similar to

the 5jets data set. For context, the bounding geometry

http://optilb.org/openlb/

(a) C

= 0.1 (b) C

= 0.5 (c) C

= 5.0

(d) Detailed glyphs

Figure 5: Energy Conversion glyphs for the 5jets data set for

several conversion thresholds C

max

. (a) - (c) An overview

of the energy conversion can be created by setting a low

threshold. (d) Zooming in, one can adjust the size and den-

sity of the glyphs to reveal the details of the energy ﬂow.

While the conversion for the left cell is heavily inﬂuenced

by the energy ﬂow from below, for the cell on the right it

is rather low, with highest conversion taking place for the

energy remaining in the cell.

of the nozzle and tube are shown. Fig. 6 shows a dis-

tribution acquired using Importance-Weighted Glyph

Placement. The density of the glyph coverage as

well as the glyph size can be adjusted interactively

to deal with potential clutter. Both color and opacity

of the glyphs depict the Conversion value. This gives

a good overview of the conversion, and also allows

for a closer examination of the conversion-heavy re-

gions in the center of the nozzle’s jet. The Conversion

(encoded in color) is particularly large right after the

nozzle exit, as well as at the “tip” of the jet. Smaller

cones and spheres at the nozzle imply a small transfer

of energy, indicating that most of the kinetic energy is

converted further along the jet.

6.4 Informal Study

For our study, we interviewed a ﬂow scientist in-

volved in the development of compressible ﬂow

solvers, working as Ph.D. student at TU Delft. On

a daily basis, he uses various visualization tools to

analyze the results. For this task, he develops ﬂow

solver algorithms within the OpenFOAM framework

customized to scalable ﬂuid-structure and ﬂuid-ﬂuid

boundary interaction. He conﬁrmed that the local-

ity assumption for energy transport made in section

4 is valid. For the visualization type showing vol-

ume renderings of the total energy, we showed him

the total energy ﬂow for the beam in cross ﬂow, as

Transportation-based Visualization of Energy Conversion

Figure 6: Our energy transport glyphs derived from time

step 49200 → 49800. The outﬂow from the inner regions of

the nozzle’s jet toward the outer layer can be immediately

derived. Also the energy ’dissipating’ right after exiting the

nozzle is rather low (small cone diameters in glyph) the con-

version is very high (red color).

seen in Fig. 3(a). He appreciated that outﬂow through

the top boundary condition, accounting for the loss of

energy over the domain, can be extracted from the vi-

sualization. However, the domain scientist identiﬁed

this visualization inadequately visualizes the energy

ﬂow behavior around the prominent region of inter-

est, the structure in the ﬂuid ﬂow. This is addressed

in the Glyph Coverage image (Fig. 3(c)), which we

presented as a solution. The scientist really liked

the clear visualization and the easy interpretation of

the glyph, and had no problem extracting informa-

tion about energy ﬂow, when asked about the infor-

mation gained. He also was intrigued about the vol-

ume visualization of the conversion ratio. In sum-

mary, the expert was clearly interested in the visu-

alization, and appreciated the fact that the informa-

tion is easily gained from both visualization types.

He was especially intrigued by the Sparse Rendering

/ Importance-Weighted Glyph Placement variants for

allowing both an option for overview while enabling

a detailed examination for more interesting features.

Overall, he expects the visualization in general to be

very valuable, and will yield a signiﬁcant improve-

ment for the analysis of ﬂow in general.

6.5 Timings

There are several steps to compute during the exe-

cution of the algorithm. The timings for calculating

the input energy distributions (i.e. the kinetic and in-

ternal energy from state variables) can be neglected

Table 1: Timings for the different steps performed for

preparing the problem and running the minimum cost ﬂow

algorithm. Once the ﬂuxes have been determined, the var-

ious mappings can be generated and manipulated interac-

tively. The relevant section is given in parentheses. Un-

mentioned timings were negligible ( 10ms).

Data elements Beam 5jets nozzle

Number of nodes 28938 573440 211248

Number of edges 319665 6382336 5116252

Process Graph

Normalization (4.3) 0.7 s 3.85 s 1.41 s

Centers (4.4) 1.02 s 18.25 s 14.43 s

Build graph (4.4) 0.42 s 8.49 s 6.55 s

Run solver (4.4) 10.31 s 624.10 s 557.30 s

Generate Mappings

Total Energy (5.2) 0.05 s 0.92 s -

Energy Conversion (5.2) 0.06 s 1.38 s 1.51 s

Weighted Glyphs (5.3) - 2.13 s 2.10 s

( 1s). As the mesh may be unstructured with ar-

bitrary cells, and varying over time (as is the case for

the beam in cross ﬂow), cell centers need to be recom-

puted for each time step, taking the arithmetic mean

of the vertices deﬁning a cell. The normalization of

the total energy to exactly match contained energy be-

tween source and destination time step, as explained

in section 4, is then performed. Next, the nodes and

edges for the minimum cost ﬂow algorithm are allo-

cated and established. When the graph is established

the algorithm can be executed to solve the problem.

Finally, one or more visualization types can be chosen

to be performed on the algorithm’s result. Timings

were taken for all data sets and the individual steps

performed, the results are given in Tab. 1. The tim-

ings are clearly dominated by the minimum cost ﬂow

algorithm, which has a high complexity in the num-

ber of nodes and edges. Once calculated, the various

mappings can be produced interactively. The proto-

type implementation has various ways for improving

performance, in particular using additional physical

constraints like ﬂuid parameters to reduce the number

of edges needed for the solution of the minimum cost

ﬂow problem.

7 CONCLUSION

We presented a technique to visualize the transport of

and conversion between internal and kinetic energy

from arbitrary compressible ﬂow simulations. For this

purpose, we modeled our energy transportation prob-

lem as a graph, and solved it via a minimum cost

ﬂow solver, inherently respecting energy conserva-

tion. Here, we explicitly considered of various sim-

ulation parameters like boundary conditions and en-

IVAPP 2017 - International Conference on Information Visualization Theory and Applications

ergy transport mechanisms. Based on the resulting

ﬂux, we then derived a local measure for the conver-

sion between energy forms using the distribution of

internal and kinetic energy. We ﬁnally employed dif-

ferent mapping approaches, and evaluated our tech-

nique by means of different simulation data sets and

feedback by a domain scientist. In particular, we in-

troduced glyphs to speciﬁcally depict convergence.

By controlling glyph placement via the conversion ra-

tio, we minimized occlusion problems by focusing on

regions that are generally of interest. For future work,

we aim to further study and evaluate our approach

with additional types of simulations, as well as data

from measurements obtained via experiments. An-

other promising improvement to enhance the quality

of the total energy ﬂux estimation, is deriving edge

weights from given physical information, e.g. using

the (easy to calculate) momentum ﬂux to favor certain

edges. For the Importance-Weighted Glyph Place-

ment, glyph density and size could be automatically

adjusted in a view-dependent fashion.

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Transportation-based Visualization of Energy Conversion