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Interactive Visualization of transient 3D
Electromagnetic and n-dimensional Parameter
Spaces in Virtual Reality
Martin Hafner, Stefan Böhmer, François Henrotte and Kay Hameyer
RWTH Aachen University, Institute of Electrical Machines
Schinkelstraße 4, D-52056 Aachen, Germany,
Martin.Hafner@iem.rwth-aachen.de
1 Abstract
Purpose - Nowadays many parameter studies for the design and optimization
of electromagnetic devices are carried out by means of 2D and 3D nonlinear
finite element models. Through optimization algorithms selecting one design as
optimal with respect to the chosen cost function, the user does not gain any
intuitive clue of the interrelations existing between design parameters, although
numerous computations have been performed across the whole parameter space
of the system.
Design/methodology/approach - This paper presents a visualization ap-
proach for n-dimensional parameter spaces of FE solutions in virtual reality
and the corresponding interpolation methods for enabling navigation through
it.
Findings - The solution of an arbitrary electromagnetic FE problem is catego-
rizes with respect to possible changes, due to chosen design parameters, within
the solution itself and variations in the underlying mesh in order to find ap-
propriate interpolation methods for the visualization of each type of parameter
space.
Practical implications - The implementation bases on modifying the popular
VTK framework.
Originality/value - Different solution approaches for the visualization of an
interpolation between arbitrary different meshes are presented as well, but the
problems remains unsolved and requires further research.
Keywords - Virtual reality, Visualization, Parameter space methods, VTK,
Scientific visualization
Paper type - Research paper
2 Introduction
Along with the development of computers, electromagnetic finite element (FE)
simulation software has attained nowadays a high level of sophistication. In
many real-life engineering situations, the problem is no longer only to be able
1
to simulate, but also to be able to interpret correctly and efficiently the huge
amount of numerical information generated by the simulation.
Complex FE simulations are routinely used for design purposes of electric
machines. With standard methods,e.g. Differential Evolution or Simulated
Annealing, the optimization process is performed blindly, like a black box, with
no prior knowledge of the system. At the end of the process, the engineer
has gained no intuitive understanding of the existing interrelations between
the different design parameters, and he must trust the optimum found by the
algorithm. A number of optimization techniques attempt to build along the
way some global knowledge about the objective function, and to perform some
sensitivity analysis, but it is seldom pointed out how much this construction is
precisely an aspect of technical design where the global cognitive capacities of
the human brain could definitely be worth being more explicitly exploited with
the help of virtual reality (VR) tools.
Therefore, the aim of this work is to visualize a complex nonlinear FE so-
lution system, which is parameterized through a number of control variables,
in virtual reality. This allows users to intuitively experience the sensitivity of
response under specific parameter changes in the multi-parameter space. In
case of a dynamic response, time can be chosen as the coordinate in a mono-
dimensional parameter space and the user can sweep back and forth in time,
with a cursor for instance, in order to analyze in detail the dynamic evolution
of a quantity of interest.
All FE simulations reported in this paper have been done with the finite ele-
ment package iMOOSE (van Riesen, Monzel, Kaehler, Schlensok & Henneberger
2004) whose numerical results were converted into the VTK data structure
(Kitware 2006).
3 The visualization pipeline of VTK
The Visualization Toolkit VTK (Schroeder, Martin & Lorensen 2006) is an
open source platform independent software library for 3D computer graphics,
image processing and visualization. It is applied for scientific purposes in various
fields of research (Joshi, Scheinost, Vives, Spencer, Staib & Papademetris 2008),
(Sustersic, Kandemir, Phoha & Schmiedekamp 2008), (Badesa, Pinto, Sabater,
Azorin, Sofrony & Cardenas 2009). This software package is capable of scalable
parallel processing (Ahrens, Law, Schroeder, Martin, Inc & Papka 2000) and is
also used for supercomputing visualizations (Ahrens, Lo, Nouanesengsy, Patch-
ett & McPherson 2008). The working principle of VTK is based on visualization
pipelines Fig. 1(a). These visualization networks are constructed by connecting
data objects, representing and providing access to data, and process or filter
objects that operate on those data objects (Schroeder, Avila & Hoffman 2000).
Each pipeline object has an internal state control to detect when a reexecute
command is necessary. Generally data update is performed from a source to a
drain object, as shown in the figure.
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4 Methodology
4.1 Requirements
Requirements on parameter space visualization arise from the data and the tasks
to be fulfilled. The aim is to easily compare and evaluate the results of multiple
FE computations with a changing set of parameter. More precisely, there is
one FE dataset for each set of parameter values. This leads to a multidimen-
sional parameter space with no restriction in dimension or in the number of
available solutions. For better comparison and an intuitive recognition of the
observed phenomena within the solution parameter space, an interpolation of
the discrete steps should be performed to gain a continuous parameter space.
This interpolation functionality has to consider two different cases occurring in
electromagnetic FEA.
•Solution Interpolation - By changing material characteristic or excita-
tion conditions, the magnitude and direction of the obtained field solution
differs, but remain defined on the same mesh. We only have to differ be-
tween continuous but discretized parameters (e.g. current density) and
non-continuous values (steel properties).
•Mesh Interpolation - A variation of geometric parameters, such as teeth
or yoke width typically, leads to FE results, where the underlying mesh
changes between two different parameter values. Here, a mesh to mesh in-
terpolation is required to map between field nodes which do not necessarily
have the same global coordinates.
The method proposed here, also works for FE models with motion, where a
particular part of the mesh is rotated or translated in function of time. An
application of arbitrary hp-adaption techniques within the FEA is unaffected
by the visualization procedure, but requires further processing within the inter-
polation, see 5.3, 5.4.
Finally, the tool has to be integrated seamlessly into the existing inter-
face (Schoning & Hameyer 2008), (Hafner, Schoening, Antczak, Demenko &
Hameyer 2009) between iMOOSE and VTK, but should also be able to run
independently of iMOOSE in conjunction with any other VTK task. Further-
more maintainability and compatibility regarding future releases of VTK are
two important demands.
4.2 Design and Implementation
Given the fact that VTK is used for visualization and that there is an exist-
ing, recently published, implementation for time support (Biddiscombe, Geveci,
Martin, Moreland & Thompson 2007) (which essentially is a mono-dimensional
parameter space), it is considered to extend this implementation concept to hold
a multidimensional parameter space. The time support in VTK is realized by
attaching time specific meta information to the pipeline architecture which can
be passed up and down the pipeline, see Fig. 1(b). Hereby filters are enabled
to request specific time steps from a previous filter or source in the pipeline and
can pass the data down the pipeline. By activating the SnapToStep function-
ality, the interpolation between solutions is deactivated for performance issues
3
Source Data Data
Filter Drain
(a) Classical visualization pipeline of VTK.
Data
Update Parameter
Data RenderingInterpolator SnapToStep
Data
DataSource
(b) Pipeline concept with time meta informations.
Figure 1: Different pipeline concepts of VTK.
and the navigation is limited to in memory available cases. This time support
represents exactly the desired behavior for the special case with only one param-
eter. So the idea of our design is to replace the scalar time by a vector whose
size equals the dimension of the parameter space, respectively the number of
parameters.
Figure 2 shows a two dimensional parameter space, where a time-transient
FE computation is carried out for a series of different excitation currents. The
crosses symbolize the computed FE solutions corresponding to a unique param-
eter set (t,J). The path represents the interactive navigation through the solu-
tions in virtual reality. At step 1
a new FE solution is loaded and projected in
VR. At step 2
and 3
a solution interpolation is required, hence the underlying
mesh is constant. A navigation through time at steps 4
and 5
requires both a
mesh and a solution interpolation for a mono-dimensional direction (time axis).
The same operation in bi-direction is given in 6
.
1
2
3
45
6
J [ A
mm2]
t [s]
Figure 2: Interactive navigation through a 2D parameter space.
4
5 Implementation details
5.1 Pipeline architecture
Coming up with version 5.0 VTK got a new pipeline architecture (Aylward,
Barre, Cedilnik, Hoffman, Ibanez, King, Martin, Moreland, Schroeder & Squillacote
2006) - the official documentation of each class can found in (VTK Online
Documentation 2010). The old pipeline concept handled both pipeline execu-
tion and algorithm operation in every filter or data source, allowing computing
only in forward direction. Performing an update or refresh command on the
render in Fig. 1(b) provokes the rendere to reload the data output of the Snap-
ToStep filter - no further request etc. are initiated. The new architecture splits
the pipeline in two separate classes. VtkAlgorithm is responsible for providing
and manipulation of data and vtkExecutive connects the different filter of the
pipeline and handles execution and communication. Following this approach it
is possible to transfer additional information up and down the pipeline by using
the executives. Furthermore all the data transfer between two objects in the
pipeline is handled by the corresponding executives. This enables the executives
to change the transmitted data before passing it to the next filter. In the VTK
5.0 concept, the exemplary refresh command of the render is communicated
backwards down to the DataSource and afterwards thought all contributing
filters back to the renderer.
VTK time support (Biddiscombe et al. 2007) uses the new architecture to
attach information about given or requested time steps and time range to the
communication in the pipeline. It is realized by extending the pipeline operation
by the following requests:
•REQUEST_INFORMATION communicates information about time steps
and time range down the pipeline.
•REQUEST_UPDATE_EXTENT communicates desired time steps up
the pipeline.
•REQUEST_DATA generates data in the filter and populates it down the
pipeline.
In this paper, this approach is extended to fulfill the requirements of a mul-
tidimensional parameter space. As sketched in 4.2, the idea is to replace the
scalar time by a vectorial representation t
t7−→
p1
:
pn
, (1)
where each entry piin this vector represents one dimension of the parameter
space and the value of piindicates a specific point within the field range of the
space dimension. This results in a vector holding n elements in the case of a
n-dimensional parameter space.
The VTK pipeline requests of the time support are designed to handle mul-
tiple time steps within one request. Based on this characteristic, the enhanced
n-D time vector, holding one parameter set, can be transcipted into a sequence
of time steps. By this, the basing pipelining functionality of VTK can be used
5
without modification. The time mapping and its reverse has to be performed
in each filter which manipulates the timing vector t, e.g. SnapToStep, or
processes data in function of t, e.g. Interpolator, cf. Fig. 1(b).
5.2 Source object
The new object DataSource is the first element in the execution pipeline of
Fig.1(b). This object is a container for all the solutions of a previously performed
FE computation. The solutions are internally stored in a vtkTemporalDataSet
which simply represents a sequence of data sets. Therefore, this class requires
the time to sequence transcription to identify every single solution by its unique
parameter set t.
5.3 Interpolator
Say one has got a FE model Ωdepending on ndesign parameters spanning a
n-dimensional parameter space Λ, and assume the model has been evaluated
for a cloud of points λi∈Λ. For each sampled point λi, the solution of the
FE problem for a physical quantity νhas been visualized over the domain of
analysis as a colormap of the solution:
v(x), x ∈Ω(2)
From a functional point of view, the displayed solution consists of 2 elements:
the evaluation point x∈Ωand the value v(x)associated by the model to the
point.
For the purpose of a smooth visualization of the solution v(x)when navi-
gating the parameter space Λ, interpolation methods between several solution
maps must be applied.
In order to formalize this problem mathematically one has to distinguish
amongst the ndesign parameters between the geometrical parameters ςi, i =
1, .., m and the non-geometrical parameters µj=m, .., n, so that the solution of
the FE problem can be written
v(x(ςi), µj). (3)
The interpolation is now obtained by the combination of a number of neigh-
boring pre-computed solutions. For the sake of simplicity, we assume only the
closest available solutions, represented by ςA
i, µA
jand ςB
i, µB
jrespectively,
are used. The intermediary states between them, assuming an affine linear
combination, are
ςi(ζ) = ςA
i+ζςB
i−ςA
i, ζ ∈[0,1] (4)
µj(ζ) = µA
j+ζµB
j−µA
j(5)
and the purpose of the interpolation method is to provide an approximate color
map
˜v(ζ)∼
=v(x(ςi(ζ)) , µj(ζ)) . (6)
This setting with only two solutions and a linear affine interpolation is rather
simple but it suffices to explore the main difficulties encountered. The µ-
parameters can be treated by blending techniques.
˜vblend = (1 −ζ)vx, µA
j+ζx, µB
j(7)
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This operation is implemented in 5.4.
If the mesh is not fixed during the FE computations (ς-parameters), for
example due to rotation, geometric parameter variation or mesh refinement
techniques, an additional mesh interpolation has to be performed. This inter-
polation consists of two tasks:
•Vertex correspondence: A mapping between the vertices (nodes) of
the two data sets has to be computed.
•Vertex path: After the mapping is found the displacement of each vertex
from its initial position to the new one has to be done.
Amongst the ς-parameters, one has to further distinguish between those that
represent a rigid body motion (angular position of the rotor for instance) from
those that describe a deformation of the underlying geometry.
In the first case, the vertex correspondence depends explicitly and exclusively
on the design parameter ς, and it can be treated rather easily. If one is aiming
at an intuitive access to the visualized field quantities and a minor loss accuracy
is acceptable in this case, the vertex path can be interpolated linearly between
the two given vertices, since the discretization of rotation is a small fraction of
an electrical period. This is also implemented in 5.4. If this simplification is not
sufficient, one can apply advanced approaches which try to preserve the mesh
structure, e.g. the intrinsic solution of (Sederberg, Gao, Wang & Mu 1993),
which modify the angles and edge lengths of the mesh. In case of an rotating FE-
part, the proposed approach can be simplified, since the rotation transformation
for each time step is explicitly known.
In the second case, morphing techniques must be applied:
˜vmorph =v(1 −ζ)xςA
i+ζx ςB
i, µj(8)
A general solution, which does not require any further information about
the mesh or of its solution is given by Sederberg et al. (Sederberg & Greenwood
1992), who use a physically based model: one of the meshes is modeled as
a construct of wire. Then a vertex mapping is chosen which minimizes the
amount of work needed to transform this mesh to the second one by bending
and stretching. A more sophisticated solution is presented in (Tian, He, Cai &
Feng 2006), where arbitrary meshed geometries are transformed into each other.
A satisfying solution for the visualization by smooth interpolation between such
types of solution is still an open question.
To demonstrate this situation, Fig. 3 shows the image blending between
two almost identical FE simulations, where the tooth head varies. Even if this
method only consists of a cost effective pixel-wise addition of the rendered solu-
tion, tendencies are slightly distinguishable. Since FE electromagnetic governing
equations are usually first-order vector potential formulations, leading to zero-
order magnetic flux density solutions, an application of time-consuming field
projection would lead to a visualization of comparable quality.
5.4 Enhanced VTK Interpolator
Interpolation is necessary to navigate smoothly through the parameter space
because containing data is given by discrete points representing different param-
eter sets. Therefor a new filter called Interpolator is build based on the VTK
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(a) Present FE solution ONE (b) Image blending 33%
(c) Image blending 66% (d) Present FE solution TWO
Figure 3: Image blending between two FE solutions with a tooth head variation.
time support interpolator. Interpolator is placed in the execution pipeline
and changes parameter information in the REQUEST_INFORMATION phase
by deleting the parameter points and only passing the parameter range down
the pipeline. Thereby, the next filter object in the pipeline is able to request
any point in the parameter space. If this point is not equal to one of the given
discrete points the interpolator generates it on demand.
The multidimensional parameter space interpolation is carried out by the
inverse distance weighting (IDW) (Shepard 1968), as sketched in case of 2D in
Fig. 4 to compute the function value at x, y by the surrounding points 0-3. The
value of a desired point is computed by a linearly weighted (affine) combination
of the surrounding discrete parameter points. Weighting is done by applying a
function of the inverse distance between the two points. Formally the method
is given by
v(x) = PN
i=0
vi
di
PN
i=0
1
di
, (9)
where xdenotes the desired point, v(x)the value of the desired point, vithe
value of the given point xiand dithe distance between xand xiin the n-
dimensional space. Ndenotes the number of neighbor points which equals 2n
in a regular n-dimensional grid. A intermediate position between two points in
one dimension is simply identified as weighted combination of the two points.
The advantage of this simple identification is, that the navigation in higher
dimensional spaces can be based on exactly the same principle. Moreover, this
allows a combination with arbitrary user interfaces or navigation models, such
as the high sophisticated time model of (Wolter 2010).
8
x1
x0
y1
v0
v2v3
v1
d3
d2
X-Axis
Y-Axis
x
y
y0
d1
d0
Figure 4: A exemplary 2D-Interpolation by IDW.
6 Conclusion and further prospects
In this paper, the popular open-source framework VTK is applied and expanded
to visualize multi-parameter spaces of finite element solutions. The correspond-
ing VTK filter chains are presented - necessary adaption within the VTK struc-
ture are proposed. To navigate smoothly through the parameter space, which
assists an intuitive interpretation of the ongoing changes within the FE solution,
a VTK based mesh and solution interpolation is applied. In present state, this
interpolation fails in case of a variation of the CAD model and consequently of
the underlying mesh, since this requires a mapping function between geomet-
rically different discretization. Even if several approaches from literature are
discussed, the development of an generally applicable solution for that problem
is current state of research.
To improve the immersion of such techniques, multi-parameter visualizations
should be projected in CAVE (Cave Automatic Virtual Environment) or CAVE-
like environments. In a further step, this would require an integration of a
virtual reality library, e.g. VISTA (van Reimersdahl, Kuhlen, Gerndt, Henrichs
& Bischof 2000). Further, the implementation of FEA restarting facilities would
enable to perform simulations as a user’s reaction within the VR scene. The
implementation of an adequate input device for the navigation through the
multi-parameter spaces would also be of assistance.
7 ACKNOWLEDGEMENT
This research was funded by the SEED Fund project "‘Interactive Data Visual-
ization in 3D Electromagnetics"’ of the Jülich Aachen Research Alliance (JARA)
(JARA - Jülich Aachen Research Alliance - Jara 2010). We are grateful to all
members of the Virtual Reality Group of the RWTH Aachen University headed
by Prof. Kuhlen for their support.
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