The paper considers the problem of evaluating the state of a generalized computational experiment in the context of a general problem of creating methods for adaptive planning and control of a generalized computational experiment in mathematical modeling of real physical processes. A generalized computational experiment implies multiple solution of the numerical simulation problem for various sets of values of defining model parameters. As a method for assessing a generalized computational experiment state, it is proposed to visualize a set of experimental data specifying this state then followed by analysis of the resulting visual image. An approach to visualization of a generalized computational experiment state is proposed based on constructing a visual map. The concept of a visual map of a generalized computational experiment is introduced, and several methods of its construction are proposed. Examples of application of those methods are considered when assessing the accuracy of numerical models of the OpenFOAM software platform for a three-dimensional problem of inviscid flow around a cone.
Keywords: generalized computational experiment, generalized computational experiment state, multidimensional data, visualization, visual analytics, visual map, approximation, problem of flow around a cone, OpenFOAM.
In the tasks
of mathematical modeling of physical processes a computational experiment plays
an important role. In the general case it consists in a series of calculations
with varying the defining parameters of the model. In this case, the task of
the experiment is often to simultaneously investigate the influence of several
parameters on the characteristics of modeling object of interest, including
investigating their joint influence in various combinations of variation
ranges. With the development of computer technology, it became possible to
construct a so-called generalized computational experiment (GCE) [1], which
assumes parallel calculation of the same problem with different sets of parameter
values in multitask mode. Currently, there are examples of successful
construction and application of a GCE in solving problems of computational
fluid dynamics [1-2], gas dynamics [3-4], power plant design automation [5].
From the point
of view of representing the results, a generalized computational experiment is
characterized by a multidimensional array the elements of which specify the
distribution of values of simulation output parameters for a given set of input
parameters. In this case, some generalized indicators often act as output
parameters, which are the results of processing primary experimental data,
contain information about general patterns and relationships inherent in the
object of modeling, and are used for interpretation, search for patterns,
formation and testing of hypotheses. Examples of generalized indicators are the
principal components in dimension reduction problems [6], L1 and L2 error
vector norms in problems of estimating the accuracy of various numerical
methods with varying the key simulation parameters [3-4], etc.
It is obvious
that both conducting a GCE and processing and interpreting its results are very
resource-intensive tasks. Moreover, it is not possible to conduct an experiment
with all allowable combinations of models and simulation parameters. Therefore,
it is necessary to resort to GCE planning choosing a specific scenario for its
implementation. At the same time, it is advisable to build a GCE on the basis
of not a static, predetermined, but a dynamic, adaptively changing plan. The
principle of constructing such a plan can be as follows: based on the results
of a series of experiments for a certain set of values of input parameters and
processing of its results in conjunction with the results of previous series of
experiments, the current GCE state is recorded. This state must be evaluated
and analyzed in order to determine or adjust the plan for the subsequent series
of experiments. To do this, it is necessary to establish dependencies of the
output simulation parameters on the input ones and on this basis to select the
value ranges of the input parameters for which more detailed studies are
required with new or refined sets of parameters. Such situations may arise, for
example, when new patterns are discovered that require confirmation and
refinement, or if the results of some already conducted experiments do not
correspond to the expected patterns and thus require rechecking.
As a method
for evaluating a GCE state, it is proposed to visualize the experimental data
that define it, followed by analysis of the resulting visual image. The
approach based on visualization of multidimensional data has proven itself well
in the tasks of exploratory analysis and hypothesis formation [7]. Visual
analysis of a GCE state helps to visually and quickly enough detect and
highlight problematic or promising value ranges of input parameters that are
subject to more detailed study.
A number of
papers are devoted to the study of visualization problems in a generalized
computational experiment, among which, for example, [2, 3, 5] can be noted. At
the same time, these studies are mainly aimed at application of visualization
methods for the analysis and interpretation of experimental results.
Visualization methods that could be used to evaluate a GCE state in order to
clarify the scenario for its implementation are currently out of consideration.
This paper attempts to fill this gap.
The concept of a visual map of a generalized
computational experiment is introduced, which is understood as a set of visual
images characterizing the state of GCE and arranged in accordance with certain
rules. Several methods are proposed for constructing a visual map of the GCE,
the application of which
is considered by the example of evaluating the accuracy of numerical models of
the OpenFOAM software platform [8] for a three-dimensional problem of inviscid
flow around a cone.
Earlier in the
article [9], the authors considered the problem of adaptive planning and
control of a generalized computational experiment and proposed the general
structure of the GCE control model. It was shown that the key concept for this
model is the concept of the state of a generalized computational experiment.
The state of the GCE is determined by a set of computational experiments
already carried out and is set by a multidimensional array of experimental data
obtained with the current partition of the space of defining parameters, a set
of generalized indicators obtained as a result of processing this array, as
well as a set of patterns identified on the basis of analysis and
interpretation of these indicators.
With regard to
the problem of constructing a visual image of the GCE, the following formal
representation of the state of the GCE can be proposed, which can be considered
as a refinement and concretization of the formal representation proposed in [9]
in the context of the visualization problem.
Let us assume
that within the framework of a GCE on a set of models
M
= {m1,
m2, …,
mNm), where
Nm
is the number
of models,
Nk
computational experiments were carried out within which
the input simulation parameters
P
= {p1,
p2,
…,
pNp} were varied, where
Np
is the number of input
simulation parameters. As a result of the computational experiments, the output
parameters
S
= {s1,
s2, …,
sNs}
were determined, where
Ns
is the number of output parameters. As noted
above, the output parameters can be generalized indicators, which are the
results of processing the primary experimental data. Each computational
experiment
k
was carried out for a fixed set of input simulation
parameters
Pk
= {pk,1,
pk,2,
…,
pk,Nk}. In this case, situations are possible when, for a
fixed set of simulation parameters, computational experiments were not
performed on all models. The results of the performed computational experiments
form a set
R
= {rk,m,v}, where
k
is
the number of the computational experiment (1 ≤
k
≤
Nk),
m
is the number of the model
(1 ≤
m
≤
Nm),
v
is the
number of the output parameter (1 ≤
v
≤
Ns).
Thus, the
state of the GCE is determined by a given partition of the space of values of
the input parameters of modeling from the set
P, the set of values of
the output parameters included in the set
S, as well as the results of a
series of computational experiments that form the set
R.
The visual
image of the state of the GCE can be presented in the form of a set of visual
elements on a plane or in space, linked in a certain way with each other. Each
visual element reflects a subset of the set
R
of GCE results and is
characterized by a set of visual features. In this work, we will restrict
ourselves to considering the following visual signs:
•
coordinates
on the plane (x,
y) or in space (x,
y,
z);
•
Shape;
•
Size;
•
Color.
At the same
time, we note that in addition to the listed features, it is possible to use
others, such as color saturation, orientation, texture, etc.
With a large
number of modeling parameters, a separate visual image of the state of the GCE
is able to reflect only a certain part of the set
R
of its results.
Thus, to visualize the state of the GCE, it is necessary to construct a set of
interconnected visual images. We will call such set
avisual map of a
generalized computational experiment.
To construct a
visual map of the GCE, you first need to prepare the data, which consists in
transforming the initial representation of the state of the GCE to a set of
dependencies of the following type.
For the
two-dimensional case:
F(x)
= <
y,
Shape,
Size,
Color
>,
or
F(x,
y) = <
Shape,
Size,
Color
>
For the
three-dimensional case:
F(x)
= <
y,
z,
Shape,
Size,
Color
>,
or
F(x,
y) = <
z,
Shape,
Size,
Color
>,
or
F(x,
y,
z) = <
Shape,
Size,
Color
>.
In this case,
the arguments of the function
F
are determined primarily by the input
parameters of the modeling, and the sets of its values are primarily associated
with the output parameters, as well as with the characteristics of the
experimental results for which the visual image is constructed. Some methods
for such a transformation are proposed in this work and will be discussed in
the next section.
The resulting
set of visual images can be subjected to preliminary analysis using well-known
methods and visual analytics tools. This approach assumes that each visual
image is analyzed separately, with subsequent comparison and integration of the
results of such analysis. We will call this approach interactive.
To obtain a
holistic visual presentation, a variety of visual images are converted into a
visual map, which in particular involves the search for a suitable
transformation method, for example, in the form of a certain set of rules for
layout visual images. The approach to the visual analysis of the state of GCE
based on the construction of a visual map will be called a complex one.
Let us
visualize the state of the GCE using a series of two-dimensional approximating
curves of the results of computational experiments. Considering the fact that
there can be several input simulation parameters, let us apply the following
algorithm (Fig. 1) to construct two-dimensional approximating curves for one
model
m.
1.
Fix sequentially each
input parameter
pi. Values of this parameter determine the
values of
x
pairs (x; y) for which we will further carry out
approximation.
2.
Determine set
T
= {td}
of all possible combinations of the remaining input parameters
pj
(p
≠
pi),
where 1 ≤
d
≤
Nti, Nti
is the
number of possible combinations of the remaining input parameters for a fixed
parameter
pi.
3.
For each such combination
td,
obtain
Nsd
dependences of the output parameters
sv
on the parameter
pi(it is assumed that not all combinations
of input values could be obtained for all output parameters).
4.
Carry out construction of
the functional dependence using approximating functions. To do this, first
determine the number –
Nfi,d,v
and the form of possible
approximating functions
for each resulting
parameter
sv, a combination of the input parameters
d
and a fixed input parameter
i. The values of the resulting parameters
sv(the corresponding
rk,m,v
are selected from the set
R)
specify the values of
y
in pairs (x; y) for which we will carry
out the approximation.
5.
For each approximating
function
fi,d,v,j
for the resulting parameter
sv
and each combination
td
of input parameters, a graph of
functional dependence on the parameter
pi
construct and
approximation accuracy is determined
ei,d,v,j.
6.
Visually compare of graph
shapes is carried out for a fixed parameter
pi
for different
approximating functions, and deviations and patterns are revealed. Among other
things, shapes of the curves obtained for different resulting parameters are
compared.
7.
Choose the following fixed
input parameter:
i
=
i
+ 1 and go to step 2.
8.
If all the input
parameters are exhausted, then the algorithm is completed.
The presented
algorithm is repeated for each model.
Fig. 1. Flowchart of
visualization algorithm for a GCE state using a series of approximating
graphs
The method
based on application of this algorithm makes it possible to visually identify
deviations in the results of experiments and determine patterns; however, it
does not reflect a GCE state as a whole, since it forms not a single visual
image but a series of visual images. In other words, this method implements an
interactive approach to visual analysis of the GCE state and can be considered
as an intermediate stage in the process of constructing a visual GCE map.
For the
transition from a series of visual images to a single visual map, it is
proposed to use the following method. The obtained characteristics of
approximation accuracy
ei,t,v,j
are summarized in tables:
for each pair of a fixed input parameter
i
and an output parameter
v,
we obtain one table, the columns of which are approximating functions
fj,v,
and the rows are specific values of the remaining parameters. The cells of this
table are the corresponding values of approximation accuracy –
ei,t,v,j.
For each row of these tables, let us define characterizing values by the
following methods:
1.
Minimum accuracy of
approximation (one parameter).
2.
Maximum accuracy of
approximation (one parameter).
3.
Average accuracy of
approximation and root-mean-square deviation of different methods of
approximation from the mean (two parameters).
Visualization
of the obtained characteristics will be carried out using two-dimensional dot
plots. On this graph, for each resulting parameter
v, the corresponding
points will be of the same color. For different resulting parameters
v,
the colors will be different. We will also visualize different input parameters
i
using different colors (different types of markers can be also used).
The ordinal number of the table rows will be used as the value of the point
along the abscissa axis in a two-dimensional visualization. The characterizing
values will be used as values along the ordinate axis. When using the 3rd
method for determining the characteristic values, the average accuracy will
determine the value along the ordinate axis, and the root-mean-square deviation
will determine the size of the point.
As a result, a
set of dot plots will again be obtained, but their number is already significantly
less than in the previous approach, and various layout methods can be applied
to this set of visual images, such as overlay, horizontal and vertical
alignment, etc. At the same time, to give the resulting image more visual
expressiveness, you can additionally use various types of transformation of
coordinate grids – stretching, rotation, mirroring.
Thus, on one
two-dimensional dot plot, we can reflect a GCE state for one specific model.
Analyzing the relative position (and the size of the points for the third
method of determining the characteristic values) of the points, it is possible
to visually determine for which input parameters there are problems with the
approximation accuracy, and as a consequence, errors in carrying out
computational experiments are possible, and for which it is not yet possible to
determine the best approximation methods and, consequently, determine the value
ranges of the input parameters for which it makes sense to carry out additional
computational experiments in intermediate values.
Let us
consider application of the proposed methods of constructing a visual map for a
GCE carried out within evaluating the accuracy of solvers of the OpenFOAM
platform (in OpenFOAM terminology, solvers are software modules in which
various numerical models of mechanics of continua are implemented [8]) for
the three-dimensional problem of inviscid flow around a cone [4, 10] (Fig.
2). Solvers rhoCentralFoam, pisoCentralFoam, sonicFoam will be considered as
models
M.
Fig. 2.
The density distribution and the streamline on the cone surface in a
supersonic flow angle of attack
The simulation
input parameters (P) are Mach number (Ma), cone half-angle (Betta,
in degrees), and angle of attack (Angle, in degrees). The output
parameters (S) of computational experiments are: the results of
calculating deviation norms L1 and L2 of the numerical calculation from the
analytical solution.
As
approximating functions, we will use the following:
•
linear
(y
=
ax
+
b);
•
exponential
(y
=
aebx);
•
logarithmic
(y
=
a
ln(x) +
b;
•
quadratic
(y
=
ax
2
+
bx
+
c).
As the
approximation accuracy, we will use the value of the approximation reliability
R2
[11].
As a result of
the approximation of the obtained data, 178 approximating functions were
constructed for each solver, considering the fact that some types of
approximations in specific cases could not be carried out. For example, for the
quadratic approximation, in the case of only two points, the graph was
expressed in a line and therefore was not taken into account. Also, for the
logarithmic function, it was impossible to determine approximating functions
for the cases when the value of the parameter
x
could be equal to 0. The
visualization results of the obtained approximating functions are partially
shown in Fig. 3-5. For convenience, one graph displays the curves for all
output parameters (norms L1 and L2).
Fig. 3.
An example of visualization of approximating
functions for rhoCentralFoam solver with a fixed input parameter – angle
of attack
Fig. 4.
An example of visualization of
approximating functions for psioCentralFoam solver with a fixed input
parameter – cone half-angle
Fig. 5.
An example of visualization of
approximating functions for sonicFoam solver with a fixed input parameter –
Mach number
After
constructing a series of diagrams and determining the approximation accuracy,
their mean values and root-mean-square deviations were calculated
(Tables 1-9). Bubble charts were used for visual map constructing,
herewith a different bubble color means belonging to different data series (a
combination of a fixed input parameter and an output parameter), and the size
of a point (bubble) characterizes the root-mean-square deviation of the
approximation accuracy by different types of curves. Fig. 6-8 show thus
obtained visual map of GCE state for different solvers. The map consists of
three bubble charts, each one of which summarizes the results of evaluating the
accuracy of the corresponding solver by representing them as a single visual
image.
Table 1. Characterizing values of approximation for
rhoCentralFoam solver for Ma fixed parameter
Input parameters combination values
Comb. number
L1
L2
Min
Max
Avg
RMSD
Min
Max
Avg
RMSD
Betta
Angle
10
0
1
0,94309
1
0,97756
0,00177
0,96078
1
0,97931
0,00078
10
5
2
0,94501
1
0,97771
0,00164
0,96456
1
0,97964
0,00067
15
0
3
0,95181
1
0,98145
0,00130
0,96382
1
0,98245
0,00066
15
5
4
0,95938
1
0,98295
0,00096
0,96914
1
0,98295
0,00049
15
10
5
0,96687
1
0,98423
0,00078
0,97589
1
0,98439
0,00037
20
0
6
0,97009
1
0,98661
0,00069
0,97306
1
0,98468
0,00038
20
5
7
0,97103
1
0,98600
0,00065
0,97530
1
0,98478
0,00035
20
10
8
0,95286
1
0,98151
0,00124
0,96410
1
0,98312
0,00065
Table 2. Characterizing values of approximation for
rhoCentralFoam solver for Betta fixed parameter
Input parameters combination values
Comb. number
L1
L2
Min
Max
Avg
RMSD
Min
Max
Avg
RMSD
Ma
Angle
3
0
1
0,95043
1
0,98336
0,00161
0,95677
1
0,98652
0,00126
3
5
2
0,95835
1
0,98699
0,00117
0,96930
1
0,99076
0,00063
3
10
3
1
1
1
0
1
1
1
0
5
0
4
0,97991
1
0,99376
0,00026
0,98595
1
0,99512
0,00012
5
5
5
0,96365
1
0,98749
0,00087
0,98076
1
0,99376
0,00024
5
10
6
1
1
1
0
1
1
1
0
7
0
7
0,94123
1
0,97376
0,00216
0,97109
1
0,98996
0,00056
7
5
8
0,95094
1
0,97905
0,00154
0,97709
1
0,99178
0,00035
7
10
9
1
1
1
0
1
1
1
0
Table 3. Characterizing values of approximation for
rhoCentralFoam solver for Angle fixed parameter
Input parameters combination values
Comb. number
L1
L2
Min
Max
Avg
RMSD
Min
Max
Avg
RMSD
Ma
Betta
3
10
1
1
1
1
0
1
1
1
0
3
15
2
0,85490
1
0,90400
0,01383
0,93488
1
0,96118
0,00236
3
20
3
0,18749
1
0,46021
0,43708
0,98542
1
0,99207
0,00011
5
10
4
1
1
1
0
1
1
1
0
5
15
5
0,97911
1
0,98748
0,00024
0,97777
1
0,98816
0,00025
5
20
6
0,05399
1
0,36938
0,59651
0,92132
1
0,95017
0,00376
7
10
7
1
1
1
0
1
1
1
0
7
15
8
0,96513
1
0,97780
0,00074
0,99184
1
0,99594
0,00003
7
20
9
0,91134
1
0,94293
0,00490
0,91572
1
0,94759
0,00418
Table 4. Characterizing values of approximation for
pisoCentralFoam solver for Ma fixed parameter
Input parameters combination values
Comb. number
L1
L2
Min
Max
Avg
RMSD
Min
Max
Avg
RMSD
Betta
Angle
10
0
1
0,96572
1
0,98198
0,00102
0,96733
1
0,98194
0,00055
10
5
2
0,95111
1
0,97941
0,00132
0,97196
1
0,98231
0,00047
15
0
3
0,96671
1
0,98576
0,00069
0,96975
1
0,98576
0,00046
15
5
4
0,97213
1
0,98645
0,00066
0,97739
1
0,98607
0,00030
15
10
5
0,97034
1
0,98600
0,00071
0,97381
1
0,98663
0,00034
20
0
6
0,97305
1
0,98788
0,00052
0,98156
1
0,98771
0,00023
20
5
7
0,97348
1
0,98835
0,00054
0,97972
1
0,98775
0,00023
20
10
8
0,96188
1
0,98500
0,00084
0,96349
1
0,98492
0,00073
Table 5. Characterizing values of approximation for
pisoCentralFoam solver for Betta fixed parameter
Input parameters combination values
Comb. number
L1
L2
Min
Max
Avg
RMSD
Min
Max
Avg
RMSD
Ma
Angle
3
0
1
0,86753
1
0,93004
0,01010
0,90982
1
0,96023
0,00477
3
5
2
0,95472
1
0,98539
0,00136
0,96162
1
0,98745
0,00098
3
10
3
1
1
1
0
1
1
1
0
5
0
4
0,82152
1
0,88742
0,02032
0,97279
1
0,98998
0,00050
5
5
5
0,93841
1
0,97293
0,00235
0,97770
1
0,99217
0,00033
5
10
6
1
1
1
0
1
1
1
0
7
0
7
0,61822
1
0,73936
0,09566
0,94428
1
0,97283
0,00200
7
5
8
0,94427
1
0,97283
0,00200
0,97481
1
0,98972
0,00045
7
10
9
1
1
1
0
1
1
1
0
Table 6. Characterizing values of approximation for
pisoCentralFoam solver for Angle fixed parameter
Input parameters combination values
Comb. number
L1
L2
Min
Max
Avg
RMSD
Min
Max
Avg
RMSD
Ma
Betta
3
10
1
1
1
1
0
1
1
1
0
3
15
2
0,91702
1
0,94593
0,00439
0,87596
1
0,92025
0,00958
3
20
3
0,10339
1
0,40438
0,53217
0,99539
1
0,99778
0,00001
5
10
4
1
1
1
0
1
1
1
0
5
15
5
0,98832
1
0,99256
0,00008
0,97153
1
0,98409
0,00042
5
20
6
0,44460
1
0,63462
0,20036
0,98909
1
0,99411
0,00006
7
10
7
1
1
1
0
1
1
1
0
7
15
8
0,59444
1
0,73036
0,10906
0,98688
1
0,99292
0,00009
7
20
9
0,95681
1
0,97358
0,00107
0,97079
1
0,98366
0,00044
Table 7. Characterizing values of approximation for
sonicFoam solver for Ma fixed parameter
Input parameters combination values
Comb. number
L1
L2
Min
Max
Avg
RMSD
Min
Max
Avg
RMSD
Betta
Angle
10
0
1
0,93393
1
0,98130
0,00303
0,94069
1
0,98329
0,00244
10
5
2
0,83358
1
0,92504
0,01578
0,94642
1
0,98478
0,00197
15
0
3
0,94378
1
0,98416
0,00218
0,95478
1
0,98693
0,00139
15
5
4
0,94249
1
0,98381
0,00229
0,95059
1
0,98562
0,00166
15
10
5
0,95377
1
0,98653
0,00145
0,95639
1
0,98714
0,00128
20
0
6
0,94410
1
0,98423
0,00216
0,95462
1
0,98681
0,00140
20
5
7
0,95508
1
0,98701
0,00137
0,96682
1
0,98993
0,00072
20
10
8
0,94389
1
0,98367
0,00216
0,93180
1
0,97619
0,00303
Table 8. Characterizing values of approximation for sonicFoam
solver for Betta fixed parameter
Input parameters combination values
Comb. number
L1
L2
Min
Max
Avg
RMSD
Min
Max
Avg
RMSD
Ma
Angle
3
0
1
0,95646
1
0,98717
0,00130
0,95243
1
0,98572
0,00154
3
5
2
0,95830
1
0,98732
0,00118
0,95756
1
0,98654
0,00120
3
10
3
1
1
1
0
1
1
1
0
5
0
4
0,97308
1
0,99112
0,00048
0,97513
1
0,99194
0,00041
5
5
5
0,77291
1
0,86953
0,02695
0,95457
1
0,98051
0,00134
5
10
6
1
1
1
0
1
1
1
0
7
0
7
0,97176
1
0,98968
0,00054
0,97402
1
0,99071
0,00045
7
5
8
0,97764
1
0,99180
0,00034
0,98942
1
0,99606
0,00007
7
10
9
1
1
1
0
1
1
1
0
Table 9. Characterizing values of approximation for
sonicFoam solver for Angle fixed parameter
Input parameters combination values
Comb. number
L1
L2
Min
Max
Avg
RMSD
Min
Max
Avg
RMSD
Ma
Betta
3
10
1
1
1
1
0
1
1
1
0
3
15
2
0,96358
1
0,98025
0,00068
0,99746
1
0,99906
0
3
20
3
0,89731
1
0,93560
0,00630
0,99195
1
0,99672
0,00004
5
10
4
1
1
1
0
1
1
1
0
5
15
5
0,96283
1
0,97789
0,00077
0,99861
1
0,99953
0
5
20
6
0,97636
1
0,98561
0,00032
0,95714
1
0,97520
0,00099
7
10
7
1
1
1
0
1
1
1
0
7
15
8
0,99736
1
0,99886
0
0,99584
1
0,99829
0,00001
7
20
9
0,66821
1
0,77960
0,07287
0,99866
1
0,99946
0
When
constructing visual maps in Fig. 6-8, the overlay layout method was used. This
made it possible to reflect several different combinations of the values of the
input and output parameters in one diagram. If you need to focus on each
combination separately, then you can use graph alignment as a layout method
(table layout: left column – L1 norm, right column – L2 norm, rows – fixed
input parameter). As a result, we get a visual map consisting of three sets of
graphs shown in Fig. 9-11.
Fig. 6. Visual map of the GCE for rhoCentralFoam
solver (overlay method)
Fig. 7. Visual map of the GCE for pisoCentralFoam
solver (overlay method)
Fig. 8. Visual map of the GCE for sonicFoam solver
(overlay method)
Fig. 9. Visual map of the GCE for rhoCentralFoam
solver (table layout)
Fig. 10. Visual map of the GCE for pisoCentralFoam
solver (table layout)
Fig. 11. Visual map of the GCE for sonicFoam solver
(table layout)
Analyzing the
obtained visual images, it can be noted that, in most cases, the curve shapes
for norms L1 and L2 are similar for a fixed value of the input parameter and
the corresponding combinations of the remaining input parameters. However, in
some cases deviations are observed. In particular, for sonicFoam solver (Fig.
5) with a fixed input parameter – Mach number and cone half-angle – 10° and
angle of attack – 5°, a significantly greater curvature of the
approximating curve for the L1 norm is observed, as well as the osculation of curves
for the L1 and L2 norms. Similar anomalies were also observed for the same
solver and the L1 norm in two other cases:
1.
Fixed
input parameter – angle of attack, Mach number – 5, half-angle – 10°.
2.
Fixed
input parameter – half-angle, Mach number – 5, and angle of attack – 5°.
These
anomalies suggest that at Mach number of 5, half-angle of 10°, and angle of
attack of 5° for sonicFoam solver, an error could have been made in the
computational experiment or in the calculation of the L1 norm. Further
clarification of this fact by the authors of the computational experiment
confirmed the presence of a technical error related to tabulating the results
of the computational experiments for the given combination of input parameters
– the corresponding value of the L2 norm fell into the table cell for the L1
norm by mistake.
Analyzing the visual
map for solvers (Fig. 6-8), it can also be noticed that for:
•
2
cases for rhoCentralFoam the applied approximation methods give rather
different results in terms of flow (2 large bubbles) and all of them are for
the L1 norm and the angle of attack as fixed parameter, which may mean that
some approximation methods are not suitable in this case.
•
4
cases for pisoCentralFoam the applied approximation methods give rather
different results in terms of flow (4 large bubbles) and all of them are for
the L1 norm (3 of them – for the angle of attack as fixed parameter and one –
for the half-angle as fixed parameter). This also may mean that some
approximation methods are not suitable in this case.
•
3
cases for sonicFoam the applied approximation methods give rather different
results in terms of flow (3 large bubbles) and all of them are for the L1 norm.
Moreover, for the sonicFoam solver, this corresponds to the following
situations:
2.
Fixed input
parameter – half-angle, Mach number – 5, angle of attack – 5°.
3.
Fixed
input parameter – angle of attack, Mach number – 7, half-angle – 20°.
The first two
cases correspond to an already discovered problem by visualizing the
approximating curves. The third case is typical for other computational
experiments and may indicate the need for an additional series of experiments
for intermediate values of the corresponding parameters in order to clarify the
nature of the dependence and, possibly, to correct the list of types of
approximating curves for sonicFoam solver.
The paper
considers the problem of evaluating the state of a generalized computational
experiment and the methods for construct visual maps of a GCE it visual
analysis. An approach is proposed to construct visual maps of GCE based on the
sequential applying of two methods: visualization of a series of dependencies
of the output simulation parameters on the input ones for a given set of
approximating functions and visualization of approximation parameters for
different value ranges of the input parameters with different layout methods.
Due to the use
of the proposed construct visual maps methods, it was been possible to identify
experiments that have signs of errors as well as the value ranges of the input
parameters for which it is advisable to conduct additional experiments for
intermediate values. These circumstances make it possible to correct the
further plan of conducting computational experiments.
In combination
with other methods for verifying GCE data [12], the presented methods can be of
great help to researchers in planning and performing computational experiments.
Their software implementation will allow creating a reliable and efficient
visualization tool that can be used in a wide range GCEs.
It should also
be noted that application of the above visual map constructing methods could
only be possible with a sufficient number of already performed computational
experiments within a GCE, since to construct the approximating curves, at least
2 points are required with a fixed input parameter.
3.
Alekseev,
A., Bondarev, A., Galaktionov, V., Kuvshinnikov, A., Shapiro, L.: On Applying
of Generalized Computational Experiment to Numerical Methods Verification. In:
CEUR Workshop Proceedings of GraphiCon 2020, vol. 2744 (2020). doi:
10.51130/graphicon-2020-2-3-19
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Bondarev,
A.E., Kuvshinnikov, A.E.: Analysis of the Accuracy of OpenFOAM Solvers for the
Problem of Supersonic Flow Around a Cone. In: Shi, Y. et al. (eds.) ICCS 2018,
LNCS, vol. 10862. pp. 221–230. Springer, Cham (2018). doi:
10.1007/978-3-319-93713-7_18
5.
Andreev,
S.V., et al.: A Computational Technology for Constructing the Optimal Shape of
a Power Plant Blade Assembly Taking into Account Structural Constraints.
Programming and Computer Software, 43(6), 345–352 (2017). doi:
10.1134/S0361768817060020
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Gorban,
A.N., Kegl, B., Wunsch, D., Zinovyev, A.Y. (eds.): Principal Manifolds for Data
Visualisation and Dimension Reduction, Springer-Verlag Berlin Heidelberg
(2007). doi: 10.1007/978-3-540-73750-6
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Zakharova,
A.A., Podvesovskii, A.G., Shklyar, A.V.: Visual and Cognitive Interpretation of
Heterogeneous Data. In: Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci.,
XLII-2/W12, pp. 243-247 (2019). doi: 10.5194/isprs-archives-XLII-2-W12-243-2019
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Free CFD Software. The OpenFOAM Foundation, https://openfoam.org, last accessed
2021/07/12.
9.
Zakharova
A., Korostelyov D., Podvesovskii A.: Evaluating State Effectiveness in Control
Model of a Generalized Computational Experiment. In: Kravets A.G. et. al.
(eds.): Creativity in Intelligent Technologies and Data Science. CIT&DS
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Cham (2021). doi: 10.1007/978-3-030-87034-8_16
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Andreev,
S.V., Bondarev, A.E., Bondareva, N.A.: Stereoscopic Construction of Textual
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