3.3 Numerical Example II
3.3
83
Numerical Example II
As a further exemplary test problem we consider the multi-objective optimization of a heat
exchanger configuration with respect to four dimensions in the objectives space, these are
the pressure drop between the inlet and the outlet channel, the maximum temperature
across the outlet, the total area occupied by the fluid, and the outflow velocity, where the
last one is measured as indicated in figure (3.49) (and will be denoted as outflow velocity
as it is actually a velocity measured at a specific point of interest). After describing the
problem and the corresponding flow model we investigate the performance of the proposed
method and the influence of its characteristic parameters.
The considered heat exchanger configuration consists of 4 hot objects at temperature
Th = 520◦ C passed by a fluid within a rectangular box with inlet and outlet channels.
The configuration is illustrated in figures (3.49) and (3.50) together with the numerical
grid. The ambient temperature is Ta = 220◦ C which is prescribed at the inlet and the
outer walls. The fluid properties are the same as in the previous example and chosen such
that the Reynolds number based on the inlet velocity and channel height is Re = 40 and
the Prandtl number is Pr = 6.70.
Inflow
Heat unit 1,
4 design variables applied
Outflow velocity
measurement
Outflow
Figure 3.49: Cutout of Flow Geometry.
The design variation is possible via the shape of the hot objects, defining 18 design
variables as indicated in figure (3.50). There is a strong interaction between the objectives,
such that the problem is representative for a conflicting multi-objective optimization.
84
3. Multi-Objective Optimization
4 design variables
on each heat unit
2 additional design variables
at each edge of geometry
Figure 3.50: 18 design variables as indicated.
3.3.1
Geometry representation
We aim to find a geometry design which provides the optimal condition for the flow
process. This is achieved by parametrizing the shape such that the design can be easily
modified in an experimental way. The deformation of the geometry is obtained via splines
which make it possible to express the connecting shape in parametric form [127] (see
section 2.5). The geometry representation with respect to the heating objects and the
mesh is adapted automatically. The spatial discretization employs 124 928 control volumes
on the finest grid and 5 grid levels are used in the multi-grid method (the third grid is
shown in figures (3.49), (3.50)). The design variables are restricted to upper and lower
limits such that there can not occur invalid shapes in the decision variable space.
The configuration consists of four hot units determined by four design variables each
(see figure (3.50)) plus two design variables on the lower left and upper right part of the
geometry. By this technique, we convert a shape optimization task to a parameter value
optimization task. Figure (3.50) is an exemplary deformed grid and indicates the location
of design variables.
3.3.2
Numerical Results
We again divide the numerical experiments in two scenarios, each of which entails several
simulation runs. In the first scenario, we chose a high probability for individuals to
undergo mutation and recombination where the second scenario slows down this effect
3.3 Numerical Example II
85
until population diversity is only driven by recombination. The settings for the two
scenarios are summarized in table 3.3.
Scenario
1
2
Mutation type
independent-bit
no mutation
Mutation/Bit-turn
probabilities
0.5/0.3
0/0
Recombination
type
Uniform crossover
one-point crossover
Recombination
probability
0.9
0.7
Table 3.3: Scenario Simulation Settings
The variation operators work with random values, initialized with a seed. This seed, and
so the initial population are the same for each optimization strategy. Hence, the first and
all succeeding generations are going in the direction the selector suggests to. The main
advantage using the PISAlib [168] software is that it allows a separation between variator
and selector. So we are able to apply different selector schemes as NSGA-II, SPEA2
and FEMO are applied to the same optimization problem, implemented in a separate
routine. The data flow communicates over text files which all algorithms share access.
Our investigations can also be seen to promote using the presented software concept.
The aim is to find a parameter combination indicating Pareto alternative designs with
as few solver runs as possible. Here, each flow evaluation requires a demanding numerical process involving 124 928 control volumes on the fifth grid of a multigrid algorithm.
In more realistic configurations which are probably more complex, or have more control
volumes as in three dimensional configurations, the whole optimization process would
rapidly become infeasible. Therefore, our investigation yield a recommendation of parameter combination that would be most successful for this kind of problem. It is not
necessary to use thousands of function evaluations or hundreds of generations - though,
it is important to set up the appropriate configuration of those parameters.
In figure (3.51), we indicate vectors volmin , volmax , pdmin , pdmax , tempmin , and tempmax ,
which correspond to the minimum and maximum volume, pressure drop and temperature,
respectively. Figures (3.51-3.58) and also (3.64-3.68) are 3-D scatter plots which are again
illustrated in figures (3.59-3.63) and (3.69-3.74) as 2-D scatter plots. The corresponding
temperature distributions are given in figures (3.83-3.90). Ich each scatter plot, the legend
states the algorithms employed.
Figures (3.51), (3.53), (3.55), (3.57), and (3.58) are 3-D plots between pressure drop,
volume, and temperature. Figures (3.52), (3.54), and (3.56) are the corresponding plots
between pressure drop, outflow velocity and temperature. In figure (3.51), the algorithms
NSGA-II and SPEA2 are shown as indicated in the legend. It can be seen that in both
cases, the Pareto front consists of only few elements and the front is quite well exploited,
i.e. the spread along the Pareto front is satisfyingly depicted. Figure (3.53) shows the final
No.
1
2
Number of
generations
20
12
Initial
population
30
40
Number of
offsprings
15
30
Function
evaluations
329-360
256-274
Tournament
size
6
6
Corresp.
figures
3.51-3.63, 3.83-3.90
3.64-3.74
Table 3.4: Parameters for simulation runs for scenarios 1 and 2
86
3. Multi-Objective Optimization
SPEA2
Femo
NSGA
SPEA
380
380
360
Temperature
Temperature
360
340
vol
min
320
temp = pd
300
max
320
300
280
0.135
0.145
34
= vol
min
0.14
36
min
pd
0.13
260
38
temp
280
260
26
max
340
max
28
30
34
Pressure Drop
36
0.155
30
Pressure Drop
Volume
32
0.15
32
3
0.16
28
26
38
Figure 3.52: 3-D scatter plot scenario 1,
pressure drop, outflow velocity and temperature
Figure 3.51: 3-D scatter plot scenario 1, volume, pressure drop and temperature
Semo
Femo
SPEA
Femo
NSGA
380
Temperature
380
2.75
2.8
360
2.85
340
Temperature
360
340
320
300
2.9
320
280
2.95
300
3
280
26
3.05
28
30
3.1
32
34
Outflow Velocity
0.165
36
3.15
Pressure Drop
Figure 3.53: 3-D scatter plot scenario
1, volume, pressure drop and temperature
Volume
260
40
35
30
Pressure Drop
25
0.18
0.17
0.16
0.15
0.14
Outflow Velocity
Figure 3.54: 3-D scatter plot scenario
1, pressure drop, outflow velocity and
temperature
0.13
3.3 Numerical Example II
87
Femo
NSGA
Femo
NSGA−II
380
340
380
320
360
300
Temperature
Temperature
360
280
260
2.7
2.8
2.9
Volume
3
3.1
3.2
26
30
28
36
34
32
38
340
2.7
320
2.8
300
2.9
280
3
260
3.1
0.13
0.135
Pressure Drop
Figure 3.55: 3-D scatter plot scenario
1, volume, pressure drop and temperature
0.14
0.145
0.15
0.155
0.16
Volume
3.2
Outflow Velocity
Figure 3.56: 3-D scatter plot scenario
1, volume, outflow velocity and temperature
Femo
SPEA
Femo
NSGA
Semo
SPEA
380
26
360
Temperature
30
400
32
350
300
34
250
3.2
36
3.1
3
2.9
Volume
2.8
38
Figure 3.57: 3-D scatter plot scenario
1, volume, pressure drop and temperature
Pressure
Drop
Temperature
28
340
320
2.7
300
2.8
280
260
26
2.9
3
28
30
Volume
3.1
32
Pressure Drop
34
36
38
3.2
Figure 3.58: 3-D scatter plot scenario
1, volume, pressure drop and temperature
88
3. Multi-Objective Optimization
38
380
SPEA
Femo
36
360
34
340
Temperature
Pressure Drop
Femo
SPEA
32
320
30
300
28
280
26
2.8
2.85
2.9
2.95
3
3.05
3.1
3.15
260
2.8
3.2
2.85
2.9
2.95
Volume
3
3.05
3.1
3.15
3.2
Volume
Figure 3.59: Scatter plot scenario 1,
volume vs. pressure drop
Figure 3.60: Scatter plot scenario 1,
volume vs. temperature
0.165
380
SPEA
Femo
SPEA
Femo
0.16
360
340
Temperature
Outflow Velocity
0.155
0.15
0.145
320
300
0.14
280
0.135
0.13
2.8
2.85
2.9
2.95
3
3.05
3.1
Volume
Figure 3.61: Scatter plot scenario 1,
volume vs. outflow velocity
3.15
3.2
260
26
28
30
32
34
36
Pressure Drop
Figure 3.62: Scatter plot scenario 1,
pressure drop vs. temperature
38
3.3 Numerical Example II
89
Femo
SPEA
360
350
Temperature
0.16
0.155
Outflow Velocity
NSGA−II
SPEA2
370
0.165
0.15
340
330
320
310
300
290
280
2.7
0.145
2.8
0.14
2.9
Volume
0.135
3
3.1
3.2
0.13
26
28
30
32
34
36
38
25
30
35
40
45
Pressure Drop
Pressure Drop
Figure 3.63: Scatter plot scenario 1,
pressure drop vs. outflow velocity
Figure 3.64: 3-D scatter plot scenario
2: volume, pressure drop and temperature
generation of the algorithms Semo and Femo and, on contrary to figure (3.51), for both
algorithms it can be seen that there are many individuals in the final generation which
are crowded in a certain part of the Pareto output space. A payoff between the objectives
can be suspected but not affirmed. Figures (3.55) and (3.57) compare Femo and NSGAII, and also Femo and SPEA2 where it is both times quite obvious that NSGA-II and
SPEA2 outperforms Femo in terms of spreading along the alternatives and having fewer
final solutions. Figure (3.58) is then a final plot of all algorithms employed. Drawing the
outflow velocity against volume and temperature in figures (3.52), (3.54), and (3.56), it
can be seen that Femo fails in giving a comparable performance as NSGA-II does. Figures
(3.59)-(3.63) are the corresponding 2-D plots. Figure (3.59) exhibits Pareto alternatives
between volume and pressure drop, figure (3.62) between pressure drop and temperature
(temperature is to be maximized), and figure (3.63) between pressure drop and flow
velocity. We conclude here the same identification and characteristics of the Pareto front
approximation as in the 3-D case. Figure (3.60), on the other hand, does not definitely
show a Pareto front arising.
The observations made shall now be confirmed by another simulation scenario, in which we
work with less generations but more offsprings in each generation. Furthermore, we do not
use mutation and lower the recombination probability. In figures (3.64)-(3.68), we show
the same 3-D scatter plots as for the first scenario. In all of these, the Pareto front is neatly
elaborated. In figure (3.64) the trade-off between objectives is most convincing. In figures
(3.65)-(3.67) it is observed that the algorithms SPEA2 and NSGA-II outperform Semo
and Femo in aspects of spreading along the Pareto front. Figure (3.68) plots pressure drop
and flow velocity against temperature, it is obvious that SPEA2 gives less non-dominated
individuals as the Femo algorithm.
The corresponding 2-D scatter plots are given in figures (3.69)-(3.74). Figures (3.69),
90
3. Multi-Objective Optimization
380
Femo
Semo
380
Temperature
340
360
Temperature
Semo
Femo
NSGA
360
340
320
300
320
300
280
280
260
260
2.7
2.7
2.8
2.8
2.9
2.9
38
36
3
Volume
Volume
34
3
3.1
32
3.1
30
3.2
3.2
28
28
26
Pressure Drop
26
Figure 3.65: 3-D scatter plot scenario
2: volume, pressure drop and temperature
32
30
36
34
40
38
Pressure Drop
Figure 3.66: 3-D scatter plot scenario
2: volume, pressure drop and temperature
Semo
Femo
SPEA
380
SPEA2
Femo
360
380
320
360
Temperature
Temperature
340
300
280
260
340
320
300
280
260
45
2.7
2.8
40
2.9
0.12
0.13
35
3
Volume
3.2
0.14
Pressure Drop
3.1
25
30
35
40
Pressure Drop
Figure 3.67: 3-D scatter plot scenario
2: volume, pressure drop and temperature
45
30
0.15
0.16
25
Outflow Velocity
0.17
Figure 3.68: 3-D scatter plot scenario
2: pressure drop, outflow velocity and
temperature
3.3 Numerical Example II
91
44
380
SPEA2
Femo
SPEA
Femo
42
360
40
340
Temperature
Pressure Drop
38
36
34
32
320
300
30
280
28
26
2.7
2.75
2.8
2.85
2.9
2.95
3
3.05
3.1
3.15
260
2.7
3.2
2.75
2.8
2.85
Volume
2.9
2.95
3
3.05
3.1
3.15
3.2
Volume
Figure 3.69: Scatter plot scenario 2:
volume vs. pressure drop
Figure 3.70: Scatter plot scenario 2:
volume vs. temperature
0.17
380
SPEA
Femo
Femo
SPEA
0.165
360
0.155
340
Temperature
Outflow Velocity
0.16
0.15
0.145
0.14
320
300
0.135
280
0.13
0.125
2.7
2.75
2.8
2.85
2.9
2.95
3
3.05
3.1
Volume
Figure 3.71: Scatter plot scenario 2:
volume vs. outflow velocity
3.15
3.2
260
26
28
30
32
34
36
38
40
Pressure Drop
Figure 3.72: Scatter plot scenario 2:
pressure drop vs. temperature
42
44
92
3. Multi-Objective Optimization
0.17
44
NSGA
SPEA
0.165
42
0.16
40
0.155
38
Pressure Drop
Outflow Velocity
SPEA
Femo
0.15
0.145
36
34
0.14
32
0.135
30
0.13
28
0.125
26
28
30
32
34
36
38
40
42
26
2.7
44
2.75
2.8
2.85
Pressure Drop
2.9
2.95
3
3.05
3.1
3.15
3.2
Volume
Figure 3.73: Scatter plot scenario 2:
pressure drop vs. outflow velocity
Figure 3.74: Scatter plot scenario 2:
volume vs. pressure drop
(3.72), (3.73), and (3.74) indicate the Pareto front between volume and pressure drop,
pressure drop and temperature, pressure drop and flow velocity, and volume and pressure
drop. Figures (3.70) and (3.71), on the other hand, are more blurry and do not conclude a Pareto front between volume and temperature and volume and outflow velocity,
respectively.
Scenario 1
Scenario 2
Scenario 1
Scenario 2
380
2.7
2.75
Temperature
320
2.85
360
280
2.9
340
2.95
320
3
300
3.05
280
3.1
260
25
340
300
2.8
380
Temperature
360
3.15
30
35
40
45
260
Volume
Volume
3
3.2
Pressure Drop
Figure 3.75: 3-D scatter plot: SPEA2
on both scenarios
26
28
30
32
34
36
38
40
Pressure Drop
Figure 3.76: 3-D scatter plot: NSGA-II
on both scenarios
Figures (3.75)-(3.78), and (3.79)-(3.82) indicate the results from an algorithm compared
to the corresponding scenarios. In scenario 2, we used less overall function evaluations as
in the first scenario (see table 3.4). From the given plots it can be seen that both scenarios
yield a similar simulation result for the corresponding algorithms. Thus, scenario 2 should
3.3 Numerical Example II
93
Scenario 1
Scenario 2
Scenario 1
Scenario 2
380
380
360
340
Temperature
Temperature
360
320
300
280
340
320
300
0.18
280
36
260
2.7
34
2.8
0.16
260
25
32
2.9
30
3
Volume
3.2
30
Outflow Velocity
35
Pressure Drop
28
3.1
0.14
40
Figure 3.77: 3-D scatter plot: Femo on
both scenarios
0.12
45
Pressure Drop
26
Figure 3.78: 3-D scatter plot: SPEA2
on both scenarios
44
0.175
Scenario 1
Scenario 2
42
0.17
40
0.165
38
0.16
Outflow Velocity
Pressure Drop
Scenario 1
Scenario 2
36
34
0.155
0.15
32
0.145
30
0.14
28
0.135
26
2.7
2.75
2.8
2.85
2.9
2.95
3
3.05
3.1
Volume
Figure 3.79: Scatter plot: SPEA2 both
scenarios, volume vs. pressure drop
3.15
3.2
0.13
2.7
2.75
2.8
2.85
2.9
2.95
3
3.05
3.1
Volume
Figure 3.80: Scatter plot: SPEA2 both
scenarios, volume vs. outflow velocity
3.15
3.2
94
3. Multi-Objective Optimization
380
Scenario 1
Scenario 2
0.16
Scenario 1
Scenario 2
360
0.155
Outflow Velocity
Temperature
340
320
300
0.145
0.14
280
260
26
0.15
0.135
28
30
32
34
36
Pressure Drop
Figure 3.81: Scatter plot: NSGA-II
both scenarios, pressure drop vs. temperature
38
40
0.13
26
28
30
32
34
36
38
40
Pressure Drop
Figure 3.82: Scatter plot: NSGA-II
both scenarios, pressure drop vs. outflow velocity
be preferred since it allows a lower computational overhead. Figure (3.80) thereby exhibits
a non-convex Pareto front which is described by SPEA2 on scenario 1, but not as clearly
with scenario 2. All other plots show that both scenarios yield similar results: SPEA2 and
NSGA-II outperform Femo and Semo in either scenario setting in aspects of spreading
along the Pareto front.
Finally, figures (3.83)-(3.90) are exemplary flow configurations as indicated in scatter plot
(3.51). For each flow configuration, we first show the temperature and also the flow
velocity distribution. With Reynolds number Re = 40, buoyancy effects can be neglected
and thus convection drives the heating effects. Streamlines are drawn as to indicate the
flow field. It is quite interesting to note that the configuration with maximal heat rods,
i.e. minimum covered flow region (figures (3.83) and (3.84)) is not the configuration with
maximum heating performance (figures (3.89) and (3.90)). It can also be confirmed that
the configuration with minimum pressure drop is the configuration with smallest covered
flow region (figures (3.85) and (3.86)). Figures (3.87) and (3.88) shows the individual with
lowest heat performance and, in comparing with configuration (3.89), (3.90), it can be
concluded that forcing the flow as circulating around the outer region increases heating
performance. This furthermore goes along with a high pressure drop value.
From the observations made, the conflict between objective functions pressure drop and
temperature increase can clearly be seen in different designs. The Pareto fronts between
fluid area and pressure drop as well as between pressure drop and temperature are also
clearly visible. It can be observed that the algorithms SPEA2 and NSGA-II outperform
the “simple” algorithms Semo and Femo in each simulation run. Especially the diversity
operators are ensuring a well spread of solutions along the design alternatives. Thus,
sharpness and diversity between design alternatives is most convincing. Femo and Semo
exploit the design regions where they get crowded, but do not spread sufficiently along
3.3 Numerical Example II
95
the Pareto front.
An improved approximation capability from a larger number of function evaluations (scenario 1 vs. scenario 2) cannot be observed in our examples. From our point of view,
scenario 2 with no mutation is most appropriate for the problems considered. The comparison of simulation scenarios in figures (3.75-3.82) indicate that scenario 2 provides
good, if not better performance than scenario 1 since it requires less function evaluations.
Each flow evaluation involves several minutes computing time according to our numerical
setup and depending on the actual grid deformation. The lower the number of function
evaluations needed, the better the overall optimization evaluation.
Figure 3.83: Temperature distribution for individual volmin .
Figure 3.84: Flow velocity distribution for individual volmin .