MODULE TITLE CONTROL SYSTEMS AND AUTOMATIONTOPIC TITLE.docx
final
1. Importance of Multi-Fidelity Analysis in
Multi-Physics Problems
Deep Atkare
Advisor: Dr. Ramana V. Grandhi
November 21, 2016
Wright State University
Department of Mechanical Engineering
3. 1 Abstract
When analyzing any engineering problem it is usually practiced to simplify the
physics by making assumptions so that problem is solvable in a reasonable
amount of time. Sometimes the simplifications and assumptions made are so
broad that the solution of the problem is too far from the real physics system
behavior. The motive of this presentation is to show the importance of choosing
fidelities while analyzing the problem. We are considering heat transfer analyses
with few different examples considering conduction, convection and radiation as
multi-fidelities and their effects on the results obtained. Multi-fidelity plays an
important role in run time of design optimization problems, therefore it is impor-
tant to acknowledge its use.
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4. 2 Introduction
While carrying out simulations of different domains some assumptions made are
so broad that the solution is too far from the actual result. Similarly, while doing
an optimization problem often multi-physics comes into picture and assumptions
are made to reduce the complexity of the problem.
These multi-physics in a system gives rise to multi-fidelity in the simulation
and it is an important aspect which cannot be neglected at any cost for any prob-
lem. These multi-fidelity models are then sub-divided into high-fidelity and low-
fidelity models based on the degree of active fidelities in the simulation. Often
times results of low-fidelity simulation gives deviation from the high-fidelity simu-
lation which can be neglected at the cost of time-saving. High-fidelity simulations
takes longer as compared to the low-fidelity simulations. In this study we will
be investigating the importance of multi-fidelity in a multi-physics problem using
both high-fidelity and low-fidelity simulations and compare the results obtained.
We are considering heat transfer as our domain of focus and the fidelities
involve heat transfer due to conduction, convection to ambiance, cavity-radiation
and radiation to ambiance. All the mentioned phenomena are the basics of heat
transfer and the equations given below shows how they occur in the mathematical
form. The equation shown below represents multi-mode heat transfer.
qtotal = qconduction + qconvection + qcavity−radiation + qradiation
⇒ qtotal = −k T + hA(T − T∞) + σAF(T4
1 − T4
2 ) + σAT4
We will be simulating few examples using ABAQUS package with various dif-
ferent cases (i.e high-fidelity and low-fidelity simulations) and record the temper-
ature behavior and find out the dominant fidelity in that particular example and
use it in low-fidelity simulations.
3
5. 3 Example 1 : Rectangular Electronic Board
3.1 Problem Statement
This is a problem of combined conduction-convection-radiation from a discretely
and non-identically heated rectangular electronic board as discussed in [1].
As shown in Figure 1 the problem geometry considered is a vertical rectangu-
lar electronic board of height L and width W (L × W = 20cm × 10cm). The board is
provided with three non-identical discrete heat sources embedded in it. The heat
sources are located centrally and axially in the descending order of their dimen-
sions with the largest (Lh1 × Lh1 = 5cm × 5cm), medium (Lh2 × Lh2 = 3cm × 3cm)
and smallest (Lh3 × Lh3 = 1cm × 1cm). The thermal conductivity of the board is
varied between k = 0.25 and 10 W/m K. The board is open to atmosphere for
convection and radiation heat transfer. The cooling medium is air at character-
istic temperature T∞ = 25◦C, the convection heat transfer coefficient h = 5 - 100
W/m2K (with lower limit and upper limit implying the asymptotic free and forced
convection, respectively). And also assumed to be radiatively transparent with
surface emissivity of board = 0.05 to 0.85 (with lower and upper limit implying
good reflector like polished aluminum and good emitter like black paint, respec-
tively). The rate of volumetric heat generation in each of the heat sources is qv
(qv = 5 × 104W/m3). The heat generated in the three heat sources is conducted
both axially and transversely in the board and is subsequently foregone by the
board by combined modes of convection and radiation.
3.2 Simulation Details
Reading the problem statement we can conclude that we have a multi-fidelity
model with conduction, convection and radiation as fidelities. From the given
conditions in the problem statement we can consider the model as plane-stress
condition (i.e no heat flux flowing along the z-direction).
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6. Figure 1: Schematic of Rectangular Electronic Board
Considering the geometry and the loading conditions given in the problem
statement we consider only half model for analysis. To run the simulations we
choose ABAQUS package. Since we are considering heat transfer along x and y-
direction only we used 2D heat transfer elements (i.e element - DC2D4 - 4-node
linear heat transfer quadrilateral). Considering all the heat sources active and
using lower limits of all the parameters (k = 0.25W/m.K, h = 5W/m2.K & = 0.05)
with high-fidelity model we do mesh convergence study as shown in Figure 2.
After doing the mesh convergence study we ran simulations for every case
possible (i.e high fidelity and low fidelity). A parametric study involving varying all
the parameters like conductivity of material, convection heat transfer coefficient
and emissivity of the surface and plot the results.
For low-fidelity models, fidelities like convection and radiation were switched
off in different models to observe the effects of fidelities on the model. Low-fidelity
5
7. Figure 2: Mesh Convergence
models have conduction-convection and conduction-radiation models. Here con-
duction is the mode of heat transfer within the material to have a temperature
gradient in the board in x-y plane.
Low-fidelity simulations have convection to ambiance and radiation to am-
biance as the fidelities with outer surrounding temperature of 25 ◦C.
3.3 Results and Discussions
Carry out the simulations as described in the previous subsection results were
plotted. The temperature contour is shown in Figure 3.
Temperature along the symmetric-axis of the board is plotted with different
values of emissivities as shown in Figure 4.
As shown in Figure 4 the temperature profiles has three local peaks, one per
heat source, with the maximum of the three peaks occurring at the biggest heat
source location. In the example it is observed that the temperature drop is ap-
proximately 22% when the surface emissivity is changed from 0.05 to 0.85. The
results obtained matches with the results of [1].
It is observed that the convection is more dominant in cooling the electronic
6
8. Figure 3: Temperature Contour Plot
Figure 4: Local Temperature Distribution along Axis of Symmetry
7
9. Figure 5: Effect of Convection and Radiation on model individually
board and the effect of each fidelity is observed individually as shown in Figure 5.
If the graph is considered it can be observed that the maximum temperature re-
mains approximately same even if the heat transfer coefficient value is increased
from 50 W/m2.K. That is, the contribution of convection after certain point is lim-
ited and does not help in cooling the board even if the heat transfer coefficient is
increased. Whereas, as shown in Figure 5 radiation causes a significant amount
of temperature drop if the emissivity of the surface is more than 0.45. Figure 5
shows that when only radiation is considered and the surface emissivity is varied
from 0.05 to 0.85 the maximum temperature drop is 60.47 %, whereas, if only
convection is considered and varied from 5 to 100 W/m2.K then the maximum
temperature drop is 45.18 %.
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10. 4 Example 2 : Engine Exhaust Manifold
4.1 Problem Statement
In this example an engine exhaust manifold from [2] has been used to observe
multi-fidelity from heat-transfer point of view. An engine exhaust manifold has
hot exhaust gases coming in from 4 different ports from engine cylinders. The
exhaust has one outlet and four inlet ports as shown in Figure 6.
Figure 6: Engine Exhaust Manifold
The example given in [2] the manifold has hot exhaust gas as heat source
and the heat transfer occurs through convection (heat transfer coefficient, h =
0.0005 W/mm2.◦C ). Cavity radiation occurs at the interior of the manifold with
surface emissivity of = 0.77. The material taken into consideration for manifold
is Gray Cast Iron with thermal conductivity, k = 4.5 × 10−2W/mm.◦C, density
ρ = 7800 × 10−9kg/mm3 and Specific Heat, C = 460J/kg.◦C.
There is convection to ambiance and radiation to ambiance occurring from the
outer surface of the manifold. The heat transfer coefficient of the outer surface,
h∞ = 0.0001W/mm2.◦C with the outer surrounding temperature is T∞ = 30◦C. The
9
11. Figure 7: Engine Exhaust Manifold with Boundary Conditions
radiation to ambiance uses the same value of surface emissivity of = 0.77. The
hot exhaust gases create a heat flux applied to the interior tube surfaces. In
this example this effect is modeled using a surface-based film condition, with a
constant temperature of 816◦C and a film condition of 500×10−6W/mm2.◦C. A tem-
perature boundary condition of 355◦C is applied at the flange surfaces attached
to the cylinder head, and a temperature boundary condition of 122◦C is applied at
the flange surfaces attached to the exhaust as shown in Figure 7.
4.2 Simulation Details
Here we have convection, cavity-radiation, convection to ambiance and radiation
to ambiance as fidelities. Due to the fourth-order dependence of the radiation flux
on the surface temperatures, this example problem is intrinsically nonlinear.
Observing the geometry and the boundary conditions of the example we use 3D
element (DC3D8, 8-node linear heat transfer brick and DC3D6, 6-node linear tri-
10
12. Figure 8: Temperature Contour for Case 1
angular prism heat transfer element) in ABAQUS. The element used for meshing
is linear hexahedral and wedge heat transfer elements. The convection phenom-
ena is model using surface film condition on the faces of elements on the interior
and exterior of the manifold. Cavity radiation is modeled using cavity radiation
interaction on interior face of manifold and radiation to ambiance is modeled on
the exterior face of the manifold using surface radiation interaction.
The main motive behind these simulations is to observe the maximum temper-
ature occurring in the manifold due to hot exhaust gases. Based on the problem
description the model was simulated with varies cases depending upon the fi-
delities. Convection at the interior face of the manifold is the heat source so it
is present in every simulation case. Various low-fidelity and high-fidelity models
were analyzed as follows:
Case 1: Combined Case (high-fidelity model): In this case all the fidelities
are active, that is, fidelities like convection at interior of manifold for heat source,
convection and radiation to ambiance and Cavity radiation.
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13. Case 2: No Radiation to ambiance: In this case all the fidelities are active
except radiation to outer surrounding from exterior of the manifold.
Case 3: No Convection to ambiance: In this case all the fidelities are active
except convection to outer surrounding from exterior of the manifold.
Case 4: No Cavity-radiation: In this case all the fidelities are active except
cavity-radiation. The calculation of view factors at the interior faces of the mani-
fold for calculation of heat flux through cavity radiation is not done in this case,
this makes the simulation faster and reduces the run time of analysis.
Case 5: No Cavity-radiation and no radiation to ambiance: In this case the
simulation consists only convection to ambiance for cooling down the manifold
through exterior. Removal of radiation interaction makes this a linear simulation.
Case 6: No Cavity-radiation and no convection to ambiance: This simula-
tion consists of radiation to ambiance only for cooling down the manifold.
All the cases were simulated and high temperature region was found and the
maximum value was recorded. Figure 8 shows the high temperature region. The
temperature contour plot is similar for all the other cases but the value of maxi-
mum temperature changes.
4.3 Results and Discussions
The maximum temperature was recorded from every case simulation and the
high temperature region was identified as shown in Figure 8. The maximum
temperatures of each case is shown in Table 1. This maximum temperature is
recorded at an interior point where the inlet ports make junction with outlet port.
Case 1 Case 2 Case 3 Case 4 Case 5 Case 6
Max.
Temp(◦
C )
649.619 677.462 710.781 675.73 712.453 747.21
Dev. (%) 0 4.286 9.415 4.019 9.672 15.023
Table 1: Simulation Results
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14. First column in Table 1 shows the high-fidelity results and rest of the columns
show low-fidelity results. First column shows the maximum temperature reached
when all the fidelities are active i.e the cooling takes place through convection and
radiation both. Case 4 (i.e fourth column of Table 1) shows a closer value to the
high-fidelity result as compared to the other low-fidelity results. Second row of the
Table 1 shows the deviation of the maximum temperature from the high-fidelity
result (i.e Case 1). Case 4 (i.e no Cavity radiation) gives a closer result to Case 1
with a deviation of 4%, the difference between the Case 1 and Case 4 temperature
value is of 26◦C which is not a big deviation when it comes to exhaust manifold.
Figure 9: Temperature Contour
Case 1 takes approximately 11 sec. to run whereas, Case 4 takes 6 sec. for
analysis. If an optimization problem is formulated which runs the simulation for
number of times then with low-fidelity (Case 4) results we can save a significant
amount of time.
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15. 5 Example 3 : Heating by Laser
5.1 Problem Statement
This example is taken from an ASTM standard method. In this method an envi-
ronmental chamber is used to heat a thin sample of semi-transparent material
to determine the thermophysical properties as described in [3]. The experimental
setup is shown in Figure 10.
Figure 10: Physical Setup
In this example instead of determining the thermophysical properties of the
material the process will be used to observe the effect of fidelities on the material
with known properties. The properties of the material are: density, ρ = 3690kg/m3,
Specific heat, C = 677.5J/kg.K, thermal conductivity, k = 5W/m.K and emissivity,
= 1 (graphite coating). The above properties are of ceramic like Alumina. The
sample is of size 1 × 1 × 0.1 cm.
The sample is heated is heated in an chamber to a high temperature until
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16. the thermal equilibrium is met with the chamber environment. After the thermal
equilibrium is reached, a laser pulse heats the sample’s top surface for 5 sec and
the sample is to be analyzed for 10 sec. The laser is applied at the center of the
sample. The laser is of a Gaussian distribution as shown in Figure 11.
Figure 11: (a) Sample Geometry and (b) simulation domain.
The sample loses heat by radiation and convection through all surfaces with a
heat transfer coefficient, h = 5W/m2.K and surface emissivity of = 1. The sample
is initially heated to a temperature as shown in Table 2. And reaching a thermal
equilibrium with the heating chamber the laser heat influx values are shown in
Table 1 (second column). The laser beam has a radius of 0.2 cm.
Initial Temp.(K) Heat Influx (W)
300 0.007
800 0.073
1300 0.150
1800 0.250
2300 0.420
Table 2: Initial Temperature and Heat Input
The effect of Henyey-Greenstein phase function for scattering characteristic in
porous ceramic is neglected, hence, the results will vary from the [3].
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17. 5.2 Simulation Details
After considering the geometry and the loading conditions of the example, the
simulation considers only quarter of the geometry. Therefore, the sample size is
0.5 × 0.5 × 0.1cm.
Since the simulation involves only heat transfer analysis the element type se-
lected is 3D heat transfer element (DC3D8, 8-node linear heat transfer brick) as
shown in Figure 13. Here the fidelities are convection to ambiance and radia-
tion to ambiance. Since this is a transient analysis, the effect of just conduction
can also be observed by deactivating radiation and convection to ambiance. So
in total there are 3 fidelities which will be considered for low-fidelity and high-
fidelity analyses. Based on the problem statement a mesh convergence study was
performed as shown in Figure 12.
Figure 12: Mesh Convergence Study
The convection occurs from all the surfaces except from the two surfaces which
comes in symmetric planes of the geometry. Except those two surfaces every
surface has surface-film convection and surface-radiation defined. Based on the
fidelities available in the example, different cases which includes combination of
16
18. Figure 13: Mesh
fidelities or individual fidelities (low-fidelity models) were run. The objective of
the simulations is to find the effect of fidelities on temperature contour so, the
maximum temperature rise was recorded after every simulation. Based on the
fidelities available there were different cases made according to combination as
follows:
Combined Case: In this case all the fidelities are active, that is, heat transfer
by conduction, convection to ambiance and radiation to ambiance (i.e high-fidelity
simulation).
Conduction Only: In this case we observe the effect of conduction itself by
deactivating convection to ambiance and radiation to ambiance. Since these anal-
yses are transient we can see the effect of conduction by considering all the sur-
faces as insulated boundary condition. (a low-fidelity simulation)
Conduction-Convection: In this case just radiation to ambiance is neglected
and rest of the surfaces (except symmetric faces) are given surface-film condition
to see the effect of convection to ambiance. (a low-fidelity simulation)
Conduction-Radiation: In this case we neglect convection to ambiance and
apply radiation to ambiance by using surface-radiation condition on all the faces
of the geometry except the symmetric faces. (a low-fidelity simulation)
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19. 5.3 Results and Discussions
For all the cases temperature at specified points are recorded and the temper-
ature rise considered for results comparison. After running the simulations the
maximum temperature was recorded at the locations A-C, located at the posi-
tions (0,0,0)cm, (0,0.2,0) cm and (0.2,0,0) cm respectively, as shown in Figure
14. Table 3 shows the results of the simulations. To check the effect of fideli-
ties, different high-fidelity and low-fidelity cases were run. The temperature con-
Figure 14: Temperature Contour Plot with Reading Locations
tour for case with initial temperature as 1800 K and all the fidelities active (i.e
conduction-convection-radiation) is shown in Figure 14. The Gaussian distribu-
tion laser beam heats up the center more intensely as compared to the other
region, therefore the point A has more temperature rise as compared to point B
and C. Since, point B and C are symmetric they have same temperature rise.
Figure 15 shows the temperature plot with respect to time of point A for different
cases (i.e high-fidelity and low-fidelity simulations).
Temperature Rise, ∆T (K)
Initial Temp.(K) Heat Influx (W) Combined Case Conduction only Conduction-Convection Conduction-Radiation
300 0.007 0.35 0.355 0.353 0.353
800 0.073 3.202 3.707 3.683 3.22
1300 0.150 4.61 7.62 7.57 4.63
1800 0.250 4.76 12.69 12.61 4.77
2300 0.420 4.8 21.33 21.19 4.8
Table 3: Maximum Temperature Rise for every case at point A
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20. Figure 15: Temperature Rise History
Figure 15 shows the temperature rise history for initial temperature of 1800
K and corresponding heat input. The results shown in Table 3 and Figure 15
shows that the temperature rise is more when radiation to ambiance is neglected,
this implies that cooling through radiation is important, from Figure 15 it is evi-
dent that the low-fidelity case i.e Conduction-Radiation case agrees with the high-
fidelity simulation (Combined Case) with less deviation. Table 4 shows the percent
deviation of the results from the Combined Case (i.e high-fidelity) results.
% Deviation of Temperature Rise
Initial Temp.(K) Heat Influx (W) Conduction only Conduction-Convection Conduction-Radiation
300 0.007 1.428 0.857 0.857
800 0.073 15.771 15.021 0.562
1300 0.150 65.292 64.208 0.433
1800 0.250 166.59 164.91 0.210
2300 0.420 344.375 341.458 0
Table 4: % Deviation of Temperature Rise from High-fidelity Simulation
From the results obtained it can be concluded that the radiation plays an
important role in cooling down the sample and more importantly, radiation cannot
be neglected in simulations since it will lead us far from the real temperature
value. If we observe the values of Table 4 carefully, it can be concluded that the
low-fidelity simulation (i.e Conduction-Radiation Case) gives result with less %
deviation as the initial temperature increases.
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21. 6 Conclusions
In Example 1 (Rectangular Electronic Board) it can be observed that convection is
the dominant fidelity if the surface emissivity is less than 0.45. But if the surface
emissivity is more than 0.45 then both of the fidelities plays an important role in
determining the maximum temperature of the board.
In Example 2 (Engine Exhaust Manifold) it can be concluded that low-fidelity
model (i.e Case 4- no cavity radiation) gives results with a deviation of 4% from
the high-fidelity model with saving a good amount of time. So to save time, cavity
radiation can be neglected as the difference in temperature is not high as com-
pared to the maximum temperature reached due to exhaust gases.
In Example 3 (Heating by Laser) it can be concluded that low-fidelity model (i.e
Conduction-Radiation Case) gives results with negligible % deviation from high-
fidelity model (i.e Combined Case). In this example, radiation to ambiance plays
an important role therefore, it cannot be neglected at any cost, whereas, con-
vection to ambiance is does not play a vital role in the simulation so it can be
neglected to make the high-fidelity model to a low-fidelity one.
From the simulations carried out for the study it can concluded that the low-
fidelity simulations agrees with high-fidelity with little percent deviation but can
reduce a significant amount of time. If the nature of the simulation is known then
the high-fidelity simulation can be easily converted into low-fidelity simulation
with little compromise.
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22. Bibliography
[1] S. Sudhakar Babu, C. Gururaja Rao, & A. V. Narasimha Rao. (2014). Para-
metric Studies on Combined Conduction-Convection-Radiation from a Discretely
and Non-Identically Heated Rectangular Electronic Board. International Jour-
nal of Advanced Mechanical Engineering, Volume 4, 527-534.
[2] Abaqus Example Problems Guide, ABAQUS 6.14
[3] O. Wellele, H.R.B. Orlande, N. Ruperti Jr., M.J. Colaco, and A. Delmas. Cou-
pled Conduction–Radiation in Semi-Transparent Materials at High Tempera-
tures. Journal of Physics and Chemistry of Solids 67 (2006): 2230-2240
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