IRJET- Estimation of Boundary Shape using Inverse Method in a Functionally Graded Material in Two Dimensional Problem

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 05 Issue: 12 | Dec 2018 www.irjet.net p-ISSN: 2395-0072
© 2018, IRJET | Impact Factor value: 7.211 | ISO 9001:2008 Certified Journal | Page 168
ESTIMATION OF BOUNDARY SHAPE USING INVERSE METHOD IN A
FUNCTIONALLY GRADED MATERIAL IN TWO DIMENSIONAL PROBLEM
Abbas Bazargani 1, Reza Amini 1, Mehdi Kashfi1
1Mr. CFD LLC., Tbilisi, Georgia
---------------------------------------------------------------------***----------------------------------------------------------------------
Abstract – In this paper, conjugated gradient method
(CGM) based inverse algorithm for transient heat conduction
problem in a functionally graded material is performed to
estimate the unknown boundary shape when measurement or
observing the body shape is impossible or needs complex,
accurate and expensive equipment. For example, when the
unknown boundary placed in high temperature or
unreachable environment. The governing equations are
derived by using finite element method as a systematic and
efficient theory. It is assumed that the material properties are
smoothly changed based on mixture law. To determine the
accuracy of the estimated data, problems with different
boundary conditions, materials and boundary shapes was
studied and all of them proved that the methods which was
used in this paper, are completely compatible with the exact
solution. Finally, the effect of parameters, such as the number
of thermocouples, themeasurement error,and volumefraction
index and substituting the materials which was used in FGM
stuff, was discussed and compared.
Key words: Estimationofboundaryshape;Inversemethod;
Functionally graded materials; Conjugate gradient method;
finite element method; Inverse heat conduction
problem(IHCP)
1. INTRODUCTION
Functionally graded materials (FGMs) are a new generation
of composites which have found extensive applications in
different industries because of their favorable and
continuously varying physical and thermal properties[1–4].
Hence, these layered components are likely to play a great
role in the construction of advanced structures, such as
supersonic and hypersonic space vehicles and nuclear
industries. Usually, these structural elements operate in a
high temperature environment which inevitably induces
some thermal stresses that can change their mechanical
behavior [5–7] or cause a catastrophic failure of materials.
Hence, an accurate and efficient determination of their
thermal characteristic (boundary heat fluxes and
temperature distributions) is of great interest for
engineering design and manufacture. On the other hand, in
some heat transfer problems the operational process for
direct measurement of the physical parameters is either
quite complicated or the measurement process
corresponding to it requires sophisticated and expensive
instruments. In such situations, satisfactory estimation
results can be obtained using an inverse method in
conjunction with simple instruments without disturbingthe
processes. For this purpose, transient temperature
measurements taken at various boundary pointsofthe body
can be used for the estimation of the required quantities, and
for this particular project the boundary shape. However,
difficulties associated with the implementation of inverse
analysis should also be recognized.
It is well-known that inverse problems are mathematically
ill-posed; that is, a small change in the input data can result
in enormous change in the computed solution at an
inaccessible part of the boundary [8]. Hence, the inverse
methods require efficient optimization tools for their
solutions. The use of the adjoin equation approach coupled
with the conjugate gradient method [9–14] appears to be
very powerful for solving inverse heatconductionproblems.
In this method, the regularization procedure is performed
during the iterative processes and thus, determination of
optimal regularization conditions is not necessary.
The inverse heat conduction problem (IHCP) has been
widely used in different practical engineeringproblemssuch
as estimation of surface conditions, initial conditions,
thermal properties and the boundary shape of a body, from
known information at some predefined positions. For
example, direct measurement or observing the boundary
shape at the surface of a wall in general [15-18] and
specifically subjected to fire, theoutersurfaceofa vehicle re-
entry or the inside surface of a combustion chamber is
extremely difficult or needs expensive instruments [19].
Over the past decades, studies shows that numerical
simulations [20-22] and experimental tests [23-25] have
been consistence in different majors including heat transfer
[26-30], energy [31], material [32-36], and other methods
[37-40].
In the previous work by Huang and Chao a transient inverse
geometry problem in identifying the unknown irregular
boundary configurations from external measurements
(either direct or infrared type) has been solved based on the
boundary element method, i.e. In that work the boundary
shape was a function of time. Thatapproachcouldbeapplied
to nondestructive evaluation (NDE) techniques and other
such as the interface geometry identification for the phase
change problems [41]. Kazemzade and Daneshmand also
developed a shape identification scheme to determine the
shape of the inaccessible parts of a 2-dimentional object
made of functionally graded material using the measured
temperatures on its accessible parts. They used the
smoothed fixed grid finite element method which wasa new
approach based on the non-boundary-fitted meshes and the
gradient smoothing technique were used for the solution of
direct problem and shape sensitivity analysis. [42]

y
x
Pure metal
Pure ceramic
𝑙 𝑦
𝑙 𝑥
f(x,t)
To the best of the authors’ knowledge, there are few works
on the inverse heat conduction analysis of FGMs [1, 2, 43–
48]. This motivates us to consider the inverse transient heat
conduction problems of 2 dimensional body shape made of
FG material here. However, it should be mentioned that
there are some valuable works on the inverse heat
conduction analysis of isotropic materials; see, for example,
references [49–55].
2. Mathematical modeling
The domain under consideration is a functionally graded
plate that two side of the plate is isolated and the other two
side is held in constant temperature which is represented in
figure 1 (a) and also in figure 1 (b) placement of the ceramic
and metal is shown. The unknown boundary shape is
estimated by determining the temperature on the opposite
side of the plate. Hence, we should first solve the direct
problem.
Fig. 1 (a): Geometry and the boundary condition,
discussed in this paper.
Fig. 1 (b): Placement of pure metal and ceramic on the
boundary.
2.1 The direct problem
At the first step, we assume the unknown boundary shape as
known. Then the temperature on the opposite side is
determined due to it. For this purpose and toaccuratelydoit,
transientheattransferequationwithoutheatgenerationtwo-
dimensional equation in FG materials is hired and the
following equation is written as:
  t
T
ycy
y
Tyk
yx
Tyk
x 















 )()()()( 
(1)
where   tyxTT ,, ,   y  ,   yCC  and   ykk 
are respectively, temperature, mass density, specific heat
capacity and thermal conductivity of an arbitrary material
point of the plate.
The boundary and initial conditions are as follows,
respectively,
0
),0(



x
yT (2)
0
)0,(



y
xT (3)
kylT 300),(  (4)
kfxT 100),(  (5)
kyxT 50)0,,(  (6)
The effective material properties of FG plate constituents
(metal and ceramic) such as density   , specific heat
capacity C and thermal conductivity  k are obtained by
using the power law distribution, without loss of generality
of the formulation and the method of solution. Hence, a
typical effective material property ‘G’ is obtained as
     n
cVcGn
mVmGyG 
(7)
Where the subscripts c and m refer to the ceramic and metal
constituents, respectively; also, 






yl
y
cV is the volume
fraction and n denotes the volume fraction index, which is a
real positive number.
In this paper, finite element method (FEM) is applied to
discretize the spatialdifferentialequationasasystematicand
efficient method. After findingtheweakformfortheequation
number (1), applying by part integration and also using
Green’ theorem, the above equation is obtained:















e
dsnwq
e
dsyn
y
T
wykxn
x
T
wyk )()(
(8)
On the other side, in each element of the finite element
meshes, we assume that:
     


n
i
yxe
iNti
eTtyxeT
1
,,,
(9)

Shape function is shown as
e
iN , and also n is the number of
nodes of the e-th element and
i
eT is temperature of the i-th
node from the e-th element. Finally the governing equation
will be:
0
1
)(
1
)()(






 
























i
eQe dxdy
t
e
jTN
j
jNicN
e
jT
N
i y
jN
y
iN
yk
x
jN
x
iN
yk

(10)
After assembling the element matrices and satisfying the
boundary and initial condition, equation (10) changes to:
       QTMTK   (11)
Where [K], [M] and {Q} are respectively hardness and mass
matrix and force vector.
And also for time derivatives, backward finite difference
method is used as an unconditionally stable method. After
applying the method and rearrangingtheequation,theabove
equation is obtained:
            ss
TMtQKtMT 
 11 (12)
where
s
T is temperaturein the s-th timestep.Assumingthe
initial conditions and solving the above equation, quantity of
the temperature for the next time step is obtained.
2.2. Creating elements and geometry of solvent
procedure
Since the purpose of this paper is to estimate the boundary
shape, the boundary shape changes in every time step. As a
matter of fact, in every step the whole meshes should be
terminated and rebuilt in the next timestep[56-59].Actually
we’re facing a moving mesh. On the other side, since the
unknown boundary is curved shape, so the elements are
irregular.For creating thematricesofhardnessandmassand
also the force vector, the irregular physicaldomainshouldbe
mapping to computational regular domain. As an efficient
method for mapping the domain, the Gauss method is used.
At this point, x and y direction changes to  and 
 























yx
yx
g
 



































i
i
i
i
N
N
G
y
N
x
N (13)
    1
 gG
  ddgdxdy det
If Ni considers as shape function, the following equation is
achieved:
 
   
   

























 



















iNjN
GGGG
jN
iN
GGGG
jN
iN
GG
jN
iN
GG
dxdy
y
jN
y
iN
x
jN
x
iN
ij
e
K
2221121122211211
2
22
2
12
2
21
2
11
  ddgdet
(14
)
2.3 The inverse problem
In the inverse problem, the boundary shape is assumed tobe
unknown, while all the other effectiveparametersforsolvent
are known. Moreover, the temperature at some suitable
locations on the other side of the plate in an arbitrary time t
are considered to be available.
2.4 Sensitivity problem
The governing differential equations of the sensitivity
problems are obtained from the original direct problem
defined by Equations (1)-(6). For this purpose, perturbing
 txf , to    txftxf ,,  , then  txT , change to
   txTtxT ,,  ,respectively.Substitutingthesenewvalues
of the field variables in Equation (1) and neglecting the
higher order terms, the governing differential equations of
the sensitivity problems are obtained as
     
t
TT
c
y
TT
k
yx
TT
k
x 
























(15)
In a similar manner, theboundaryconditions(2)-(6)become,
respectively,
0x : 0


x
T (16)
xlx  : 0T (17)
0t : 0T (18)
0y : 0


y
T (19)
 txfy , :
y
T
fT



(20)

2.5 Adjoin problem and gradient equation
In order to derive the governing differential equations of
adjoin problem, Lagrange multipliers approach is adopted.
Using this approach, the new functional is defined
   

 

 





























dxdydt
t
T
c
y
T
k
yx
T
k
x
dxdtxxYTJ
ft
t
l
x
m


0 0
2ˆ
(21)
In the adjoin problem, the values of the field variables are
specified at the final time jtt instead of the initial time t=0,
as in the traditional initial value problems. However, by
defining a new timevariableas )( tjt  ,thisproblemiseasily
transformed to a standard initial value problem. Then, with
perturbing  txf , to    txftxf ,,  and respectively
 txT , to    txTtxT , and also by-parting the
integration, the above equation is achieved:
0






















t
c
y
k
yx
k
x


 (22)
And the boundary conditions become:
0x : 0


x
 (23)
lx  : 0 (24)
0y :    mxxYT
y
k 




2
(25)
jtt  : 0 (26)
 txfy , : 0 (27)
And the gradient of the abovefunctional can be presentedas:
 
)(
,ˆ
xfy
y
T
y
ktxJ






 (28)
2.6 conjugated gradient method for minimization
In this paper, for finding the temperature on the surface, the
number of M thermocouple is placed on it. The
thermocouplessensethetemperatureas  tmY whichmeans
the temperature for mx in the time of t.     tmxYtmY ,
the process of solving the inverse problem continues till the
above functional minimizes:
        


t
dttYtTtxfJ f
t
M
m
mm
0
1
2
,
(29)
In which  tmT is known as the estimated temperature for
mx in the time of t. Based on the Conjugated gradient
method, the unknown quantity  txf , is estimated using an
iterative process as follows,
     txptxftxf rrrr
,,,1

,
r = 0,1,2,3, …
(30)
Where r is the search step sizein goingfromtheiterationr
to the iteration r+1and  t
r
P is thedirection of descent (i.e.
search direction) given by,
     txPtxJtxP rrrr
,,ˆ, 1
  ,
(31)
Which is a conjugation of the gradient
direction

















r
f
Jr
J
ˆ'ˆ attheiterationrandthedirectionof
descent  tPr 1
at the iteration r-1. The conjugate
coefficient  at the iteration r is determined from the
following relation,
  
  











f x
f x
t l r
t l r
r
dxdt
dxdt
J
J
0 0
21
0 0
2
ˆ
ˆ

,
0
0

(32)
It can be seen that the method degenerate to the steepest
descent method when 0
r
 for any r in Eq. (31). The
convergence of the above iterative procedure in minimizing
the functional Jˆ is demonstrated previously [9].
After substituting the equation (30) in (29), the functional of
 1ˆ r
fJ for iteration r+1 is written as bellow:

       



t
dttYpfTfJ f
t
M
m
m
rrr
m
r
0
1
2
1 ˆˆ 
(33)
After writing the derivation of the above equation toward
r
 and also making it equivalent to zero, we reach to bellow
equation as search step size:
      
   
 







f
f
t
t
M
m
m
t
t
M
m
mmm
r
dttT
dttTtYtT
0
1
2
0
1

(34)
2.7 Stopping criterion
The stopping criterion to terminate the iteration processes
depends on the measurement errors. If the temperature at
the surface can be measured accurately (i.e. without any
measurement error) then as the value of the objective
functional converge to a very small number, the
computational procedure is stopped, i.e.
   
tfJ r 1 (35)
where is a small specified number. But, sincethemeasured
temperaturedatacontains some measurementerrors,inthis
study the discrepancy principle is adopted as the stopping
criterion to terminate the iteration procedure. Based on this
criterion, the temperature residuals is approximated as,
   tYtT mm  (36)
Where σ is the standard deviation of the temperature
measurements, whichisassumedtobeconstant.Substituting
Eq. (36) into Eq. (29), then the stopping criteria which
should be used in inequality (35) is obtained as,
jtM 2
  (37)
2.8 Computational procedure
The computational procedure for the solutionofthisinverse
problem using conjugate gradient method may be
summarized as follows:
i. Set  txf , as known for the first iteration
ii. Solve the direct problem given by equation (1)-(6)
for  tyxT ,,
iii. Examine the stopping criterion given by equation
(35) with ε given by equation (37). Continue if not
satisfied
iv. Solve the adjoin problem given by equation (22)-
(27) for  tyx ,,
v. Compute the gradient of the functional
r
J ˆ from
equation(28)
vi. Compute the conjugate coefficient
r
 and direction
of descent
r
P from equation (32) and (31),
respectively.
vii. Set
r
pf  , and solve the sensitivity problem
given by equation (15)-(20) for  txT ,
viii. Compute the search step size
r
 from equation
(34).
ix. Compute the new estimation for  tx
r
f ,
1
from
equation (30) and return to step II.
3. Numerical results and discussion
In this section, the accuracy of the proposed inverse
algorithm in predicting the boundary shape on the
functionally graded plate is investigated. For this purpose, 8
distinct example of FG platewith unknown boundary shapes
are investigated. Also in this section the effect of parameters,
such as the number of thermocouples, the measurement
error, and volume fraction index and substituting the
materials which was used in FGM stuff are discussed and
compared with the exact solution. The number of
thermocouples in the y-direction are the same as number of
the nodes. The total time is assumed to be  sft 40 and
the value of  st 1 is used in all the solved examples. The
material properties of the FG fins for different examples are
presented in Table 1. Initial guess fortheunknownboundary
for all the examples is 0
0
f .
Table 1 Thermo-physical properties of the materials.
Material C(J/kgK) k (W/mK)  (kg/m3)
Iron 448 80 7850
Al2O3 775 30.7 3970
Aluminum 900 247 2712
Al2O3 775 30.7 3970
To compare the results for situations involving random
measurement errors, the normally distributed uncorrelated
errors with zero mean and constant standard deviation, ,
are used. The simulated vector of inexact measurement data
Y can be expressed as
      exactYY (38)
where Y represents the exact values of the temperature of
the direct problem at different times and  is the vector of
random errors with zero mean and a specified standard

deviation. In all the solved examples, a value of 01.0 is
used for the case of .0
In the solved examples, the average error is evaluated
based on the following formulation,
   
 
   %1001
1 1 ,
,ˆ,
%







JM
M
i
J
j jtixf
jtixfjtixf
Error
(39)
In which  jtixf , is the estimated equation for the
unknown boundary shape and  jtixf ,ˆ is the exact
equation. And also M is the number of thermocouples in x-
direction and J+1 is the number of time nodes.
For making sure that the method and the solvent process is
correct, we compare the results with the Sakurai’s results.
[30] For this purpose, table 2 is shown.
Table 2. Comparison between results of Sukurai and this
paper.
x Present study Sakurai (30)
0.1 7.865 7.863
0.2 6.231 6.226
0.3 4.911 4.901
0.4 3.658 3.652
0.5 2.695 2.691
0.6 1.934 1.928
0.7 1.247 1.238
0.8 0.836 0.831
0.9 0.421 0.416
The material whichwasusedfortheexamplesareAluminaas
ceramic and Iron as metal, except example 64.
In the first and second example the equation of unknown
boundary shapes are respectively, xy  2 and
 xy sin5.1  . Also 20M and 2n . The estimated
boundary shape for 3,1,5.0,0 is shown and compared
with the exact shape in figure 2 and 3. The computational
time efforts and the percentage of average errors (evaluated
using Eq. (39)) for the above solved examples are given in
Table 3.
Table 3. The computational time efforts and percentage of
errors for the first two examples.
(a) σ = 0 (b) σ = 0.5
(c) σ = 1 (d) σ = 3
Fig. 2 (a)-(d): Comparison between the estimated and
exact shape of boundary with equation xy  2 for
different standard deviation (example 1).
σ
Example
(1)
Example
(2)
0.0
Number of
iteration
14 22
Error % 0.61 1.42
0.5
Number of
iteration
8 14
Error % 3.56 4.77
1
Number of
iteration
11 8
Error % 4.71 6.30
3
Number of
iteration
6 11
Error % 6.58 9.06

(b) σ =0.5(a) σ = 0
(d) σ = 3(c) σ = 1
exact shape of boundary with equation  xy sin5.1 
for different standard deviation (Ex. 2).
(b) M=10(a) M=5
(d) M=25(c) M=20
exact shape of boundary for different number of
thermocouples with equation  xy sin5.1  (Ex. 3).
In the third example, theeffectofnumberofthermocouplesis
shown in figure 4 and discussed. The equation of unknown
boundary shape is  xy sin5.1  and also 2.0 and
2n . The purpose of this example is finding the
thermocouple number, which has the minimum deviation
with the exact answer.
In the fourth example, the effect of substituting the materials
which was used in FGM stuff is discussed and compared in
figure 5. The equation of unknown boundary shape
is xy 25.1  and also 2.0 and 2n . The purpose of
this example is comparing the results when the place of the
pure ceramic and metal is substituted.
(b)
(a)
Fig. 5 (a)-(b): Comparison between the estimated and
exact shape of boundary with equation xy 25.1  when
a) metal is on y=0 (down) b) ceramic is on y=0 (down)
(example 4).
In the fifth example, the effect of variable value for volume
fraction index is presented in figure 6. The equation of
unknown boundary shape is  xy 2sin5.1  and also
2.0 and 10M .
(b) n=0.5
(a) n=0
(d) n=1.5(c) n=1

(f) n=4(e) n=3
Fig. 6 (a)-(f): Comparison between the estimated and
exact shape of boundary with equation  xy 2sin5.1 
for various volume fraction indexes (example 5).
In the sixth example, different FG materials are used to
compare the results. The equation of unknown boundary
shape is  xy 2sin5.1  and also 2.0 and 10M . In
the both figure 7 (a) and 7(b) Alumina is used as ceramic.
Iron is used for the first part and Aluminum for the second
part is hired as metal.
(a) (b)
Fig. 7 (a)-(b): Comparison between the estimated and
exact shape of boundary with equation  xy 2sin5.1 
when the FG plate consists of a) Iron and Alumina b)
Aluminum and Alumina (example 6).
In the seventh and 8th example new boundary condition is
performed, as figure 8 and also in the new boundary
condition: 1001  TT , 3002  TT , 1003  TT .The
equation of unknown boundary shapes are respectively,
xy  2 and  xy sin5.1  .Also 20M and 2n .The
estimated boundaryshapefor 0 isshownandcompared
with the exact figure in figure 9 and 10.
Fig. 8 Geometry and the boundary condition for example 7
and 8.
Fig. 9: Comparison between the estimated and exact shape
of boundary with equation xy  2 when 0 for new
boundary condition (example 7).
Fig. 10: Comparison between the estimated and exact
shape of boundary with equation  xy sin5.1  when
0 for new boundary condition (example 8).
4. Conclusion
Inverse transient heat conduction problems of
functionally graded (FG) plate is presented. The equation of
boundary shape is estimated by using the measured
temperatureat the otherside’splate.Toaccuratelymodelthe
heat conduction phenomena, the non-Fourier heat transfer
equation is used. The conjugate gradient method (CGM) is
employed for the optimization procedure and the finite
element method is applied to solve the governingdifferential
equations. From the solved examples, it is revealed that the
presented approach has the ability to predict the unknown
boundary shape with good accuracy and low computational
time efforts.

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