Research ArticleSolving of Two-Dimensional Unsteady-State Heat-TransferInverse Problem Using Finite Difference Method and ModelPrediction Control Method
Shoubin Wang and Rui Ni
School of Control and Mechanical Engineering Tianjin Chengjian University Tianjin 300384 China
Correspondence should be addressed to Shoubin Wang wsbin800126com
Received 23 January 2019 Revised 4 April 2019 Accepted 13 June 2019 Published 22 July 2019
Academic Editor Basil M Al-Hadithi
Copyright copy 2019 Shoubin Wang and Rui Ni This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The InverseHeat Conduction Problem (IHCP) refers to the inversion of the internal characteristics or thermal boundary conditionsof a heat transfer system by using other known conditions of the system and according to some information that the system canobserve It has been extensively applied in the fields of engineering related to heat-transfer measurement such as the aerospaceatomic energy technology mechanical engineering and metallurgyThe paper adopts Finite DifferenceMethod (FDM) andModelPredictive Control Method (MPCM) to study the inverse problem in the third-type boundary heat-transfer coefficient involved inthe two-dimensional unsteady heat conduction systemThe residual principle is introduced to estimate the optimized regularizationparameter in the model prediction control method thereby obtaining a more precise inversion result Finite difference method(FDM) is adopted for direct problem to calculate the temperature value in various time quanta of needed discrete point as well asthe temperature field verification by time quantum while inverse problem discusses the impact of differentmeasurement errors andmeasurement point positions on the inverse result As demonstrated by empirical analysis the proposed method remains highlyprecise despite the presence of measurement errors or the close distance of measurement point position from the boundary angularpoint angle
1 Introduction
The Inverse Heat Conduction Problem is able to retrieve theunknown parameters such as boundary conditions materialthermophysical parameters [1ndash3] internal heat sources andboundary geometry by measuring the temperature infor-mation at the boundary or at some point in the heat-transfer system [4 5]The research of inverse heat conductionproblem has a very wide application background It hasbeen applied in almost all fields of scientific engineeringincluding power engineering aerospace engineering met-allurgical engineering biomedical engineering mechanicalmanufacturing chemical engineering nuclear physics mate-rial processing geometry optimization of equipment andnondestructive testing In view of the inverse problem ofheat-transfer experts and scholars have done quite a lotof research [6ndash11] Duda identified the heat flux in two-dimensional transient heat conduction and reconstructed the
transient temperature field by utilizing the finite elementmethod (FEM) and Levenberg Marquardt method in ANSYSMultiphysics software The method mentioned above wasapplied to the identification of aerodynamic heating on anatmospheric reentry capsule [12] Beck raised the conceptof sensitivity coefficient and introduced it into the inverseproblem thereby successfully obtaining the steady-state andunsteady-state heat conduction application [13ndash15] Luo etal proposed the decentralized fuzzy inference method appli-cable to unsteady IHCP by dispersion and coordination ofmeasurement information on the time domain on the basisof researching steady IHCP by using the decentralized fuzzyinference method [16] Qian et al solved the unsteady IHCPby using the SFSM and conjugate gradient method whichsufficiently demonstrated effectiveness of these twomethodsanalyzed and compared the advantages and disadvantages ofthese two methods [17] Jian Su et al solved the heat conduc-tion inverse problem of one transient heat-transfer coefficient
HindawiComplexityVolume 2019 Article ID 7432138 12 pageshttpsdoiorg10115520197432138
2 Complexity
by employing Alifanovrsquos iterative regularization algorithm[18] Blanc G et al investigated the one-dimensional transientinverse problem finding that residual principle can optimizethe key parameter in the heat conduction problem [19] WuZhaochun studied the measurement point arrangement pat-tern during the solving process of the two-dimensional heatconduction inverse problem with DSC method accordinglymaking some rational suggestions regarding the measure-ment points [20] Lesnic et al identified the thermophysicalparameters in one-dimensional transient heat conductionproblems by using the BEM [21] Ershova et al used the resid-ual error principles in the Tikhonov regularization methodand completed the crystal phonon inspection identificationtasks [22] Zhao Luyao combined the particle swarm opti-mization (PSO) and conjugate gradient method and appliedthe combined method to the inversion of the heat-transfercoefficient of one-dimensional unsteady-state system It hasbeen reported that themethod exhibits high preciseness [23]Li et al researched IHCP by using the BEM and identifiedthe irregular boundaries [24 25] Zhou et al solved the heatconductivity coefficient in the two-dimensional transientinverse problems by using the BEM and gradient regulariza-tion method and obtained the relatively accurate inversionresults [26] Li Yanhao resolved the heat-flow problem foundin the two-dimensional transient inverse problem by usingthe model prediction control algorithm and inversion resultwas relatively precise [27] Fan Jianxue also adopted themodel prediction control algorithm to solve the heat-transfercoefficient in the inner wall of two-dimensional transientsteam drum and achieved good calculation results [28]
Regarding the boundary heat transfer in the heat conduc-tion system in the paper FDM is adopted to solve the directproblem of the two-dimensional unsteady-state heat con-duction without internal heat source and model predictioncontrol method is used to solve the inverse problem Besidesresidual principle is introduced to optimize the regularizationparameter during the inversion process thereby improvingthe efficiency of inversion in terms of speed and time
2 Unsteady-State Direct Problem
The Inverse Heat Conduction Problem usually involves themultiple deduction of the forward problem and its inversionaccuracy is directly affected by the calculation accuracy ofthe forward problem Positive problem refers to the solutionof historical temperature field through given boundary con-ditions initial temperature and thermal conductivity differ-ential equation Common solutions are Lattice BoltzmannMethod Finite Volume Method Adomain DecompositionMethod Boundary Element Method and Finite DifferenceMethod
LBM (Lattice Boltzmann Method) [29] is a mesoscopicresearchmethod based onmolecular kinetics which can welldescribe the complex and small interfaces in porousmedia Itis widely used in small-scale numerical simulation of porousmedia and other objects with complex interface structures
The basic idea of FVM (Finite VolumeMethod) [30] is todivide the computational region into a series of nonrepeatedcontrol volumes and make each grid point have a control
volume around it A set of discrete equations is obtained byintegrating the differential equations to be resolved with eachcontrol volume It is commonly used in the case of discreteand complex grids
ADM (Adomain Decomposition Method) [31] is aresearch method that decomposes the true solution of anequation into the sum of the components of several solutionsand then tries to find the components of the solutions andmake the sum of the components of the solutions approxi-mate the true solution with any desired high precision But itcan only obtain accurate results under the condition that theenergy conservation equation is not nonlinear
BEM (Boundary Element Method) [32] is a researchmethod which divides elements on the boundary of thedefined domain and approximates the boundary conditionsby functions satisfying the governing equations The basicadvantage of BEM is that it can reduce the dimensionality ofthe problem but when it comes to solving the basic solutionof the problem the process of solving the basic solution isgenerally complicated
In this paper a square rectangular plate is selected asan experimental physical model which is a very commonphysical model The finite-difference method [33] can reducethe amount of calculation required by other researchmethodsfor the positive problem and it is also convenient to query thetemperature change curves of the required measuring pointsin the negative problem
21 Mathematical Model The mathematical model of two-dimensional unsteady-state heat conduction without internalheat source is expressed as follows
where Γ1 is the Dirichlet (first-type) boundary conditionΓ2 is the Neumann (second-type) boundary condition Γ3 isthe Robin (third-type) boundary condition and Γ = Γ1 +Γ2 + Γ3 is the boundary of the whole region Ω 120572 denotesthe thermal diffusivity and 120572 = 120582119888120588 119888 120588 and 120582 denote thespecific heat the density and the heat conduction coefficientof the object respectively T represents the temperature 119879 isthe temperature given by the Dirichlet boundary condition119879119891 is the environment temperature and 1198790 is the initialtemperature 119902 refers to the heat flux 119899 refers to the boundaryouter normal vector and ℎ refers to the surface heat-transfercoefficient
22 Discretization and Difference Scheme The discrete rulesof two-dimensional unsteady-state heat conduction problem
Complexity 3
without internal heat source in geometry and time domainare as follows
Assuming that after the domain of uniformdiscretizationthe step length of x-axis is Δ119909 = 119909119894+1 minus 119909119894 and that of y-axisis Δy = 119910119895+1 minus 119910119895 obviously 119909119894 = 119894Δ119909 119910119895 = 119895Δ119910 and 119894 119895 =0 1 2 3
n(119899 = 0 1 2 3 ) is used to uniformly discretizethe time domain t≧0 and the step length between twotime moments Δ119905 = 119905119899+1 minus 119905119899 119905119899 = 119899Δ119905 where(119894 119895) sdot sdot sdot (119909119894 119910119895) 119899 sdot sdot sdot 119905119899 119879119899119894119895 sdot sdot sdot 119899 and the temperature at node(119894 119895) in the time moment 119899 is 119879(119909119894 119910119895 119905119899)
The explicit difference array of the two-dimensionalunsteady-state heat conduction without internal heat sourceis expressed as follows
Applying the first heat conduction equation in (1) to node(119894 119895) at the time moment of 119899 the equation can be rewrittenas
The partial differential in the two sides of (2) can beapproximated by difference quotient The temperature itemin the right of the equal sign can be approximated by first-order time forward difference quotient
(120597119879120597119905 )119899
119894119895
= (119879119899+1119894119895 minus 119879119899119894119895)Δ119905 + 119874 (Δ119905) (3)
The second-order partial differential in the left of theequal sign can be approximated by the central differencequotient
(12059721198791205971199092 )119899
119894119895
= (119879119899119894+1119895 minus 2119879119899119894119895 + 119879119899119894minus1119895)(Δ119909)2 + 119874 (Δ1199092) (4)
(12059721198791205971199102 )119899
119894119895
= (119879119899119894119895+1 minus 2119879119899119894119895 + 119879119899119894119895minus1)(Δ119910)2 + 119874 (Δ1199102) (5)
Substituting (3) (4) and (5) into (2) we can get thedifference equation of (2)
119879119899119894+1119895 minus 2119879119899119894119895 + 119879119899119894minus1119895(Δ119909)2 + 119879119899119894119895+1 minus 2119879119899119894119895 + 119879119899119894119895minus1(Δ119910)2= 1120572 119879
119899+1119894119895 minus 119879119899119894119895Δ119905
(6)
Equation (6) is the difference equation of heat conductionequation and the truncation error [34] is 119874(Δ119905 + Δ1199092Δ1199102)
Assuming that Δ119909 = Δ119910 = Δ and substituting it into (6)we can obtain
119879119899119894+1119895 + 119879119899119894minus1119895 + 119879119899119894119895minus1 minus (4 minus 11198650)119879119899119894119895 =119879119899+11198941198951198650 (7)
where 1198650 refers to the Fourier coefficient and 1198650 =120572Δ119905(Δ119909)2 = 120572Δ119905(Δ119910)2 = 120572Δ119905Δ2
The stability condition of explicit finite difference equa-tion of two-dimensional unsteady-state heat conductionwithout internal heat source is in interior node 1198650 le 14in boundary node 1198650 le 1[2(2 + 119861119894)] in boundary angularpoint 1198650 le 1[4(1 + 119861119894)]23 Boundary Conditions First-type boundary conditionie the temperature is given In general when the FDM isused for calculation it shall be processed as in the momentof the initial the boundary node temperature is 1198792 then theboundary node temperature remains at 119879
In the second- and third-type boundary condition it isnecessary to set virtual node outside the boundary to makethe boundary node into interior node The node numberingis shown in Figure 2
Second-type boundary condition ie the heat flowboundary is givenminus120582(120597119879120597119899) = 119902 Setting boundary 119909 = 0 asthe given heat flow boundary condition and keeping it stablethe second-type boundary condition can be expressed asminus120582(120597119879120597119909) = 119902 Using central difference quotient to replacethe first-order partial differential equation
minus1205821198791198992 minus 119879119899210158402Δ = 119902 (8)
(8) is rewritten as
11987911989921015840 = 1198791198992 minus 2Δ120582 119902 (9)
Node 1 is changed into interior node and (9) is substitutedinto the interior node explicit difference equation of (7) to get
1198791198992 + 11987911989921015840 + 1198791198994 + 1198791198990 minus (4 minus 11198650)1198791198991 = 11198650119879119899+11 (10)
Substituting (9) into (10) we can obtain
119879119899+11 = 1198650 [1198791198990 + 21198791198992 + 1198791198994 minus 2Δ120582 119902 + ( 11198650 minus 4)1198791198991] (11)
Third-type boundary condition ie the heat transferboundary is given 120582(120597119879120597119899) = ℎ(119879 minus 119879119891) Setting boundary119909 = 0 as the given heat transfer boundary condition andkeeping it stable the third-type boundary condition can beexpressed as 120582(120597119879120597119909) = ℎ(119879 minus 119879119891) Using central differencequotient to replace the first-order partial differential equation
120582 (1198791198992 minus 11987911989921015840)(2Δ) = ℎ (1198791198991 minus 119879119891) (12)
(12) is rewritten as
11987911989921015840 = 1198791198992 minus 2119861119894 (1198791198991 minus 119879119891) (13)
Node 1 is changed into interior node and (13) is substi-tuted into the interior node explicit difference equation of (7)to get
1198791198992 + 11987911989921015840 + 1198791198994 + 1198791198990 minus (4 minus 11198650)1198791198991 = 11198650119879119899+11 (14)
4 Complexity
Substituting (13) into (14) we can obtain
119879119899+11= 1198650 [1198791198990 + 21198791198992 + 1198791198994 + 2119861119894119879119891 + ( 11198650 minus 4 minus 2119861119894)1198791198991]
(15)
where 119861119894 is the Biot number 119861119894 = ℎΔ120582 the truncationerror of the second- and third-type boundary condition is119874(Δ2)
Adiabatic boundary condition is 119879 = 0 Similarly thesecond- and third-type boundary condition is
Boundary angular point is 0 node Virtual nodes 1rsquo and3rsquo are set in the symmetric position of node 1 and node 3respectively and the central different quotient is applied inthe 119909- and 119910-direction respectively
120582 (1198791198993 minus 11987911989931015840)(2Δ) = ℎ (1198791198990 minus 119879119891)120582 (1198791198991 minus 11987911989911015840)(2Δ) = ℎ (1198791198990 minus 119879119891)
(17)
Equation (17) is rewritten as
11987911989931015840 = 1198791198993 minus 2119861119894 (1198791198990 minus 119879119891)11987911989911015840 = 1198791198991 minus 2119861119894 (1198791198990 minus 119879119891) (18)
Node 1 is changed into interior node and (17) and (18) aresubstituted into (7) to get
1198791198993 + 11987911989931015840 + 1198791198991 + 11987911989911015840 minus (4 minus 11198650)1198791198990 = 11198650119879119899+10119879119899+10 = 21198650 [1198791198991 + 1198791198993 + 2119861119894119879119891 + ( 121198650 minus 2 minus 2119861119894)1198791198990]
(19)
By (7) (11) (15) (16) and (19) the temperature value inany point of the model can be obtained
24 Mathematical Model about the Heat Transfer Processof Rectangular Plate Figure 3 shows the model of two-dimensional unsteady-state heat conduction system withoutinternal heat source The rectangle plate in Figure 3 isadopted boundary 1198631 1198632 1198633 is for heat insulation and 1198634is the third-type boundary condition ℎ is the heat-transfercoefficient Then (1) can be changed by the correspondingmathematical model as follows
25 Direct Problem Verification Figures 4 5 6 and 7display the temperature field distribution when 119905 =50 100 150 200(119904) Figure 8 is the curve of measuring pointswith time The simulation result of direct problem solvingcan demonstrate the rationality of explicit finite differencewhich is convenient for performing the inversion algorithmof inverse problem
The length Lx and width Ly of the plate is 02119898 Theheat conductivity coefficient 120582 = 47(119882119898 lowast 119870) thermaldiffusivity 119886 = 128 lowast 10minus6(1198982119904) initial temperature 1198790 =20∘119862 environment temperature119879119891 = 50∘119862 and heat-transfercoefficient ℎ = 2000(1198821198982 lowast 119870)3 Unsteady-State Problem
Predictive control is a model-based control algorithm whichfocuses on the function of the model rather than the formof the model Compared with other control methods itscharacteristics are reflected in the use of rolling optimizationand rolling implementation of the control mode to achievethe control effect but also did not give up the traditionalcontrol feedback Therefore the predictive control algorithmis based on the future dynamic behavior of the processmodel prediction system under a certain control effect usesthe rolling optimization to obtain the control effect underthe corresponding constraint conditions and performancerequirements and corrects the prediction of future dynamicbehavior in the rolling optimization process by detecting real-time information
31 PredictionModel of Inverse Problem The step response ofheat-transfer coefficient in the1198634 boundary of direct problemmodel is taken as the prediction model of inverse problemThe increment of heat-transfer coefficient in the boundary infuture time ie Δℎ119896+119875 is used to predict the temperature at119878 in the 1198634 boundary in the moment of 119877 ie 119879119896+119875 where119875 = 0 1 2 119877 minus 1 and 119877 is the future time step
According to the principle of linear superposition [35]after loading 119877 group of increment Δℎ119896+119875 on system sincethe time moment of 119896 the temperature at 119878 ie 119879119896+119875 isobtained
Δ120601119884 = 120601119884+1 minus 120601119884 (119884 = 0 1 sdot sdot sdot 119877 minus 1) (24)
The step response system function from the timemoment119905119896 to 119905119896+119877minus1 is defined as the impact of heat-transfer coefficienton 119879(119909 119910 119905) After the derivation of ℎ119896 by (20) the stepresponse equation can be obtained as follows
12059721198671205971199092 + 12059721198671205971199102 = 1120572 120597119867120597119905 (isin Ω 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 0 (isin 1198631 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 0 (isin 1198632 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119910 = 0 (isin 1198633 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 119879119896 minus 119879119891 minus ℎ119867 (119909 119910 119905)
(isin Γ3 119905119896 le 119905 le 119905119896+119877minus1)119867 = 1198670 (isin Ω 119905 = 119905119896)
(25)
From (25) it can be seen that119867 is related to ℎ Thereforewhile solving inverse problem it is necessary to use theexplicit difference algorithm used in the direct problemsolving and keep updating the calculation of ℎ
The corresponding discrete value 119867119896+119877minus1 is obtained by(25) hence the dynamic step response coefficient 120601119877 isfurther determined as
120601119877 = 119867119896+119877minus1 (26)
32 Rolling Optimization of Inverse Problem Measurementvalue and predictive value of temperature can be seen inthe time range from 119905119896 to 119905119896+119877minus1 According to the finiteoptimization 119877 parameters to be inverted are obtained sothe predictive value can be as close as possible to the mea-surement value in the future time domain at this momentthe quadratic performance index of system can be launched
min 119869 (ℎ) = (119879119898119890119886 minus 119879119901119903119890)Τ (119879119898119890119886 minus 119879119901119903119890) + ℎΤ120579ℎ (27)
where 120579 is the regularization parameter matrix
and 120572 is the regularization parameterAfter the derivation of ℎ based on (27) and making119889119869(ℎ)119889ℎ = 0 the optimal control rate can be obtained as
ℎ = (119860Τ119860 + 120579)minus1 119860Τ [119879119898119890119886 minus 11987911990111990311989010038161003816100381610038161003816ℎ=0] (29)
The optimal heat-transfer coefficient at moment 119896 can beobtained following (29)
ℎ = R (119860Τ119860 + 120579)minus1 119860Τ [119879119898119890119886 minus 11987911990111990311989010038161003816100381610038161003816ℎ=0] (30)
where R = [1 0 0]Τ There is no need to presetthe function form for the inverse problem calculation by theabove optimization algorithm
33 Regularization Parameter The residual principle [36ndash38] is introduced to calculate the optimal regularizationparameter aiming to reduce the impact of measurementerrors on the inversion results
To invert the boundary heat-transfer coefficient it isnecessary to firstly solve the direct problem using the predic-tive value of heat-transfer coefficient to get the temperaturecalculation at 119878 in the 119896 moment 119879119896119878 Besides in the casethat the temperature measurement value at 119878 119879119896119898119890119886 seesmeasurement error the temperature measurement value canbe expressed by the actual temperature plus themeasurementerror
where 120596 is the random number of standard normal in therange and 120590 is the standard deviation of measurement valuewhich is expressed as
120590 = radic 1119870 minus 1119896sum119909=1
(119879119896119898119890119886 minus 119879119896119886119888119905)2 (32)
where the constant K is the number of iterations
6 Complexity
The residual of heat-transfer coefficient in the wholeinversion time domain is defined as
119878 (120572) = radic 1119870 minus 1119896sum119896=1
(ℎ119896119886119888119905 minus ℎ119896119878)2 (33)
In (33) ℎ119896119886119888119905 and ℎ119896119878 are the actual value and inversion valueof heat-transfer coefficient respectively
Since ℎ119896119878 is unknown it is available to calculate thetemperaturemeasurement value at 119878 (119879119896119878 ) using ℎ119896119878 with directproblem algorithm thus the temperature residual in theinversion time domain can be obtained
119878 (120572) = radic 1119870 minus 1119896sum119896=1
(119879119896119898119890119886 minus 1198791198961198781)2 (34)
In ideal condition
ℎ119896119886119888119905 = ℎ119896119878 (35)
Similarily
1198791198961198781 = 119879119896119878 (36)
From the residual principle the regularization parameteris the optimal when both (35) and (36) are satisfied
119878119879 (120572) = 120590 (37)
34 Solving Procedure of Inverse Problem (1) Select the initialpredictive value of the heat-transfer coefficient ℎ119896119901 at a timemoment to perform inversion(2)Obtain the temperature calculation values inmeasure-ment point S at R time moments after that moment based onℎ119896119901 and (20)(3) Calculate the optimal regularization parameter 120572based on (37)(4)Assume the heat-transfer coefficient in the initial stageof inversion ℎ = 0 and obtain the step response matrix 119860 byEq (25)(5) Confirm the heat-transfer coefficient at the timemoment ℎ119896 according to (30) and then use the direct problemalgorithm to reconstruct the temperature field when the heat-transfer coefficient is ℎ119896(6) Following the time direction backward change thevalue in the initial stage of inversion and repeat steps (4) and(5) then get the inverse value of heat-transfer coefficient atdifferent time moments
4 Numerical Experiment and Analysis
Numerical experiments are performed to validate whetherthe proposed method is effective with the focus on analyzingthe impact of differentmeasurement errors andmeasurementpoint positions on the inversion result Also the inversionresult obtained in the condition without measurement erroris compared with the practical result which verifies theprecision of the proposed method
Γ1
Γ2
Γ3
Ω
Figure 1 Heat conduction model
4
2
1rsquo
3rsquo
2rsquo
0 3
1
Figure 2 Boundary node
D4
y
D3
D2D1
x
hTf
ΩΩ
Figure 3 The model of two-dimensional unsteady-state heat con-duction system without internal heat source
The two-dimensional plate heat transfer model (Figure 1)used in the above-mentioned direct problem is adoptedIn the simulation example the length Lx and width Ly ofthe plate is 02119898 The heat conductivity coefficient 120582 =47(119882119898lowast119870) thermal diffusivity119886 = 128lowast10minus6(1198982119904) initialtemperature 1198790 = 20∘119862 environment temperature 119879119891 =50∘119862 and heat-transfer coefficient ℎ = 2000(1198821198982 lowast119870) Thepurpose is to obtain the actual heat-transfer coefficient of theboundary D4
Complexity 7
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
21
22
23
24
25
26
27
28
29
30
Figure 4 The temperature field in the t=50s
016
017
018
019
020
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
Y
Figure 5 The temperature field in the t=100s
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
016
017
018
019
Y
Figure 6 The temperature field in the t=150s
41 Impact When the Measurement Error Is Zero Given themeasurement error 120590 = 0000 when the measurement pointis in the 119871 = 01119898 of D4 boundary and the future time step119877 = 5 the inversion result is shown in Figure 9
Figure 9 displays that except the transitory vibration inthe initial stage The inversion value is basically identical tothe practical value demonstrating the effectiveness of theinversion algorithm
8 Complexity
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
34
Figure 7 The temperature field in the t=200s
20
21
22
23
24
25
26
27
28
29
30
L1=01mL2=0004m
L3=0008ml1=0001m
50 100 150 200 2500Time (s)
Tem
pera
ture
(∘C)
Figure 8 The curve of the measuring point with time
42 Impact of Measurement Error Given the future time step119877 = 5 and the measurement point is in the 119871 = 01119898 of D4boundary the inversion results when the measurement erroris 120590 = 0001 120590 = 0005 and 120590 = 001 are displayed in Figures10 11 and 12 respectively
According to Table 1 and Figures 10 11 and 12 smallerrelative measurement error contributes to better inversionresults And enlarging measurement error will worsen theinversion results and aggravate the fluctuation
43 Impact of Measurement Point Position Given measure-ment error 120590 = 0001 and future time step 119877 = 5 theinversion results when the measurement point is in the 119871 =0004119898 0008119898 of D4 boundary and when the measurementpoint is 119897 = 0001119898 from the D4 boundary are shown inFigures 13 14 and 15 respectively
Analyze the contents of Table 2 and Figures 13 and 14The explicit FDM is used for direct problem when the
measurement point in boundary is closer to the boundaryangular point which however imposes a little impact on theinversion result Despite the increased relative average errorthe proposed method still exhibits a better ability to track theexact solution of heat-transfer coefficient and the inversionresult is relatively precise In Figure 15 considering that theposition of the measuring point is 0001m away from theboundary and the initial time temperature of the position is20 the temperature cannot change for a period of time sothe inversion result fluctuates greatly in the initial stage andincreases when the distance of measurement point positionfrom the boundary angular point becomes farther
5 Conclusion
The boundary heat-transfer coefficient of the two-dimensional unsteady heat conduction system is inversed bythe FDM and model prediction control method By solving
Complexity 9
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 9 The heat transfer coefficient of the measuring point without error
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 10 The heat transfer coefficient of the measuring point with 120590 = 0001
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 11 The heat transfer coefficient of the measuring point with 120590 = 0005
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 12 The heat transfer coefficient of the measuring point with 120590 = 001
and analyzing the algorithm example it demonstratesthat the proposed methods have higher accuracy in theinversion process Model predictive control method focuseson the model function rather than the structural form sothat we only need to know the step response or impulseresponse of the object we can directly get the predictionmodel and skip the derivation process It absorbs the idea
of optimization control and replaces global optimization byrolling time-domain optimization combined with feedbackcorrection which avoids a lot of calculation required byglobal optimization and constantly corrects the influencecaused by uncertain factors in the system At the sametime by discussing the impacts of error free measuringpoint positions and measuring errors on the results it
10 Complexity
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
1000150020002500300035004000
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 13 The heat transfer coefficient of the measuring point at the boundary 119871 = 0008119898
0500
10001500200025003000350040004500
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 14 The heat transfer coefficient of the measuring point at the boundary 119871 = 0004119898
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
100015002000250030003500
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 15 The heat transfer coefficient of the measuring point at the distance from the boundary 119897 = 0001119898
Table 1 Relative average errors of inversion result under differentmeasurement errors given 119871 = 01119898 and 119877 = 5Measurement error120590 0001 0005 001
Relative average error120578 701 1129 1601
Table 2 Relative average errors of inversion result in differentmeasurement point positions given 120590 = 0001 and 119877 = 5Measurement point 119871 = 0004119898 119871 = 0008119898 119897 = 0001119898Relative average error120578 765 706 2105
demonstrates that the obtained inversion results except theearly oscillation can better represent the stability of the exactsolution
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interestregarding the publication of this paper
Authorsrsquo Contributions
ShoubinWang andRuiNi contributed to developing the ideasof this research All of the authors were involved in preparingthis manuscript
Acknowledgments
This work was financially supported by the National KeyFoundation for Exploring Scientific Instrument of China
Complexity 11
(2013YQ470767) Tianjin Municipal Education CommissionProject for Scientific Research Items (2017KJ059) and Tian-jin Science and Technology Commissioner Project (18JCT-PJC62200 18JCTPJC64100)
References
[1] W Shoubin Z Li S Xiaogang and J Huangchao ldquoInversionof thermal conductivity in two-dimensional unsteady-stateheat transfer system based on boundary element method anddecentralized fuzzy inferencerdquo Complexity vol 2018 Article ID8783946 9 pages 2018
[2] S Wang Y Deng and X Sun ldquoSolving of two-dimensionalunsteady inverse heat conduction problems based on boundaryelementmethod and sequential function specificationmethodrdquoComplexity vol 2018 Article ID 6741632 11 pages 2018
[3] SWang H Jia X Sun and L Zhang ldquoResearch on the recogni-tion algorithm concerning geometric boundary regarding heatconduction based on BEM and CGMrdquoMathematical Problemsin Engineering vol 2018 Article ID 3723949 13 pages 2018
[4] S Wang L Zhang X Sun et al ldquoSolution to two-dimensionalsteady inverse heat transfer problems with interior heat sourcebased on the conjugate gradient methodrdquo Mathematical Prob-lems in Engineering vol 2017 Article ID 2861342 9 pages 2017
[5] S Wang H Jia X Sun et al ldquoTwo-dimensional steady-state boundary shape inversion of CGM-SPSO algorithm ontemperature informationrdquo Advances in Materials Science andEngineering vol 2017 Article ID 2461498 12 pages 2017
[6] Z Tianyu and D Changhong ldquoCompound control systemdesign based on adaptive backstepping theoryrdquo Journal ofBeijing University of Aeronautics and A vol 39 no 7 pp 902ndash906 2013
[7] P Duda ldquoA general method for solving transient multidimen-sional inverse heat transfer problemsrdquo International Journal ofHeat and Mass Transfer vol 93 pp 665ndash673 2016
[8] B Li and L Liu ldquoAn algorithm for geometry boundary identifi-cation of heat conduction problem based on boundary elementdiscretizationrdquo Proceedings of the CSEE vol 28 no 20 pp 38ndash43 2008
[9] C-Y Yang ldquoBoundary prediction of bio-heat conduction ina two-dimensional multilayer tissuerdquo International Journal ofHeat and Mass Transfer vol 78 no 7 pp 232ndash239 2014
[10] P Duda ldquoNumerical and experimental verification of twomethods for solving an inverse heat conduction problemrdquoInternational Journal of Heat and Mass Transfer vol 84 pp1101ndash1112 2015
[11] V M Luchesi and R T Coelho ldquoAn inverse method to estimatethe moving heat source inmachining processrdquoAppliedermalEngineering vol 45-46 pp 64ndash78 2012
[12] PDuda ldquoAmethod for transient thermal load estimation and itsapplication to identification of aerodynamic heating on atmo-spheric reentry capsulerdquo Aerospace Science and Technology vol51 pp 26ndash33 2016
[13] J V Beck B Blackwell and C R Clair Inverse HeatConductionIll-Posed Problems A Wiley-Interscience Publica-tion 1985
[14] T RHsu N S Sun G G Chen and Z L Gong ldquoFinite elementformulation for two-dimensional inverse heat conduction anal-ysisrdquo Journal of Heat Transfer vol 114 no 3 p 553 1992
[15] A A Tseng and F Z Zhao ldquoMultidimensional inverse tran-sient heat conduction problems by direct sensitivity coefficient
method using a finite-element schemerdquo Numerical Heat Trans-fer Part B Fundamentals vol 29 no 3 pp 365ndash380 1996
[16] L Zhaoming Further Studies on Fuzzy Inference Method forIn-verse Heat Transfer Problems Chongqing University 2014
[17] W Q Qian Y Zhou K F He J Y Yuan and J D HuangldquoEstimation of surface heat flux for nonlinear inverse heatconduction problemrdquo Acta Aerodynamica Sinica vol 30 no 2pp 145ndash150 2012
[18] J Su and G F Hewitt ldquoInverse heat conduction problemof estimating time-varying heattransfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004
[19] G Blanc J V Beck and M Raynaud ldquoSolution of the inverseheat conduction problemwith a time-variable number of futuretemperaturesrdquo Numerical Heat Transfer Part B Fundamentalsvol 32 no 4 pp 437ndash451 1997
[21] D Lesnic L Elliott and D B Ingham ldquoIdentification of thethermal conductivity and heat capacity in unsteady nonlin-ear heat conduction problems using the boundary elementmethodrdquo Journal of Computational Physics vol 126 no 2 pp410ndash420 1996
[22] A A Ershova and A I Sidikova ldquoUncertainty estimation ofthe method based on generalized residual principle for therestore task of the spectral density of crystalsrdquo Vestn Yuzhno-UralGosUn-taSerMatemMekhFiz vol 2015 pp 25ndash30 2015
[23] L Zhao Research on Fluid Temperature Inversion Algorithms forOne-Dimensional Unsteady Convection Heat Transfer HarbinInstitute of Technology 2017
[24] B Li and L Liu ldquoAn algorithm for geometry boundary identi-fication of heat conduction problem based on boundary ele-ment discretizationrdquo Proceedings of the CSEE vol 28 no 20pp 38ndash43 2008
[25] B Li and L Liu ldquoGeometry boundary identification of unsteadyheat conduction based on dual reciprocity boundary elementmethodrdquo Proceedings of the CSEE vol 29 no 5 pp 66ndash71 2009
[26] H Zhou X Xu X Li and H Chen ldquoIdentification oftemperature-dependent thermal conductivity for 2-D transientheat conduction problemsrdquo Applied Mathematics and Mechan-ics vol 29 no 1 pp 55ndash68 2014
[27] L Yanhao Model Predictive Inverse Method for Heat TransferProcess and Application Chongqing University 2017
[28] J Fan Reconstruction of Boiler Drum Temperature Field Basedon Inverse Heat Transfer Problem Chongqing University 2017
[29] H Ya-fen et alMechanism and Characteristics of Heat Conduc-tion in Nanocomposites Harbin Institute of Technology 2013
[30] Q Yue-ping M Jun J Jing-yan Y Xiao-bin and L WeildquoUnsteady heat transfer problems with finite volume methodrdquoJournal of Liaoning Technical University Natural Science vol 32no 05 pp 577ndash581 2013
[31] C-H Chiu and C-K Chen ldquoApplications of adomianrsquos decom-position procedure to the analysis of convective-radiative finsrdquoJournal of Heat Transfer vol 125 no 2 pp 312ndash316 2003
[32] C Sheng Direct and Inverse Heat Conduction Problems Solvingby the Boundary Element Method Hunan University 2007
[33] MMKhader AM Eid andAMMegahed ldquoNumerical stud-ies using FDM for viscous dissipation and thermal radiationeffects on the slip flow and heat transfer due to a stretchingsheet embedded in a porous medium with variable thickness
12 Complexity
and variable thermal conductivityrdquoNewTrends inMathematicalSciences vol 4 no 1 p 38 2016
[34] S Chuanzhuo and Z Tiande ldquoA simple method for calculatingtruncation error of numerical integral formulardquo Journal ofMathematis for Technology vol 2 1994
[35] Q Jixin Z Jun and X Zuhu Predictive Control ChemicalIndustry Press 2007
[36] L Zhang H Chen G Wang and Z Luo ldquoSolving transientinverse heat conduction problems based on optimal number offuture time stepsrdquo Proceedings of the CSEE vol 32 no 2 pp99ndash103 2012
[37] N Buong T T Huong and N T Thuy ldquoA quasi-residualprinciple in regularization for a common solution of a system ofnonlinearmonotone ill-posed equationsrdquoRussianMathematicsvol 60 no 3 pp 47ndash55 2016
[38] K A Woodbury and S K Thakur ldquoRedundant data and futuretimes in the inverse heat conduction problemrdquo Inverse Problemsin Science and Engineering vol 2 no 4 pp 319ndash333 1996
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2 Complexity
by employing Alifanovrsquos iterative regularization algorithm[18] Blanc G et al investigated the one-dimensional transientinverse problem finding that residual principle can optimizethe key parameter in the heat conduction problem [19] WuZhaochun studied the measurement point arrangement pat-tern during the solving process of the two-dimensional heatconduction inverse problem with DSC method accordinglymaking some rational suggestions regarding the measure-ment points [20] Lesnic et al identified the thermophysicalparameters in one-dimensional transient heat conductionproblems by using the BEM [21] Ershova et al used the resid-ual error principles in the Tikhonov regularization methodand completed the crystal phonon inspection identificationtasks [22] Zhao Luyao combined the particle swarm opti-mization (PSO) and conjugate gradient method and appliedthe combined method to the inversion of the heat-transfercoefficient of one-dimensional unsteady-state system It hasbeen reported that themethod exhibits high preciseness [23]Li et al researched IHCP by using the BEM and identifiedthe irregular boundaries [24 25] Zhou et al solved the heatconductivity coefficient in the two-dimensional transientinverse problems by using the BEM and gradient regulariza-tion method and obtained the relatively accurate inversionresults [26] Li Yanhao resolved the heat-flow problem foundin the two-dimensional transient inverse problem by usingthe model prediction control algorithm and inversion resultwas relatively precise [27] Fan Jianxue also adopted themodel prediction control algorithm to solve the heat-transfercoefficient in the inner wall of two-dimensional transientsteam drum and achieved good calculation results [28]
Regarding the boundary heat transfer in the heat conduc-tion system in the paper FDM is adopted to solve the directproblem of the two-dimensional unsteady-state heat con-duction without internal heat source and model predictioncontrol method is used to solve the inverse problem Besidesresidual principle is introduced to optimize the regularizationparameter during the inversion process thereby improvingthe efficiency of inversion in terms of speed and time
2 Unsteady-State Direct Problem
The Inverse Heat Conduction Problem usually involves themultiple deduction of the forward problem and its inversionaccuracy is directly affected by the calculation accuracy ofthe forward problem Positive problem refers to the solutionof historical temperature field through given boundary con-ditions initial temperature and thermal conductivity differ-ential equation Common solutions are Lattice BoltzmannMethod Finite Volume Method Adomain DecompositionMethod Boundary Element Method and Finite DifferenceMethod
LBM (Lattice Boltzmann Method) [29] is a mesoscopicresearchmethod based onmolecular kinetics which can welldescribe the complex and small interfaces in porousmedia Itis widely used in small-scale numerical simulation of porousmedia and other objects with complex interface structures
The basic idea of FVM (Finite VolumeMethod) [30] is todivide the computational region into a series of nonrepeatedcontrol volumes and make each grid point have a control
volume around it A set of discrete equations is obtained byintegrating the differential equations to be resolved with eachcontrol volume It is commonly used in the case of discreteand complex grids
ADM (Adomain Decomposition Method) [31] is aresearch method that decomposes the true solution of anequation into the sum of the components of several solutionsand then tries to find the components of the solutions andmake the sum of the components of the solutions approxi-mate the true solution with any desired high precision But itcan only obtain accurate results under the condition that theenergy conservation equation is not nonlinear
BEM (Boundary Element Method) [32] is a researchmethod which divides elements on the boundary of thedefined domain and approximates the boundary conditionsby functions satisfying the governing equations The basicadvantage of BEM is that it can reduce the dimensionality ofthe problem but when it comes to solving the basic solutionof the problem the process of solving the basic solution isgenerally complicated
In this paper a square rectangular plate is selected asan experimental physical model which is a very commonphysical model The finite-difference method [33] can reducethe amount of calculation required by other researchmethodsfor the positive problem and it is also convenient to query thetemperature change curves of the required measuring pointsin the negative problem
21 Mathematical Model The mathematical model of two-dimensional unsteady-state heat conduction without internalheat source is expressed as follows
where Γ1 is the Dirichlet (first-type) boundary conditionΓ2 is the Neumann (second-type) boundary condition Γ3 isthe Robin (third-type) boundary condition and Γ = Γ1 +Γ2 + Γ3 is the boundary of the whole region Ω 120572 denotesthe thermal diffusivity and 120572 = 120582119888120588 119888 120588 and 120582 denote thespecific heat the density and the heat conduction coefficientof the object respectively T represents the temperature 119879 isthe temperature given by the Dirichlet boundary condition119879119891 is the environment temperature and 1198790 is the initialtemperature 119902 refers to the heat flux 119899 refers to the boundaryouter normal vector and ℎ refers to the surface heat-transfercoefficient
22 Discretization and Difference Scheme The discrete rulesof two-dimensional unsteady-state heat conduction problem
Complexity 3
without internal heat source in geometry and time domainare as follows
Assuming that after the domain of uniformdiscretizationthe step length of x-axis is Δ119909 = 119909119894+1 minus 119909119894 and that of y-axisis Δy = 119910119895+1 minus 119910119895 obviously 119909119894 = 119894Δ119909 119910119895 = 119895Δ119910 and 119894 119895 =0 1 2 3
n(119899 = 0 1 2 3 ) is used to uniformly discretizethe time domain t≧0 and the step length between twotime moments Δ119905 = 119905119899+1 minus 119905119899 119905119899 = 119899Δ119905 where(119894 119895) sdot sdot sdot (119909119894 119910119895) 119899 sdot sdot sdot 119905119899 119879119899119894119895 sdot sdot sdot 119899 and the temperature at node(119894 119895) in the time moment 119899 is 119879(119909119894 119910119895 119905119899)
The explicit difference array of the two-dimensionalunsteady-state heat conduction without internal heat sourceis expressed as follows
Applying the first heat conduction equation in (1) to node(119894 119895) at the time moment of 119899 the equation can be rewrittenas
The partial differential in the two sides of (2) can beapproximated by difference quotient The temperature itemin the right of the equal sign can be approximated by first-order time forward difference quotient
(120597119879120597119905 )119899
119894119895
= (119879119899+1119894119895 minus 119879119899119894119895)Δ119905 + 119874 (Δ119905) (3)
The second-order partial differential in the left of theequal sign can be approximated by the central differencequotient
(12059721198791205971199092 )119899
119894119895
= (119879119899119894+1119895 minus 2119879119899119894119895 + 119879119899119894minus1119895)(Δ119909)2 + 119874 (Δ1199092) (4)
(12059721198791205971199102 )119899
119894119895
= (119879119899119894119895+1 minus 2119879119899119894119895 + 119879119899119894119895minus1)(Δ119910)2 + 119874 (Δ1199102) (5)
Substituting (3) (4) and (5) into (2) we can get thedifference equation of (2)
119879119899119894+1119895 minus 2119879119899119894119895 + 119879119899119894minus1119895(Δ119909)2 + 119879119899119894119895+1 minus 2119879119899119894119895 + 119879119899119894119895minus1(Δ119910)2= 1120572 119879
119899+1119894119895 minus 119879119899119894119895Δ119905
(6)
Equation (6) is the difference equation of heat conductionequation and the truncation error [34] is 119874(Δ119905 + Δ1199092Δ1199102)
Assuming that Δ119909 = Δ119910 = Δ and substituting it into (6)we can obtain
119879119899119894+1119895 + 119879119899119894minus1119895 + 119879119899119894119895minus1 minus (4 minus 11198650)119879119899119894119895 =119879119899+11198941198951198650 (7)
where 1198650 refers to the Fourier coefficient and 1198650 =120572Δ119905(Δ119909)2 = 120572Δ119905(Δ119910)2 = 120572Δ119905Δ2
The stability condition of explicit finite difference equa-tion of two-dimensional unsteady-state heat conductionwithout internal heat source is in interior node 1198650 le 14in boundary node 1198650 le 1[2(2 + 119861119894)] in boundary angularpoint 1198650 le 1[4(1 + 119861119894)]23 Boundary Conditions First-type boundary conditionie the temperature is given In general when the FDM isused for calculation it shall be processed as in the momentof the initial the boundary node temperature is 1198792 then theboundary node temperature remains at 119879
In the second- and third-type boundary condition it isnecessary to set virtual node outside the boundary to makethe boundary node into interior node The node numberingis shown in Figure 2
Second-type boundary condition ie the heat flowboundary is givenminus120582(120597119879120597119899) = 119902 Setting boundary 119909 = 0 asthe given heat flow boundary condition and keeping it stablethe second-type boundary condition can be expressed asminus120582(120597119879120597119909) = 119902 Using central difference quotient to replacethe first-order partial differential equation
minus1205821198791198992 minus 119879119899210158402Δ = 119902 (8)
(8) is rewritten as
11987911989921015840 = 1198791198992 minus 2Δ120582 119902 (9)
Node 1 is changed into interior node and (9) is substitutedinto the interior node explicit difference equation of (7) to get
1198791198992 + 11987911989921015840 + 1198791198994 + 1198791198990 minus (4 minus 11198650)1198791198991 = 11198650119879119899+11 (10)
Substituting (9) into (10) we can obtain
119879119899+11 = 1198650 [1198791198990 + 21198791198992 + 1198791198994 minus 2Δ120582 119902 + ( 11198650 minus 4)1198791198991] (11)
Third-type boundary condition ie the heat transferboundary is given 120582(120597119879120597119899) = ℎ(119879 minus 119879119891) Setting boundary119909 = 0 as the given heat transfer boundary condition andkeeping it stable the third-type boundary condition can beexpressed as 120582(120597119879120597119909) = ℎ(119879 minus 119879119891) Using central differencequotient to replace the first-order partial differential equation
120582 (1198791198992 minus 11987911989921015840)(2Δ) = ℎ (1198791198991 minus 119879119891) (12)
(12) is rewritten as
11987911989921015840 = 1198791198992 minus 2119861119894 (1198791198991 minus 119879119891) (13)
Node 1 is changed into interior node and (13) is substi-tuted into the interior node explicit difference equation of (7)to get
1198791198992 + 11987911989921015840 + 1198791198994 + 1198791198990 minus (4 minus 11198650)1198791198991 = 11198650119879119899+11 (14)
4 Complexity
Substituting (13) into (14) we can obtain
119879119899+11= 1198650 [1198791198990 + 21198791198992 + 1198791198994 + 2119861119894119879119891 + ( 11198650 minus 4 minus 2119861119894)1198791198991]
(15)
where 119861119894 is the Biot number 119861119894 = ℎΔ120582 the truncationerror of the second- and third-type boundary condition is119874(Δ2)
Adiabatic boundary condition is 119879 = 0 Similarly thesecond- and third-type boundary condition is
Boundary angular point is 0 node Virtual nodes 1rsquo and3rsquo are set in the symmetric position of node 1 and node 3respectively and the central different quotient is applied inthe 119909- and 119910-direction respectively
120582 (1198791198993 minus 11987911989931015840)(2Δ) = ℎ (1198791198990 minus 119879119891)120582 (1198791198991 minus 11987911989911015840)(2Δ) = ℎ (1198791198990 minus 119879119891)
(17)
Equation (17) is rewritten as
11987911989931015840 = 1198791198993 minus 2119861119894 (1198791198990 minus 119879119891)11987911989911015840 = 1198791198991 minus 2119861119894 (1198791198990 minus 119879119891) (18)
Node 1 is changed into interior node and (17) and (18) aresubstituted into (7) to get
1198791198993 + 11987911989931015840 + 1198791198991 + 11987911989911015840 minus (4 minus 11198650)1198791198990 = 11198650119879119899+10119879119899+10 = 21198650 [1198791198991 + 1198791198993 + 2119861119894119879119891 + ( 121198650 minus 2 minus 2119861119894)1198791198990]
(19)
By (7) (11) (15) (16) and (19) the temperature value inany point of the model can be obtained
24 Mathematical Model about the Heat Transfer Processof Rectangular Plate Figure 3 shows the model of two-dimensional unsteady-state heat conduction system withoutinternal heat source The rectangle plate in Figure 3 isadopted boundary 1198631 1198632 1198633 is for heat insulation and 1198634is the third-type boundary condition ℎ is the heat-transfercoefficient Then (1) can be changed by the correspondingmathematical model as follows
25 Direct Problem Verification Figures 4 5 6 and 7display the temperature field distribution when 119905 =50 100 150 200(119904) Figure 8 is the curve of measuring pointswith time The simulation result of direct problem solvingcan demonstrate the rationality of explicit finite differencewhich is convenient for performing the inversion algorithmof inverse problem
The length Lx and width Ly of the plate is 02119898 Theheat conductivity coefficient 120582 = 47(119882119898 lowast 119870) thermaldiffusivity 119886 = 128 lowast 10minus6(1198982119904) initial temperature 1198790 =20∘119862 environment temperature119879119891 = 50∘119862 and heat-transfercoefficient ℎ = 2000(1198821198982 lowast 119870)3 Unsteady-State Problem
Predictive control is a model-based control algorithm whichfocuses on the function of the model rather than the formof the model Compared with other control methods itscharacteristics are reflected in the use of rolling optimizationand rolling implementation of the control mode to achievethe control effect but also did not give up the traditionalcontrol feedback Therefore the predictive control algorithmis based on the future dynamic behavior of the processmodel prediction system under a certain control effect usesthe rolling optimization to obtain the control effect underthe corresponding constraint conditions and performancerequirements and corrects the prediction of future dynamicbehavior in the rolling optimization process by detecting real-time information
31 PredictionModel of Inverse Problem The step response ofheat-transfer coefficient in the1198634 boundary of direct problemmodel is taken as the prediction model of inverse problemThe increment of heat-transfer coefficient in the boundary infuture time ie Δℎ119896+119875 is used to predict the temperature at119878 in the 1198634 boundary in the moment of 119877 ie 119879119896+119875 where119875 = 0 1 2 119877 minus 1 and 119877 is the future time step
According to the principle of linear superposition [35]after loading 119877 group of increment Δℎ119896+119875 on system sincethe time moment of 119896 the temperature at 119878 ie 119879119896+119875 isobtained
Δ120601119884 = 120601119884+1 minus 120601119884 (119884 = 0 1 sdot sdot sdot 119877 minus 1) (24)
The step response system function from the timemoment119905119896 to 119905119896+119877minus1 is defined as the impact of heat-transfer coefficienton 119879(119909 119910 119905) After the derivation of ℎ119896 by (20) the stepresponse equation can be obtained as follows
12059721198671205971199092 + 12059721198671205971199102 = 1120572 120597119867120597119905 (isin Ω 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 0 (isin 1198631 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 0 (isin 1198632 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119910 = 0 (isin 1198633 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 119879119896 minus 119879119891 minus ℎ119867 (119909 119910 119905)
(isin Γ3 119905119896 le 119905 le 119905119896+119877minus1)119867 = 1198670 (isin Ω 119905 = 119905119896)
(25)
From (25) it can be seen that119867 is related to ℎ Thereforewhile solving inverse problem it is necessary to use theexplicit difference algorithm used in the direct problemsolving and keep updating the calculation of ℎ
The corresponding discrete value 119867119896+119877minus1 is obtained by(25) hence the dynamic step response coefficient 120601119877 isfurther determined as
120601119877 = 119867119896+119877minus1 (26)
32 Rolling Optimization of Inverse Problem Measurementvalue and predictive value of temperature can be seen inthe time range from 119905119896 to 119905119896+119877minus1 According to the finiteoptimization 119877 parameters to be inverted are obtained sothe predictive value can be as close as possible to the mea-surement value in the future time domain at this momentthe quadratic performance index of system can be launched
min 119869 (ℎ) = (119879119898119890119886 minus 119879119901119903119890)Τ (119879119898119890119886 minus 119879119901119903119890) + ℎΤ120579ℎ (27)
where 120579 is the regularization parameter matrix
and 120572 is the regularization parameterAfter the derivation of ℎ based on (27) and making119889119869(ℎ)119889ℎ = 0 the optimal control rate can be obtained as
ℎ = (119860Τ119860 + 120579)minus1 119860Τ [119879119898119890119886 minus 11987911990111990311989010038161003816100381610038161003816ℎ=0] (29)
The optimal heat-transfer coefficient at moment 119896 can beobtained following (29)
ℎ = R (119860Τ119860 + 120579)minus1 119860Τ [119879119898119890119886 minus 11987911990111990311989010038161003816100381610038161003816ℎ=0] (30)
where R = [1 0 0]Τ There is no need to presetthe function form for the inverse problem calculation by theabove optimization algorithm
33 Regularization Parameter The residual principle [36ndash38] is introduced to calculate the optimal regularizationparameter aiming to reduce the impact of measurementerrors on the inversion results
To invert the boundary heat-transfer coefficient it isnecessary to firstly solve the direct problem using the predic-tive value of heat-transfer coefficient to get the temperaturecalculation at 119878 in the 119896 moment 119879119896119878 Besides in the casethat the temperature measurement value at 119878 119879119896119898119890119886 seesmeasurement error the temperature measurement value canbe expressed by the actual temperature plus themeasurementerror
where 120596 is the random number of standard normal in therange and 120590 is the standard deviation of measurement valuewhich is expressed as
120590 = radic 1119870 minus 1119896sum119909=1
(119879119896119898119890119886 minus 119879119896119886119888119905)2 (32)
where the constant K is the number of iterations
6 Complexity
The residual of heat-transfer coefficient in the wholeinversion time domain is defined as
119878 (120572) = radic 1119870 minus 1119896sum119896=1
(ℎ119896119886119888119905 minus ℎ119896119878)2 (33)
In (33) ℎ119896119886119888119905 and ℎ119896119878 are the actual value and inversion valueof heat-transfer coefficient respectively
Since ℎ119896119878 is unknown it is available to calculate thetemperaturemeasurement value at 119878 (119879119896119878 ) using ℎ119896119878 with directproblem algorithm thus the temperature residual in theinversion time domain can be obtained
119878 (120572) = radic 1119870 minus 1119896sum119896=1
(119879119896119898119890119886 minus 1198791198961198781)2 (34)
In ideal condition
ℎ119896119886119888119905 = ℎ119896119878 (35)
Similarily
1198791198961198781 = 119879119896119878 (36)
From the residual principle the regularization parameteris the optimal when both (35) and (36) are satisfied
119878119879 (120572) = 120590 (37)
34 Solving Procedure of Inverse Problem (1) Select the initialpredictive value of the heat-transfer coefficient ℎ119896119901 at a timemoment to perform inversion(2)Obtain the temperature calculation values inmeasure-ment point S at R time moments after that moment based onℎ119896119901 and (20)(3) Calculate the optimal regularization parameter 120572based on (37)(4)Assume the heat-transfer coefficient in the initial stageof inversion ℎ = 0 and obtain the step response matrix 119860 byEq (25)(5) Confirm the heat-transfer coefficient at the timemoment ℎ119896 according to (30) and then use the direct problemalgorithm to reconstruct the temperature field when the heat-transfer coefficient is ℎ119896(6) Following the time direction backward change thevalue in the initial stage of inversion and repeat steps (4) and(5) then get the inverse value of heat-transfer coefficient atdifferent time moments
4 Numerical Experiment and Analysis
Numerical experiments are performed to validate whetherthe proposed method is effective with the focus on analyzingthe impact of differentmeasurement errors andmeasurementpoint positions on the inversion result Also the inversionresult obtained in the condition without measurement erroris compared with the practical result which verifies theprecision of the proposed method
Γ1
Γ2
Γ3
Ω
Figure 1 Heat conduction model
4
2
1rsquo
3rsquo
2rsquo
0 3
1
Figure 2 Boundary node
D4
y
D3
D2D1
x
hTf
ΩΩ
Figure 3 The model of two-dimensional unsteady-state heat con-duction system without internal heat source
The two-dimensional plate heat transfer model (Figure 1)used in the above-mentioned direct problem is adoptedIn the simulation example the length Lx and width Ly ofthe plate is 02119898 The heat conductivity coefficient 120582 =47(119882119898lowast119870) thermal diffusivity119886 = 128lowast10minus6(1198982119904) initialtemperature 1198790 = 20∘119862 environment temperature 119879119891 =50∘119862 and heat-transfer coefficient ℎ = 2000(1198821198982 lowast119870) Thepurpose is to obtain the actual heat-transfer coefficient of theboundary D4
Complexity 7
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
21
22
23
24
25
26
27
28
29
30
Figure 4 The temperature field in the t=50s
016
017
018
019
020
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
Y
Figure 5 The temperature field in the t=100s
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
016
017
018
019
Y
Figure 6 The temperature field in the t=150s
41 Impact When the Measurement Error Is Zero Given themeasurement error 120590 = 0000 when the measurement pointis in the 119871 = 01119898 of D4 boundary and the future time step119877 = 5 the inversion result is shown in Figure 9
Figure 9 displays that except the transitory vibration inthe initial stage The inversion value is basically identical tothe practical value demonstrating the effectiveness of theinversion algorithm
8 Complexity
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
34
Figure 7 The temperature field in the t=200s
20
21
22
23
24
25
26
27
28
29
30
L1=01mL2=0004m
L3=0008ml1=0001m
50 100 150 200 2500Time (s)
Tem
pera
ture
(∘C)
Figure 8 The curve of the measuring point with time
42 Impact of Measurement Error Given the future time step119877 = 5 and the measurement point is in the 119871 = 01119898 of D4boundary the inversion results when the measurement erroris 120590 = 0001 120590 = 0005 and 120590 = 001 are displayed in Figures10 11 and 12 respectively
According to Table 1 and Figures 10 11 and 12 smallerrelative measurement error contributes to better inversionresults And enlarging measurement error will worsen theinversion results and aggravate the fluctuation
43 Impact of Measurement Point Position Given measure-ment error 120590 = 0001 and future time step 119877 = 5 theinversion results when the measurement point is in the 119871 =0004119898 0008119898 of D4 boundary and when the measurementpoint is 119897 = 0001119898 from the D4 boundary are shown inFigures 13 14 and 15 respectively
Analyze the contents of Table 2 and Figures 13 and 14The explicit FDM is used for direct problem when the
measurement point in boundary is closer to the boundaryangular point which however imposes a little impact on theinversion result Despite the increased relative average errorthe proposed method still exhibits a better ability to track theexact solution of heat-transfer coefficient and the inversionresult is relatively precise In Figure 15 considering that theposition of the measuring point is 0001m away from theboundary and the initial time temperature of the position is20 the temperature cannot change for a period of time sothe inversion result fluctuates greatly in the initial stage andincreases when the distance of measurement point positionfrom the boundary angular point becomes farther
5 Conclusion
The boundary heat-transfer coefficient of the two-dimensional unsteady heat conduction system is inversed bythe FDM and model prediction control method By solving
Complexity 9
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 9 The heat transfer coefficient of the measuring point without error
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 10 The heat transfer coefficient of the measuring point with 120590 = 0001
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 11 The heat transfer coefficient of the measuring point with 120590 = 0005
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 12 The heat transfer coefficient of the measuring point with 120590 = 001
and analyzing the algorithm example it demonstratesthat the proposed methods have higher accuracy in theinversion process Model predictive control method focuseson the model function rather than the structural form sothat we only need to know the step response or impulseresponse of the object we can directly get the predictionmodel and skip the derivation process It absorbs the idea
of optimization control and replaces global optimization byrolling time-domain optimization combined with feedbackcorrection which avoids a lot of calculation required byglobal optimization and constantly corrects the influencecaused by uncertain factors in the system At the sametime by discussing the impacts of error free measuringpoint positions and measuring errors on the results it
10 Complexity
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
1000150020002500300035004000
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 13 The heat transfer coefficient of the measuring point at the boundary 119871 = 0008119898
0500
10001500200025003000350040004500
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 14 The heat transfer coefficient of the measuring point at the boundary 119871 = 0004119898
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
100015002000250030003500
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 15 The heat transfer coefficient of the measuring point at the distance from the boundary 119897 = 0001119898
Table 1 Relative average errors of inversion result under differentmeasurement errors given 119871 = 01119898 and 119877 = 5Measurement error120590 0001 0005 001
Relative average error120578 701 1129 1601
Table 2 Relative average errors of inversion result in differentmeasurement point positions given 120590 = 0001 and 119877 = 5Measurement point 119871 = 0004119898 119871 = 0008119898 119897 = 0001119898Relative average error120578 765 706 2105
demonstrates that the obtained inversion results except theearly oscillation can better represent the stability of the exactsolution
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interestregarding the publication of this paper
Authorsrsquo Contributions
ShoubinWang andRuiNi contributed to developing the ideasof this research All of the authors were involved in preparingthis manuscript
Acknowledgments
This work was financially supported by the National KeyFoundation for Exploring Scientific Instrument of China
Complexity 11
(2013YQ470767) Tianjin Municipal Education CommissionProject for Scientific Research Items (2017KJ059) and Tian-jin Science and Technology Commissioner Project (18JCT-PJC62200 18JCTPJC64100)
References
[1] W Shoubin Z Li S Xiaogang and J Huangchao ldquoInversionof thermal conductivity in two-dimensional unsteady-stateheat transfer system based on boundary element method anddecentralized fuzzy inferencerdquo Complexity vol 2018 Article ID8783946 9 pages 2018
[2] S Wang Y Deng and X Sun ldquoSolving of two-dimensionalunsteady inverse heat conduction problems based on boundaryelementmethod and sequential function specificationmethodrdquoComplexity vol 2018 Article ID 6741632 11 pages 2018
[3] SWang H Jia X Sun and L Zhang ldquoResearch on the recogni-tion algorithm concerning geometric boundary regarding heatconduction based on BEM and CGMrdquoMathematical Problemsin Engineering vol 2018 Article ID 3723949 13 pages 2018
[4] S Wang L Zhang X Sun et al ldquoSolution to two-dimensionalsteady inverse heat transfer problems with interior heat sourcebased on the conjugate gradient methodrdquo Mathematical Prob-lems in Engineering vol 2017 Article ID 2861342 9 pages 2017
[5] S Wang H Jia X Sun et al ldquoTwo-dimensional steady-state boundary shape inversion of CGM-SPSO algorithm ontemperature informationrdquo Advances in Materials Science andEngineering vol 2017 Article ID 2461498 12 pages 2017
[6] Z Tianyu and D Changhong ldquoCompound control systemdesign based on adaptive backstepping theoryrdquo Journal ofBeijing University of Aeronautics and A vol 39 no 7 pp 902ndash906 2013
[7] P Duda ldquoA general method for solving transient multidimen-sional inverse heat transfer problemsrdquo International Journal ofHeat and Mass Transfer vol 93 pp 665ndash673 2016
[8] B Li and L Liu ldquoAn algorithm for geometry boundary identifi-cation of heat conduction problem based on boundary elementdiscretizationrdquo Proceedings of the CSEE vol 28 no 20 pp 38ndash43 2008
[9] C-Y Yang ldquoBoundary prediction of bio-heat conduction ina two-dimensional multilayer tissuerdquo International Journal ofHeat and Mass Transfer vol 78 no 7 pp 232ndash239 2014
[10] P Duda ldquoNumerical and experimental verification of twomethods for solving an inverse heat conduction problemrdquoInternational Journal of Heat and Mass Transfer vol 84 pp1101ndash1112 2015
[11] V M Luchesi and R T Coelho ldquoAn inverse method to estimatethe moving heat source inmachining processrdquoAppliedermalEngineering vol 45-46 pp 64ndash78 2012
[12] PDuda ldquoAmethod for transient thermal load estimation and itsapplication to identification of aerodynamic heating on atmo-spheric reentry capsulerdquo Aerospace Science and Technology vol51 pp 26ndash33 2016
[13] J V Beck B Blackwell and C R Clair Inverse HeatConductionIll-Posed Problems A Wiley-Interscience Publica-tion 1985
[14] T RHsu N S Sun G G Chen and Z L Gong ldquoFinite elementformulation for two-dimensional inverse heat conduction anal-ysisrdquo Journal of Heat Transfer vol 114 no 3 p 553 1992
[15] A A Tseng and F Z Zhao ldquoMultidimensional inverse tran-sient heat conduction problems by direct sensitivity coefficient
method using a finite-element schemerdquo Numerical Heat Trans-fer Part B Fundamentals vol 29 no 3 pp 365ndash380 1996
[16] L Zhaoming Further Studies on Fuzzy Inference Method forIn-verse Heat Transfer Problems Chongqing University 2014
[17] W Q Qian Y Zhou K F He J Y Yuan and J D HuangldquoEstimation of surface heat flux for nonlinear inverse heatconduction problemrdquo Acta Aerodynamica Sinica vol 30 no 2pp 145ndash150 2012
[18] J Su and G F Hewitt ldquoInverse heat conduction problemof estimating time-varying heattransfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004
[19] G Blanc J V Beck and M Raynaud ldquoSolution of the inverseheat conduction problemwith a time-variable number of futuretemperaturesrdquo Numerical Heat Transfer Part B Fundamentalsvol 32 no 4 pp 437ndash451 1997
[21] D Lesnic L Elliott and D B Ingham ldquoIdentification of thethermal conductivity and heat capacity in unsteady nonlin-ear heat conduction problems using the boundary elementmethodrdquo Journal of Computational Physics vol 126 no 2 pp410ndash420 1996
[22] A A Ershova and A I Sidikova ldquoUncertainty estimation ofthe method based on generalized residual principle for therestore task of the spectral density of crystalsrdquo Vestn Yuzhno-UralGosUn-taSerMatemMekhFiz vol 2015 pp 25ndash30 2015
[23] L Zhao Research on Fluid Temperature Inversion Algorithms forOne-Dimensional Unsteady Convection Heat Transfer HarbinInstitute of Technology 2017
[24] B Li and L Liu ldquoAn algorithm for geometry boundary identi-fication of heat conduction problem based on boundary ele-ment discretizationrdquo Proceedings of the CSEE vol 28 no 20pp 38ndash43 2008
[25] B Li and L Liu ldquoGeometry boundary identification of unsteadyheat conduction based on dual reciprocity boundary elementmethodrdquo Proceedings of the CSEE vol 29 no 5 pp 66ndash71 2009
[26] H Zhou X Xu X Li and H Chen ldquoIdentification oftemperature-dependent thermal conductivity for 2-D transientheat conduction problemsrdquo Applied Mathematics and Mechan-ics vol 29 no 1 pp 55ndash68 2014
[27] L Yanhao Model Predictive Inverse Method for Heat TransferProcess and Application Chongqing University 2017
[28] J Fan Reconstruction of Boiler Drum Temperature Field Basedon Inverse Heat Transfer Problem Chongqing University 2017
[29] H Ya-fen et alMechanism and Characteristics of Heat Conduc-tion in Nanocomposites Harbin Institute of Technology 2013
[30] Q Yue-ping M Jun J Jing-yan Y Xiao-bin and L WeildquoUnsteady heat transfer problems with finite volume methodrdquoJournal of Liaoning Technical University Natural Science vol 32no 05 pp 577ndash581 2013
[31] C-H Chiu and C-K Chen ldquoApplications of adomianrsquos decom-position procedure to the analysis of convective-radiative finsrdquoJournal of Heat Transfer vol 125 no 2 pp 312ndash316 2003
[32] C Sheng Direct and Inverse Heat Conduction Problems Solvingby the Boundary Element Method Hunan University 2007
[33] MMKhader AM Eid andAMMegahed ldquoNumerical stud-ies using FDM for viscous dissipation and thermal radiationeffects on the slip flow and heat transfer due to a stretchingsheet embedded in a porous medium with variable thickness
12 Complexity
and variable thermal conductivityrdquoNewTrends inMathematicalSciences vol 4 no 1 p 38 2016
[34] S Chuanzhuo and Z Tiande ldquoA simple method for calculatingtruncation error of numerical integral formulardquo Journal ofMathematis for Technology vol 2 1994
[35] Q Jixin Z Jun and X Zuhu Predictive Control ChemicalIndustry Press 2007
[36] L Zhang H Chen G Wang and Z Luo ldquoSolving transientinverse heat conduction problems based on optimal number offuture time stepsrdquo Proceedings of the CSEE vol 32 no 2 pp99ndash103 2012
[37] N Buong T T Huong and N T Thuy ldquoA quasi-residualprinciple in regularization for a common solution of a system ofnonlinearmonotone ill-posed equationsrdquoRussianMathematicsvol 60 no 3 pp 47ndash55 2016
[38] K A Woodbury and S K Thakur ldquoRedundant data and futuretimes in the inverse heat conduction problemrdquo Inverse Problemsin Science and Engineering vol 2 no 4 pp 319ndash333 1996
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Submit your manuscripts atwwwhindawicom
Complexity 3
without internal heat source in geometry and time domainare as follows
Assuming that after the domain of uniformdiscretizationthe step length of x-axis is Δ119909 = 119909119894+1 minus 119909119894 and that of y-axisis Δy = 119910119895+1 minus 119910119895 obviously 119909119894 = 119894Δ119909 119910119895 = 119895Δ119910 and 119894 119895 =0 1 2 3
n(119899 = 0 1 2 3 ) is used to uniformly discretizethe time domain t≧0 and the step length between twotime moments Δ119905 = 119905119899+1 minus 119905119899 119905119899 = 119899Δ119905 where(119894 119895) sdot sdot sdot (119909119894 119910119895) 119899 sdot sdot sdot 119905119899 119879119899119894119895 sdot sdot sdot 119899 and the temperature at node(119894 119895) in the time moment 119899 is 119879(119909119894 119910119895 119905119899)
The explicit difference array of the two-dimensionalunsteady-state heat conduction without internal heat sourceis expressed as follows
Applying the first heat conduction equation in (1) to node(119894 119895) at the time moment of 119899 the equation can be rewrittenas
The partial differential in the two sides of (2) can beapproximated by difference quotient The temperature itemin the right of the equal sign can be approximated by first-order time forward difference quotient
(120597119879120597119905 )119899
119894119895
= (119879119899+1119894119895 minus 119879119899119894119895)Δ119905 + 119874 (Δ119905) (3)
The second-order partial differential in the left of theequal sign can be approximated by the central differencequotient
(12059721198791205971199092 )119899
119894119895
= (119879119899119894+1119895 minus 2119879119899119894119895 + 119879119899119894minus1119895)(Δ119909)2 + 119874 (Δ1199092) (4)
(12059721198791205971199102 )119899
119894119895
= (119879119899119894119895+1 minus 2119879119899119894119895 + 119879119899119894119895minus1)(Δ119910)2 + 119874 (Δ1199102) (5)
Substituting (3) (4) and (5) into (2) we can get thedifference equation of (2)
119879119899119894+1119895 minus 2119879119899119894119895 + 119879119899119894minus1119895(Δ119909)2 + 119879119899119894119895+1 minus 2119879119899119894119895 + 119879119899119894119895minus1(Δ119910)2= 1120572 119879
119899+1119894119895 minus 119879119899119894119895Δ119905
(6)
Equation (6) is the difference equation of heat conductionequation and the truncation error [34] is 119874(Δ119905 + Δ1199092Δ1199102)
Assuming that Δ119909 = Δ119910 = Δ and substituting it into (6)we can obtain
119879119899119894+1119895 + 119879119899119894minus1119895 + 119879119899119894119895minus1 minus (4 minus 11198650)119879119899119894119895 =119879119899+11198941198951198650 (7)
where 1198650 refers to the Fourier coefficient and 1198650 =120572Δ119905(Δ119909)2 = 120572Δ119905(Δ119910)2 = 120572Δ119905Δ2
The stability condition of explicit finite difference equa-tion of two-dimensional unsteady-state heat conductionwithout internal heat source is in interior node 1198650 le 14in boundary node 1198650 le 1[2(2 + 119861119894)] in boundary angularpoint 1198650 le 1[4(1 + 119861119894)]23 Boundary Conditions First-type boundary conditionie the temperature is given In general when the FDM isused for calculation it shall be processed as in the momentof the initial the boundary node temperature is 1198792 then theboundary node temperature remains at 119879
In the second- and third-type boundary condition it isnecessary to set virtual node outside the boundary to makethe boundary node into interior node The node numberingis shown in Figure 2
Second-type boundary condition ie the heat flowboundary is givenminus120582(120597119879120597119899) = 119902 Setting boundary 119909 = 0 asthe given heat flow boundary condition and keeping it stablethe second-type boundary condition can be expressed asminus120582(120597119879120597119909) = 119902 Using central difference quotient to replacethe first-order partial differential equation
minus1205821198791198992 minus 119879119899210158402Δ = 119902 (8)
(8) is rewritten as
11987911989921015840 = 1198791198992 minus 2Δ120582 119902 (9)
Node 1 is changed into interior node and (9) is substitutedinto the interior node explicit difference equation of (7) to get
1198791198992 + 11987911989921015840 + 1198791198994 + 1198791198990 minus (4 minus 11198650)1198791198991 = 11198650119879119899+11 (10)
Substituting (9) into (10) we can obtain
119879119899+11 = 1198650 [1198791198990 + 21198791198992 + 1198791198994 minus 2Δ120582 119902 + ( 11198650 minus 4)1198791198991] (11)
Third-type boundary condition ie the heat transferboundary is given 120582(120597119879120597119899) = ℎ(119879 minus 119879119891) Setting boundary119909 = 0 as the given heat transfer boundary condition andkeeping it stable the third-type boundary condition can beexpressed as 120582(120597119879120597119909) = ℎ(119879 minus 119879119891) Using central differencequotient to replace the first-order partial differential equation
120582 (1198791198992 minus 11987911989921015840)(2Δ) = ℎ (1198791198991 minus 119879119891) (12)
(12) is rewritten as
11987911989921015840 = 1198791198992 minus 2119861119894 (1198791198991 minus 119879119891) (13)
Node 1 is changed into interior node and (13) is substi-tuted into the interior node explicit difference equation of (7)to get
1198791198992 + 11987911989921015840 + 1198791198994 + 1198791198990 minus (4 minus 11198650)1198791198991 = 11198650119879119899+11 (14)
4 Complexity
Substituting (13) into (14) we can obtain
119879119899+11= 1198650 [1198791198990 + 21198791198992 + 1198791198994 + 2119861119894119879119891 + ( 11198650 minus 4 minus 2119861119894)1198791198991]
(15)
where 119861119894 is the Biot number 119861119894 = ℎΔ120582 the truncationerror of the second- and third-type boundary condition is119874(Δ2)
Adiabatic boundary condition is 119879 = 0 Similarly thesecond- and third-type boundary condition is
Boundary angular point is 0 node Virtual nodes 1rsquo and3rsquo are set in the symmetric position of node 1 and node 3respectively and the central different quotient is applied inthe 119909- and 119910-direction respectively
120582 (1198791198993 minus 11987911989931015840)(2Δ) = ℎ (1198791198990 minus 119879119891)120582 (1198791198991 minus 11987911989911015840)(2Δ) = ℎ (1198791198990 minus 119879119891)
(17)
Equation (17) is rewritten as
11987911989931015840 = 1198791198993 minus 2119861119894 (1198791198990 minus 119879119891)11987911989911015840 = 1198791198991 minus 2119861119894 (1198791198990 minus 119879119891) (18)
Node 1 is changed into interior node and (17) and (18) aresubstituted into (7) to get
1198791198993 + 11987911989931015840 + 1198791198991 + 11987911989911015840 minus (4 minus 11198650)1198791198990 = 11198650119879119899+10119879119899+10 = 21198650 [1198791198991 + 1198791198993 + 2119861119894119879119891 + ( 121198650 minus 2 minus 2119861119894)1198791198990]
(19)
By (7) (11) (15) (16) and (19) the temperature value inany point of the model can be obtained
24 Mathematical Model about the Heat Transfer Processof Rectangular Plate Figure 3 shows the model of two-dimensional unsteady-state heat conduction system withoutinternal heat source The rectangle plate in Figure 3 isadopted boundary 1198631 1198632 1198633 is for heat insulation and 1198634is the third-type boundary condition ℎ is the heat-transfercoefficient Then (1) can be changed by the correspondingmathematical model as follows
25 Direct Problem Verification Figures 4 5 6 and 7display the temperature field distribution when 119905 =50 100 150 200(119904) Figure 8 is the curve of measuring pointswith time The simulation result of direct problem solvingcan demonstrate the rationality of explicit finite differencewhich is convenient for performing the inversion algorithmof inverse problem
The length Lx and width Ly of the plate is 02119898 Theheat conductivity coefficient 120582 = 47(119882119898 lowast 119870) thermaldiffusivity 119886 = 128 lowast 10minus6(1198982119904) initial temperature 1198790 =20∘119862 environment temperature119879119891 = 50∘119862 and heat-transfercoefficient ℎ = 2000(1198821198982 lowast 119870)3 Unsteady-State Problem
Predictive control is a model-based control algorithm whichfocuses on the function of the model rather than the formof the model Compared with other control methods itscharacteristics are reflected in the use of rolling optimizationand rolling implementation of the control mode to achievethe control effect but also did not give up the traditionalcontrol feedback Therefore the predictive control algorithmis based on the future dynamic behavior of the processmodel prediction system under a certain control effect usesthe rolling optimization to obtain the control effect underthe corresponding constraint conditions and performancerequirements and corrects the prediction of future dynamicbehavior in the rolling optimization process by detecting real-time information
31 PredictionModel of Inverse Problem The step response ofheat-transfer coefficient in the1198634 boundary of direct problemmodel is taken as the prediction model of inverse problemThe increment of heat-transfer coefficient in the boundary infuture time ie Δℎ119896+119875 is used to predict the temperature at119878 in the 1198634 boundary in the moment of 119877 ie 119879119896+119875 where119875 = 0 1 2 119877 minus 1 and 119877 is the future time step
According to the principle of linear superposition [35]after loading 119877 group of increment Δℎ119896+119875 on system sincethe time moment of 119896 the temperature at 119878 ie 119879119896+119875 isobtained
Δ120601119884 = 120601119884+1 minus 120601119884 (119884 = 0 1 sdot sdot sdot 119877 minus 1) (24)
The step response system function from the timemoment119905119896 to 119905119896+119877minus1 is defined as the impact of heat-transfer coefficienton 119879(119909 119910 119905) After the derivation of ℎ119896 by (20) the stepresponse equation can be obtained as follows
12059721198671205971199092 + 12059721198671205971199102 = 1120572 120597119867120597119905 (isin Ω 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 0 (isin 1198631 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 0 (isin 1198632 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119910 = 0 (isin 1198633 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 119879119896 minus 119879119891 minus ℎ119867 (119909 119910 119905)
(isin Γ3 119905119896 le 119905 le 119905119896+119877minus1)119867 = 1198670 (isin Ω 119905 = 119905119896)
(25)
From (25) it can be seen that119867 is related to ℎ Thereforewhile solving inverse problem it is necessary to use theexplicit difference algorithm used in the direct problemsolving and keep updating the calculation of ℎ
The corresponding discrete value 119867119896+119877minus1 is obtained by(25) hence the dynamic step response coefficient 120601119877 isfurther determined as
120601119877 = 119867119896+119877minus1 (26)
32 Rolling Optimization of Inverse Problem Measurementvalue and predictive value of temperature can be seen inthe time range from 119905119896 to 119905119896+119877minus1 According to the finiteoptimization 119877 parameters to be inverted are obtained sothe predictive value can be as close as possible to the mea-surement value in the future time domain at this momentthe quadratic performance index of system can be launched
min 119869 (ℎ) = (119879119898119890119886 minus 119879119901119903119890)Τ (119879119898119890119886 minus 119879119901119903119890) + ℎΤ120579ℎ (27)
where 120579 is the regularization parameter matrix
and 120572 is the regularization parameterAfter the derivation of ℎ based on (27) and making119889119869(ℎ)119889ℎ = 0 the optimal control rate can be obtained as
ℎ = (119860Τ119860 + 120579)minus1 119860Τ [119879119898119890119886 minus 11987911990111990311989010038161003816100381610038161003816ℎ=0] (29)
The optimal heat-transfer coefficient at moment 119896 can beobtained following (29)
ℎ = R (119860Τ119860 + 120579)minus1 119860Τ [119879119898119890119886 minus 11987911990111990311989010038161003816100381610038161003816ℎ=0] (30)
where R = [1 0 0]Τ There is no need to presetthe function form for the inverse problem calculation by theabove optimization algorithm
33 Regularization Parameter The residual principle [36ndash38] is introduced to calculate the optimal regularizationparameter aiming to reduce the impact of measurementerrors on the inversion results
To invert the boundary heat-transfer coefficient it isnecessary to firstly solve the direct problem using the predic-tive value of heat-transfer coefficient to get the temperaturecalculation at 119878 in the 119896 moment 119879119896119878 Besides in the casethat the temperature measurement value at 119878 119879119896119898119890119886 seesmeasurement error the temperature measurement value canbe expressed by the actual temperature plus themeasurementerror
where 120596 is the random number of standard normal in therange and 120590 is the standard deviation of measurement valuewhich is expressed as
120590 = radic 1119870 minus 1119896sum119909=1
(119879119896119898119890119886 minus 119879119896119886119888119905)2 (32)
where the constant K is the number of iterations
6 Complexity
The residual of heat-transfer coefficient in the wholeinversion time domain is defined as
119878 (120572) = radic 1119870 minus 1119896sum119896=1
(ℎ119896119886119888119905 minus ℎ119896119878)2 (33)
In (33) ℎ119896119886119888119905 and ℎ119896119878 are the actual value and inversion valueof heat-transfer coefficient respectively
Since ℎ119896119878 is unknown it is available to calculate thetemperaturemeasurement value at 119878 (119879119896119878 ) using ℎ119896119878 with directproblem algorithm thus the temperature residual in theinversion time domain can be obtained
119878 (120572) = radic 1119870 minus 1119896sum119896=1
(119879119896119898119890119886 minus 1198791198961198781)2 (34)
In ideal condition
ℎ119896119886119888119905 = ℎ119896119878 (35)
Similarily
1198791198961198781 = 119879119896119878 (36)
From the residual principle the regularization parameteris the optimal when both (35) and (36) are satisfied
119878119879 (120572) = 120590 (37)
34 Solving Procedure of Inverse Problem (1) Select the initialpredictive value of the heat-transfer coefficient ℎ119896119901 at a timemoment to perform inversion(2)Obtain the temperature calculation values inmeasure-ment point S at R time moments after that moment based onℎ119896119901 and (20)(3) Calculate the optimal regularization parameter 120572based on (37)(4)Assume the heat-transfer coefficient in the initial stageof inversion ℎ = 0 and obtain the step response matrix 119860 byEq (25)(5) Confirm the heat-transfer coefficient at the timemoment ℎ119896 according to (30) and then use the direct problemalgorithm to reconstruct the temperature field when the heat-transfer coefficient is ℎ119896(6) Following the time direction backward change thevalue in the initial stage of inversion and repeat steps (4) and(5) then get the inverse value of heat-transfer coefficient atdifferent time moments
4 Numerical Experiment and Analysis
Numerical experiments are performed to validate whetherthe proposed method is effective with the focus on analyzingthe impact of differentmeasurement errors andmeasurementpoint positions on the inversion result Also the inversionresult obtained in the condition without measurement erroris compared with the practical result which verifies theprecision of the proposed method
Γ1
Γ2
Γ3
Ω
Figure 1 Heat conduction model
4
2
1rsquo
3rsquo
2rsquo
0 3
1
Figure 2 Boundary node
D4
y
D3
D2D1
x
hTf
ΩΩ
Figure 3 The model of two-dimensional unsteady-state heat con-duction system without internal heat source
The two-dimensional plate heat transfer model (Figure 1)used in the above-mentioned direct problem is adoptedIn the simulation example the length Lx and width Ly ofthe plate is 02119898 The heat conductivity coefficient 120582 =47(119882119898lowast119870) thermal diffusivity119886 = 128lowast10minus6(1198982119904) initialtemperature 1198790 = 20∘119862 environment temperature 119879119891 =50∘119862 and heat-transfer coefficient ℎ = 2000(1198821198982 lowast119870) Thepurpose is to obtain the actual heat-transfer coefficient of theboundary D4
Complexity 7
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
21
22
23
24
25
26
27
28
29
30
Figure 4 The temperature field in the t=50s
016
017
018
019
020
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
Y
Figure 5 The temperature field in the t=100s
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
016
017
018
019
Y
Figure 6 The temperature field in the t=150s
41 Impact When the Measurement Error Is Zero Given themeasurement error 120590 = 0000 when the measurement pointis in the 119871 = 01119898 of D4 boundary and the future time step119877 = 5 the inversion result is shown in Figure 9
Figure 9 displays that except the transitory vibration inthe initial stage The inversion value is basically identical tothe practical value demonstrating the effectiveness of theinversion algorithm
8 Complexity
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
34
Figure 7 The temperature field in the t=200s
20
21
22
23
24
25
26
27
28
29
30
L1=01mL2=0004m
L3=0008ml1=0001m
50 100 150 200 2500Time (s)
Tem
pera
ture
(∘C)
Figure 8 The curve of the measuring point with time
42 Impact of Measurement Error Given the future time step119877 = 5 and the measurement point is in the 119871 = 01119898 of D4boundary the inversion results when the measurement erroris 120590 = 0001 120590 = 0005 and 120590 = 001 are displayed in Figures10 11 and 12 respectively
According to Table 1 and Figures 10 11 and 12 smallerrelative measurement error contributes to better inversionresults And enlarging measurement error will worsen theinversion results and aggravate the fluctuation
43 Impact of Measurement Point Position Given measure-ment error 120590 = 0001 and future time step 119877 = 5 theinversion results when the measurement point is in the 119871 =0004119898 0008119898 of D4 boundary and when the measurementpoint is 119897 = 0001119898 from the D4 boundary are shown inFigures 13 14 and 15 respectively
Analyze the contents of Table 2 and Figures 13 and 14The explicit FDM is used for direct problem when the
measurement point in boundary is closer to the boundaryangular point which however imposes a little impact on theinversion result Despite the increased relative average errorthe proposed method still exhibits a better ability to track theexact solution of heat-transfer coefficient and the inversionresult is relatively precise In Figure 15 considering that theposition of the measuring point is 0001m away from theboundary and the initial time temperature of the position is20 the temperature cannot change for a period of time sothe inversion result fluctuates greatly in the initial stage andincreases when the distance of measurement point positionfrom the boundary angular point becomes farther
5 Conclusion
The boundary heat-transfer coefficient of the two-dimensional unsteady heat conduction system is inversed bythe FDM and model prediction control method By solving
Complexity 9
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 9 The heat transfer coefficient of the measuring point without error
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 10 The heat transfer coefficient of the measuring point with 120590 = 0001
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 11 The heat transfer coefficient of the measuring point with 120590 = 0005
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 12 The heat transfer coefficient of the measuring point with 120590 = 001
and analyzing the algorithm example it demonstratesthat the proposed methods have higher accuracy in theinversion process Model predictive control method focuseson the model function rather than the structural form sothat we only need to know the step response or impulseresponse of the object we can directly get the predictionmodel and skip the derivation process It absorbs the idea
of optimization control and replaces global optimization byrolling time-domain optimization combined with feedbackcorrection which avoids a lot of calculation required byglobal optimization and constantly corrects the influencecaused by uncertain factors in the system At the sametime by discussing the impacts of error free measuringpoint positions and measuring errors on the results it
10 Complexity
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
1000150020002500300035004000
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 13 The heat transfer coefficient of the measuring point at the boundary 119871 = 0008119898
0500
10001500200025003000350040004500
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 14 The heat transfer coefficient of the measuring point at the boundary 119871 = 0004119898
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
100015002000250030003500
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 15 The heat transfer coefficient of the measuring point at the distance from the boundary 119897 = 0001119898
Table 1 Relative average errors of inversion result under differentmeasurement errors given 119871 = 01119898 and 119877 = 5Measurement error120590 0001 0005 001
Relative average error120578 701 1129 1601
Table 2 Relative average errors of inversion result in differentmeasurement point positions given 120590 = 0001 and 119877 = 5Measurement point 119871 = 0004119898 119871 = 0008119898 119897 = 0001119898Relative average error120578 765 706 2105
demonstrates that the obtained inversion results except theearly oscillation can better represent the stability of the exactsolution
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interestregarding the publication of this paper
Authorsrsquo Contributions
ShoubinWang andRuiNi contributed to developing the ideasof this research All of the authors were involved in preparingthis manuscript
Acknowledgments
This work was financially supported by the National KeyFoundation for Exploring Scientific Instrument of China
Complexity 11
(2013YQ470767) Tianjin Municipal Education CommissionProject for Scientific Research Items (2017KJ059) and Tian-jin Science and Technology Commissioner Project (18JCT-PJC62200 18JCTPJC64100)
References
[1] W Shoubin Z Li S Xiaogang and J Huangchao ldquoInversionof thermal conductivity in two-dimensional unsteady-stateheat transfer system based on boundary element method anddecentralized fuzzy inferencerdquo Complexity vol 2018 Article ID8783946 9 pages 2018
[2] S Wang Y Deng and X Sun ldquoSolving of two-dimensionalunsteady inverse heat conduction problems based on boundaryelementmethod and sequential function specificationmethodrdquoComplexity vol 2018 Article ID 6741632 11 pages 2018
[3] SWang H Jia X Sun and L Zhang ldquoResearch on the recogni-tion algorithm concerning geometric boundary regarding heatconduction based on BEM and CGMrdquoMathematical Problemsin Engineering vol 2018 Article ID 3723949 13 pages 2018
[4] S Wang L Zhang X Sun et al ldquoSolution to two-dimensionalsteady inverse heat transfer problems with interior heat sourcebased on the conjugate gradient methodrdquo Mathematical Prob-lems in Engineering vol 2017 Article ID 2861342 9 pages 2017
[5] S Wang H Jia X Sun et al ldquoTwo-dimensional steady-state boundary shape inversion of CGM-SPSO algorithm ontemperature informationrdquo Advances in Materials Science andEngineering vol 2017 Article ID 2461498 12 pages 2017
[6] Z Tianyu and D Changhong ldquoCompound control systemdesign based on adaptive backstepping theoryrdquo Journal ofBeijing University of Aeronautics and A vol 39 no 7 pp 902ndash906 2013
[7] P Duda ldquoA general method for solving transient multidimen-sional inverse heat transfer problemsrdquo International Journal ofHeat and Mass Transfer vol 93 pp 665ndash673 2016
[8] B Li and L Liu ldquoAn algorithm for geometry boundary identifi-cation of heat conduction problem based on boundary elementdiscretizationrdquo Proceedings of the CSEE vol 28 no 20 pp 38ndash43 2008
[9] C-Y Yang ldquoBoundary prediction of bio-heat conduction ina two-dimensional multilayer tissuerdquo International Journal ofHeat and Mass Transfer vol 78 no 7 pp 232ndash239 2014
[10] P Duda ldquoNumerical and experimental verification of twomethods for solving an inverse heat conduction problemrdquoInternational Journal of Heat and Mass Transfer vol 84 pp1101ndash1112 2015
[11] V M Luchesi and R T Coelho ldquoAn inverse method to estimatethe moving heat source inmachining processrdquoAppliedermalEngineering vol 45-46 pp 64ndash78 2012
[12] PDuda ldquoAmethod for transient thermal load estimation and itsapplication to identification of aerodynamic heating on atmo-spheric reentry capsulerdquo Aerospace Science and Technology vol51 pp 26ndash33 2016
[13] J V Beck B Blackwell and C R Clair Inverse HeatConductionIll-Posed Problems A Wiley-Interscience Publica-tion 1985
[14] T RHsu N S Sun G G Chen and Z L Gong ldquoFinite elementformulation for two-dimensional inverse heat conduction anal-ysisrdquo Journal of Heat Transfer vol 114 no 3 p 553 1992
[15] A A Tseng and F Z Zhao ldquoMultidimensional inverse tran-sient heat conduction problems by direct sensitivity coefficient
method using a finite-element schemerdquo Numerical Heat Trans-fer Part B Fundamentals vol 29 no 3 pp 365ndash380 1996
[16] L Zhaoming Further Studies on Fuzzy Inference Method forIn-verse Heat Transfer Problems Chongqing University 2014
[17] W Q Qian Y Zhou K F He J Y Yuan and J D HuangldquoEstimation of surface heat flux for nonlinear inverse heatconduction problemrdquo Acta Aerodynamica Sinica vol 30 no 2pp 145ndash150 2012
[18] J Su and G F Hewitt ldquoInverse heat conduction problemof estimating time-varying heattransfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004
[19] G Blanc J V Beck and M Raynaud ldquoSolution of the inverseheat conduction problemwith a time-variable number of futuretemperaturesrdquo Numerical Heat Transfer Part B Fundamentalsvol 32 no 4 pp 437ndash451 1997
[21] D Lesnic L Elliott and D B Ingham ldquoIdentification of thethermal conductivity and heat capacity in unsteady nonlin-ear heat conduction problems using the boundary elementmethodrdquo Journal of Computational Physics vol 126 no 2 pp410ndash420 1996
[22] A A Ershova and A I Sidikova ldquoUncertainty estimation ofthe method based on generalized residual principle for therestore task of the spectral density of crystalsrdquo Vestn Yuzhno-UralGosUn-taSerMatemMekhFiz vol 2015 pp 25ndash30 2015
[23] L Zhao Research on Fluid Temperature Inversion Algorithms forOne-Dimensional Unsteady Convection Heat Transfer HarbinInstitute of Technology 2017
[24] B Li and L Liu ldquoAn algorithm for geometry boundary identi-fication of heat conduction problem based on boundary ele-ment discretizationrdquo Proceedings of the CSEE vol 28 no 20pp 38ndash43 2008
[25] B Li and L Liu ldquoGeometry boundary identification of unsteadyheat conduction based on dual reciprocity boundary elementmethodrdquo Proceedings of the CSEE vol 29 no 5 pp 66ndash71 2009
[26] H Zhou X Xu X Li and H Chen ldquoIdentification oftemperature-dependent thermal conductivity for 2-D transientheat conduction problemsrdquo Applied Mathematics and Mechan-ics vol 29 no 1 pp 55ndash68 2014
[27] L Yanhao Model Predictive Inverse Method for Heat TransferProcess and Application Chongqing University 2017
[28] J Fan Reconstruction of Boiler Drum Temperature Field Basedon Inverse Heat Transfer Problem Chongqing University 2017
[29] H Ya-fen et alMechanism and Characteristics of Heat Conduc-tion in Nanocomposites Harbin Institute of Technology 2013
[30] Q Yue-ping M Jun J Jing-yan Y Xiao-bin and L WeildquoUnsteady heat transfer problems with finite volume methodrdquoJournal of Liaoning Technical University Natural Science vol 32no 05 pp 577ndash581 2013
[31] C-H Chiu and C-K Chen ldquoApplications of adomianrsquos decom-position procedure to the analysis of convective-radiative finsrdquoJournal of Heat Transfer vol 125 no 2 pp 312ndash316 2003
[32] C Sheng Direct and Inverse Heat Conduction Problems Solvingby the Boundary Element Method Hunan University 2007
[33] MMKhader AM Eid andAMMegahed ldquoNumerical stud-ies using FDM for viscous dissipation and thermal radiationeffects on the slip flow and heat transfer due to a stretchingsheet embedded in a porous medium with variable thickness
12 Complexity
and variable thermal conductivityrdquoNewTrends inMathematicalSciences vol 4 no 1 p 38 2016
[34] S Chuanzhuo and Z Tiande ldquoA simple method for calculatingtruncation error of numerical integral formulardquo Journal ofMathematis for Technology vol 2 1994
[35] Q Jixin Z Jun and X Zuhu Predictive Control ChemicalIndustry Press 2007
[36] L Zhang H Chen G Wang and Z Luo ldquoSolving transientinverse heat conduction problems based on optimal number offuture time stepsrdquo Proceedings of the CSEE vol 32 no 2 pp99ndash103 2012
[37] N Buong T T Huong and N T Thuy ldquoA quasi-residualprinciple in regularization for a common solution of a system ofnonlinearmonotone ill-posed equationsrdquoRussianMathematicsvol 60 no 3 pp 47ndash55 2016
[38] K A Woodbury and S K Thakur ldquoRedundant data and futuretimes in the inverse heat conduction problemrdquo Inverse Problemsin Science and Engineering vol 2 no 4 pp 319ndash333 1996
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Boundary angular point is 0 node Virtual nodes 1rsquo and3rsquo are set in the symmetric position of node 1 and node 3respectively and the central different quotient is applied inthe 119909- and 119910-direction respectively
120582 (1198791198993 minus 11987911989931015840)(2Δ) = ℎ (1198791198990 minus 119879119891)120582 (1198791198991 minus 11987911989911015840)(2Δ) = ℎ (1198791198990 minus 119879119891)
(17)
Equation (17) is rewritten as
11987911989931015840 = 1198791198993 minus 2119861119894 (1198791198990 minus 119879119891)11987911989911015840 = 1198791198991 minus 2119861119894 (1198791198990 minus 119879119891) (18)
Node 1 is changed into interior node and (17) and (18) aresubstituted into (7) to get
1198791198993 + 11987911989931015840 + 1198791198991 + 11987911989911015840 minus (4 minus 11198650)1198791198990 = 11198650119879119899+10119879119899+10 = 21198650 [1198791198991 + 1198791198993 + 2119861119894119879119891 + ( 121198650 minus 2 minus 2119861119894)1198791198990]
(19)
By (7) (11) (15) (16) and (19) the temperature value inany point of the model can be obtained
24 Mathematical Model about the Heat Transfer Processof Rectangular Plate Figure 3 shows the model of two-dimensional unsteady-state heat conduction system withoutinternal heat source The rectangle plate in Figure 3 isadopted boundary 1198631 1198632 1198633 is for heat insulation and 1198634is the third-type boundary condition ℎ is the heat-transfercoefficient Then (1) can be changed by the correspondingmathematical model as follows
25 Direct Problem Verification Figures 4 5 6 and 7display the temperature field distribution when 119905 =50 100 150 200(119904) Figure 8 is the curve of measuring pointswith time The simulation result of direct problem solvingcan demonstrate the rationality of explicit finite differencewhich is convenient for performing the inversion algorithmof inverse problem
The length Lx and width Ly of the plate is 02119898 Theheat conductivity coefficient 120582 = 47(119882119898 lowast 119870) thermaldiffusivity 119886 = 128 lowast 10minus6(1198982119904) initial temperature 1198790 =20∘119862 environment temperature119879119891 = 50∘119862 and heat-transfercoefficient ℎ = 2000(1198821198982 lowast 119870)3 Unsteady-State Problem
Predictive control is a model-based control algorithm whichfocuses on the function of the model rather than the formof the model Compared with other control methods itscharacteristics are reflected in the use of rolling optimizationand rolling implementation of the control mode to achievethe control effect but also did not give up the traditionalcontrol feedback Therefore the predictive control algorithmis based on the future dynamic behavior of the processmodel prediction system under a certain control effect usesthe rolling optimization to obtain the control effect underthe corresponding constraint conditions and performancerequirements and corrects the prediction of future dynamicbehavior in the rolling optimization process by detecting real-time information
31 PredictionModel of Inverse Problem The step response ofheat-transfer coefficient in the1198634 boundary of direct problemmodel is taken as the prediction model of inverse problemThe increment of heat-transfer coefficient in the boundary infuture time ie Δℎ119896+119875 is used to predict the temperature at119878 in the 1198634 boundary in the moment of 119877 ie 119879119896+119875 where119875 = 0 1 2 119877 minus 1 and 119877 is the future time step
According to the principle of linear superposition [35]after loading 119877 group of increment Δℎ119896+119875 on system sincethe time moment of 119896 the temperature at 119878 ie 119879119896+119875 isobtained
Δ120601119884 = 120601119884+1 minus 120601119884 (119884 = 0 1 sdot sdot sdot 119877 minus 1) (24)
The step response system function from the timemoment119905119896 to 119905119896+119877minus1 is defined as the impact of heat-transfer coefficienton 119879(119909 119910 119905) After the derivation of ℎ119896 by (20) the stepresponse equation can be obtained as follows
12059721198671205971199092 + 12059721198671205971199102 = 1120572 120597119867120597119905 (isin Ω 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 0 (isin 1198631 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 0 (isin 1198632 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119910 = 0 (isin 1198633 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 119879119896 minus 119879119891 minus ℎ119867 (119909 119910 119905)
(isin Γ3 119905119896 le 119905 le 119905119896+119877minus1)119867 = 1198670 (isin Ω 119905 = 119905119896)
(25)
From (25) it can be seen that119867 is related to ℎ Thereforewhile solving inverse problem it is necessary to use theexplicit difference algorithm used in the direct problemsolving and keep updating the calculation of ℎ
The corresponding discrete value 119867119896+119877minus1 is obtained by(25) hence the dynamic step response coefficient 120601119877 isfurther determined as
120601119877 = 119867119896+119877minus1 (26)
32 Rolling Optimization of Inverse Problem Measurementvalue and predictive value of temperature can be seen inthe time range from 119905119896 to 119905119896+119877minus1 According to the finiteoptimization 119877 parameters to be inverted are obtained sothe predictive value can be as close as possible to the mea-surement value in the future time domain at this momentthe quadratic performance index of system can be launched
min 119869 (ℎ) = (119879119898119890119886 minus 119879119901119903119890)Τ (119879119898119890119886 minus 119879119901119903119890) + ℎΤ120579ℎ (27)
where 120579 is the regularization parameter matrix
and 120572 is the regularization parameterAfter the derivation of ℎ based on (27) and making119889119869(ℎ)119889ℎ = 0 the optimal control rate can be obtained as
ℎ = (119860Τ119860 + 120579)minus1 119860Τ [119879119898119890119886 minus 11987911990111990311989010038161003816100381610038161003816ℎ=0] (29)
The optimal heat-transfer coefficient at moment 119896 can beobtained following (29)
ℎ = R (119860Τ119860 + 120579)minus1 119860Τ [119879119898119890119886 minus 11987911990111990311989010038161003816100381610038161003816ℎ=0] (30)
where R = [1 0 0]Τ There is no need to presetthe function form for the inverse problem calculation by theabove optimization algorithm
33 Regularization Parameter The residual principle [36ndash38] is introduced to calculate the optimal regularizationparameter aiming to reduce the impact of measurementerrors on the inversion results
To invert the boundary heat-transfer coefficient it isnecessary to firstly solve the direct problem using the predic-tive value of heat-transfer coefficient to get the temperaturecalculation at 119878 in the 119896 moment 119879119896119878 Besides in the casethat the temperature measurement value at 119878 119879119896119898119890119886 seesmeasurement error the temperature measurement value canbe expressed by the actual temperature plus themeasurementerror
where 120596 is the random number of standard normal in therange and 120590 is the standard deviation of measurement valuewhich is expressed as
120590 = radic 1119870 minus 1119896sum119909=1
(119879119896119898119890119886 minus 119879119896119886119888119905)2 (32)
where the constant K is the number of iterations
6 Complexity
The residual of heat-transfer coefficient in the wholeinversion time domain is defined as
119878 (120572) = radic 1119870 minus 1119896sum119896=1
(ℎ119896119886119888119905 minus ℎ119896119878)2 (33)
In (33) ℎ119896119886119888119905 and ℎ119896119878 are the actual value and inversion valueof heat-transfer coefficient respectively
Since ℎ119896119878 is unknown it is available to calculate thetemperaturemeasurement value at 119878 (119879119896119878 ) using ℎ119896119878 with directproblem algorithm thus the temperature residual in theinversion time domain can be obtained
119878 (120572) = radic 1119870 minus 1119896sum119896=1
(119879119896119898119890119886 minus 1198791198961198781)2 (34)
In ideal condition
ℎ119896119886119888119905 = ℎ119896119878 (35)
Similarily
1198791198961198781 = 119879119896119878 (36)
From the residual principle the regularization parameteris the optimal when both (35) and (36) are satisfied
119878119879 (120572) = 120590 (37)
34 Solving Procedure of Inverse Problem (1) Select the initialpredictive value of the heat-transfer coefficient ℎ119896119901 at a timemoment to perform inversion(2)Obtain the temperature calculation values inmeasure-ment point S at R time moments after that moment based onℎ119896119901 and (20)(3) Calculate the optimal regularization parameter 120572based on (37)(4)Assume the heat-transfer coefficient in the initial stageof inversion ℎ = 0 and obtain the step response matrix 119860 byEq (25)(5) Confirm the heat-transfer coefficient at the timemoment ℎ119896 according to (30) and then use the direct problemalgorithm to reconstruct the temperature field when the heat-transfer coefficient is ℎ119896(6) Following the time direction backward change thevalue in the initial stage of inversion and repeat steps (4) and(5) then get the inverse value of heat-transfer coefficient atdifferent time moments
4 Numerical Experiment and Analysis
Numerical experiments are performed to validate whetherthe proposed method is effective with the focus on analyzingthe impact of differentmeasurement errors andmeasurementpoint positions on the inversion result Also the inversionresult obtained in the condition without measurement erroris compared with the practical result which verifies theprecision of the proposed method
Γ1
Γ2
Γ3
Ω
Figure 1 Heat conduction model
4
2
1rsquo
3rsquo
2rsquo
0 3
1
Figure 2 Boundary node
D4
y
D3
D2D1
x
hTf
ΩΩ
Figure 3 The model of two-dimensional unsteady-state heat con-duction system without internal heat source
The two-dimensional plate heat transfer model (Figure 1)used in the above-mentioned direct problem is adoptedIn the simulation example the length Lx and width Ly ofthe plate is 02119898 The heat conductivity coefficient 120582 =47(119882119898lowast119870) thermal diffusivity119886 = 128lowast10minus6(1198982119904) initialtemperature 1198790 = 20∘119862 environment temperature 119879119891 =50∘119862 and heat-transfer coefficient ℎ = 2000(1198821198982 lowast119870) Thepurpose is to obtain the actual heat-transfer coefficient of theboundary D4
Complexity 7
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
21
22
23
24
25
26
27
28
29
30
Figure 4 The temperature field in the t=50s
016
017
018
019
020
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
Y
Figure 5 The temperature field in the t=100s
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
016
017
018
019
Y
Figure 6 The temperature field in the t=150s
41 Impact When the Measurement Error Is Zero Given themeasurement error 120590 = 0000 when the measurement pointis in the 119871 = 01119898 of D4 boundary and the future time step119877 = 5 the inversion result is shown in Figure 9
Figure 9 displays that except the transitory vibration inthe initial stage The inversion value is basically identical tothe practical value demonstrating the effectiveness of theinversion algorithm
8 Complexity
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
34
Figure 7 The temperature field in the t=200s
20
21
22
23
24
25
26
27
28
29
30
L1=01mL2=0004m
L3=0008ml1=0001m
50 100 150 200 2500Time (s)
Tem
pera
ture
(∘C)
Figure 8 The curve of the measuring point with time
42 Impact of Measurement Error Given the future time step119877 = 5 and the measurement point is in the 119871 = 01119898 of D4boundary the inversion results when the measurement erroris 120590 = 0001 120590 = 0005 and 120590 = 001 are displayed in Figures10 11 and 12 respectively
According to Table 1 and Figures 10 11 and 12 smallerrelative measurement error contributes to better inversionresults And enlarging measurement error will worsen theinversion results and aggravate the fluctuation
43 Impact of Measurement Point Position Given measure-ment error 120590 = 0001 and future time step 119877 = 5 theinversion results when the measurement point is in the 119871 =0004119898 0008119898 of D4 boundary and when the measurementpoint is 119897 = 0001119898 from the D4 boundary are shown inFigures 13 14 and 15 respectively
Analyze the contents of Table 2 and Figures 13 and 14The explicit FDM is used for direct problem when the
measurement point in boundary is closer to the boundaryangular point which however imposes a little impact on theinversion result Despite the increased relative average errorthe proposed method still exhibits a better ability to track theexact solution of heat-transfer coefficient and the inversionresult is relatively precise In Figure 15 considering that theposition of the measuring point is 0001m away from theboundary and the initial time temperature of the position is20 the temperature cannot change for a period of time sothe inversion result fluctuates greatly in the initial stage andincreases when the distance of measurement point positionfrom the boundary angular point becomes farther
5 Conclusion
The boundary heat-transfer coefficient of the two-dimensional unsteady heat conduction system is inversed bythe FDM and model prediction control method By solving
Complexity 9
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 9 The heat transfer coefficient of the measuring point without error
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 10 The heat transfer coefficient of the measuring point with 120590 = 0001
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 11 The heat transfer coefficient of the measuring point with 120590 = 0005
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 12 The heat transfer coefficient of the measuring point with 120590 = 001
and analyzing the algorithm example it demonstratesthat the proposed methods have higher accuracy in theinversion process Model predictive control method focuseson the model function rather than the structural form sothat we only need to know the step response or impulseresponse of the object we can directly get the predictionmodel and skip the derivation process It absorbs the idea
of optimization control and replaces global optimization byrolling time-domain optimization combined with feedbackcorrection which avoids a lot of calculation required byglobal optimization and constantly corrects the influencecaused by uncertain factors in the system At the sametime by discussing the impacts of error free measuringpoint positions and measuring errors on the results it
10 Complexity
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
1000150020002500300035004000
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 13 The heat transfer coefficient of the measuring point at the boundary 119871 = 0008119898
0500
10001500200025003000350040004500
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 14 The heat transfer coefficient of the measuring point at the boundary 119871 = 0004119898
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
100015002000250030003500
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 15 The heat transfer coefficient of the measuring point at the distance from the boundary 119897 = 0001119898
Table 1 Relative average errors of inversion result under differentmeasurement errors given 119871 = 01119898 and 119877 = 5Measurement error120590 0001 0005 001
Relative average error120578 701 1129 1601
Table 2 Relative average errors of inversion result in differentmeasurement point positions given 120590 = 0001 and 119877 = 5Measurement point 119871 = 0004119898 119871 = 0008119898 119897 = 0001119898Relative average error120578 765 706 2105
demonstrates that the obtained inversion results except theearly oscillation can better represent the stability of the exactsolution
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interestregarding the publication of this paper
Authorsrsquo Contributions
ShoubinWang andRuiNi contributed to developing the ideasof this research All of the authors were involved in preparingthis manuscript
Acknowledgments
This work was financially supported by the National KeyFoundation for Exploring Scientific Instrument of China
Complexity 11
(2013YQ470767) Tianjin Municipal Education CommissionProject for Scientific Research Items (2017KJ059) and Tian-jin Science and Technology Commissioner Project (18JCT-PJC62200 18JCTPJC64100)
References
[1] W Shoubin Z Li S Xiaogang and J Huangchao ldquoInversionof thermal conductivity in two-dimensional unsteady-stateheat transfer system based on boundary element method anddecentralized fuzzy inferencerdquo Complexity vol 2018 Article ID8783946 9 pages 2018
[2] S Wang Y Deng and X Sun ldquoSolving of two-dimensionalunsteady inverse heat conduction problems based on boundaryelementmethod and sequential function specificationmethodrdquoComplexity vol 2018 Article ID 6741632 11 pages 2018
[3] SWang H Jia X Sun and L Zhang ldquoResearch on the recogni-tion algorithm concerning geometric boundary regarding heatconduction based on BEM and CGMrdquoMathematical Problemsin Engineering vol 2018 Article ID 3723949 13 pages 2018
[4] S Wang L Zhang X Sun et al ldquoSolution to two-dimensionalsteady inverse heat transfer problems with interior heat sourcebased on the conjugate gradient methodrdquo Mathematical Prob-lems in Engineering vol 2017 Article ID 2861342 9 pages 2017
[5] S Wang H Jia X Sun et al ldquoTwo-dimensional steady-state boundary shape inversion of CGM-SPSO algorithm ontemperature informationrdquo Advances in Materials Science andEngineering vol 2017 Article ID 2461498 12 pages 2017
[6] Z Tianyu and D Changhong ldquoCompound control systemdesign based on adaptive backstepping theoryrdquo Journal ofBeijing University of Aeronautics and A vol 39 no 7 pp 902ndash906 2013
[7] P Duda ldquoA general method for solving transient multidimen-sional inverse heat transfer problemsrdquo International Journal ofHeat and Mass Transfer vol 93 pp 665ndash673 2016
[8] B Li and L Liu ldquoAn algorithm for geometry boundary identifi-cation of heat conduction problem based on boundary elementdiscretizationrdquo Proceedings of the CSEE vol 28 no 20 pp 38ndash43 2008
[9] C-Y Yang ldquoBoundary prediction of bio-heat conduction ina two-dimensional multilayer tissuerdquo International Journal ofHeat and Mass Transfer vol 78 no 7 pp 232ndash239 2014
[10] P Duda ldquoNumerical and experimental verification of twomethods for solving an inverse heat conduction problemrdquoInternational Journal of Heat and Mass Transfer vol 84 pp1101ndash1112 2015
[11] V M Luchesi and R T Coelho ldquoAn inverse method to estimatethe moving heat source inmachining processrdquoAppliedermalEngineering vol 45-46 pp 64ndash78 2012
[12] PDuda ldquoAmethod for transient thermal load estimation and itsapplication to identification of aerodynamic heating on atmo-spheric reentry capsulerdquo Aerospace Science and Technology vol51 pp 26ndash33 2016
[13] J V Beck B Blackwell and C R Clair Inverse HeatConductionIll-Posed Problems A Wiley-Interscience Publica-tion 1985
[14] T RHsu N S Sun G G Chen and Z L Gong ldquoFinite elementformulation for two-dimensional inverse heat conduction anal-ysisrdquo Journal of Heat Transfer vol 114 no 3 p 553 1992
[15] A A Tseng and F Z Zhao ldquoMultidimensional inverse tran-sient heat conduction problems by direct sensitivity coefficient
method using a finite-element schemerdquo Numerical Heat Trans-fer Part B Fundamentals vol 29 no 3 pp 365ndash380 1996
[16] L Zhaoming Further Studies on Fuzzy Inference Method forIn-verse Heat Transfer Problems Chongqing University 2014
[17] W Q Qian Y Zhou K F He J Y Yuan and J D HuangldquoEstimation of surface heat flux for nonlinear inverse heatconduction problemrdquo Acta Aerodynamica Sinica vol 30 no 2pp 145ndash150 2012
[18] J Su and G F Hewitt ldquoInverse heat conduction problemof estimating time-varying heattransfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004
[19] G Blanc J V Beck and M Raynaud ldquoSolution of the inverseheat conduction problemwith a time-variable number of futuretemperaturesrdquo Numerical Heat Transfer Part B Fundamentalsvol 32 no 4 pp 437ndash451 1997
[21] D Lesnic L Elliott and D B Ingham ldquoIdentification of thethermal conductivity and heat capacity in unsteady nonlin-ear heat conduction problems using the boundary elementmethodrdquo Journal of Computational Physics vol 126 no 2 pp410ndash420 1996
[22] A A Ershova and A I Sidikova ldquoUncertainty estimation ofthe method based on generalized residual principle for therestore task of the spectral density of crystalsrdquo Vestn Yuzhno-UralGosUn-taSerMatemMekhFiz vol 2015 pp 25ndash30 2015
[23] L Zhao Research on Fluid Temperature Inversion Algorithms forOne-Dimensional Unsteady Convection Heat Transfer HarbinInstitute of Technology 2017
[24] B Li and L Liu ldquoAn algorithm for geometry boundary identi-fication of heat conduction problem based on boundary ele-ment discretizationrdquo Proceedings of the CSEE vol 28 no 20pp 38ndash43 2008
[25] B Li and L Liu ldquoGeometry boundary identification of unsteadyheat conduction based on dual reciprocity boundary elementmethodrdquo Proceedings of the CSEE vol 29 no 5 pp 66ndash71 2009
[26] H Zhou X Xu X Li and H Chen ldquoIdentification oftemperature-dependent thermal conductivity for 2-D transientheat conduction problemsrdquo Applied Mathematics and Mechan-ics vol 29 no 1 pp 55ndash68 2014
[27] L Yanhao Model Predictive Inverse Method for Heat TransferProcess and Application Chongqing University 2017
[28] J Fan Reconstruction of Boiler Drum Temperature Field Basedon Inverse Heat Transfer Problem Chongqing University 2017
[29] H Ya-fen et alMechanism and Characteristics of Heat Conduc-tion in Nanocomposites Harbin Institute of Technology 2013
[30] Q Yue-ping M Jun J Jing-yan Y Xiao-bin and L WeildquoUnsteady heat transfer problems with finite volume methodrdquoJournal of Liaoning Technical University Natural Science vol 32no 05 pp 577ndash581 2013
[31] C-H Chiu and C-K Chen ldquoApplications of adomianrsquos decom-position procedure to the analysis of convective-radiative finsrdquoJournal of Heat Transfer vol 125 no 2 pp 312ndash316 2003
[32] C Sheng Direct and Inverse Heat Conduction Problems Solvingby the Boundary Element Method Hunan University 2007
[33] MMKhader AM Eid andAMMegahed ldquoNumerical stud-ies using FDM for viscous dissipation and thermal radiationeffects on the slip flow and heat transfer due to a stretchingsheet embedded in a porous medium with variable thickness
12 Complexity
and variable thermal conductivityrdquoNewTrends inMathematicalSciences vol 4 no 1 p 38 2016
[34] S Chuanzhuo and Z Tiande ldquoA simple method for calculatingtruncation error of numerical integral formulardquo Journal ofMathematis for Technology vol 2 1994
[35] Q Jixin Z Jun and X Zuhu Predictive Control ChemicalIndustry Press 2007
[36] L Zhang H Chen G Wang and Z Luo ldquoSolving transientinverse heat conduction problems based on optimal number offuture time stepsrdquo Proceedings of the CSEE vol 32 no 2 pp99ndash103 2012
[37] N Buong T T Huong and N T Thuy ldquoA quasi-residualprinciple in regularization for a common solution of a system ofnonlinearmonotone ill-posed equationsrdquoRussianMathematicsvol 60 no 3 pp 47ndash55 2016
[38] K A Woodbury and S K Thakur ldquoRedundant data and futuretimes in the inverse heat conduction problemrdquo Inverse Problemsin Science and Engineering vol 2 no 4 pp 319ndash333 1996
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Δ120601119884 = 120601119884+1 minus 120601119884 (119884 = 0 1 sdot sdot sdot 119877 minus 1) (24)
The step response system function from the timemoment119905119896 to 119905119896+119877minus1 is defined as the impact of heat-transfer coefficienton 119879(119909 119910 119905) After the derivation of ℎ119896 by (20) the stepresponse equation can be obtained as follows
12059721198671205971199092 + 12059721198671205971199102 = 1120572 120597119867120597119905 (isin Ω 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 0 (isin 1198631 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 0 (isin 1198632 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119910 = 0 (isin 1198633 119905119896 le 119905 le 119905119896+119877minus1)minus120582120597119867120597119909 = 119879119896 minus 119879119891 minus ℎ119867 (119909 119910 119905)
(isin Γ3 119905119896 le 119905 le 119905119896+119877minus1)119867 = 1198670 (isin Ω 119905 = 119905119896)
(25)
From (25) it can be seen that119867 is related to ℎ Thereforewhile solving inverse problem it is necessary to use theexplicit difference algorithm used in the direct problemsolving and keep updating the calculation of ℎ
The corresponding discrete value 119867119896+119877minus1 is obtained by(25) hence the dynamic step response coefficient 120601119877 isfurther determined as
120601119877 = 119867119896+119877minus1 (26)
32 Rolling Optimization of Inverse Problem Measurementvalue and predictive value of temperature can be seen inthe time range from 119905119896 to 119905119896+119877minus1 According to the finiteoptimization 119877 parameters to be inverted are obtained sothe predictive value can be as close as possible to the mea-surement value in the future time domain at this momentthe quadratic performance index of system can be launched
min 119869 (ℎ) = (119879119898119890119886 minus 119879119901119903119890)Τ (119879119898119890119886 minus 119879119901119903119890) + ℎΤ120579ℎ (27)
where 120579 is the regularization parameter matrix
and 120572 is the regularization parameterAfter the derivation of ℎ based on (27) and making119889119869(ℎ)119889ℎ = 0 the optimal control rate can be obtained as
ℎ = (119860Τ119860 + 120579)minus1 119860Τ [119879119898119890119886 minus 11987911990111990311989010038161003816100381610038161003816ℎ=0] (29)
The optimal heat-transfer coefficient at moment 119896 can beobtained following (29)
ℎ = R (119860Τ119860 + 120579)minus1 119860Τ [119879119898119890119886 minus 11987911990111990311989010038161003816100381610038161003816ℎ=0] (30)
where R = [1 0 0]Τ There is no need to presetthe function form for the inverse problem calculation by theabove optimization algorithm
33 Regularization Parameter The residual principle [36ndash38] is introduced to calculate the optimal regularizationparameter aiming to reduce the impact of measurementerrors on the inversion results
To invert the boundary heat-transfer coefficient it isnecessary to firstly solve the direct problem using the predic-tive value of heat-transfer coefficient to get the temperaturecalculation at 119878 in the 119896 moment 119879119896119878 Besides in the casethat the temperature measurement value at 119878 119879119896119898119890119886 seesmeasurement error the temperature measurement value canbe expressed by the actual temperature plus themeasurementerror
where 120596 is the random number of standard normal in therange and 120590 is the standard deviation of measurement valuewhich is expressed as
120590 = radic 1119870 minus 1119896sum119909=1
(119879119896119898119890119886 minus 119879119896119886119888119905)2 (32)
where the constant K is the number of iterations
6 Complexity
The residual of heat-transfer coefficient in the wholeinversion time domain is defined as
119878 (120572) = radic 1119870 minus 1119896sum119896=1
(ℎ119896119886119888119905 minus ℎ119896119878)2 (33)
In (33) ℎ119896119886119888119905 and ℎ119896119878 are the actual value and inversion valueof heat-transfer coefficient respectively
Since ℎ119896119878 is unknown it is available to calculate thetemperaturemeasurement value at 119878 (119879119896119878 ) using ℎ119896119878 with directproblem algorithm thus the temperature residual in theinversion time domain can be obtained
119878 (120572) = radic 1119870 minus 1119896sum119896=1
(119879119896119898119890119886 minus 1198791198961198781)2 (34)
In ideal condition
ℎ119896119886119888119905 = ℎ119896119878 (35)
Similarily
1198791198961198781 = 119879119896119878 (36)
From the residual principle the regularization parameteris the optimal when both (35) and (36) are satisfied
119878119879 (120572) = 120590 (37)
34 Solving Procedure of Inverse Problem (1) Select the initialpredictive value of the heat-transfer coefficient ℎ119896119901 at a timemoment to perform inversion(2)Obtain the temperature calculation values inmeasure-ment point S at R time moments after that moment based onℎ119896119901 and (20)(3) Calculate the optimal regularization parameter 120572based on (37)(4)Assume the heat-transfer coefficient in the initial stageof inversion ℎ = 0 and obtain the step response matrix 119860 byEq (25)(5) Confirm the heat-transfer coefficient at the timemoment ℎ119896 according to (30) and then use the direct problemalgorithm to reconstruct the temperature field when the heat-transfer coefficient is ℎ119896(6) Following the time direction backward change thevalue in the initial stage of inversion and repeat steps (4) and(5) then get the inverse value of heat-transfer coefficient atdifferent time moments
4 Numerical Experiment and Analysis
Numerical experiments are performed to validate whetherthe proposed method is effective with the focus on analyzingthe impact of differentmeasurement errors andmeasurementpoint positions on the inversion result Also the inversionresult obtained in the condition without measurement erroris compared with the practical result which verifies theprecision of the proposed method
Γ1
Γ2
Γ3
Ω
Figure 1 Heat conduction model
4
2
1rsquo
3rsquo
2rsquo
0 3
1
Figure 2 Boundary node
D4
y
D3
D2D1
x
hTf
ΩΩ
Figure 3 The model of two-dimensional unsteady-state heat con-duction system without internal heat source
The two-dimensional plate heat transfer model (Figure 1)used in the above-mentioned direct problem is adoptedIn the simulation example the length Lx and width Ly ofthe plate is 02119898 The heat conductivity coefficient 120582 =47(119882119898lowast119870) thermal diffusivity119886 = 128lowast10minus6(1198982119904) initialtemperature 1198790 = 20∘119862 environment temperature 119879119891 =50∘119862 and heat-transfer coefficient ℎ = 2000(1198821198982 lowast119870) Thepurpose is to obtain the actual heat-transfer coefficient of theboundary D4
Complexity 7
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
21
22
23
24
25
26
27
28
29
30
Figure 4 The temperature field in the t=50s
016
017
018
019
020
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
Y
Figure 5 The temperature field in the t=100s
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
016
017
018
019
Y
Figure 6 The temperature field in the t=150s
41 Impact When the Measurement Error Is Zero Given themeasurement error 120590 = 0000 when the measurement pointis in the 119871 = 01119898 of D4 boundary and the future time step119877 = 5 the inversion result is shown in Figure 9
Figure 9 displays that except the transitory vibration inthe initial stage The inversion value is basically identical tothe practical value demonstrating the effectiveness of theinversion algorithm
8 Complexity
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
34
Figure 7 The temperature field in the t=200s
20
21
22
23
24
25
26
27
28
29
30
L1=01mL2=0004m
L3=0008ml1=0001m
50 100 150 200 2500Time (s)
Tem
pera
ture
(∘C)
Figure 8 The curve of the measuring point with time
42 Impact of Measurement Error Given the future time step119877 = 5 and the measurement point is in the 119871 = 01119898 of D4boundary the inversion results when the measurement erroris 120590 = 0001 120590 = 0005 and 120590 = 001 are displayed in Figures10 11 and 12 respectively
According to Table 1 and Figures 10 11 and 12 smallerrelative measurement error contributes to better inversionresults And enlarging measurement error will worsen theinversion results and aggravate the fluctuation
43 Impact of Measurement Point Position Given measure-ment error 120590 = 0001 and future time step 119877 = 5 theinversion results when the measurement point is in the 119871 =0004119898 0008119898 of D4 boundary and when the measurementpoint is 119897 = 0001119898 from the D4 boundary are shown inFigures 13 14 and 15 respectively
Analyze the contents of Table 2 and Figures 13 and 14The explicit FDM is used for direct problem when the
measurement point in boundary is closer to the boundaryangular point which however imposes a little impact on theinversion result Despite the increased relative average errorthe proposed method still exhibits a better ability to track theexact solution of heat-transfer coefficient and the inversionresult is relatively precise In Figure 15 considering that theposition of the measuring point is 0001m away from theboundary and the initial time temperature of the position is20 the temperature cannot change for a period of time sothe inversion result fluctuates greatly in the initial stage andincreases when the distance of measurement point positionfrom the boundary angular point becomes farther
5 Conclusion
The boundary heat-transfer coefficient of the two-dimensional unsteady heat conduction system is inversed bythe FDM and model prediction control method By solving
Complexity 9
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 9 The heat transfer coefficient of the measuring point without error
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 10 The heat transfer coefficient of the measuring point with 120590 = 0001
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 11 The heat transfer coefficient of the measuring point with 120590 = 0005
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 12 The heat transfer coefficient of the measuring point with 120590 = 001
and analyzing the algorithm example it demonstratesthat the proposed methods have higher accuracy in theinversion process Model predictive control method focuseson the model function rather than the structural form sothat we only need to know the step response or impulseresponse of the object we can directly get the predictionmodel and skip the derivation process It absorbs the idea
of optimization control and replaces global optimization byrolling time-domain optimization combined with feedbackcorrection which avoids a lot of calculation required byglobal optimization and constantly corrects the influencecaused by uncertain factors in the system At the sametime by discussing the impacts of error free measuringpoint positions and measuring errors on the results it
10 Complexity
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
1000150020002500300035004000
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 13 The heat transfer coefficient of the measuring point at the boundary 119871 = 0008119898
0500
10001500200025003000350040004500
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 14 The heat transfer coefficient of the measuring point at the boundary 119871 = 0004119898
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
100015002000250030003500
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 15 The heat transfer coefficient of the measuring point at the distance from the boundary 119897 = 0001119898
Table 1 Relative average errors of inversion result under differentmeasurement errors given 119871 = 01119898 and 119877 = 5Measurement error120590 0001 0005 001
Relative average error120578 701 1129 1601
Table 2 Relative average errors of inversion result in differentmeasurement point positions given 120590 = 0001 and 119877 = 5Measurement point 119871 = 0004119898 119871 = 0008119898 119897 = 0001119898Relative average error120578 765 706 2105
demonstrates that the obtained inversion results except theearly oscillation can better represent the stability of the exactsolution
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interestregarding the publication of this paper
Authorsrsquo Contributions
ShoubinWang andRuiNi contributed to developing the ideasof this research All of the authors were involved in preparingthis manuscript
Acknowledgments
This work was financially supported by the National KeyFoundation for Exploring Scientific Instrument of China
Complexity 11
(2013YQ470767) Tianjin Municipal Education CommissionProject for Scientific Research Items (2017KJ059) and Tian-jin Science and Technology Commissioner Project (18JCT-PJC62200 18JCTPJC64100)
References
[1] W Shoubin Z Li S Xiaogang and J Huangchao ldquoInversionof thermal conductivity in two-dimensional unsteady-stateheat transfer system based on boundary element method anddecentralized fuzzy inferencerdquo Complexity vol 2018 Article ID8783946 9 pages 2018
[2] S Wang Y Deng and X Sun ldquoSolving of two-dimensionalunsteady inverse heat conduction problems based on boundaryelementmethod and sequential function specificationmethodrdquoComplexity vol 2018 Article ID 6741632 11 pages 2018
[3] SWang H Jia X Sun and L Zhang ldquoResearch on the recogni-tion algorithm concerning geometric boundary regarding heatconduction based on BEM and CGMrdquoMathematical Problemsin Engineering vol 2018 Article ID 3723949 13 pages 2018
[4] S Wang L Zhang X Sun et al ldquoSolution to two-dimensionalsteady inverse heat transfer problems with interior heat sourcebased on the conjugate gradient methodrdquo Mathematical Prob-lems in Engineering vol 2017 Article ID 2861342 9 pages 2017
[5] S Wang H Jia X Sun et al ldquoTwo-dimensional steady-state boundary shape inversion of CGM-SPSO algorithm ontemperature informationrdquo Advances in Materials Science andEngineering vol 2017 Article ID 2461498 12 pages 2017
[6] Z Tianyu and D Changhong ldquoCompound control systemdesign based on adaptive backstepping theoryrdquo Journal ofBeijing University of Aeronautics and A vol 39 no 7 pp 902ndash906 2013
[7] P Duda ldquoA general method for solving transient multidimen-sional inverse heat transfer problemsrdquo International Journal ofHeat and Mass Transfer vol 93 pp 665ndash673 2016
[8] B Li and L Liu ldquoAn algorithm for geometry boundary identifi-cation of heat conduction problem based on boundary elementdiscretizationrdquo Proceedings of the CSEE vol 28 no 20 pp 38ndash43 2008
[9] C-Y Yang ldquoBoundary prediction of bio-heat conduction ina two-dimensional multilayer tissuerdquo International Journal ofHeat and Mass Transfer vol 78 no 7 pp 232ndash239 2014
[10] P Duda ldquoNumerical and experimental verification of twomethods for solving an inverse heat conduction problemrdquoInternational Journal of Heat and Mass Transfer vol 84 pp1101ndash1112 2015
[11] V M Luchesi and R T Coelho ldquoAn inverse method to estimatethe moving heat source inmachining processrdquoAppliedermalEngineering vol 45-46 pp 64ndash78 2012
[12] PDuda ldquoAmethod for transient thermal load estimation and itsapplication to identification of aerodynamic heating on atmo-spheric reentry capsulerdquo Aerospace Science and Technology vol51 pp 26ndash33 2016
[13] J V Beck B Blackwell and C R Clair Inverse HeatConductionIll-Posed Problems A Wiley-Interscience Publica-tion 1985
[14] T RHsu N S Sun G G Chen and Z L Gong ldquoFinite elementformulation for two-dimensional inverse heat conduction anal-ysisrdquo Journal of Heat Transfer vol 114 no 3 p 553 1992
[15] A A Tseng and F Z Zhao ldquoMultidimensional inverse tran-sient heat conduction problems by direct sensitivity coefficient
method using a finite-element schemerdquo Numerical Heat Trans-fer Part B Fundamentals vol 29 no 3 pp 365ndash380 1996
[16] L Zhaoming Further Studies on Fuzzy Inference Method forIn-verse Heat Transfer Problems Chongqing University 2014
[17] W Q Qian Y Zhou K F He J Y Yuan and J D HuangldquoEstimation of surface heat flux for nonlinear inverse heatconduction problemrdquo Acta Aerodynamica Sinica vol 30 no 2pp 145ndash150 2012
[18] J Su and G F Hewitt ldquoInverse heat conduction problemof estimating time-varying heattransfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004
[19] G Blanc J V Beck and M Raynaud ldquoSolution of the inverseheat conduction problemwith a time-variable number of futuretemperaturesrdquo Numerical Heat Transfer Part B Fundamentalsvol 32 no 4 pp 437ndash451 1997
[21] D Lesnic L Elliott and D B Ingham ldquoIdentification of thethermal conductivity and heat capacity in unsteady nonlin-ear heat conduction problems using the boundary elementmethodrdquo Journal of Computational Physics vol 126 no 2 pp410ndash420 1996
[22] A A Ershova and A I Sidikova ldquoUncertainty estimation ofthe method based on generalized residual principle for therestore task of the spectral density of crystalsrdquo Vestn Yuzhno-UralGosUn-taSerMatemMekhFiz vol 2015 pp 25ndash30 2015
[23] L Zhao Research on Fluid Temperature Inversion Algorithms forOne-Dimensional Unsteady Convection Heat Transfer HarbinInstitute of Technology 2017
[24] B Li and L Liu ldquoAn algorithm for geometry boundary identi-fication of heat conduction problem based on boundary ele-ment discretizationrdquo Proceedings of the CSEE vol 28 no 20pp 38ndash43 2008
[25] B Li and L Liu ldquoGeometry boundary identification of unsteadyheat conduction based on dual reciprocity boundary elementmethodrdquo Proceedings of the CSEE vol 29 no 5 pp 66ndash71 2009
[26] H Zhou X Xu X Li and H Chen ldquoIdentification oftemperature-dependent thermal conductivity for 2-D transientheat conduction problemsrdquo Applied Mathematics and Mechan-ics vol 29 no 1 pp 55ndash68 2014
[27] L Yanhao Model Predictive Inverse Method for Heat TransferProcess and Application Chongqing University 2017
[28] J Fan Reconstruction of Boiler Drum Temperature Field Basedon Inverse Heat Transfer Problem Chongqing University 2017
[29] H Ya-fen et alMechanism and Characteristics of Heat Conduc-tion in Nanocomposites Harbin Institute of Technology 2013
[30] Q Yue-ping M Jun J Jing-yan Y Xiao-bin and L WeildquoUnsteady heat transfer problems with finite volume methodrdquoJournal of Liaoning Technical University Natural Science vol 32no 05 pp 577ndash581 2013
[31] C-H Chiu and C-K Chen ldquoApplications of adomianrsquos decom-position procedure to the analysis of convective-radiative finsrdquoJournal of Heat Transfer vol 125 no 2 pp 312ndash316 2003
[32] C Sheng Direct and Inverse Heat Conduction Problems Solvingby the Boundary Element Method Hunan University 2007
[33] MMKhader AM Eid andAMMegahed ldquoNumerical stud-ies using FDM for viscous dissipation and thermal radiationeffects on the slip flow and heat transfer due to a stretchingsheet embedded in a porous medium with variable thickness
12 Complexity
and variable thermal conductivityrdquoNewTrends inMathematicalSciences vol 4 no 1 p 38 2016
[34] S Chuanzhuo and Z Tiande ldquoA simple method for calculatingtruncation error of numerical integral formulardquo Journal ofMathematis for Technology vol 2 1994
[35] Q Jixin Z Jun and X Zuhu Predictive Control ChemicalIndustry Press 2007
[36] L Zhang H Chen G Wang and Z Luo ldquoSolving transientinverse heat conduction problems based on optimal number offuture time stepsrdquo Proceedings of the CSEE vol 32 no 2 pp99ndash103 2012
[37] N Buong T T Huong and N T Thuy ldquoA quasi-residualprinciple in regularization for a common solution of a system ofnonlinearmonotone ill-posed equationsrdquoRussianMathematicsvol 60 no 3 pp 47ndash55 2016
[38] K A Woodbury and S K Thakur ldquoRedundant data and futuretimes in the inverse heat conduction problemrdquo Inverse Problemsin Science and Engineering vol 2 no 4 pp 319ndash333 1996
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
6 Complexity
The residual of heat-transfer coefficient in the wholeinversion time domain is defined as
119878 (120572) = radic 1119870 minus 1119896sum119896=1
(ℎ119896119886119888119905 minus ℎ119896119878)2 (33)
In (33) ℎ119896119886119888119905 and ℎ119896119878 are the actual value and inversion valueof heat-transfer coefficient respectively
Since ℎ119896119878 is unknown it is available to calculate thetemperaturemeasurement value at 119878 (119879119896119878 ) using ℎ119896119878 with directproblem algorithm thus the temperature residual in theinversion time domain can be obtained
119878 (120572) = radic 1119870 minus 1119896sum119896=1
(119879119896119898119890119886 minus 1198791198961198781)2 (34)
In ideal condition
ℎ119896119886119888119905 = ℎ119896119878 (35)
Similarily
1198791198961198781 = 119879119896119878 (36)
From the residual principle the regularization parameteris the optimal when both (35) and (36) are satisfied
119878119879 (120572) = 120590 (37)
34 Solving Procedure of Inverse Problem (1) Select the initialpredictive value of the heat-transfer coefficient ℎ119896119901 at a timemoment to perform inversion(2)Obtain the temperature calculation values inmeasure-ment point S at R time moments after that moment based onℎ119896119901 and (20)(3) Calculate the optimal regularization parameter 120572based on (37)(4)Assume the heat-transfer coefficient in the initial stageof inversion ℎ = 0 and obtain the step response matrix 119860 byEq (25)(5) Confirm the heat-transfer coefficient at the timemoment ℎ119896 according to (30) and then use the direct problemalgorithm to reconstruct the temperature field when the heat-transfer coefficient is ℎ119896(6) Following the time direction backward change thevalue in the initial stage of inversion and repeat steps (4) and(5) then get the inverse value of heat-transfer coefficient atdifferent time moments
4 Numerical Experiment and Analysis
Numerical experiments are performed to validate whetherthe proposed method is effective with the focus on analyzingthe impact of differentmeasurement errors andmeasurementpoint positions on the inversion result Also the inversionresult obtained in the condition without measurement erroris compared with the practical result which verifies theprecision of the proposed method
Γ1
Γ2
Γ3
Ω
Figure 1 Heat conduction model
4
2
1rsquo
3rsquo
2rsquo
0 3
1
Figure 2 Boundary node
D4
y
D3
D2D1
x
hTf
ΩΩ
Figure 3 The model of two-dimensional unsteady-state heat con-duction system without internal heat source
The two-dimensional plate heat transfer model (Figure 1)used in the above-mentioned direct problem is adoptedIn the simulation example the length Lx and width Ly ofthe plate is 02119898 The heat conductivity coefficient 120582 =47(119882119898lowast119870) thermal diffusivity119886 = 128lowast10minus6(1198982119904) initialtemperature 1198790 = 20∘119862 environment temperature 119879119891 =50∘119862 and heat-transfer coefficient ℎ = 2000(1198821198982 lowast119870) Thepurpose is to obtain the actual heat-transfer coefficient of theboundary D4
Complexity 7
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
21
22
23
24
25
26
27
28
29
30
Figure 4 The temperature field in the t=50s
016
017
018
019
020
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
Y
Figure 5 The temperature field in the t=100s
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
016
017
018
019
Y
Figure 6 The temperature field in the t=150s
41 Impact When the Measurement Error Is Zero Given themeasurement error 120590 = 0000 when the measurement pointis in the 119871 = 01119898 of D4 boundary and the future time step119877 = 5 the inversion result is shown in Figure 9
Figure 9 displays that except the transitory vibration inthe initial stage The inversion value is basically identical tothe practical value demonstrating the effectiveness of theinversion algorithm
8 Complexity
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
34
Figure 7 The temperature field in the t=200s
20
21
22
23
24
25
26
27
28
29
30
L1=01mL2=0004m
L3=0008ml1=0001m
50 100 150 200 2500Time (s)
Tem
pera
ture
(∘C)
Figure 8 The curve of the measuring point with time
42 Impact of Measurement Error Given the future time step119877 = 5 and the measurement point is in the 119871 = 01119898 of D4boundary the inversion results when the measurement erroris 120590 = 0001 120590 = 0005 and 120590 = 001 are displayed in Figures10 11 and 12 respectively
According to Table 1 and Figures 10 11 and 12 smallerrelative measurement error contributes to better inversionresults And enlarging measurement error will worsen theinversion results and aggravate the fluctuation
43 Impact of Measurement Point Position Given measure-ment error 120590 = 0001 and future time step 119877 = 5 theinversion results when the measurement point is in the 119871 =0004119898 0008119898 of D4 boundary and when the measurementpoint is 119897 = 0001119898 from the D4 boundary are shown inFigures 13 14 and 15 respectively
Analyze the contents of Table 2 and Figures 13 and 14The explicit FDM is used for direct problem when the
measurement point in boundary is closer to the boundaryangular point which however imposes a little impact on theinversion result Despite the increased relative average errorthe proposed method still exhibits a better ability to track theexact solution of heat-transfer coefficient and the inversionresult is relatively precise In Figure 15 considering that theposition of the measuring point is 0001m away from theboundary and the initial time temperature of the position is20 the temperature cannot change for a period of time sothe inversion result fluctuates greatly in the initial stage andincreases when the distance of measurement point positionfrom the boundary angular point becomes farther
5 Conclusion
The boundary heat-transfer coefficient of the two-dimensional unsteady heat conduction system is inversed bythe FDM and model prediction control method By solving
Complexity 9
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 9 The heat transfer coefficient of the measuring point without error
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 10 The heat transfer coefficient of the measuring point with 120590 = 0001
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 11 The heat transfer coefficient of the measuring point with 120590 = 0005
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 12 The heat transfer coefficient of the measuring point with 120590 = 001
and analyzing the algorithm example it demonstratesthat the proposed methods have higher accuracy in theinversion process Model predictive control method focuseson the model function rather than the structural form sothat we only need to know the step response or impulseresponse of the object we can directly get the predictionmodel and skip the derivation process It absorbs the idea
of optimization control and replaces global optimization byrolling time-domain optimization combined with feedbackcorrection which avoids a lot of calculation required byglobal optimization and constantly corrects the influencecaused by uncertain factors in the system At the sametime by discussing the impacts of error free measuringpoint positions and measuring errors on the results it
10 Complexity
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
1000150020002500300035004000
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 13 The heat transfer coefficient of the measuring point at the boundary 119871 = 0008119898
0500
10001500200025003000350040004500
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 14 The heat transfer coefficient of the measuring point at the boundary 119871 = 0004119898
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
100015002000250030003500
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 15 The heat transfer coefficient of the measuring point at the distance from the boundary 119897 = 0001119898
Table 1 Relative average errors of inversion result under differentmeasurement errors given 119871 = 01119898 and 119877 = 5Measurement error120590 0001 0005 001
Relative average error120578 701 1129 1601
Table 2 Relative average errors of inversion result in differentmeasurement point positions given 120590 = 0001 and 119877 = 5Measurement point 119871 = 0004119898 119871 = 0008119898 119897 = 0001119898Relative average error120578 765 706 2105
demonstrates that the obtained inversion results except theearly oscillation can better represent the stability of the exactsolution
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interestregarding the publication of this paper
Authorsrsquo Contributions
ShoubinWang andRuiNi contributed to developing the ideasof this research All of the authors were involved in preparingthis manuscript
Acknowledgments
This work was financially supported by the National KeyFoundation for Exploring Scientific Instrument of China
Complexity 11
(2013YQ470767) Tianjin Municipal Education CommissionProject for Scientific Research Items (2017KJ059) and Tian-jin Science and Technology Commissioner Project (18JCT-PJC62200 18JCTPJC64100)
References
[1] W Shoubin Z Li S Xiaogang and J Huangchao ldquoInversionof thermal conductivity in two-dimensional unsteady-stateheat transfer system based on boundary element method anddecentralized fuzzy inferencerdquo Complexity vol 2018 Article ID8783946 9 pages 2018
[2] S Wang Y Deng and X Sun ldquoSolving of two-dimensionalunsteady inverse heat conduction problems based on boundaryelementmethod and sequential function specificationmethodrdquoComplexity vol 2018 Article ID 6741632 11 pages 2018
[3] SWang H Jia X Sun and L Zhang ldquoResearch on the recogni-tion algorithm concerning geometric boundary regarding heatconduction based on BEM and CGMrdquoMathematical Problemsin Engineering vol 2018 Article ID 3723949 13 pages 2018
[4] S Wang L Zhang X Sun et al ldquoSolution to two-dimensionalsteady inverse heat transfer problems with interior heat sourcebased on the conjugate gradient methodrdquo Mathematical Prob-lems in Engineering vol 2017 Article ID 2861342 9 pages 2017
[5] S Wang H Jia X Sun et al ldquoTwo-dimensional steady-state boundary shape inversion of CGM-SPSO algorithm ontemperature informationrdquo Advances in Materials Science andEngineering vol 2017 Article ID 2461498 12 pages 2017
[6] Z Tianyu and D Changhong ldquoCompound control systemdesign based on adaptive backstepping theoryrdquo Journal ofBeijing University of Aeronautics and A vol 39 no 7 pp 902ndash906 2013
[7] P Duda ldquoA general method for solving transient multidimen-sional inverse heat transfer problemsrdquo International Journal ofHeat and Mass Transfer vol 93 pp 665ndash673 2016
[8] B Li and L Liu ldquoAn algorithm for geometry boundary identifi-cation of heat conduction problem based on boundary elementdiscretizationrdquo Proceedings of the CSEE vol 28 no 20 pp 38ndash43 2008
[9] C-Y Yang ldquoBoundary prediction of bio-heat conduction ina two-dimensional multilayer tissuerdquo International Journal ofHeat and Mass Transfer vol 78 no 7 pp 232ndash239 2014
[10] P Duda ldquoNumerical and experimental verification of twomethods for solving an inverse heat conduction problemrdquoInternational Journal of Heat and Mass Transfer vol 84 pp1101ndash1112 2015
[11] V M Luchesi and R T Coelho ldquoAn inverse method to estimatethe moving heat source inmachining processrdquoAppliedermalEngineering vol 45-46 pp 64ndash78 2012
[12] PDuda ldquoAmethod for transient thermal load estimation and itsapplication to identification of aerodynamic heating on atmo-spheric reentry capsulerdquo Aerospace Science and Technology vol51 pp 26ndash33 2016
[13] J V Beck B Blackwell and C R Clair Inverse HeatConductionIll-Posed Problems A Wiley-Interscience Publica-tion 1985
[14] T RHsu N S Sun G G Chen and Z L Gong ldquoFinite elementformulation for two-dimensional inverse heat conduction anal-ysisrdquo Journal of Heat Transfer vol 114 no 3 p 553 1992
[15] A A Tseng and F Z Zhao ldquoMultidimensional inverse tran-sient heat conduction problems by direct sensitivity coefficient
method using a finite-element schemerdquo Numerical Heat Trans-fer Part B Fundamentals vol 29 no 3 pp 365ndash380 1996
[16] L Zhaoming Further Studies on Fuzzy Inference Method forIn-verse Heat Transfer Problems Chongqing University 2014
[17] W Q Qian Y Zhou K F He J Y Yuan and J D HuangldquoEstimation of surface heat flux for nonlinear inverse heatconduction problemrdquo Acta Aerodynamica Sinica vol 30 no 2pp 145ndash150 2012
[18] J Su and G F Hewitt ldquoInverse heat conduction problemof estimating time-varying heattransfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004
[19] G Blanc J V Beck and M Raynaud ldquoSolution of the inverseheat conduction problemwith a time-variable number of futuretemperaturesrdquo Numerical Heat Transfer Part B Fundamentalsvol 32 no 4 pp 437ndash451 1997
[21] D Lesnic L Elliott and D B Ingham ldquoIdentification of thethermal conductivity and heat capacity in unsteady nonlin-ear heat conduction problems using the boundary elementmethodrdquo Journal of Computational Physics vol 126 no 2 pp410ndash420 1996
[22] A A Ershova and A I Sidikova ldquoUncertainty estimation ofthe method based on generalized residual principle for therestore task of the spectral density of crystalsrdquo Vestn Yuzhno-UralGosUn-taSerMatemMekhFiz vol 2015 pp 25ndash30 2015
[23] L Zhao Research on Fluid Temperature Inversion Algorithms forOne-Dimensional Unsteady Convection Heat Transfer HarbinInstitute of Technology 2017
[24] B Li and L Liu ldquoAn algorithm for geometry boundary identi-fication of heat conduction problem based on boundary ele-ment discretizationrdquo Proceedings of the CSEE vol 28 no 20pp 38ndash43 2008
[25] B Li and L Liu ldquoGeometry boundary identification of unsteadyheat conduction based on dual reciprocity boundary elementmethodrdquo Proceedings of the CSEE vol 29 no 5 pp 66ndash71 2009
[26] H Zhou X Xu X Li and H Chen ldquoIdentification oftemperature-dependent thermal conductivity for 2-D transientheat conduction problemsrdquo Applied Mathematics and Mechan-ics vol 29 no 1 pp 55ndash68 2014
[27] L Yanhao Model Predictive Inverse Method for Heat TransferProcess and Application Chongqing University 2017
[28] J Fan Reconstruction of Boiler Drum Temperature Field Basedon Inverse Heat Transfer Problem Chongqing University 2017
[29] H Ya-fen et alMechanism and Characteristics of Heat Conduc-tion in Nanocomposites Harbin Institute of Technology 2013
[30] Q Yue-ping M Jun J Jing-yan Y Xiao-bin and L WeildquoUnsteady heat transfer problems with finite volume methodrdquoJournal of Liaoning Technical University Natural Science vol 32no 05 pp 577ndash581 2013
[31] C-H Chiu and C-K Chen ldquoApplications of adomianrsquos decom-position procedure to the analysis of convective-radiative finsrdquoJournal of Heat Transfer vol 125 no 2 pp 312ndash316 2003
[32] C Sheng Direct and Inverse Heat Conduction Problems Solvingby the Boundary Element Method Hunan University 2007
[33] MMKhader AM Eid andAMMegahed ldquoNumerical stud-ies using FDM for viscous dissipation and thermal radiationeffects on the slip flow and heat transfer due to a stretchingsheet embedded in a porous medium with variable thickness
12 Complexity
and variable thermal conductivityrdquoNewTrends inMathematicalSciences vol 4 no 1 p 38 2016
[34] S Chuanzhuo and Z Tiande ldquoA simple method for calculatingtruncation error of numerical integral formulardquo Journal ofMathematis for Technology vol 2 1994
[35] Q Jixin Z Jun and X Zuhu Predictive Control ChemicalIndustry Press 2007
[36] L Zhang H Chen G Wang and Z Luo ldquoSolving transientinverse heat conduction problems based on optimal number offuture time stepsrdquo Proceedings of the CSEE vol 32 no 2 pp99ndash103 2012
[37] N Buong T T Huong and N T Thuy ldquoA quasi-residualprinciple in regularization for a common solution of a system ofnonlinearmonotone ill-posed equationsrdquoRussianMathematicsvol 60 no 3 pp 47ndash55 2016
[38] K A Woodbury and S K Thakur ldquoRedundant data and futuretimes in the inverse heat conduction problemrdquo Inverse Problemsin Science and Engineering vol 2 no 4 pp 319ndash333 1996
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 7
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
21
22
23
24
25
26
27
28
29
30
Figure 4 The temperature field in the t=50s
016
017
018
019
020
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
Y
Figure 5 The temperature field in the t=100s
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
016
017
018
019
Y
Figure 6 The temperature field in the t=150s
41 Impact When the Measurement Error Is Zero Given themeasurement error 120590 = 0000 when the measurement pointis in the 119871 = 01119898 of D4 boundary and the future time step119877 = 5 the inversion result is shown in Figure 9
Figure 9 displays that except the transitory vibration inthe initial stage The inversion value is basically identical tothe practical value demonstrating the effectiveness of theinversion algorithm
8 Complexity
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
34
Figure 7 The temperature field in the t=200s
20
21
22
23
24
25
26
27
28
29
30
L1=01mL2=0004m
L3=0008ml1=0001m
50 100 150 200 2500Time (s)
Tem
pera
ture
(∘C)
Figure 8 The curve of the measuring point with time
42 Impact of Measurement Error Given the future time step119877 = 5 and the measurement point is in the 119871 = 01119898 of D4boundary the inversion results when the measurement erroris 120590 = 0001 120590 = 0005 and 120590 = 001 are displayed in Figures10 11 and 12 respectively
According to Table 1 and Figures 10 11 and 12 smallerrelative measurement error contributes to better inversionresults And enlarging measurement error will worsen theinversion results and aggravate the fluctuation
43 Impact of Measurement Point Position Given measure-ment error 120590 = 0001 and future time step 119877 = 5 theinversion results when the measurement point is in the 119871 =0004119898 0008119898 of D4 boundary and when the measurementpoint is 119897 = 0001119898 from the D4 boundary are shown inFigures 13 14 and 15 respectively
Analyze the contents of Table 2 and Figures 13 and 14The explicit FDM is used for direct problem when the
measurement point in boundary is closer to the boundaryangular point which however imposes a little impact on theinversion result Despite the increased relative average errorthe proposed method still exhibits a better ability to track theexact solution of heat-transfer coefficient and the inversionresult is relatively precise In Figure 15 considering that theposition of the measuring point is 0001m away from theboundary and the initial time temperature of the position is20 the temperature cannot change for a period of time sothe inversion result fluctuates greatly in the initial stage andincreases when the distance of measurement point positionfrom the boundary angular point becomes farther
5 Conclusion
The boundary heat-transfer coefficient of the two-dimensional unsteady heat conduction system is inversed bythe FDM and model prediction control method By solving
Complexity 9
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 9 The heat transfer coefficient of the measuring point without error
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 10 The heat transfer coefficient of the measuring point with 120590 = 0001
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 11 The heat transfer coefficient of the measuring point with 120590 = 0005
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 12 The heat transfer coefficient of the measuring point with 120590 = 001
and analyzing the algorithm example it demonstratesthat the proposed methods have higher accuracy in theinversion process Model predictive control method focuseson the model function rather than the structural form sothat we only need to know the step response or impulseresponse of the object we can directly get the predictionmodel and skip the derivation process It absorbs the idea
of optimization control and replaces global optimization byrolling time-domain optimization combined with feedbackcorrection which avoids a lot of calculation required byglobal optimization and constantly corrects the influencecaused by uncertain factors in the system At the sametime by discussing the impacts of error free measuringpoint positions and measuring errors on the results it
10 Complexity
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
1000150020002500300035004000
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 13 The heat transfer coefficient of the measuring point at the boundary 119871 = 0008119898
0500
10001500200025003000350040004500
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 14 The heat transfer coefficient of the measuring point at the boundary 119871 = 0004119898
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
100015002000250030003500
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 15 The heat transfer coefficient of the measuring point at the distance from the boundary 119897 = 0001119898
Table 1 Relative average errors of inversion result under differentmeasurement errors given 119871 = 01119898 and 119877 = 5Measurement error120590 0001 0005 001
Relative average error120578 701 1129 1601
Table 2 Relative average errors of inversion result in differentmeasurement point positions given 120590 = 0001 and 119877 = 5Measurement point 119871 = 0004119898 119871 = 0008119898 119897 = 0001119898Relative average error120578 765 706 2105
demonstrates that the obtained inversion results except theearly oscillation can better represent the stability of the exactsolution
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interestregarding the publication of this paper
Authorsrsquo Contributions
ShoubinWang andRuiNi contributed to developing the ideasof this research All of the authors were involved in preparingthis manuscript
Acknowledgments
This work was financially supported by the National KeyFoundation for Exploring Scientific Instrument of China
Complexity 11
(2013YQ470767) Tianjin Municipal Education CommissionProject for Scientific Research Items (2017KJ059) and Tian-jin Science and Technology Commissioner Project (18JCT-PJC62200 18JCTPJC64100)
References
[1] W Shoubin Z Li S Xiaogang and J Huangchao ldquoInversionof thermal conductivity in two-dimensional unsteady-stateheat transfer system based on boundary element method anddecentralized fuzzy inferencerdquo Complexity vol 2018 Article ID8783946 9 pages 2018
[2] S Wang Y Deng and X Sun ldquoSolving of two-dimensionalunsteady inverse heat conduction problems based on boundaryelementmethod and sequential function specificationmethodrdquoComplexity vol 2018 Article ID 6741632 11 pages 2018
[3] SWang H Jia X Sun and L Zhang ldquoResearch on the recogni-tion algorithm concerning geometric boundary regarding heatconduction based on BEM and CGMrdquoMathematical Problemsin Engineering vol 2018 Article ID 3723949 13 pages 2018
[4] S Wang L Zhang X Sun et al ldquoSolution to two-dimensionalsteady inverse heat transfer problems with interior heat sourcebased on the conjugate gradient methodrdquo Mathematical Prob-lems in Engineering vol 2017 Article ID 2861342 9 pages 2017
[5] S Wang H Jia X Sun et al ldquoTwo-dimensional steady-state boundary shape inversion of CGM-SPSO algorithm ontemperature informationrdquo Advances in Materials Science andEngineering vol 2017 Article ID 2461498 12 pages 2017
[6] Z Tianyu and D Changhong ldquoCompound control systemdesign based on adaptive backstepping theoryrdquo Journal ofBeijing University of Aeronautics and A vol 39 no 7 pp 902ndash906 2013
[7] P Duda ldquoA general method for solving transient multidimen-sional inverse heat transfer problemsrdquo International Journal ofHeat and Mass Transfer vol 93 pp 665ndash673 2016
[8] B Li and L Liu ldquoAn algorithm for geometry boundary identifi-cation of heat conduction problem based on boundary elementdiscretizationrdquo Proceedings of the CSEE vol 28 no 20 pp 38ndash43 2008
[9] C-Y Yang ldquoBoundary prediction of bio-heat conduction ina two-dimensional multilayer tissuerdquo International Journal ofHeat and Mass Transfer vol 78 no 7 pp 232ndash239 2014
[10] P Duda ldquoNumerical and experimental verification of twomethods for solving an inverse heat conduction problemrdquoInternational Journal of Heat and Mass Transfer vol 84 pp1101ndash1112 2015
[11] V M Luchesi and R T Coelho ldquoAn inverse method to estimatethe moving heat source inmachining processrdquoAppliedermalEngineering vol 45-46 pp 64ndash78 2012
[12] PDuda ldquoAmethod for transient thermal load estimation and itsapplication to identification of aerodynamic heating on atmo-spheric reentry capsulerdquo Aerospace Science and Technology vol51 pp 26ndash33 2016
[13] J V Beck B Blackwell and C R Clair Inverse HeatConductionIll-Posed Problems A Wiley-Interscience Publica-tion 1985
[14] T RHsu N S Sun G G Chen and Z L Gong ldquoFinite elementformulation for two-dimensional inverse heat conduction anal-ysisrdquo Journal of Heat Transfer vol 114 no 3 p 553 1992
[15] A A Tseng and F Z Zhao ldquoMultidimensional inverse tran-sient heat conduction problems by direct sensitivity coefficient
method using a finite-element schemerdquo Numerical Heat Trans-fer Part B Fundamentals vol 29 no 3 pp 365ndash380 1996
[16] L Zhaoming Further Studies on Fuzzy Inference Method forIn-verse Heat Transfer Problems Chongqing University 2014
[17] W Q Qian Y Zhou K F He J Y Yuan and J D HuangldquoEstimation of surface heat flux for nonlinear inverse heatconduction problemrdquo Acta Aerodynamica Sinica vol 30 no 2pp 145ndash150 2012
[18] J Su and G F Hewitt ldquoInverse heat conduction problemof estimating time-varying heattransfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004
[19] G Blanc J V Beck and M Raynaud ldquoSolution of the inverseheat conduction problemwith a time-variable number of futuretemperaturesrdquo Numerical Heat Transfer Part B Fundamentalsvol 32 no 4 pp 437ndash451 1997
[21] D Lesnic L Elliott and D B Ingham ldquoIdentification of thethermal conductivity and heat capacity in unsteady nonlin-ear heat conduction problems using the boundary elementmethodrdquo Journal of Computational Physics vol 126 no 2 pp410ndash420 1996
[22] A A Ershova and A I Sidikova ldquoUncertainty estimation ofthe method based on generalized residual principle for therestore task of the spectral density of crystalsrdquo Vestn Yuzhno-UralGosUn-taSerMatemMekhFiz vol 2015 pp 25ndash30 2015
[23] L Zhao Research on Fluid Temperature Inversion Algorithms forOne-Dimensional Unsteady Convection Heat Transfer HarbinInstitute of Technology 2017
[24] B Li and L Liu ldquoAn algorithm for geometry boundary identi-fication of heat conduction problem based on boundary ele-ment discretizationrdquo Proceedings of the CSEE vol 28 no 20pp 38ndash43 2008
[25] B Li and L Liu ldquoGeometry boundary identification of unsteadyheat conduction based on dual reciprocity boundary elementmethodrdquo Proceedings of the CSEE vol 29 no 5 pp 66ndash71 2009
[26] H Zhou X Xu X Li and H Chen ldquoIdentification oftemperature-dependent thermal conductivity for 2-D transientheat conduction problemsrdquo Applied Mathematics and Mechan-ics vol 29 no 1 pp 55ndash68 2014
[27] L Yanhao Model Predictive Inverse Method for Heat TransferProcess and Application Chongqing University 2017
[28] J Fan Reconstruction of Boiler Drum Temperature Field Basedon Inverse Heat Transfer Problem Chongqing University 2017
[29] H Ya-fen et alMechanism and Characteristics of Heat Conduc-tion in Nanocomposites Harbin Institute of Technology 2013
[30] Q Yue-ping M Jun J Jing-yan Y Xiao-bin and L WeildquoUnsteady heat transfer problems with finite volume methodrdquoJournal of Liaoning Technical University Natural Science vol 32no 05 pp 577ndash581 2013
[31] C-H Chiu and C-K Chen ldquoApplications of adomianrsquos decom-position procedure to the analysis of convective-radiative finsrdquoJournal of Heat Transfer vol 125 no 2 pp 312ndash316 2003
[32] C Sheng Direct and Inverse Heat Conduction Problems Solvingby the Boundary Element Method Hunan University 2007
[33] MMKhader AM Eid andAMMegahed ldquoNumerical stud-ies using FDM for viscous dissipation and thermal radiationeffects on the slip flow and heat transfer due to a stretchingsheet embedded in a porous medium with variable thickness
12 Complexity
and variable thermal conductivityrdquoNewTrends inMathematicalSciences vol 4 no 1 p 38 2016
[34] S Chuanzhuo and Z Tiande ldquoA simple method for calculatingtruncation error of numerical integral formulardquo Journal ofMathematis for Technology vol 2 1994
[35] Q Jixin Z Jun and X Zuhu Predictive Control ChemicalIndustry Press 2007
[36] L Zhang H Chen G Wang and Z Luo ldquoSolving transientinverse heat conduction problems based on optimal number offuture time stepsrdquo Proceedings of the CSEE vol 32 no 2 pp99ndash103 2012
[37] N Buong T T Huong and N T Thuy ldquoA quasi-residualprinciple in regularization for a common solution of a system ofnonlinearmonotone ill-posed equationsrdquoRussianMathematicsvol 60 no 3 pp 47ndash55 2016
[38] K A Woodbury and S K Thakur ldquoRedundant data and futuretimes in the inverse heat conduction problemrdquo Inverse Problemsin Science and Engineering vol 2 no 4 pp 319ndash333 1996
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Complexity
016
017
018
019
020
Y
0 002 004 006 008 010 012 014 016 018 020X
20
22
24
26
28
30
32
34
Figure 7 The temperature field in the t=200s
20
21
22
23
24
25
26
27
28
29
30
L1=01mL2=0004m
L3=0008ml1=0001m
50 100 150 200 2500Time (s)
Tem
pera
ture
(∘C)
Figure 8 The curve of the measuring point with time
42 Impact of Measurement Error Given the future time step119877 = 5 and the measurement point is in the 119871 = 01119898 of D4boundary the inversion results when the measurement erroris 120590 = 0001 120590 = 0005 and 120590 = 001 are displayed in Figures10 11 and 12 respectively
According to Table 1 and Figures 10 11 and 12 smallerrelative measurement error contributes to better inversionresults And enlarging measurement error will worsen theinversion results and aggravate the fluctuation
43 Impact of Measurement Point Position Given measure-ment error 120590 = 0001 and future time step 119877 = 5 theinversion results when the measurement point is in the 119871 =0004119898 0008119898 of D4 boundary and when the measurementpoint is 119897 = 0001119898 from the D4 boundary are shown inFigures 13 14 and 15 respectively
Analyze the contents of Table 2 and Figures 13 and 14The explicit FDM is used for direct problem when the
measurement point in boundary is closer to the boundaryangular point which however imposes a little impact on theinversion result Despite the increased relative average errorthe proposed method still exhibits a better ability to track theexact solution of heat-transfer coefficient and the inversionresult is relatively precise In Figure 15 considering that theposition of the measuring point is 0001m away from theboundary and the initial time temperature of the position is20 the temperature cannot change for a period of time sothe inversion result fluctuates greatly in the initial stage andincreases when the distance of measurement point positionfrom the boundary angular point becomes farther
5 Conclusion
The boundary heat-transfer coefficient of the two-dimensional unsteady heat conduction system is inversed bythe FDM and model prediction control method By solving
Complexity 9
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 9 The heat transfer coefficient of the measuring point without error
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 10 The heat transfer coefficient of the measuring point with 120590 = 0001
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 11 The heat transfer coefficient of the measuring point with 120590 = 0005
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 12 The heat transfer coefficient of the measuring point with 120590 = 001
and analyzing the algorithm example it demonstratesthat the proposed methods have higher accuracy in theinversion process Model predictive control method focuseson the model function rather than the structural form sothat we only need to know the step response or impulseresponse of the object we can directly get the predictionmodel and skip the derivation process It absorbs the idea
of optimization control and replaces global optimization byrolling time-domain optimization combined with feedbackcorrection which avoids a lot of calculation required byglobal optimization and constantly corrects the influencecaused by uncertain factors in the system At the sametime by discussing the impacts of error free measuringpoint positions and measuring errors on the results it
10 Complexity
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
1000150020002500300035004000
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 13 The heat transfer coefficient of the measuring point at the boundary 119871 = 0008119898
0500
10001500200025003000350040004500
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 14 The heat transfer coefficient of the measuring point at the boundary 119871 = 0004119898
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
100015002000250030003500
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 15 The heat transfer coefficient of the measuring point at the distance from the boundary 119897 = 0001119898
Table 1 Relative average errors of inversion result under differentmeasurement errors given 119871 = 01119898 and 119877 = 5Measurement error120590 0001 0005 001
Relative average error120578 701 1129 1601
Table 2 Relative average errors of inversion result in differentmeasurement point positions given 120590 = 0001 and 119877 = 5Measurement point 119871 = 0004119898 119871 = 0008119898 119897 = 0001119898Relative average error120578 765 706 2105
demonstrates that the obtained inversion results except theearly oscillation can better represent the stability of the exactsolution
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interestregarding the publication of this paper
Authorsrsquo Contributions
ShoubinWang andRuiNi contributed to developing the ideasof this research All of the authors were involved in preparingthis manuscript
Acknowledgments
This work was financially supported by the National KeyFoundation for Exploring Scientific Instrument of China
Complexity 11
(2013YQ470767) Tianjin Municipal Education CommissionProject for Scientific Research Items (2017KJ059) and Tian-jin Science and Technology Commissioner Project (18JCT-PJC62200 18JCTPJC64100)
References
[1] W Shoubin Z Li S Xiaogang and J Huangchao ldquoInversionof thermal conductivity in two-dimensional unsteady-stateheat transfer system based on boundary element method anddecentralized fuzzy inferencerdquo Complexity vol 2018 Article ID8783946 9 pages 2018
[2] S Wang Y Deng and X Sun ldquoSolving of two-dimensionalunsteady inverse heat conduction problems based on boundaryelementmethod and sequential function specificationmethodrdquoComplexity vol 2018 Article ID 6741632 11 pages 2018
[3] SWang H Jia X Sun and L Zhang ldquoResearch on the recogni-tion algorithm concerning geometric boundary regarding heatconduction based on BEM and CGMrdquoMathematical Problemsin Engineering vol 2018 Article ID 3723949 13 pages 2018
[4] S Wang L Zhang X Sun et al ldquoSolution to two-dimensionalsteady inverse heat transfer problems with interior heat sourcebased on the conjugate gradient methodrdquo Mathematical Prob-lems in Engineering vol 2017 Article ID 2861342 9 pages 2017
[5] S Wang H Jia X Sun et al ldquoTwo-dimensional steady-state boundary shape inversion of CGM-SPSO algorithm ontemperature informationrdquo Advances in Materials Science andEngineering vol 2017 Article ID 2461498 12 pages 2017
[6] Z Tianyu and D Changhong ldquoCompound control systemdesign based on adaptive backstepping theoryrdquo Journal ofBeijing University of Aeronautics and A vol 39 no 7 pp 902ndash906 2013
[7] P Duda ldquoA general method for solving transient multidimen-sional inverse heat transfer problemsrdquo International Journal ofHeat and Mass Transfer vol 93 pp 665ndash673 2016
[8] B Li and L Liu ldquoAn algorithm for geometry boundary identifi-cation of heat conduction problem based on boundary elementdiscretizationrdquo Proceedings of the CSEE vol 28 no 20 pp 38ndash43 2008
[9] C-Y Yang ldquoBoundary prediction of bio-heat conduction ina two-dimensional multilayer tissuerdquo International Journal ofHeat and Mass Transfer vol 78 no 7 pp 232ndash239 2014
[10] P Duda ldquoNumerical and experimental verification of twomethods for solving an inverse heat conduction problemrdquoInternational Journal of Heat and Mass Transfer vol 84 pp1101ndash1112 2015
[11] V M Luchesi and R T Coelho ldquoAn inverse method to estimatethe moving heat source inmachining processrdquoAppliedermalEngineering vol 45-46 pp 64ndash78 2012
[12] PDuda ldquoAmethod for transient thermal load estimation and itsapplication to identification of aerodynamic heating on atmo-spheric reentry capsulerdquo Aerospace Science and Technology vol51 pp 26ndash33 2016
[13] J V Beck B Blackwell and C R Clair Inverse HeatConductionIll-Posed Problems A Wiley-Interscience Publica-tion 1985
[14] T RHsu N S Sun G G Chen and Z L Gong ldquoFinite elementformulation for two-dimensional inverse heat conduction anal-ysisrdquo Journal of Heat Transfer vol 114 no 3 p 553 1992
[15] A A Tseng and F Z Zhao ldquoMultidimensional inverse tran-sient heat conduction problems by direct sensitivity coefficient
method using a finite-element schemerdquo Numerical Heat Trans-fer Part B Fundamentals vol 29 no 3 pp 365ndash380 1996
[16] L Zhaoming Further Studies on Fuzzy Inference Method forIn-verse Heat Transfer Problems Chongqing University 2014
[17] W Q Qian Y Zhou K F He J Y Yuan and J D HuangldquoEstimation of surface heat flux for nonlinear inverse heatconduction problemrdquo Acta Aerodynamica Sinica vol 30 no 2pp 145ndash150 2012
[18] J Su and G F Hewitt ldquoInverse heat conduction problemof estimating time-varying heattransfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004
[19] G Blanc J V Beck and M Raynaud ldquoSolution of the inverseheat conduction problemwith a time-variable number of futuretemperaturesrdquo Numerical Heat Transfer Part B Fundamentalsvol 32 no 4 pp 437ndash451 1997
[21] D Lesnic L Elliott and D B Ingham ldquoIdentification of thethermal conductivity and heat capacity in unsteady nonlin-ear heat conduction problems using the boundary elementmethodrdquo Journal of Computational Physics vol 126 no 2 pp410ndash420 1996
[22] A A Ershova and A I Sidikova ldquoUncertainty estimation ofthe method based on generalized residual principle for therestore task of the spectral density of crystalsrdquo Vestn Yuzhno-UralGosUn-taSerMatemMekhFiz vol 2015 pp 25ndash30 2015
[23] L Zhao Research on Fluid Temperature Inversion Algorithms forOne-Dimensional Unsteady Convection Heat Transfer HarbinInstitute of Technology 2017
[24] B Li and L Liu ldquoAn algorithm for geometry boundary identi-fication of heat conduction problem based on boundary ele-ment discretizationrdquo Proceedings of the CSEE vol 28 no 20pp 38ndash43 2008
[25] B Li and L Liu ldquoGeometry boundary identification of unsteadyheat conduction based on dual reciprocity boundary elementmethodrdquo Proceedings of the CSEE vol 29 no 5 pp 66ndash71 2009
[26] H Zhou X Xu X Li and H Chen ldquoIdentification oftemperature-dependent thermal conductivity for 2-D transientheat conduction problemsrdquo Applied Mathematics and Mechan-ics vol 29 no 1 pp 55ndash68 2014
[27] L Yanhao Model Predictive Inverse Method for Heat TransferProcess and Application Chongqing University 2017
[28] J Fan Reconstruction of Boiler Drum Temperature Field Basedon Inverse Heat Transfer Problem Chongqing University 2017
[29] H Ya-fen et alMechanism and Characteristics of Heat Conduc-tion in Nanocomposites Harbin Institute of Technology 2013
[30] Q Yue-ping M Jun J Jing-yan Y Xiao-bin and L WeildquoUnsteady heat transfer problems with finite volume methodrdquoJournal of Liaoning Technical University Natural Science vol 32no 05 pp 577ndash581 2013
[31] C-H Chiu and C-K Chen ldquoApplications of adomianrsquos decom-position procedure to the analysis of convective-radiative finsrdquoJournal of Heat Transfer vol 125 no 2 pp 312ndash316 2003
[32] C Sheng Direct and Inverse Heat Conduction Problems Solvingby the Boundary Element Method Hunan University 2007
[33] MMKhader AM Eid andAMMegahed ldquoNumerical stud-ies using FDM for viscous dissipation and thermal radiationeffects on the slip flow and heat transfer due to a stretchingsheet embedded in a porous medium with variable thickness
12 Complexity
and variable thermal conductivityrdquoNewTrends inMathematicalSciences vol 4 no 1 p 38 2016
[34] S Chuanzhuo and Z Tiande ldquoA simple method for calculatingtruncation error of numerical integral formulardquo Journal ofMathematis for Technology vol 2 1994
[35] Q Jixin Z Jun and X Zuhu Predictive Control ChemicalIndustry Press 2007
[36] L Zhang H Chen G Wang and Z Luo ldquoSolving transientinverse heat conduction problems based on optimal number offuture time stepsrdquo Proceedings of the CSEE vol 32 no 2 pp99ndash103 2012
[37] N Buong T T Huong and N T Thuy ldquoA quasi-residualprinciple in regularization for a common solution of a system ofnonlinearmonotone ill-posed equationsrdquoRussianMathematicsvol 60 no 3 pp 47ndash55 2016
[38] K A Woodbury and S K Thakur ldquoRedundant data and futuretimes in the inverse heat conduction problemrdquo Inverse Problemsin Science and Engineering vol 2 no 4 pp 319ndash333 1996
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 9
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 9 The heat transfer coefficient of the measuring point without error
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 10 The heat transfer coefficient of the measuring point with 120590 = 0001
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 11 The heat transfer coefficient of the measuring point with 120590 = 0005
0500
1000150020002500300035004000
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 12 The heat transfer coefficient of the measuring point with 120590 = 001
and analyzing the algorithm example it demonstratesthat the proposed methods have higher accuracy in theinversion process Model predictive control method focuseson the model function rather than the structural form sothat we only need to know the step response or impulseresponse of the object we can directly get the predictionmodel and skip the derivation process It absorbs the idea
of optimization control and replaces global optimization byrolling time-domain optimization combined with feedbackcorrection which avoids a lot of calculation required byglobal optimization and constantly corrects the influencecaused by uncertain factors in the system At the sametime by discussing the impacts of error free measuringpoint positions and measuring errors on the results it
10 Complexity
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
1000150020002500300035004000
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 13 The heat transfer coefficient of the measuring point at the boundary 119871 = 0008119898
0500
10001500200025003000350040004500
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 14 The heat transfer coefficient of the measuring point at the boundary 119871 = 0004119898
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
100015002000250030003500
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 15 The heat transfer coefficient of the measuring point at the distance from the boundary 119897 = 0001119898
Table 1 Relative average errors of inversion result under differentmeasurement errors given 119871 = 01119898 and 119877 = 5Measurement error120590 0001 0005 001
Relative average error120578 701 1129 1601
Table 2 Relative average errors of inversion result in differentmeasurement point positions given 120590 = 0001 and 119877 = 5Measurement point 119871 = 0004119898 119871 = 0008119898 119897 = 0001119898Relative average error120578 765 706 2105
demonstrates that the obtained inversion results except theearly oscillation can better represent the stability of the exactsolution
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interestregarding the publication of this paper
Authorsrsquo Contributions
ShoubinWang andRuiNi contributed to developing the ideasof this research All of the authors were involved in preparingthis manuscript
Acknowledgments
This work was financially supported by the National KeyFoundation for Exploring Scientific Instrument of China
Complexity 11
(2013YQ470767) Tianjin Municipal Education CommissionProject for Scientific Research Items (2017KJ059) and Tian-jin Science and Technology Commissioner Project (18JCT-PJC62200 18JCTPJC64100)
References
[1] W Shoubin Z Li S Xiaogang and J Huangchao ldquoInversionof thermal conductivity in two-dimensional unsteady-stateheat transfer system based on boundary element method anddecentralized fuzzy inferencerdquo Complexity vol 2018 Article ID8783946 9 pages 2018
[2] S Wang Y Deng and X Sun ldquoSolving of two-dimensionalunsteady inverse heat conduction problems based on boundaryelementmethod and sequential function specificationmethodrdquoComplexity vol 2018 Article ID 6741632 11 pages 2018
[3] SWang H Jia X Sun and L Zhang ldquoResearch on the recogni-tion algorithm concerning geometric boundary regarding heatconduction based on BEM and CGMrdquoMathematical Problemsin Engineering vol 2018 Article ID 3723949 13 pages 2018
[4] S Wang L Zhang X Sun et al ldquoSolution to two-dimensionalsteady inverse heat transfer problems with interior heat sourcebased on the conjugate gradient methodrdquo Mathematical Prob-lems in Engineering vol 2017 Article ID 2861342 9 pages 2017
[5] S Wang H Jia X Sun et al ldquoTwo-dimensional steady-state boundary shape inversion of CGM-SPSO algorithm ontemperature informationrdquo Advances in Materials Science andEngineering vol 2017 Article ID 2461498 12 pages 2017
[6] Z Tianyu and D Changhong ldquoCompound control systemdesign based on adaptive backstepping theoryrdquo Journal ofBeijing University of Aeronautics and A vol 39 no 7 pp 902ndash906 2013
[7] P Duda ldquoA general method for solving transient multidimen-sional inverse heat transfer problemsrdquo International Journal ofHeat and Mass Transfer vol 93 pp 665ndash673 2016
[8] B Li and L Liu ldquoAn algorithm for geometry boundary identifi-cation of heat conduction problem based on boundary elementdiscretizationrdquo Proceedings of the CSEE vol 28 no 20 pp 38ndash43 2008
[9] C-Y Yang ldquoBoundary prediction of bio-heat conduction ina two-dimensional multilayer tissuerdquo International Journal ofHeat and Mass Transfer vol 78 no 7 pp 232ndash239 2014
[10] P Duda ldquoNumerical and experimental verification of twomethods for solving an inverse heat conduction problemrdquoInternational Journal of Heat and Mass Transfer vol 84 pp1101ndash1112 2015
[11] V M Luchesi and R T Coelho ldquoAn inverse method to estimatethe moving heat source inmachining processrdquoAppliedermalEngineering vol 45-46 pp 64ndash78 2012
[12] PDuda ldquoAmethod for transient thermal load estimation and itsapplication to identification of aerodynamic heating on atmo-spheric reentry capsulerdquo Aerospace Science and Technology vol51 pp 26ndash33 2016
[13] J V Beck B Blackwell and C R Clair Inverse HeatConductionIll-Posed Problems A Wiley-Interscience Publica-tion 1985
[14] T RHsu N S Sun G G Chen and Z L Gong ldquoFinite elementformulation for two-dimensional inverse heat conduction anal-ysisrdquo Journal of Heat Transfer vol 114 no 3 p 553 1992
[15] A A Tseng and F Z Zhao ldquoMultidimensional inverse tran-sient heat conduction problems by direct sensitivity coefficient
method using a finite-element schemerdquo Numerical Heat Trans-fer Part B Fundamentals vol 29 no 3 pp 365ndash380 1996
[16] L Zhaoming Further Studies on Fuzzy Inference Method forIn-verse Heat Transfer Problems Chongqing University 2014
[17] W Q Qian Y Zhou K F He J Y Yuan and J D HuangldquoEstimation of surface heat flux for nonlinear inverse heatconduction problemrdquo Acta Aerodynamica Sinica vol 30 no 2pp 145ndash150 2012
[18] J Su and G F Hewitt ldquoInverse heat conduction problemof estimating time-varying heattransfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004
[19] G Blanc J V Beck and M Raynaud ldquoSolution of the inverseheat conduction problemwith a time-variable number of futuretemperaturesrdquo Numerical Heat Transfer Part B Fundamentalsvol 32 no 4 pp 437ndash451 1997
[21] D Lesnic L Elliott and D B Ingham ldquoIdentification of thethermal conductivity and heat capacity in unsteady nonlin-ear heat conduction problems using the boundary elementmethodrdquo Journal of Computational Physics vol 126 no 2 pp410ndash420 1996
[22] A A Ershova and A I Sidikova ldquoUncertainty estimation ofthe method based on generalized residual principle for therestore task of the spectral density of crystalsrdquo Vestn Yuzhno-UralGosUn-taSerMatemMekhFiz vol 2015 pp 25ndash30 2015
[23] L Zhao Research on Fluid Temperature Inversion Algorithms forOne-Dimensional Unsteady Convection Heat Transfer HarbinInstitute of Technology 2017
[24] B Li and L Liu ldquoAn algorithm for geometry boundary identi-fication of heat conduction problem based on boundary ele-ment discretizationrdquo Proceedings of the CSEE vol 28 no 20pp 38ndash43 2008
[25] B Li and L Liu ldquoGeometry boundary identification of unsteadyheat conduction based on dual reciprocity boundary elementmethodrdquo Proceedings of the CSEE vol 29 no 5 pp 66ndash71 2009
[26] H Zhou X Xu X Li and H Chen ldquoIdentification oftemperature-dependent thermal conductivity for 2-D transientheat conduction problemsrdquo Applied Mathematics and Mechan-ics vol 29 no 1 pp 55ndash68 2014
[27] L Yanhao Model Predictive Inverse Method for Heat TransferProcess and Application Chongqing University 2017
[28] J Fan Reconstruction of Boiler Drum Temperature Field Basedon Inverse Heat Transfer Problem Chongqing University 2017
[29] H Ya-fen et alMechanism and Characteristics of Heat Conduc-tion in Nanocomposites Harbin Institute of Technology 2013
[30] Q Yue-ping M Jun J Jing-yan Y Xiao-bin and L WeildquoUnsteady heat transfer problems with finite volume methodrdquoJournal of Liaoning Technical University Natural Science vol 32no 05 pp 577ndash581 2013
[31] C-H Chiu and C-K Chen ldquoApplications of adomianrsquos decom-position procedure to the analysis of convective-radiative finsrdquoJournal of Heat Transfer vol 125 no 2 pp 312ndash316 2003
[32] C Sheng Direct and Inverse Heat Conduction Problems Solvingby the Boundary Element Method Hunan University 2007
[33] MMKhader AM Eid andAMMegahed ldquoNumerical stud-ies using FDM for viscous dissipation and thermal radiationeffects on the slip flow and heat transfer due to a stretchingsheet embedded in a porous medium with variable thickness
12 Complexity
and variable thermal conductivityrdquoNewTrends inMathematicalSciences vol 4 no 1 p 38 2016
[34] S Chuanzhuo and Z Tiande ldquoA simple method for calculatingtruncation error of numerical integral formulardquo Journal ofMathematis for Technology vol 2 1994
[35] Q Jixin Z Jun and X Zuhu Predictive Control ChemicalIndustry Press 2007
[36] L Zhang H Chen G Wang and Z Luo ldquoSolving transientinverse heat conduction problems based on optimal number offuture time stepsrdquo Proceedings of the CSEE vol 32 no 2 pp99ndash103 2012
[37] N Buong T T Huong and N T Thuy ldquoA quasi-residualprinciple in regularization for a common solution of a system ofnonlinearmonotone ill-posed equationsrdquoRussianMathematicsvol 60 no 3 pp 47ndash55 2016
[38] K A Woodbury and S K Thakur ldquoRedundant data and futuretimes in the inverse heat conduction problemrdquo Inverse Problemsin Science and Engineering vol 2 no 4 pp 319ndash333 1996
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
10 Complexity
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
1000150020002500300035004000
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 13 The heat transfer coefficient of the measuring point at the boundary 119871 = 0008119898
0500
10001500200025003000350040004500
Ｂ(７
Ｇ2lowastＥ)
20 40 60 80 100 120 140 160 180 2000Time (s)
Exact-hInverse-h
Figure 14 The heat transfer coefficient of the measuring point at the boundary 119871 = 0004119898
0 20 40 60 80 100 120 140 160 180 200Time (s)
0500
100015002000250030003500
Exact-hInverse-h
Ｂ(７
Ｇ2lowastＥ)
Figure 15 The heat transfer coefficient of the measuring point at the distance from the boundary 119897 = 0001119898
Table 1 Relative average errors of inversion result under differentmeasurement errors given 119871 = 01119898 and 119877 = 5Measurement error120590 0001 0005 001
Relative average error120578 701 1129 1601
Table 2 Relative average errors of inversion result in differentmeasurement point positions given 120590 = 0001 and 119877 = 5Measurement point 119871 = 0004119898 119871 = 0008119898 119897 = 0001119898Relative average error120578 765 706 2105
demonstrates that the obtained inversion results except theearly oscillation can better represent the stability of the exactsolution
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that they have no conflicts of interestregarding the publication of this paper
Authorsrsquo Contributions
ShoubinWang andRuiNi contributed to developing the ideasof this research All of the authors were involved in preparingthis manuscript
Acknowledgments
This work was financially supported by the National KeyFoundation for Exploring Scientific Instrument of China
Complexity 11
(2013YQ470767) Tianjin Municipal Education CommissionProject for Scientific Research Items (2017KJ059) and Tian-jin Science and Technology Commissioner Project (18JCT-PJC62200 18JCTPJC64100)
References
[1] W Shoubin Z Li S Xiaogang and J Huangchao ldquoInversionof thermal conductivity in two-dimensional unsteady-stateheat transfer system based on boundary element method anddecentralized fuzzy inferencerdquo Complexity vol 2018 Article ID8783946 9 pages 2018
[2] S Wang Y Deng and X Sun ldquoSolving of two-dimensionalunsteady inverse heat conduction problems based on boundaryelementmethod and sequential function specificationmethodrdquoComplexity vol 2018 Article ID 6741632 11 pages 2018
[3] SWang H Jia X Sun and L Zhang ldquoResearch on the recogni-tion algorithm concerning geometric boundary regarding heatconduction based on BEM and CGMrdquoMathematical Problemsin Engineering vol 2018 Article ID 3723949 13 pages 2018
[4] S Wang L Zhang X Sun et al ldquoSolution to two-dimensionalsteady inverse heat transfer problems with interior heat sourcebased on the conjugate gradient methodrdquo Mathematical Prob-lems in Engineering vol 2017 Article ID 2861342 9 pages 2017
[5] S Wang H Jia X Sun et al ldquoTwo-dimensional steady-state boundary shape inversion of CGM-SPSO algorithm ontemperature informationrdquo Advances in Materials Science andEngineering vol 2017 Article ID 2461498 12 pages 2017
[6] Z Tianyu and D Changhong ldquoCompound control systemdesign based on adaptive backstepping theoryrdquo Journal ofBeijing University of Aeronautics and A vol 39 no 7 pp 902ndash906 2013
[7] P Duda ldquoA general method for solving transient multidimen-sional inverse heat transfer problemsrdquo International Journal ofHeat and Mass Transfer vol 93 pp 665ndash673 2016
[8] B Li and L Liu ldquoAn algorithm for geometry boundary identifi-cation of heat conduction problem based on boundary elementdiscretizationrdquo Proceedings of the CSEE vol 28 no 20 pp 38ndash43 2008
[9] C-Y Yang ldquoBoundary prediction of bio-heat conduction ina two-dimensional multilayer tissuerdquo International Journal ofHeat and Mass Transfer vol 78 no 7 pp 232ndash239 2014
[10] P Duda ldquoNumerical and experimental verification of twomethods for solving an inverse heat conduction problemrdquoInternational Journal of Heat and Mass Transfer vol 84 pp1101ndash1112 2015
[11] V M Luchesi and R T Coelho ldquoAn inverse method to estimatethe moving heat source inmachining processrdquoAppliedermalEngineering vol 45-46 pp 64ndash78 2012
[12] PDuda ldquoAmethod for transient thermal load estimation and itsapplication to identification of aerodynamic heating on atmo-spheric reentry capsulerdquo Aerospace Science and Technology vol51 pp 26ndash33 2016
[13] J V Beck B Blackwell and C R Clair Inverse HeatConductionIll-Posed Problems A Wiley-Interscience Publica-tion 1985
[14] T RHsu N S Sun G G Chen and Z L Gong ldquoFinite elementformulation for two-dimensional inverse heat conduction anal-ysisrdquo Journal of Heat Transfer vol 114 no 3 p 553 1992
[15] A A Tseng and F Z Zhao ldquoMultidimensional inverse tran-sient heat conduction problems by direct sensitivity coefficient
method using a finite-element schemerdquo Numerical Heat Trans-fer Part B Fundamentals vol 29 no 3 pp 365ndash380 1996
[16] L Zhaoming Further Studies on Fuzzy Inference Method forIn-verse Heat Transfer Problems Chongqing University 2014
[17] W Q Qian Y Zhou K F He J Y Yuan and J D HuangldquoEstimation of surface heat flux for nonlinear inverse heatconduction problemrdquo Acta Aerodynamica Sinica vol 30 no 2pp 145ndash150 2012
[18] J Su and G F Hewitt ldquoInverse heat conduction problemof estimating time-varying heattransfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004
[19] G Blanc J V Beck and M Raynaud ldquoSolution of the inverseheat conduction problemwith a time-variable number of futuretemperaturesrdquo Numerical Heat Transfer Part B Fundamentalsvol 32 no 4 pp 437ndash451 1997
[21] D Lesnic L Elliott and D B Ingham ldquoIdentification of thethermal conductivity and heat capacity in unsteady nonlin-ear heat conduction problems using the boundary elementmethodrdquo Journal of Computational Physics vol 126 no 2 pp410ndash420 1996
[22] A A Ershova and A I Sidikova ldquoUncertainty estimation ofthe method based on generalized residual principle for therestore task of the spectral density of crystalsrdquo Vestn Yuzhno-UralGosUn-taSerMatemMekhFiz vol 2015 pp 25ndash30 2015
[23] L Zhao Research on Fluid Temperature Inversion Algorithms forOne-Dimensional Unsteady Convection Heat Transfer HarbinInstitute of Technology 2017
[24] B Li and L Liu ldquoAn algorithm for geometry boundary identi-fication of heat conduction problem based on boundary ele-ment discretizationrdquo Proceedings of the CSEE vol 28 no 20pp 38ndash43 2008
[25] B Li and L Liu ldquoGeometry boundary identification of unsteadyheat conduction based on dual reciprocity boundary elementmethodrdquo Proceedings of the CSEE vol 29 no 5 pp 66ndash71 2009
[26] H Zhou X Xu X Li and H Chen ldquoIdentification oftemperature-dependent thermal conductivity for 2-D transientheat conduction problemsrdquo Applied Mathematics and Mechan-ics vol 29 no 1 pp 55ndash68 2014
[27] L Yanhao Model Predictive Inverse Method for Heat TransferProcess and Application Chongqing University 2017
[28] J Fan Reconstruction of Boiler Drum Temperature Field Basedon Inverse Heat Transfer Problem Chongqing University 2017
[29] H Ya-fen et alMechanism and Characteristics of Heat Conduc-tion in Nanocomposites Harbin Institute of Technology 2013
[30] Q Yue-ping M Jun J Jing-yan Y Xiao-bin and L WeildquoUnsteady heat transfer problems with finite volume methodrdquoJournal of Liaoning Technical University Natural Science vol 32no 05 pp 577ndash581 2013
[31] C-H Chiu and C-K Chen ldquoApplications of adomianrsquos decom-position procedure to the analysis of convective-radiative finsrdquoJournal of Heat Transfer vol 125 no 2 pp 312ndash316 2003
[32] C Sheng Direct and Inverse Heat Conduction Problems Solvingby the Boundary Element Method Hunan University 2007
[33] MMKhader AM Eid andAMMegahed ldquoNumerical stud-ies using FDM for viscous dissipation and thermal radiationeffects on the slip flow and heat transfer due to a stretchingsheet embedded in a porous medium with variable thickness
12 Complexity
and variable thermal conductivityrdquoNewTrends inMathematicalSciences vol 4 no 1 p 38 2016
[34] S Chuanzhuo and Z Tiande ldquoA simple method for calculatingtruncation error of numerical integral formulardquo Journal ofMathematis for Technology vol 2 1994
[35] Q Jixin Z Jun and X Zuhu Predictive Control ChemicalIndustry Press 2007
[36] L Zhang H Chen G Wang and Z Luo ldquoSolving transientinverse heat conduction problems based on optimal number offuture time stepsrdquo Proceedings of the CSEE vol 32 no 2 pp99ndash103 2012
[37] N Buong T T Huong and N T Thuy ldquoA quasi-residualprinciple in regularization for a common solution of a system ofnonlinearmonotone ill-posed equationsrdquoRussianMathematicsvol 60 no 3 pp 47ndash55 2016
[38] K A Woodbury and S K Thakur ldquoRedundant data and futuretimes in the inverse heat conduction problemrdquo Inverse Problemsin Science and Engineering vol 2 no 4 pp 319ndash333 1996
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Complexity 11
(2013YQ470767) Tianjin Municipal Education CommissionProject for Scientific Research Items (2017KJ059) and Tian-jin Science and Technology Commissioner Project (18JCT-PJC62200 18JCTPJC64100)
References
[1] W Shoubin Z Li S Xiaogang and J Huangchao ldquoInversionof thermal conductivity in two-dimensional unsteady-stateheat transfer system based on boundary element method anddecentralized fuzzy inferencerdquo Complexity vol 2018 Article ID8783946 9 pages 2018
[2] S Wang Y Deng and X Sun ldquoSolving of two-dimensionalunsteady inverse heat conduction problems based on boundaryelementmethod and sequential function specificationmethodrdquoComplexity vol 2018 Article ID 6741632 11 pages 2018
[3] SWang H Jia X Sun and L Zhang ldquoResearch on the recogni-tion algorithm concerning geometric boundary regarding heatconduction based on BEM and CGMrdquoMathematical Problemsin Engineering vol 2018 Article ID 3723949 13 pages 2018
[4] S Wang L Zhang X Sun et al ldquoSolution to two-dimensionalsteady inverse heat transfer problems with interior heat sourcebased on the conjugate gradient methodrdquo Mathematical Prob-lems in Engineering vol 2017 Article ID 2861342 9 pages 2017
[5] S Wang H Jia X Sun et al ldquoTwo-dimensional steady-state boundary shape inversion of CGM-SPSO algorithm ontemperature informationrdquo Advances in Materials Science andEngineering vol 2017 Article ID 2461498 12 pages 2017
[6] Z Tianyu and D Changhong ldquoCompound control systemdesign based on adaptive backstepping theoryrdquo Journal ofBeijing University of Aeronautics and A vol 39 no 7 pp 902ndash906 2013
[7] P Duda ldquoA general method for solving transient multidimen-sional inverse heat transfer problemsrdquo International Journal ofHeat and Mass Transfer vol 93 pp 665ndash673 2016
[8] B Li and L Liu ldquoAn algorithm for geometry boundary identifi-cation of heat conduction problem based on boundary elementdiscretizationrdquo Proceedings of the CSEE vol 28 no 20 pp 38ndash43 2008
[9] C-Y Yang ldquoBoundary prediction of bio-heat conduction ina two-dimensional multilayer tissuerdquo International Journal ofHeat and Mass Transfer vol 78 no 7 pp 232ndash239 2014
[10] P Duda ldquoNumerical and experimental verification of twomethods for solving an inverse heat conduction problemrdquoInternational Journal of Heat and Mass Transfer vol 84 pp1101ndash1112 2015
[11] V M Luchesi and R T Coelho ldquoAn inverse method to estimatethe moving heat source inmachining processrdquoAppliedermalEngineering vol 45-46 pp 64ndash78 2012
[12] PDuda ldquoAmethod for transient thermal load estimation and itsapplication to identification of aerodynamic heating on atmo-spheric reentry capsulerdquo Aerospace Science and Technology vol51 pp 26ndash33 2016
[13] J V Beck B Blackwell and C R Clair Inverse HeatConductionIll-Posed Problems A Wiley-Interscience Publica-tion 1985
[14] T RHsu N S Sun G G Chen and Z L Gong ldquoFinite elementformulation for two-dimensional inverse heat conduction anal-ysisrdquo Journal of Heat Transfer vol 114 no 3 p 553 1992
[15] A A Tseng and F Z Zhao ldquoMultidimensional inverse tran-sient heat conduction problems by direct sensitivity coefficient
method using a finite-element schemerdquo Numerical Heat Trans-fer Part B Fundamentals vol 29 no 3 pp 365ndash380 1996
[16] L Zhaoming Further Studies on Fuzzy Inference Method forIn-verse Heat Transfer Problems Chongqing University 2014
[17] W Q Qian Y Zhou K F He J Y Yuan and J D HuangldquoEstimation of surface heat flux for nonlinear inverse heatconduction problemrdquo Acta Aerodynamica Sinica vol 30 no 2pp 145ndash150 2012
[18] J Su and G F Hewitt ldquoInverse heat conduction problemof estimating time-varying heattransfer coefficientrdquo NumericalHeat Transfer Part A Applications vol 45 no 8 pp 777ndash7892004
[19] G Blanc J V Beck and M Raynaud ldquoSolution of the inverseheat conduction problemwith a time-variable number of futuretemperaturesrdquo Numerical Heat Transfer Part B Fundamentalsvol 32 no 4 pp 437ndash451 1997
[21] D Lesnic L Elliott and D B Ingham ldquoIdentification of thethermal conductivity and heat capacity in unsteady nonlin-ear heat conduction problems using the boundary elementmethodrdquo Journal of Computational Physics vol 126 no 2 pp410ndash420 1996
[22] A A Ershova and A I Sidikova ldquoUncertainty estimation ofthe method based on generalized residual principle for therestore task of the spectral density of crystalsrdquo Vestn Yuzhno-UralGosUn-taSerMatemMekhFiz vol 2015 pp 25ndash30 2015
[23] L Zhao Research on Fluid Temperature Inversion Algorithms forOne-Dimensional Unsteady Convection Heat Transfer HarbinInstitute of Technology 2017
[24] B Li and L Liu ldquoAn algorithm for geometry boundary identi-fication of heat conduction problem based on boundary ele-ment discretizationrdquo Proceedings of the CSEE vol 28 no 20pp 38ndash43 2008
[25] B Li and L Liu ldquoGeometry boundary identification of unsteadyheat conduction based on dual reciprocity boundary elementmethodrdquo Proceedings of the CSEE vol 29 no 5 pp 66ndash71 2009
[26] H Zhou X Xu X Li and H Chen ldquoIdentification oftemperature-dependent thermal conductivity for 2-D transientheat conduction problemsrdquo Applied Mathematics and Mechan-ics vol 29 no 1 pp 55ndash68 2014
[27] L Yanhao Model Predictive Inverse Method for Heat TransferProcess and Application Chongqing University 2017
[28] J Fan Reconstruction of Boiler Drum Temperature Field Basedon Inverse Heat Transfer Problem Chongqing University 2017
[29] H Ya-fen et alMechanism and Characteristics of Heat Conduc-tion in Nanocomposites Harbin Institute of Technology 2013
[30] Q Yue-ping M Jun J Jing-yan Y Xiao-bin and L WeildquoUnsteady heat transfer problems with finite volume methodrdquoJournal of Liaoning Technical University Natural Science vol 32no 05 pp 577ndash581 2013
[31] C-H Chiu and C-K Chen ldquoApplications of adomianrsquos decom-position procedure to the analysis of convective-radiative finsrdquoJournal of Heat Transfer vol 125 no 2 pp 312ndash316 2003
[32] C Sheng Direct and Inverse Heat Conduction Problems Solvingby the Boundary Element Method Hunan University 2007
[33] MMKhader AM Eid andAMMegahed ldquoNumerical stud-ies using FDM for viscous dissipation and thermal radiationeffects on the slip flow and heat transfer due to a stretchingsheet embedded in a porous medium with variable thickness
12 Complexity
and variable thermal conductivityrdquoNewTrends inMathematicalSciences vol 4 no 1 p 38 2016
[34] S Chuanzhuo and Z Tiande ldquoA simple method for calculatingtruncation error of numerical integral formulardquo Journal ofMathematis for Technology vol 2 1994
[35] Q Jixin Z Jun and X Zuhu Predictive Control ChemicalIndustry Press 2007
[36] L Zhang H Chen G Wang and Z Luo ldquoSolving transientinverse heat conduction problems based on optimal number offuture time stepsrdquo Proceedings of the CSEE vol 32 no 2 pp99ndash103 2012
[37] N Buong T T Huong and N T Thuy ldquoA quasi-residualprinciple in regularization for a common solution of a system ofnonlinearmonotone ill-posed equationsrdquoRussianMathematicsvol 60 no 3 pp 47ndash55 2016
[38] K A Woodbury and S K Thakur ldquoRedundant data and futuretimes in the inverse heat conduction problemrdquo Inverse Problemsin Science and Engineering vol 2 no 4 pp 319ndash333 1996
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
12 Complexity
and variable thermal conductivityrdquoNewTrends inMathematicalSciences vol 4 no 1 p 38 2016
[34] S Chuanzhuo and Z Tiande ldquoA simple method for calculatingtruncation error of numerical integral formulardquo Journal ofMathematis for Technology vol 2 1994
[35] Q Jixin Z Jun and X Zuhu Predictive Control ChemicalIndustry Press 2007
[36] L Zhang H Chen G Wang and Z Luo ldquoSolving transientinverse heat conduction problems based on optimal number offuture time stepsrdquo Proceedings of the CSEE vol 32 no 2 pp99ndash103 2012
[37] N Buong T T Huong and N T Thuy ldquoA quasi-residualprinciple in regularization for a common solution of a system ofnonlinearmonotone ill-posed equationsrdquoRussianMathematicsvol 60 no 3 pp 47ndash55 2016
[38] K A Woodbury and S K Thakur ldquoRedundant data and futuretimes in the inverse heat conduction problemrdquo Inverse Problemsin Science and Engineering vol 2 no 4 pp 319ndash333 1996
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
![From Lattice Boltzmann Method to Lattice Boltzmann Flux … · From Lattice Boltzmann Method to Lattice Boltzmann Flux Solver Yan Wang 1, ... flows [8,13–15], compressible flows](https://static.documents.pub/doc/80x56/5cadf91b88c9938f4d8c0cd6/from-lattice-boltzmann-method-to-lattice-boltzmann-flux-from-lattice-boltzmann.jpg)
