Research Article Solution to Two-Dimensional Steady Inverse Heat Transfer Problems with Interior Heat Source Based on the Conjugate Gradient Method

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1 Hindawi Matematical Problems in Engineering Volume 27, Article ID , 9 pages ttps://doiorg/55/27/ Researc Article Solution to Two-Dimensional Steady Inverse Heat Transfer Problems wit Interior Heat Source Based on te Conjugate Gradient Metod Soubin Wang, Li Zang, Xiaogang Sun, 2 and Huangcao Jia Scool of Control and Mecanical Engineering, Tianjin Cengjian University, Tianjin 3384, Cina 2 Scool of Electrical Engineering and Automation, Harbin Institute of Tecnology, Harbin 5, Cina Correspondence sould be addressed to Soubin Wang; wsbin8@26com Received 5 February 27; Revised 4 May 27; Accepted 28 May 27; Publised July 27 Academic Editor: Francisco Alama Copyrigt 27 Soubin Wang et al Tis is an open access article distributed under te Creative Commons Attribution License, wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited Te compound variable inverse problem wic comprises boundary temperature distribution and surface convective eat conduction coefficient of two-dimensional steady eat transfer system wit inner eat source is studied in tis paper applying te conjugate gradient metod Te introduction of complex variable to solve te gradient matrix of te objective function obtains more precise inversion results Tis paper applies boundary element metod to solve te temperature calculation of discrete points in forward problems Te factors of measuring error and te number of measuring points zero error wic impact te measurement result are discussed and compared wit L-MM metod in inverse problems Instance calculation and analysis prove tat te metod applied in tis paper still as good effectiveness and accuracy even if measurement error exists and te boundary measurement points number is reduced Te comparison indicates tat te influence of error on te inversion solution can be minimized effectively using tis metod Introduction Inverse eat transfer problem (Inverse Heat Transfer Problems, IHTP) means inversion unknown caracteristic parameters of eat conduction objects using te internal or surface local eat measurement [, 2], suc as termal pysical parameters, termal boundary conditions, geometric boundary sape, and eat conduction coefficient Inverse eat conduction problem as a broad application prospect in nondestructive testing, geometry optimization, aerospace engineering, power engineering, mecanical engineering, construction engineering, biological engineering, metallurgical engineering, materials processing, biomedical and food engineering, and oter fields [ 2] Focusing on tis problem, domestic and foreign scolars ave done a lot of researc Zu et al studied te disadvantages of optimization algoritms and inerent space distribution caracteristics of te measurement in inverse eat conduction problems Tey proposed a decentralized fuzzy reasoning mecanism and establised a decentralized fuzzy inference system of two-dimensional steady inverse eat conduction problems [3, 4] Cui et al gave an improved conjugate gradient metod Tey introduced te complex variable derivation into te traditional conjugate gradient metod, calculated te sensitivity coefficients accurately, and identified te boundary conditions [5] Yu combined te boundary element metod wit complex variables derivation to inverse te inomogeneous material coefficients of termal conductivity [6] Wang et al applied particle swarm optimization algoritm and te least square metod for solving te inverse problems Tat improved te precision and sorted te solving time attesametime[3]yaparovastudiedteeatconduction boundary value inverse problem by solving te stability boundary based on Laplace and Fourier transformation [4] Tian combined SPSO algoritm wit te conjugate gradient metod and te fast convergence of te traditional regularized gradient algoritm and global convergence of te intelligent optimization algoritm were igligted [5] Using te boundary element metod, Zou et al analyzed two-dimensional transient conduction problems, introduced

2 2 Matematical Problems in Engineering te conjugate gradient metod to find te eat conduction coefficient, and verified te validity and stability of te metod [6] A stable differential metod was proposed to solve te inverse eat conduction problems by Baranov et al on te basis of te differential transform [7] In tis paper, te two-dimensional steady forward problems wit internaleatsourcearesolvedusingteboundaryelement metod Te existing surface convective termal dissipation is analyzed, and te conjugate gradient metod is applied to solve te inverse problems And te complex variable derivationmetodisintroducedtoincreasetesensitivity coefficient and te accuracy in te inversion 2 Termal Steady State Forward Problems 2 Te Boundary Integral Equation Te matematical model of two-dimensional steady eat conduction problems wit internal eat source is 2 T x T y 2 =f, T=T, () T n = q 2, T n + λ T=q, were, 2,and areteboundaryoftedomain, = + 2 +, f = s/λ, s is te internal eat source, is te eat conduction coefficient between solid and fluid, λ=htc, T= temp, and n is te normal vector outside te boundary q = (/λ)t f, T f is environment temperature, and q is eat flux Weigt function T is introduced into weigted residual of control equation ( 2 T f)t d = ( T n + λ T q )T d 2 ( T n q) T d (T T) T n d Green s function of Laplace equation is D (V 2 u u 2 V)d = c (V( u/ n) u( V/ n))d s,werec is te boundary curve of plane closed area D and d s is arc differential [8] To decompose te left side of te equation, it becomes ( 2 T f)t d = ( 2 T) T d ft d (3) According to Laplace Green s function, we can ave ( 2 T) T d = (T 2 T T 2 T )d T 2 T d (2) (4) Furter reduction gives ( 2 T) T d = (T T n T T n )d T 2 T d Substituting (5) for te same term in (3), (2) becomes (T T n T T n )d + T 2 T d ft d = ( T n + λ T q )T d 2 ( T n q) T d (T T) T n d Furter reduction gives T 2 T d = ( T n + λ T q )T d 2 ( T n q) T d (T T) T n d (T T n T T n )d ft d T is te basic solution of Laplace equation and it can make T +δ(r r i )=,weretepoint i refers to te unit point concentration Because T / n = q and T/ n = q, (7) can be furter reduced Te left side of (7) becomes T 2 T d = T( 2 T +δ(r r i )) Tδ (r r i )d Calculating furtermore, it is sown as follows: T( 2 T +δ(r r i )) Tδ (r r i )d = Tδ (r r i )d For te selectivity of δ function, + f(x)δ(x x )d r = f(x ) Ten te reduced result on te left side of (7) is (5) (6) (7) (8) (9) Tδ (r r i ) d = T i ()

3 Matematical Problems in Engineering 3 Substituting () into (7), T i = (q + λ T q )T d 2 (q q) T d (T T) q d (T q Tq )d ft d () Because = + 2 +, (T q Tq )d is decomposed: (T q Tq )d = (T q Tq )d 2 (T q Tq )d (T q Tq )d Substituting (2) for te same term in (), T i = ( λ TT q T +Tq )d 2 (Tq qt )d (Tq T q) d ft d Furter reduction gives T i = λ TT d +( Tq d 2 Tq d Tq d ) ( q T d 2 qt d qt d ) ft d (2) (3) (4) Te second and te tird terms are combined, wic makes T i q Td λ TT d = T qd ft d (5) Te two-dimensional steady T = (/2π) ln(/r), r = (x x i ) 2 (y y i ) 2 Movingtesourcepointi to te boundary, te integral equation at any point on te boundary can be obtained Tus, te basic solution T and te singularity of te integral T / n need to be considered Assuming tat a circular arc centering on te point i near te border and ε is te small radius of te circle, (5) can be written as T i q Td q Td ε ε λ TT d = T qd T qd ft d, ε ε (6) were ε is te new circular border and εis te boundary outside te new circular boundary in Considering te smoot boundary, te integral mean value teorem is introduced Tq d = T ε ε n ( 4πε 2 )d = T ε 4πε 2 d =T(ξ) 4πε 2 2πε2, were ξ is te point on εletε T(ξ) T i ;ten lim ( Tq d )= ε ε 2 T i, lim ( q Td )= q Td, ε ε lim ( qt d )= q Td ε ε Syntesizing te above limit results, we ave 2 T i q Td λ TT d = T qd ft d (7) (8) (9) Considering (5) and (9) simultaneously, boundary integral equation of any point in te domain or on te boundary can be found out: C i T i q Td λ TT d = T qd ft d, {, C i = { { 2 (2) 22 Calculation of te Unknown Value Based on Boundary Element Metod Discreteness of boundary integral equations: because te quantities T and q on te boundary are known, te only ting to do is divide te boundary discretely Boundary is divided into boundary units j (j =,,), were tere are units on te boundary, 2 units on te boundary 2,and 3 units on te boundary =

4 4 Matematical Problems in Engineering Te unknown points in te boundary units are called nodes For te requirements of different difference, te expressions of T and q tobefoundoutcanbewrittenas T= q= φ i (ξ) T i, φ i (ξ) q i, (2) were is te interpolation node number in a boundary unit and φ is te interpolation function Wen j is in or 2,(9) canbewrittenas T i + = q Td j + 2 T qd ft d j (22) Tis paper uses linear interpolation Because φ (ξ) = ( ξ)/ 2, φ 2 (ξ) = ( + ξ)/2 is te interpolation function of linear unit, te straigt line is used to approximate te boundary curve, and te two linearized terminal values are used to approximate te values of T and q So te node values can be converted into te following: T (ξ) =φ (ξ) T j +φ 2 (ξ) T j+ =[φ (ξ),φ 2 (ξ)]( T j T j+ ), q (ξ) =φ (ξ) q j +φ 2 (ξ) q j+ =[φ (ξ),φ 2 (ξ)]( q j q j+ ) Substituting (23a) and (23b) into (22), were T i + = ( ij () T j + ij (2) T j+ ) + 2 (g () ij q j +g (2) ij q j+ ) ft d, (e) ij = φ e q d, j (23a) (23b) (24) and we ave T i + H ij T j = + 2 G ij q j ft d (27) Similarly, wen te point locates in,(24)becomes Let 3 2 T i + [( () ij + λ gij() )T j +( ij (2) + λ g ij(2) )T j+ ]= +g (2) ij q j+ ) ft d 3 (g ij () q j H ij =( ij (2) + λ g ij (2) )+( ij () + λ g ij() ), G ij =g ij (2) +g ij (), and te boundary integral equation is were 3 2 T i + H ij T j = 3 Rearranging (3), we ave H ij T j = G ij q j G ij q j (28) (29) ft d (3) ft d, (3) { H ij, (i =j), H ij = { H ij + { 2, (i = j) (32) considering B i = ft d, (3) can be expressed as a matrix form HT = GQ B i Te boundary conditions T and Q are written as T = [ T ], Q = [ Q ],weret T 2 Q 2, Q 2 are known temperature and eat flux Equation (3) becomes [H H 2 ][ T T 2 ] = [G G 2 ][ Q Q 2 ] B i (33) Putteknownquantitiesonteleft Let g (e) ij = φ e T d, j e=(, 2) H ij = ij (2) + ij (), G ij =g ij (2) +g ij (), (25) (26) [H 2 G ][ T 2 Q ]=[G 2 H ][ Q 2 T ] B i (34) Appling te radial integration metod, domain integration caused by eat source is converted to boundary integration [9 2] B i = ft d = F r R n d, (35)

5 Matematical Problems in Engineering 5 were R is te distance between te source and te field point and F is radial integration wic is expressed as R F= ft rd r (36) Because f is a known function and te function form is simple, T is te basic function, te radial integration can be found out troug (36), and te boundary integration can be obtained from (35) Tus, (34) can be written as AX = K to solve te above equations and to find T and Q 3 Te Steady Inverse Problem 3 Te Objective Function of Inverse Problem For aforementioned eat conduction system, te eat transfer coefficient between solid and fluid and te termal conductivity λ areknownteboundarytemperaturetobefoundoutis unknown wic can be determined based on te known measured boundary temperature and te known condition of forward problem Te following objective function is defined as J(q)= [T i (q) T mea i ] 2 (37) In te objective function, q is a parameter vector wose temperatureneedstobeinversedand is te number of temperature measuring points on te boundary q={r,r 2,,r m }={T,T 2,,T m } (38) T i (q) is te calculation of measuring point i in te forward problem and T mea is te measurement of temperature Te minimum of te objective function J(q) is te parameter vector q of inverse problem 32 Conjugate Gradient Metod of Inverse Problem Conjugate gradient metod is a metod wic combines te conjugacy and te steepest descent metod It derives from perturbation principle and te inverse problem is converted into tree questions, suc as forward problem, sensitivity problem, and adjoint problem In order to solve te effects of tese tree questions, tis paper introduces te complex variable derivation metod into te traditional conjugate gradient metod, wic makes te calculation of te sensitivity coefficient accurate During te calculation, unconstrained optimization algoritm is acieved by iteration Considering tat te kt iteration point q k as been available, te (k + )t iteration calculates according to te following formula Te iterative equation solving te inverse problem by conjugate gradient metod includes q k+ =q k +α k d k, q k+ =q k α k d k, d k = { g k, { g { k +β k d k, k=, k, (39) were α k is step lengt, obtained by searcing some onedimensional line; d k is te searcing direction in wic g k = J(q k ) and β k is a scalar Different β k corresponds to different nonlinear conjugate gradient metods Te searcing step lengt α k, te conjugate coefficient β k, and gradient J(q k ) are needed to be found out Conjugate coefficient β k equals te ratio of te square of normal form between te current and te previous-stepgradient paradigm: β k = [ J (q k) J(q k )] J T (q k ), (4) J (q k ) J T (q k ) J (q k )=( J r, J r 2,, J (q k ) =2(T r i q k T i ) T i (q k ) j r j J r m+ ), (4),2,,m+ (42) If te iteration step is k+and searcing step lengt is α k, (37) J[q k+ ]= [T i(q k +α k d k ) T ] 2 gives α k = [T i (q k ) T i ] T id k [ T id k ] 2, (43) were T i =( T i / r, T i / r2,, T i / rm+ ) 33 Complex Variable Derivation Metod [22 24] For any real function f(x), a very small imaginary part is added to te real variable x It is expressed in complex function f(x + i) and its Taylor series is f (x+i) =f(x) +if (x) 2 2 f (x) +o( 3 ) (44) As is very small, df dx Im (f (x+i)) =, d 2 f 2[f(x) Re (f (x+i))] = dx2 2, (45) were T i (q k )/ r j is sensitivity coefficient and its matrix form is T (r) T (r) T (r) r r 2 r m [ ] T r = T 2 (r) r [ T (r) [ r T 2 (r) r 2 T (r) r 2 T 2 (r) r m (46) T (r)] r m ] Te complex variable derivation metod is used to calculate te matrix of sensitivity coefficient

6 6 Matematical Problems in Engineering Im (T (r +)) T Im (T 2 (r +)) r = [ Im (T (r +)) [ Im (T (r 2 + )) Im (T 2 (r 2 + )) Im (T (r 2 +)) Im (T (r m +)) Im (T 2 (r m +)) Im (T (r m +)) (47) ] ] Te complex variable derivation metod is introduced into te traditional conjugate gradient metod, wic makes te calculation of te sensitivity coefficient accurate and avoids te sensitivity problem and te adjoint problem 34 Solving Process of te Inverse Problem Initialization: k=, q,r =,andletε be a small positive number Solveteforwardproblemgivenby(),calculatetemperature T i, and judge if te following condition is true: J(q)= [T i (q) T mea i ] 2 <ε (48) Te equation means te condition required by te instruction DO WHILE as not been satisfied Calculating te gradient J using (4) Calculating te searcing-down direction d k using (27) Calculating te searcing step lengt α k using (43) Calculating te new estimation using (34) Solving te forward problem given by () and obtaining T i : End do k=k+ (49) 4 Te Instance Calculation and Analysis To testify te availability of te aforementioned metod, te simulation experiment aving two sets of data was designed Te experiment discussed te effect of te testing points number and te measuring error on te inversion results and compared CGM and L-MM Te scematic diagram is sown as Figure In te experiment, te concrete slab s tickness is δ = 3 m, and te tickness of casting concrete slab between two wooden templates is δ = δ 2 = 2 m, and coefficient of termal conductivity is λ = λ 2 = 6 w/(mk), te air temperature is T f = 293 K, and DandD3areteconvective eat conduction boundary conditions Te eat transfer coefficients are i = 8W/(m 2 K) and e = 2W/(m 2 K) Te termal conductivity coefficient is λ = 5 W/(mK) and te power density of inner eat source is s = W/m 3 (similar to even-distributed and constant in sort time) Te real temperature distribution of te boundary D4 to be solved is y = x x2 (5) 3 According to te real temperatures, te calculated temperatures T i of testing points on te known boundary D2are countedtrougteforwardquestion T mea i ε= =T i +ωσ, σ 2, (5) were ω is standard-normal-distributed random number and σ is te standard deviation of measuring temperatures 4 Te Influence of Measuring Points umber on Inversion Wen te measurement error is zero, T f = 293 K, i = 8 W/(m 2 K), and e = 2W/(m 2 K) Te termal conductivity coefficient is λ = 5 W/(mK) and te power density of inner eat source is s = W/m 3 Teinfluenceof measuring points number on inversion is sown as Figure 2 Wen te measuring point =5,9,or2,teinversion results reveal tat te more points are measured, te more accuracy of inversion can be acieved 42TeInfluenceofteMeasuringErroronInversion = 2, T f = 293 K, i =8W/(m 2 K), and e =2W/(m 2 K) Te termal conductivity coefficient is λ = 5 W/(mK), and te power density of inner eat source is s = W/m 3 Te influence of different measuring error on inversion is sownasfigure3astemeasuringstandarddeviation σ =, 2, 4, te inversion results reveal tat te metod can obtain better inversion results for a relatively small measurement standard deviation 43 Te Comparison between CGM and L-MM Tis paper applies CGM and L-MM to calculate te inversion and compare wit eac oter under tree conditions: σ =, =9,andσ=, =2,and=2, σ = 2 (oter conditions are te same as te aforementioned) Te inversions under te conditions σ=, =9, and σ=, =2are sown as Figures 4 and 5 Figures 4 and 5

7 Matematical Problems in Engineering 7 D D D3 y (K) D4 296 Figure : Model of te instance x (m) Exact CGM L-MM Figure 4: Te inversion under te condition: σ=and =9 y (K) =2 =9 =5 x (m) Figure 2: Te influence of measuring points number on inversion y (K) x (m) Exact CGM L-MM Figure 5: Te inversion under te condition: σ=and =2 38 y (K) σ= σ = 2 σ = 4 x (m) Figure 3: Te influence of different measuring error on inversion sow tat te inversions using CGM and L-MM are similar and bot ave ig accuracy Bot of tem are satisfied Te inversions under te conditions σ = 2, =9are sown as Figure 6 Comparing wit Figure 5, te inversions using L-MM cause obvious fluctuation wen te measuring error increases, wile te inversions using CGM are satisfied Tat means CGM as te better stability 5 Conclusion Tis paper applies CGM based on complex variable derivation to study te multivariable inverse problem wic combines boundary distribution wit convection coefficient of two-dimensional steady system wit inner eat source Te inversions applying CGM and L-MM are compared Te

8 8 Matematical Problems in Engineering y (K) Exact CGM L-MM x (m) Figure 6: Te inversion under te condition: σ = 2 and =2 influence of measuring points number and measuring error on inversion is tested Te simulation reveals tat applying boundary element metod and conjugate gradient metod to solve te inverse eat conduction problem is successful and as good stability Conflicts of Interest Te autors declare tat tey ave no conflicts of interest Acknowledgments Tis work was financially supported by te ational Key Foundation for Exploring Scientific Instrument of Cina (23YQ47767), Tianjin Science and Tecnology Committee for Science and Tecnology Development Strategy Researc Project (5ZLZLZF35), Tianjin Science and Tecnology Commissioner Project (6JCTPJC53), Science and Tecnology Plan Project of Tianjin Science and Tecnology Committee (6ZLZXZF27), and te 3t Five-Year Plan (26 22) of Science Education Project in Tianjin City (HE7) References [] C Seng, Direct And Inverse Heat Conduction Problems Solving by Te Boundary Element Metod,HunanUniversity,27 [2]KAWoodbury,JVBeck,andHajafi, Filtersolutionof inverse eat conduction problem using measured temperature istory as remote boundary condition, International Heat and Mass Transfer,vol72,pp39 47,24 [3] L Zu, G Wang, and H Cen, Estimating steady multivariables inverse eat conduction problem by using conjugate gradient metod, Proceedings of te Cinese Society of Electrical Engineering, vol 3, no 8, pp 58 6, 2 [4] L Zu, Fuzzy Inverse for Two-Dimensional Steady Heat Conduction System And Application, Congqing University [5] M Cui, W-W Duan, and X-W Gao, Conjugate Gradient Metod Based on Complex-variable-differentiation Metod and Its Application for Identification of Boundary Conditions in Inverse Heat Conduction Problem, CIESC Journal, vol 9, supplement, pp 6, 25 [6] X Yu, Iverse Analysis of Termal Conductivities in on- Homogeneous Heat Conductions Using Boundary Element Metod, Dalian University of Tecnology, 23 [7] Z Qian, Optimal modified metod for a fractional-diffusion inverse eat conduction problem, Inverse Problems in Science and Engineering,vol8,no4,pp52 533,2 [8] B T Joansson, D Lesnic, and T Reeve, A metod of fundamental solutions for te radially symmetric inverse eat conduction problem, International Communications in Heat and Mass Transfer,vol39,no7,pp ,22 [9] S Tapaswini, S Cakraverty, and D Beera, umerical solution of te imprecisely defined inverse eat conduction problem, Cinese Pysics B, vol 24, no 5, ArticleID523, 25 [] P Duda, umerical and experimental verification of two metods for solving an inverse eat conduction problem, International Heat and Mass Transfer, vol84,pp 2, 25 [] Y B Wang, J Ceng, J akagawa, and M Yamamoto, A numerical metod for solving te inverse eat conduction problem witout initial value, Inverse Problems in Science and Engineering,vol8,no5,pp655 67,2 [2] Q Xue and w Wei, Parameters identification of non-linear inverse eat conduction problem, Engineering Mecanics, vol 27, no 8, pp 5 9, 2 [3] L Wang, L Mei, and J Huang, Inverse eat conduction problem based on least squares prediction, CIESC Journal, vol 67, supplement, pp 3, 26 [4] Yaparova, umerical metods for solving a boundary-value inverse eat conduction problem, Inverse Problems in Science and Engineering,vol22,no5,pp ,24 [5] Tian, umerical Metods for Te PDE-Based Inverse Problems and Applications, Jiangnan University, 22 [6] H Zou, X Xu, X Li, and H Cen, Identification of temperature-dependent termal conductivity for 2-d transient eat conduction problems, Applied Matematics and Mecanics, vol2,no35,pp34 35,24 [7] V L Baranov, A A Zasyad ko, and G A Frolov, Integrodifferential metod of solving te inverse coefficient eat conduction problem, Engineering Pysics and Termopysics,vol83,no,pp6 7,2 [8] H Wu, Heat Conduction Problem Solving By Boundary Element Metod, ational Defence Industry Press, Beijing, Cina, 28 [9] L J M Jesus, C A Cimini, and E L Albuquerque, Application of te radial integration metod into dynamic formulation of anisotropic sallow sells using boundary element metod, Key Engineering Materials,vol627,pp ,25 [2] X W Gao, Te radial integration metod for evaluation of domain integrals wit boundary-only discretization, Engineering Analysis wit Boundary Elements, vol 26, no, pp 95 96, 22 [2] SQu,SLi,H-RCen,andZQu, Radialintegrationboundary element metod for acoustic eigenvalue problems, Engineering Analysis wit Boundary Elements, vol37,supplement 6-7, pp 43 5, 23

9 Matematical Problems in Engineering 9 [22] M-W Liu, Y-R Zeng, and Y-F Zang, A new inversion metod of rock-soils parameters based on complex-variabledifferentiation metod, Cinese Computational Mecanics, vol 26, no 5, pp , 29 [23]JLynessandCBMoler, umericaldifferentiationof analytic functions, SIAM Journal on umerical Analysis, vol 4, pp 22 2, 967 [24] S Kim, J Ryu, and M Co, umerically generated tangent stiffness matrices using te complex variable derivative metod for nonlinear structural analysis, Computer Metods in Applied Mecanics and Engineering,vol2,no-4,pp43 43,2

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