Estimation of heat transfer parameters by using trained pod-rbf and grey wolf optimizer

Nội dung text: Estimation of heat transfer parameters by using trained pod-rbf and grey wolf optimizer

1. Vietnam Journal of Mechanics, VAST, Vol.42, No. 4 (2020), pp. 401 – 414 DOI: ESTIMATION OF HEAT TRANSFER PARAMETERS BY USING TRAINED POD-RBF AND GREY WOLF OPTIMIZER Minh Ngoc Nguyen1,2,∗, Nha Thanh Nguyen1,2, Thien Tich Truong1,2 1Ho Chi Minh City University of Technology, Vietnam 2Vietnam National University Ho Chi Minh City, Vietnam ∗E-mail: nguyenngocminh@hcmut.edu.vn Received: 27 April 2020 / Published online: 16 December 2020 Abstract. The article presents a numerical model for estimation of heat transfer parame- ters, e.g. thermal conductivity and convective coefficient, in two-dimensional solid bodies under steady-state conduction. This inverse problem is stated as an optimization prob- lem, in which input is reference temperature data and the output is the design variables, i.e. the thermal properties to be identified. The search for optimum design variables is conducted by using a recent heuristic method, namely Grey Wolf Optimizer. During the heuristic search, direct heat conduction problem has to be solved several times. The set of heat transfer parameters that lead to smallest error rate between computed temperature field and reference one is the optimum output of the inverse problem. In order to acceler- ate the process, the model order reduction technique Proper-Orthogonal-Decomposition (POD) is used. The idea is to express the direct solution (temperature field) as a linear combination of orthogonal basis vectors. Practically, a majority of the basis vectors can be truncated, without losing much accuracy. The amplitude of this reduced-order approxi- mation is then further interpolated by Radial Basis Functions (RBF). The whole scheme, named as trained POD-RBF, is then used as a surrogate model to retrieve the heat transfer parameters. Keywords: inverse analysis, Grey Wolf Optimizer, heat transfer parameters identification, Proper Orthogonal Decomposition (POD), Radial Basis Function (RBF). 1. INTRODUCTION In direct heat transfer analysis, distribution of temperature within a conducting do- main is determined given known boundary conditions and thermal properties. In con- trast, based on the knowledge of temperature history within a conducting body, inverse heat transfer analysis is used to determine the thermal properties and/or boundary con- ditions. The estimated quantities of inverse heat transfer analysis are very sensitive to the inaccuracy of input data. Mathematically, the problem is ill-posed [1]. Unfortunately, © 2020 Vietnam Academy of Science and Technology
2. 402 Minh Ngoc Nguyen, Nha Thanh Nguyen, Thien Tich Truong noise in measurement of temperature is not avoidable. Therefore, development of com- putational schemes which can overcome the issue of ill-posedness has attracted much attention from researchers. Inverse analysis has been widely used in heat transfer to identify heat flux [2–4], boundary conditions [5–7] and unknown thermal properties such as conductivity and convective coefficient [8–11]. Basically, the problem is described as minimization of the error rate between computed temperature and measured data. The design variables are the unknown quantities to be determined. For solution of optimization problem, ei- ther gradient-based or non-gradient-based methods can be used. The gradient-based approaches [3,4,8] usually involve sensitivity analysis, i.e. the computation of derivative of objective function with respect to the sought variables. However, derivation of objec- tive function as an explicit function of design variables is usually not a trivial task. An- other drawback is that the gradient-based approach may fall into local optimum. On the other hand, the non-gradient-based methods do not require sensitivity analysis. Instead, various heuristic algorithms are used such as Genetic Algorithm [11], Particle Swarm Optimization [2], Differential Evolution [12], Firefly Algorithm [7], Cuckoo Search [13] and so on. Although each algorithm has a different strategy, they commonly employ a group of M agents which search N rounds in the admissible solution space to find the optimum one, i.e. the unknown quantities to be estimated. Indeed, it is common knowledge that there exists no algorithm which is superior to the others in all types of problems. Nevertheless, the attractiveness of GWO algorithm comes from the fact that it has small number of user-defined parameters to control the balance of exploitation (local search) and exploration (global search). In this work, the recently proposed Grey Wolf Optimizer (GWO) [14] is used to solve the optimization problem to identify the thermal parameters, e.g. heat conductivity and convective coefficient. The algorithm has been widely applied in many fields such as machine learning [15,16], electric engineering [17], earthquake engineering [18], image processing [19], path planning [20]. However, to the best knowledge of the authors, GWO has not been investigated in inverse heat transfer analysis. During the search for optimum solution, the direct heat transfer problems have to be solved many times to evaluate temperature field. The difference between the com- puted temperature and reference one, i.e. the objective function, is then determined. The process is time-consuming and needs to be accelerated. The model order reduction tech- nique Proper Orthogonal Decomposition (POD) has been successfully employed in direct heat transfer problems [21–24]. The core idea is to find a set of orthogonal vectors (POD bases) using singular value decomposition, which is then utilized to approximate the temperature field. Temperature is expressed as a linear combination of POD basis and associated amplitudes. Usually, this linear combination can be truncated, thus the prob- lem size is reduced, while high accuracy is still attained. Ostrowski et al. [9, 25] pointed out that POD also acts as a filter to lessen the influence of noise in measured temper- ature data, improving the stability of inverse heat transfer analysis. Consequently, the benefit of the employment of POD in inverse heat transfer problems is two-fold: acceler- ation of computational process and regularization method to treat the ill-posedness. The amplitude vectors in POD approximation is then further interpolated using Radial Basis
3. Estimation of heat transfer parameters by using trained POD-RBF and Grey Wolf Optimizer 403 Functions (RBF), which are defined as functions of thermal parameters, resulting in the trained POD-RBF surrogate model [9, 26–28]. In this paper, the trained POD-RBF is coupled with GWO to develop a numerical model to identify thermal conductivity and convective coefficient, in two-dimensional solid bodies under steady-state conduction. The paper is organized as follows. Immediately after the Introduction, a brief review of GWO is presented in Section 2. Section 3 is reserved for trained POD-RBF in identifica- tion of thermal properties. In Section 4, a numerical example is presented and discussed in details, demonstrating the numerical scheme. Finally, conclusions and remarks are given in the last Section. Nomenclature (units are given in square bracket) Symbol Definition Symbol Definition α The alpha wolf (i.e. the δ The delta wolf (i.e. the search search agent that has the agent that has the third best best fitness in the whole fitness in the whole search) search) β The beta wolf (i.e. the search agent that has the second best fitness in the whole search) T [K] Temperature h [W/(m2K)] Convective heat transfer co- efficient q [W/m2] Heat flux k [W/(mK)] Thermal conductivity Ta [K] Ambient temperature Tsnap Snapshot matrix p Vector of thermal properties (i.e. h and k in the current work) Φ Orthogonal basis vectors 2. GREY WOLF OPTIMIZER (GWO) GWO is a bio-inspired optimization technique recently proposed by Mirjalili et al. [14]. In an attempt to mimic the social hierarchy of grey wolf, the fitness of wolves after each iteration is sorted in ascending order (in the context of minimization problem, the wolf with lowest value of objective function is the fittest). The three fittest solutions are named the alpha (α), the beta (β), and the delta (δ), respectively. The rest of the population is called omegas. With the hypothesis that the leadership hierarchy of grey wolf also applies in hunting process, the algorithm updates the position of an ordinary grey wolf
4. 404 Minh Ngoc Nguyen, Nha Thanh Nguyen, Thien Tich Truong (i.e the omegas) at the current iteration t + 1, ~X (t + 1), by the last known positions of the best candidates, i.e. the alpha, beta and delta wolf 1   ~X (t + 1) = ~X + ~X + ~X , (1) 3 1 2 3 ~ ~ ~ where X1, X2, X3 are some points surrounding the positions of three dominant wolves ~ ~ ~ (denoted by Xα, Xβ, Xδ) ~ ~ ~ ~  ~ ~  X1 = Xα − aα · Dα, Dα = cα · Xα − X (t) , (2) ~ ~ ~ ~  ~ ~  X2 = Xβ − aβ · Dβ, Dβ = cβ · Xβ − X (t) , (3) ~ ~ ~ ~  ~ ~  X3 = Xδ − aδ · Dδ, Dδ = cδ · Xδ − X (t) , (4) The numbers ai and ci (i = α, β, δ) are calculated by ai = 2s · r1 − s, (5) ci = 2r2, (6) where r1 and r2 are random real values ranging from 0 to 1. Parameter s gradually de- creases from some pre-defined value smax (in [14], smax is set to 2) to zero with respect to the number of iterations  t  s = smax 1 − , (7) tmax with tmax being the pre-set maximum number of iterations. The value of controlling parameter s has influence on ai in Eq. (5), which is key for a wolf to decide whether it approaches or run away from the three leading wolves (the alpha, beta and delta). Particularly, if |ai| 1, the wolf runs away to explore the space far from the leaders, with a hope to discover a more attractive prey. This option allows exploration, i.e. the global search, in order to avoid being trapped in local optimum. Gao and Zhao [29] argue that the equal weights in Eq. (1) do not reflect the rank of the three dominant wolves. The individual roles of the alpha, beta and delta are the same, despite the fact that alpha is closest to the prey (in the context of optimization problem). Instead, more weights should be assigned to the alpha in order to enhance local search. Furthermore, the weights should also follow a descending order: ω1 ≥ ω2 ≥ ω3 ≥ 0. Based on the above reasoning, they propose the following calculation of the weights 1 ω = cos θ, ω = sin θ · cos φ, ω = 1 − ω − ω , (8) 1 2 2 3 1 2 where 2 1 1 θ = · arccos · arctan t and φ = · arctan t. (9) π 3 2 The second argument of Gao and Zhao [29] is that in the beginning of the search, the wolves should be encouraged to go for a global search, while in long term, local
5. Estimation of heat transfer parameters by using trained POD-RBF and Grey Wolf Optimizer 405 search should be more emphasized. Therefore the controlling parameter s is suggested to decline exponentially, instead of linearly as in Eq. (7)  10t  s = smax · exp − . (10) tmax 3. TRAINED POD-RBF FOR IDENTIFICATION OF THERMAL PROPERTIES 3.1. Governing equations of direct heat transfer problems in two-dimensional do- mains Let us consider a two-dimensional solid body Ω being bounded by Γ. When there is no heat sink/source, the governing equation of steady-state heat transfer in the body Ω is written by ∇ · (k∇T) = 0, (11) where T is the temperature and k is the thermal conductivity. Without consideration of heat radiation, the boundary conditions are given as follows T = T¯, on Γ1: Dirichlet boundary, (12) (k∇T) · n = q¯, on Γ2: Neumann boundary, (13) (k∇T) · n = h (Ta − T) , on Γ3: convection boundary. (14) In Eqs. (12)–(14), T¯ is the prescribed temperature; q¯ is the prescribed heat flux; n is the outward normal unit vector of the boundary; Ta is the ambient temperature and h is the convective heat transfer coefficient. After some mathematical manipulation, the partial differential equation (11) is trans- formed into weak formulation as follows Z Z Z (δ∇T) k∇TdΩ − q¯δTdΓ − h (Ta − T) δTdΓ = 0. (15) Ω Γ Γ 3.2. Training data and reference data Given the same domain geometry and boundary conditions, the training data are temperature values obtained from solution of direct problem, corresponding to known thermal properties, i.e. thermal conductivity k and convective coefficient h. One set of (k, h) is connected to one set of training temperature data. In fact, the training data can be obtained by measurement, given that the number of experiments and the number of sampling points are large enough. Another option is that a finite element model can be developed for generation of training data. Reference data are temperature values collected at some certain points in problem domain (usually on the boundaries). Thermal properties that lead to reference data are not known a priori and have to be identified by inverse analysis. In this paper, the ref- erence data are also taken from finite element solution of the direct steady-state heat transfer. At each point, a noise of 5% is added to finite element solution to mimic that of measurement.
6. 406 Minh Ngoc Nguyen, Nha Thanh Nguyen, Thien Tich Truong 3.3. Model order reduction by Proper Orthogonal Decomposition The training data can be arranged as an m-by-n matrix Tsnap, in which n is the number of data sets (one data set corresponds to one set of parameters (k, h)), and m is the number of points where the data are collected. In this work, nodal values of temperature at all nodes obtained by direct solution of finite element analysis are taken as the training data. Following the terminology used in literatures [21–24,30], each column of training data is called a snapshot, and the matrix of training data itself is called the snapshot matrix   Tsnap = T1 T2 Ti Tn . (16) A singular value decomposition applied on Tsnap reads T Tsnap = ΦDV , (17) where Φ (size m-by-m) and V (size n-by-n) are orthogonal matrices, and D is a rectangular matrix of size m-by-n. In matrix D, only the values along the diagonal are non-negative, which are called singular values, while the rest are all zeroes. In practice, the singular values are sorted in descending order, i.e. λ1 ≥ λ2 ≥ ≥ λr ≥ 0, r = min(m, n). Denote A = DVT, Eq. (17) can be rewritten as Tsnap = ΦA. (18) By Eq. (18), the snapshot matrix is expressed as a linear combination, in which Φ is the set of orthogonal basis vectors and A stores the associated amplitudes. Taking the advantage that the singular values in D drop quickly to zero, the snapshot can be approximated with up to l terms, with l ≤ r, without losing much accuracy Tsnap ≈ Φ¯ A¯ , (19) in which the set of truncated orthogonal basis vectors Φ¯ is the first l columns of Φ. The set of truncated amplitudes is calculated by T A¯ = Φ¯ Tsnap. (20) Similarly to [23], the “cumulative energy coefficient” is defined as l ∑ λi i=1 e (l) = r . (21) ∑ λj j=1 The “truncated energy” is then calculated by ε = 1 − e (l) . (22) Simply by setting the expected value of ε, e.g. ε = 10−8, the l number of POD basis vectors can be selected.
7. Estimation of heat transfer parameters by using trained POD-RBF and Grey Wolf Optimizer 407 3.4. Approximation of the amplitudes by Radial Basis Function (RBF) Let the amplitudes in Eq. (20) be function of thermal properties, the following linear combination can be written for each column of A¯ ¯a = ¯a (p) = B · f (p) , (23) in which B stores the unknown coefficients; p is the vector of thermal properties; and f is the vector of n Radial Basis Functions (corresponding to n sets of parameters mentioned in Section 3.2)  T f (p) = f1 (p) f2 (p) fi (p) fn (p) . (24) Various types of RBF have been introduced in literatures. Curious readers are re- ferred to [31] for details. Here, the recently proposed quartic polynomial radial basis is employed [32] 2 3 4 i fi (p) = 1 − 6ri + 8ri − 3ri , where ri = p − p . (25) Requiring that Eq. (23) holds for all the snapshots in the training data, the following matrix equation is obtained A¯ = A¯ (p) = B · F (p) , (26) where   2 1  n 1  1 f1 p , p f1 p , p      1 2 n 2
     f2 p , p 1 . . . f2 p , p  F =   . (27)  . . . .   . . .    1 n 2 n
    fn p , p fn p , p . . . 1 The n sets of thermal properties in Eq. (24), i.e. p1, p2, , pn, are the sets used to get training data and thus are all known. Therefore, matrices F and B can be easily computed. When POD basis Φ¯ and the matrix of coefficients B are known, the POD-RBF system has been trained. For an arbitrary set p, e.g. the one generated by the optimization algorithm, the temperature values can be quickly retrieved by Tretrieved = Φ¯ · B · f (p) . (28) 4. NUMERICAL EXAMPLES Let us consider a steady-state heat transfer problem in a complicated domain as pre- sented in Fig.1. The width of the three fins are the same. Temperature on the right surface is prescribed by T = 300 K. Heat convection takes place on the left surface with ambient 2 temperature Ta = 200 K and convective coefficient is h W/(m K). The other boundaries are all insulated. Thermal conductivity within the domain is k W/(mK). Inward heat flux is applied on the curved surface of the middle fin is q = 20000 W/m2. Parameters h and k will be identified by the proposed trained POD-RBF system. The finite element model, which is used to generate the training data, is verified by a convergence study. Three levels of quadrilateral mesh are considered: 219 elements (272
8. ESTIMATION OF HEAT TRANSFER PARAMETERS BY INVERSE ANALYSIS USING 7 TRAINED POD-RBF 408 Minh Ngoc Nguyen, Nha Thanh Nguyen, Thien Tich Truong nodes), 546 elements (622 nodes), 1984 elements (2125 nodes). The “equivalent thermal energy” is defined as Z U = (∇T)T k (∇T) dΩ (29) ESTIMATION OF HEAT TRANSFER PARAMETERS BY INVERSE ANALYSIS USING 7 Ω TRAINED POD-RBF Figure 1. Geometry and boundary conditions. Dimensions are in meter. 8 Minh Ngoc Nguyen, Nha Thanh Nguyen and Thien Tich Truong convergence would be recorded for other values of h and k. Therefore, it is acceptable to use the mesh of 546 quadrilateral elements to generate the training data. The training data, i.e. the snapshot matrix defined in Equation (16), is generated by finite element analysis (FEA) of direct problems, using the following sets: k = 1, 6, 11, , 196, 201 W/(m K) and h = 1, 6, 11, , 196, 201 W/(m2 K). In fact, the lower bound and upper bound of design parameters shall be guessed. Uniform discretization of design space is a basic and common approach. In order to reduce the number of training data, the Taguchi’s method for design of experiments can be employed, as presented by [33]. However, this method is not within the scope of the current work. Temperature at 10 points (marked by dots in Figure 3) are taken as reference data, with k = 87.25 W/(m K) and h = 103.5 W/(m2 K). Minimization of error rate between reference temperature and the values retrieved by the trained POD-RBF is the objective of the optimization block using Grey Wolf Optimizer. In order to mimic measurement error, 5% noise is added into the finite element solution, i.e. Figure 1. Geometry and boundary conditionsthe “measured”. DimensionsFigure 2temperature .are Convergence in meter. atof eachequivalent point thermal is assumed energy (seeto be Equation within (29)) the withrange respect 095 toTT theFEM number of 105 nodes T FEM . Fig. 1. Geometry and boundaryReference conditions temperatureThe (di-finite elementat Fig.each model, 2point. Convergence iswhich the isaveraged used to of generate value equivalent of the 5 training“measurements”. thermal data, is en-verified Details by a areconvergence presented mensions are in meter)in Tablestudy. 1. A comparisonThree levels ofstudyergy quadrilateral is (see conducted Eq. mesh (29 between are)) withconsidered: two respect variants 219 elements to of the Grey number(272 Wolf nodes), Optimizer: 546 elements the original (622 one as describednodes), 1984 in [14] elements, denoted (2125 by nodes). GWO The, and “equivalent theof improved nodes thermal one, energy” denoted is definedby VW- asGWO . In VW-GWO, variable weights (Equation (8)) and the exponential-decay control parameter (Equation (10)) are used. T UTkT d (29) 2 The value of h is h = 100 W/(m K) and that of k is k = 100 W/(mK). TheThe convergence value of h is h = 100 W/(m2 K) and that of k is k = 100 W/(m K). The convergence of the equivalent of the equivalent thermal energythermal with energy respect with respect to number of nodes is displayed in Figure 2. Due to the lack of analytical solution, the result obtained by a fine mesh of 4464 elements (4675 nodes) is used as reference to to number of nodes is displayedevaluate in Fig. the2. numerical Due error. It is observed that with the mesh of 546 elements (see Figure 3), numerical to the lack of analytical solution,error the is result only 1.2 ob- %. In linear heat transfer analysis, which is the case being considered, the same tained by a fine mesh of 4464 elements (4675 nodes) is used as reference to evaluate the nu- merical error. It is observed that with the mesh of 546 elements (see Fig.3), numerical error is only 1.2%. In linear heat transfer analysis, Figure 2. Convergencewhich of equivalent is the thermal case energy being (see Equation considered, (29)) with respect the same to the number of nodes The finite elementconvergence model, which wouldis used to generate be recorded the training for data, other is verified val- by a convergence study. Three levels of quadrilateral mesh are considered: 219 elements (272 nodes), 546 elements (622 nodes), 1984 elementsues (2125 of hnodes).and Thek. Therefore, “equivalent thermal it is acceptableenergy” is defined to as use the mesh of 546 quadrilateralT elements to gen- UTkT d (29) erate the training data. FigureFig. 3. Finite 3 element mesh and location of 10 reference points The value of h is h = 100 W/(mThe2 trainingK) and that of data, k is k = i.e. 100 theW/(m snapshot K). The convergence matrix of the equivalent. Finite element mesh and location of thermal energy withdefined respect to number in Eq. of( nodes16), is displayed generated in Figure by 2 finite. Due to ele-the lack of analyticalTable 10 1. referenceThe 10 reference points points solution, the result mentobtained analysis by a fine mesh (FEA) of 4464 of direct elements problems, (4675 nodes)using is used as reference to evaluate the numerical error. It is observed that withk the mesh of 546Points elements (seeCoordinates Figure 3), numericalRef. Temperatureh [K] Ref. Temperature [K] error is only 1.2 %.the In linear following heat transfer sets: analysis,= 1, which 6, is11, the. case. . , 196,being considered, 201 W/(mK) the(FEA same and solution =, without 1, 6, noise) 11, . . .(FEA , 196, solution, 201 5% noise) W/(m2K). In fact, the lower bound and upper bound of design parameters shall be P1 [0.05, 0.4] 253.2039 256.8404 guessed. Uniform discretization of design space is a basic and common approach. In P2 [0.45, 0.4] 300.0839 296.5855 P3 [0.1, 0.2] 261.5149 256.1581 P4 [0.4, 0.2] 300.8999 296.8543 P5 [0.35, 0.1] 313.1507 315.2422
9. Estimation of heat transfer parameters by using trained POD-RBF and Grey Wolf Optimizer 409 order to reduce the number of training data, the Taguchi’s method for design of exper- iments can be employed, as presented by [33]. However, this method is not within the scope of the current work. Temperature at 10 points (marked by dots in Fig.3) are taken as reference data, with k = 87.25 W/(mK) and h = 103.5 W/(m2K). Minimization of error rate between reference temperature and the values retrieved by the trained POD-RBF is the objective of the op- timization block using Grey Wolf Optimizer. In order to mimic measurement error, 5% noise is added into the finite element solution, i.e. the “measured” temperature at each point is assumed to be within the range 0.95TFEM ≤ T ≤ 1.05TFEM. Reference temper- ature at each point is the averaged value of 5 “measurements”. Details are presented in Tab.1. A comparison study is conducted between two variants of Grey Wolf Optimizer: the original one as described in [14], denoted by GWO, and the improved one, denoted by VW-GWO. In VW-GWO, variable weights (Eq. (8)) and the exponential-decay control parameter (Eq. (10)) are used. Figure 3. Finite element mesh and location of 10 reference points Table 1. The 10 reference points Ref. Temperature [K] Ref. Temperature [K] Points Coordinates (FEA solution, without noise) (FEA solution, 5% noise) P1 [0.05, 0.4] 253.2039 256.8404 P2 [0.45, 0.4] 300.0839 296.5855 P3 [0.1, 0.2] 261.5149 256.1581 P4 [0.4, 0.2] 300.8999 296.8543 P5 [0.35, 0.1] 313.1507 315.2422 P6 [0.15, 0.1] 299.1058 291.6881 P7 [0.15, 0] 298.4134 298.1384 P8 [0.35, 0] 311.9706 312.8024 P9 [0.2, 0.2] 361.0809 357.3288 P10 [0.3, 0.2] 361.2115 356.1655 Two cases are consider: (a) Reference data are obtained without noise and (b) Refer- ence data are obtained with 5% noise. For each case, the inverse analysis is run 10 times by both GWO and VW-GWO. In all cases, the number of grey wolves is 10. Results are presented in Tab.2. It is observed that for both cases (i.e. zero noise and 5% noise in reference data), VW-GWO exhibits better performance than GWO. Although the mean values of estimated k and h are almost the same, the standard deviation in VW- GWO is much lower. For comparison, the results obtained by Genetic Algorithm (GA) are also presented. Agreement between the three algorithms can be observed, although the performance of GA is slightly behind. The possible reason is that the information of the best agents are taken into account by the two GWO variants, but not by GA. For case (a), i.e. zero noise, the values of k and h by the surrogate model are almost equal to the true ones. For case (b), i.e. 5% noise in reference data, error rates of the mean
10. 410 Minh Ngoc Nguyen, Nha Thanh Nguyen, Thien Tich Truong values of estimated k and h, compared with the correct ones (i.e. k = 87.25 and h = 103.5), are 4.87% and 5.40%, respectively. These error rates are very close to the noise existed in reference data. The above results have demonstrated the accuracy of inverse analysis using trained POD-RBF and GWO. Table 2. Parameters estimated by the proposed model in both cases (a) and (b). The true values of k and h are: k = 87.25 W/(mK) and h = 103.5 W/(m2K). Results obtained by Genetic Algorithm (GA) are also presented k h Mean Standard deviation Mean Standard deviation 10 Minh Ngoc Nguyen, Nha Thanh Nguyen and Thien Tich Truong GWO 87.2527 0.0350 103.5016 0.0772 Case (a) value of objective function) repeatedly does not change within many iterations (e.g. 50 iterations), the VW-GWO 87.2506optimization process 0.0023 can be considered as 103.5006 being converged and thus 0.0011 can be terminated. (zero noise) GA 86.5116 4.5390 105.2710 6.0082 GWO 91.5073 0.0743 108.8373 0.1280 Case (b) VW-GWO 91.4993 0.0084 108.8626 0.0267 (5% noise) GA 81.09318 6.6859 111.7144 5.8421 Figs.4 and5 present the mean convergence curves of 10 runs, each run with 10 agents, achieved by GWO, VW-GWO amd GA for case (a) and case (b), respectively. In both cases, the optimization process using VW-GWO tends to converge with much less iterations than GWO. Fig.4 clearly exhibits the efficiency of VW-GWO, as best fitness 10 quicklyMinh Ngoc drops Nguyen, to Nha zero Thanh after Nguyen more and Thien than Tich Truong 50 iterations. After 100 iterations, the best fitness value of objective function)obtained repeatedly by does GWO not change is still within higher many iterations than that (e.g. by50 iterations), VW-GWO. the Similar observation is recorded optimization process canin be Fig. considered5 for caseas being (b). converged VW-GWO and thus requirescan beFigure termina much4. ted.Convergence smaller curve obtained number by GWO of iterationand VW-GWO than for case GWO (a): zero to noise in reference data Figure 4. ConvergenceFig. 4curve. Convergence obtained by GWO curveand VW- obtainedGWO for case by(a):Figure GWOzero noise5. Convergence in referenceFig. 5curve. data Convergence obtained by GWO curve and VW obtained-GWO for case by (b): GWO 5% noise in reference data and VW-GWO for case (a): zero noise in refer- and VW-GWO for case (b): 5% noise in refer- ence data 5. CONCLUSIONence AND data OUTLOOKS In this paper, a trained POD-RBF system is coupled with Grey Wolf Optimizer to develop a surrogate model for estimation of thermal parameters. It is demonstrated that the proposed numerical scheme yields reliable output. When there is no noise in reference data, the error rate between predicted thermal parameters and the true ones is almost zero. When noise is included in the reference data, the parameters are predicted with an error rate within the range of noise. Comparison between two variants of Grey Wolf Optimizer, i.e. the original one (namely GWO) and the improved one (namely VW-GWO) has been conducted. It is shown that by using VW-GWO, Figure 5. Convergence curve obtained by GWO and VW-GWO for case (b): 5% noise in reference data 5. CONCLUSION AND OUTLOOKS In this paper, a trained POD-RBF system is coupled with Grey Wolf Optimizer to develop a surrogate model for estimation of thermal parameters. It is demonstrated that the proposed numerical scheme yields reliable output. When there is no noise in reference data, the error rate between predicted thermal parameters and the true ones is almost zero. When noise is included in the reference data, the parameters are predicted with an error rate within the range of noise. Comparison between two variants of Grey Wolf Optimizer, i.e. the original one (namely GWO) and the improved one (namely VW-GWO) has been conducted. It is shown that by using VW-GWO,
11. Estimation of heat transfer parameters by using trained POD-RBF and Grey Wolf Optimizer 411 reach convergence. Computational time for each iteration is not much difference between GWO and VW-GWO. Therefore, with higher rate of convergence, there is potential to save elapsed time by using VW-GWO. The number of necessary iterations is not known beforehand. It is possible to define a lower limit for the number of iterations. After that limit, if fitness value (i.e. the value of objective function) repeatedly does not change within many iterations (e.g. 50 iterations), the optimization process can be considered as being converged and thus can be terminated. 5. CONCLUSION AND OUTLOOKS In this paper, a trained POD-RBF system is coupled with Grey Wolf Optimizer to de- velop a surrogate model for estimation of thermal parameters. It is demonstrated that the proposed numerical scheme yields reliable output. When there is no noise in reference data, the error rate between predicted thermal parameters and the true ones is almost zero. When noise is included in the reference data, the parameters are predicted with an error rate within the range of noise. Comparison between two variants of Grey Wolf Optimizer, i.e. the original one (namely GWO) and the improved one (namely VW-GWO) has been conducted. It is shown that by using VW-GWO, the convergence rate of the optimizing process is in- creased. Therefore, less number of iterations is required and as a result, computational time can be potentially saved. There are still many issues left open. Improving computational efficiency of the opti- mization process is a constant demand. For the POD-RBF block, the size of training data would increase with respect to the number of the parameters to be identified. Loosely speaking, if identification of 1 parameter needs N samples, then identification of d pa- rameters would need Nd samples. Special technique is necessary to handle with a large and multi-dimensional data. Experiments could be involved in both the preparation of training data and the collection of reference data. However, a large number of data is usually required for training. Therefore, a numerical data generator might be more prac- tical. On the other hand, the numerical model has to be verified before it can be used for generation of training data. The reference data in practice shall be obtained from mea- surement. Obviously, the more number of sensors are placed, the more information could be gained. Unfortunately, in most of the cases, the number of sensors cannot be large due to the cost issues. Therefore, it is necessary to optimize the number of sensors and the positions where the sensors are located [34, 35]. This is also an interesting research topic which can be employed together with inverse analysis. ACKNOWLEDGMENT We acknowledge the support of time and facilities from Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for this study. REFERENCES [1] M. N. Ozisik¨ and H. R. B. Orlande. Inverse heat transfer: fundamentals and applications. Taylor & Francis, (2000).
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14. 414 Minh Ngoc Nguyen, Nha Thanh Nguyen, Thien Tich Truong [30] A. Chatterjee. An introduction to the proper orthogonal decomposition. Current Science, (2000), pp. 808–817. [31] G. R. Liu. Meshfree methods: moving beyond the finite element method. Taylor & Francis, second edition, (2010). [32] C. H. Thai, V. N. V. Do, and H. Nguyen-Xuan. An improved Moving Kriging-based mesh- free method for static, dynamic and buckling analyses of functionally graded isotropic and sandwichESTIMATION plates. OF HEATEngineering TRANSFER Analysis PARAMETERS with BoundaryBY INVERSE Elements ANALYSIS, 64USING, (2016), 11 pp. 122–136. POD-RBF . [33] S.the U. converge Hamimnce rate and of the R. optimizing P. Singh. process Taguchi-based is increased. Therefore, design less number of experiments of iterations is in train- ingrequired POD-RBF and as a result surrogate, computational model time can for be inversepotentially saved. material modelling using nanoinden- tation. ThereInverse are still Problems many issues in left Science open. Improving and Engineering computational, efficiency25, (3), of the (2017), optimization pp. 363–381. is a constant demand. For the POD-RBF block, the. size of training data would increase with [34] C.respect Leyder, to the V. number Dertimanis, of the parameters A. Frangi, to be identified E. Chatzi,. Loosely andspeaking, G. if Lombaert. identification of Optimal 1 sen- sorparameter placement needs N methods samples, then and identification metrics–comparison of d parameters would and implementationneed Nd samples. Special on a timber technique is necessary to handle with a large and multi-dimensional data. Experiments could be involved framein both structure. the preparationStructure of training anddata and Infrastructure the collection of Engineering reference data., However,14, (7), a large (2018), number pp. of 997–1010. is usually required for training. Therefore, a numerical. data generator might be more practical. On [35] D.the Dinh-Cong, other hand, the H. numerical Dang-Trung, model has and to be T. verified Nguyen-Thoi. before it can An be efficientused for generation approach of training for optimal sen- sordata. placement The reference and data damage in practice identification shall be obtained in from laminated measurement. composite Obviously, structures. the more numberAdvances in En- gineeringof sensors Software are placed, the more information could be gained. Unfortunately, in most of the cases, the number of sensors ,cannot119, (2018),be large pp.due to 48–59. the cost issues. Therefore, it is necessary to optimize the . number of sensors and the positions where the sensors are located [34, 35]. This is also an interesting research topic which can be employed togetherAPPENDIX with inverse analysis. A The flow chart of the proposed procedure for inverse heat transfer analysis is given in Fig. A.1. APPENDIX The flow chart of the proposed procedure for inverse heat transfer analysis is given in Figure 6. Figure 6. Flow chart Fig. A.1. Flow chart