Nonconventional Technologies Review no. 2/2011 INDUCTION HEATING DEVICE METAMODELING APPROACH

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1 INDUCTION HEATING DEVICE METAMODELING APPROACH MASTEI Daniela 1, VLADU Ecaterina 2, NOVAC Mihaela 3 1 University of Oradea, Romania, mdani@uoradea.ro 2 University of Oradea, Romania, evladu@uoradea.ro 3 University of Oradea, Romania, mnovac@uoradea.ro ABSTRACT: In the context of the optimal design of electromagnetic devices, complex and computational expensive models based on finite element method (FEM) are to be used, which leads to almost impracticable robust design or optimization. Alternative to the expensive models, metamodels offer fast predictions which might facilitate these processes. This paper investigates the applicability of metamodels in the design of electromagnetic devices, particularly in induction heating processes. It explores the use of Kriging interpolation based models in a procedure where the expensive model is based on the FEM method. KEYWORDS: induction heating, modeling, metamodeling, surrogate modeling, Kriging interpolation, finite element method. 1. INTRODUCTION Computer modeling of induction heating devices allows analyzing and designing the induction heating system fulfilling conditions of improved efficiency and desired temperature profile. Induction heating modeling involves different areas of physics such as heat transfer and electromagnetism. Some material properties depend on the heat, which requires coupling of these phenomena. Optimal design of induction heating devices usually requires optimization of multimodal, nonlinear s which leads to time-expensive design process. However, due to this computational cost, metamodeling techniques have become indispensable. The basic idea is to build approximations of expensive models (which will be referred to as computational models) with approximate analytical models that are more efficient in terms of computing time and the necessary hardware resources. Approximation aims to achieve a global metamodel as accurately as possible, at reasonable cost, and the models are called metamodels, response surface models or surrogate models, their running time being smaller several times than the running time of the computational model [12]. 2. THE COMPUTATIONAL MODEL This paper uses as computational model a 2D numerical model intended to perform 43 simulations with different values of the design parameters.the model was built by using FEM method to solve the coupled electromagnetic and thermal problem, based on the Matlab Partial Differential Equation (PDE) Toolbox. This paragraph briefly presents the computational model [10]. The considered problem involves the induction heating process of a cylindrical worpiece. The heater is double sided, heating in longitudinal field without magnetic core. The device has a cylindrical symmetry so that the problem is reduced to a 2Dproblem in the plane Orz (Fig.1). We ignore the eddy currents in the coil. The domain for the magnetic field can be reduced to a quarter of the device bounded by a boundary at a finite distance from the device. For the thermal field we consider the worpiece as the analysis domain. The penetration depth of the magnetic field in the worpiece imposes the overlapping domains for the two fields. The numerical model is considered in cylindrical coordinates with the axis Or and Oz. The mathematical model of the electromagnetic field is given by relation (1) and of the thermal field by (2): rot( 1 jωa rota) + = J μ ρ s (1) γc T t + div(λgradt) = p (2) Where A is the magnetic vector potential, J s is the current density, μ is the magnetic permeability, ω = 2πf, f is the frequency, T is

2 the temperature, ρ is the resistivity, λ is the thermal conductivity, γc is the specific caloric capacity and p is the density of the power produced by Joule effect which is given by relation (3): p = ω 2 A 2 (3) ρ z M Q O R S Fig. 1. The considered magnetic field domain In relation (1), the second term is the induced current density, which has a non null value only in the worpiece. The magnetic field domain considered is presented in Fig. 1, which is a quarter of the device. The MN and NP lines are approximated to be the infinite boundary, where the potential vector A has null value. The other magnetic field boundaries are the Oz axis, which is parallel with the magnetic field lines and the axis Or, which is normal on the magnetic field lines. For the thermal field, the domain analysis is the worpiece. On the boundaries OS and OQ, the thermal flux has null value. Instead, on the boundaries QR and RS convection and radiation losses occur, by satisfying the boundary condition given by relation (4) λ T = α(t T n a ) (4) Where T a is the ambient temperature. The input values that are supposed to be nown are: the frequency of the power inductor, the average current density in the inductor relative to the depth of penetration, the geometrical dimensions and the physical properties of the worpiece and of the inductor, the initial temperature of the worpiece, the average final temperature. The cylindrical piece heated in the inductor is built from steel, has a diameter of 80 mm and 800 mm in length. The design parameters are: frequency f of the supplied voltage, the inductor length Li and the air-gap h between the piece and the inductor. P r N 44 The mathematical model for coupled fields involves relations (1), (2) and (3). Combining these equations yields a coupled system of non-linear equations. To solve this system we develop a numerical model based on the finite-element method. After the geometry is defined, a discrete spatial mesh is defined by a number of nodes. The unnown values of the problem are the temperature T and the magnetic vector potential, [18]. The thermal field affects many material properties that influence the electromagnetic field in a direct or indirect way. These thermal dependencies occur as part of nonlinear coefficients in the electromagnetic equations. An important temperature dependent characteristic, directly affecting the electrical current distribution, is the electrical conductivity ς. The magnetic properties of metallic materials with a small hysteresis loop, such as ferromagnetic laminations, have a temperature dependent permeability. All those materials have, apart from the saturation phenomenon, a transition temperature named Curie temperature T C, beyond which the magnetic properties vanish. The electromagnetic field depends directly on the temperature distribution through material properties, [4]. The main problem is related to the time step choice in the simulation. If a very small but stable time step, related to the magnetic field dynamics is used, yields an extremely long computation time. If a large time step related to the thermal time step is used, the problem requires special, expensive integration methods in order to obtain a stable computation. The considered method assumes that one of the sub-problems is in a steady state. The magnetic field is recomputed after every time step. The implemented algorithm consists in 3 major steps repeated until the worpiece is heated to the final temperature: a. Define the geometry and generate the mesh; isolate the triangles in the worpiece. On a distance equal to the one of penetration depth generate a set of triangles corresponding to the initial ones, having the sides on the worpiece, for the thermal process. b. Solve magnetic problem. c. Solve thermal problem; adjust the time step; Update material properties.

3 Finally, compute the efficiency and the temperature gap in the worpiece. Electrical efficiency is defined in relation (5): η = P wp P tot (5) Where P tot is the total active power supplied to the system P wp active power dissipated in the worpiece. The temperature gap is given by relation (6): T gap = T max T min (6) Where T max is the maximum temperature and T min the minimum temperature in the worpiece. Although adapting the time step was considered, the simulation time is considerable, especially if the model is intended to be used in some optimization processes. Consequently, there is great interest in techniques that facilitate the construction of surrogate models, while minimizing the computational cost and maximizing model accuracy. The next paragraph introduces some surrogate modeling techniques, emphasizing the models based on Kriging interpolation. 3. SURROGATE MODELING Constructing a surrogate model involves selecting the experiments performed on the computational model, deciding the model type and tuning it with the available data, as shown in Fig. 2. Selecting experiments Selecting surrogate model type Input Data intrare Computation al model Tuning the surrogate model based on available data Surrogate model Data evaluation result Fig. 2. Constructing a surrogate model Surrogate model construction uses predictions based on limited information and simplifying assumptions [8], [3], [9]. One of the main simplifying assumptions supposes that the describing the model is continuous and smooth. Methods such as radial basis, support vector 45 regression, Kriging interpolation and others are based on such assumptions. Additional assumptions can be used regarding the form of the, for example by applying a polynomial regression. Surrogate modeling is building a model denoted f(x) using available data x i and evaluation of the tuning errors ε(x) for the model f. Using the surrogate model, a prediction in x is defined by (7): f p (x) = f(x) + ε (x) (7) To build the surrogate model, in the literature there are several types of possible models. This section will present some significant models, which are addressed in [8], [3] and [9] Response Surface Methodology (RSM) The models used in RSM approximation are polynomials of low degree. For a small curvature, a first order polynomial can be used (8): f(x) = β 0 + i=1 β i x i (8) For more significant curvatures, second order polynomials are used, which includes all available interactions between two factors (9): f(x) = β 0 + i=1 β i x i + i=1 β ii x 2 i + j =1,i<j β ij i=1 x i x j (9) The unnown coefficients of polynomials of (8) and (9) are usually determined by least square fitting. The disadvantage of RSM consists in the fact that, for complex nonlinear models high order polynomials are to be used [15], [14], [11], [13], [7] and [3] Artificial Neural Networs A neural networ consists of neurons, which are multiple linear regression models fitted with a nonlinear. Neurons are assembled into complex structures that define the networ architecture. The networ weights are the unnown values to be found. For modeling through artificial neural networs, firstly their architecture is to be defined. Secondly, networ training and adjusting weights is performed. In [5], an adaptive modeling algorithm based on neural networs is presented, compared with other methods on a number of problems Kriging interpolation Kriging is an interpolation method which uses statistical models that will produce,

4 besides the prediction, some other results such as standard prediction error etc. Kriging [3], assumes that the data comes from a stationary stochastic process (some variants of the method also assume a normal distribution of data). Kriging is an interpolation method named after the engineer DG Krige from South Africa, the inventor of the method. Being f an unnown real valued, Kriging interpolation f(x 0 ) predicted in x 0, based on the nown values in the points x 1, x 2,...,x n which are different from x 0 is given by (10): n f(x 0 ) = i=1 λ i (x 0 )f(x i ) (10) where λ i (x 0 ) are unown coefficients to be found by means of a stationar random F(X i ). The estimation error is the difference between the estimated and the true value in the same point (11): r i = f i f i (11) The average error for a number of estimates is given by (12): m r = 1 r i=1 i = 1 f i=1 i f i (12) In [6], [1], [2] the theoretical basis of Kriging interpolation is presented. There are several types of Kriging interpolation having different degrees of complexity, such as: - Normal Kriging; - Simple Kriging; - Universal Kriging. Simple Kriging is based on the assumption that the mean and covariance of F(x) are nown, while Normal Kriging considers in addition that the mean value of n F(x) is null and i=1 λ i = 1. Universal Kriging allows the user to specify a polynomial deviation from the mean in the data related to the process to be modeled Optimizations based on surrogate models. In the last decade has been shown that surrogate modeling has a decisive role in supporting optimal design in engineering activities [8], [17] and [16]. Optimization based on surrogate modeling is a sequential process consisting essentially in the following steps: 1. Choice of optimization variables, based on their importance as determined by preliminary experiments; 2. Setting values for the variables that 46 become inputs for the computational model (expensive model) based on a predefined strategy and performance evaluation; 3. Selecting the type of the surrogate model; 4. Building the surrogate model using the outputs of the computational model (available information); 5. Optimization using the surrogate model to find new points of interest for the considered computational model; 6. Tuning the surrogate model taing into account the additional information available; 7. Return to step 4 until a stopping criterion is fulfilled. Optimal inductor design problem can be solved using the surrogate model DACE toolbox In order to build a surrogate model for the induction heating device presented in paragraph 2, the DACE toolbox (Design and Analysis of Computer Experiments) was used. DACE is a Matlab toolbox specialized in Kriging interpolation, [2]. Using the data obtained from experiments, the toolbox builds a model of type universal Kriging which can be used as a surrogate model for the computation model. The surrogate model can be used to predict the model outputs for new entries for which the outputs are not nown from the experiments. The toolbox returns also the estimation error for each input, allowing to evaluate the quality of the generated model. The DACE toolbox modeling is named dacefit. Calling this with arguments specifying the available data (x i and f i ), a polynomial of -degree that models the bias of the data and the correlation of the model, will return the Kriging model in the form of a complex data structure, as shown in Fig. 3. DACE toolbox implements the following correlation s: linear, Gaussian, exponential, spleen and spherical. The bias of the data is implemented by polynomial of degree 0, 1 and 2. The estimation of the DACE Toolbox is named predictor. Calling this with an argument specifying the Kriging model previously generated and the points where data is to be estimated, the toolbox returns the estimated values and the corresponding estimation error denoted er i., as presented in Fig. 4.

5 x i, values for the variables f i, the values in x i degree polinomial Correlation Fig. 3. The DACE toolbox modeling Kriging type model x i, values for estimation points DACE Toolbox (dacefit ) DACE Toolbox (predictor funcion) Kriging type surrogate model Estimated values f i Estimation errors er i Fig. 4. The DACE toolbox predicting 4. BUILDING THE SURROGATE MODEL FOR THE INDUCTION HEATING DEVICE The inductor model resumed in paragraph 2 is used, with two objectives, Efficiency and T. The design variables are the frequency (f) and the inductor length/2 (L ind ). The range of these parameters is [500 Hz, 1500 Hz] for the frequency f and [0.40 m, 0.50 m] for L ind. These domains were divided each into 5 disjoint intervals, resulting 6 values for each parameter. A number of 6 2 = 36 combinations of parameters and corresponding values for the outputs were obtained by using the computation model (the numerical model implemented in Matlab by using the PDE toolbox), as presented in Table 1. Examining the values in Table 1 it can be seen that the two objectives tae values in different ranges. For this reason, a scaling operation is applied. Table 1. Data evaluation result f L ind Efficiency 500 0,40 0, , ,40 0, , ,40 0, , ,40 0, , ,40 0, , ,40 0, , ,42 0, , ,42 0, , ,42 0, , ,42 0,825 96, ,42 0,823 73, ,42 0, , ,44 0, , ,44 0,821 83, ,44 0, , ,44 0, , ,44 0, , ,44 0, , ,46 0,809 72, ,46 0, , ,46 0, , ,46 0, , ,46 0, , ,46 0, , ,48 0, , ,48 0, , ,48 0, , ,48 0, , ,48 0, , ,48 0, , ,50 0, , ,50 0, , ,50 0, , ,50 0, , ,50 0, , ,50 0, ,84 47

6 After scaling, the data having different upper and lower limits will tae values in the same range. Scaling can be done in different ways, the most common being orthogonal scaling. In this wor a scaling method which maps the values of the two considered objectives in the interval [0, 1], given by relations (13) and (14) was used: Gaussian, since it leads to the highest mean square errors of predictions for all polynomial s used. Generating further models, the most advantageous result are obtained for the polynomial of degree 4 and the spleen correlation. For these values, in the range of the design parameters, surrogate models were generated. T scal _i = Eff scal _i = T i T min T max T min (13) Eff i Eff min Eff max Eff min (14) Applying scaling for the data presented in Table 1 leads to the data presented in Table 2, where: T i is the unscaled value for the temperature gap; T scal _i is the corresponding scaled value. Similarly, Eff i is the unscaled value for the efficiency and Eff scal _i is the corresponding scaled value. Using the DACE toolbox to which we have added additional polynomial models of degree 3, 4 and 5 the surrogate model for the inductor heating device was generated. Some of the results are presented in Table 2. Table 2. Generating the surrogate models Pol. Correlation max err max model for err bias of for Eff for T the data Gr. 0 Linear 0, ,16450 Gr. 0 Gaussian 0, ,61329 Gr. 0 Exp. 0, ,18335 Gr. 0 Spleen 0, ,10749 Gr. 0 Spherical 0, ,12102 Gr. 1 Linear 0, ,17304 Gr. 1 Gaussian 0, ,47153 Gr. 1 Exp. 0, ,21428 Gr. 1 Spleen 0, ,09185 Gr. 1 Spherical 0, ,11682 Gr. 2 Linear 0, ,13250 Gr. 2 Gaussian 0, ,27007 Gr. 2 Exp 0, ,14012 Analyzing the obtained results in Table 2, it can be seen that the most disadvantageous correlation for the problem is the 48 Fig. 4. The Efficiency output for the surrogate model Figure 5 and 6 present the Efficiency output of the generated surrogate model, respectively the Temperature gap T output. Fig. 5. The Temperature gap output for the surrogate model In these figures, the blac points represent the initial available data assessed with the computation model. It can be seen that most of the points belong to the model s surface, except some insulated points. The predicted mean square error to the Efficiency is presented in figure 6.

7 optimal design tass result a minimal number of expensive evaluations. For the development of the metamodel an existing tool DACE has been used. The results suggest the potential of such modeling approach to be used as a tool for electromagnetic device design. Future wor includes using the obtained model in optimal design of the induction heating device by means of a multiobjective evolutionary search. REFERENCE [1] CHAO-YI L.: neurobio/land/oldstudentprojects/cs490-94to95/ clang/riging.html Fig. 6. The predicted error for the Efficiency objective In figure 7, the mean square error to the temperature gap is presented [2] LOPHAVEN S. N., NIELSEN H. B., SØNDERGAARD J.: Dace a Matlab Kriging Toolbox. Informatics and Mathematical modeling, Technical University of Denmar, 2002 [3] DONG Z.: A comparative study of metamodeling methods considering sample quality merits, Springer-Verlag, pp , Fig. 7. The predicted error for the temperature gap 5. CONCLUSION This wor provides an overview of several Surrogate-Based Optimization methods. The DACE toolbox is used to perform surrogate modeling of the induction heating device. The obtained model compares well against a reference design obtained by FEM method, so that the riging surrogate model is able to approximate accurately the two considered objective s: efficiency and temperature gap. By this way, in future 49 [4] DRIESEN J., RONNIE J. M.: Methodologies for Coupled Transient Electromagnetic-Thermal Finite-Element Modeling of Electrical Energy Transducers. IEEE, 2002 [5] GORISSEN D., HENDRICKX W., DHAENE T.:, Adaptive Global Metamodeling with Neural Networs, ESANN'2007 proceedings - European Symposium on Artificial Neural Networs Bruges (Belgium), 2007 [6] Groundwater Modeling System (GMS Software): Interpolation/Interpolation_Schemes/Kriging/ [7] HAFTKA R. T., E SCOTT. P., and CRUZ J. R.: Optimization and Experiments: A Survey. Applied Mechanics Review, 51(7), , 1998 [8] JIN R., CHEN W., SIMPSON, T. W.: Comparative studies of metamodeling techniques under multiple modelling criteria. Structural and Multidisciplinary Optimization, vol.23, pp.1 13, 2001 [9] KLEIJNEN J. P.C.: Kriging metamodeling in simulation: A review, European Journal of Operational Research 192 (2009) , [10] MASTEI, D.: Simulation of the coupled electromagnetic and thermal field in induction

8 heating processes. Journal of Electrical and Electronics Engineering (JEEE) vol 2, in press, Oradea, 2011 [11] NACEUR H., GUO Y. Q., BEN-ELECHI S.: Response surface methodology for design of sheet forming parameters to control springbac effects. Computers and Structures, vol. 84, pp , 2006 [12] NIELSEN, H.B.: Surrogate modelling by Kriging, [13] ORR M. J. L., Introduction to Radial Basis Function Networs, rbf/rbf.html [14] REDHE M., FORSBERG J., JANSSON T., MARKLUND P. O., NILSSON L.: Using the response surface methodology and the D- optimality criterion in crashworthiness related problems - an analysis of the surface approximation error versus the number of evaluations. Structural and Multidisciplinary Optimization, vol. 24, pp , 2002 [15] ROUX W. J., STANDER N., HAFTKA R. T: Response surface approximations for structural optimization. International Journal for Numerical Methods in Engineering, vol. 42, pp , 1998 [16] SIMPSON T. W.,, BOOKER A. GHOSH J., D., GIUNTA A. A., KOCH P. N., YANG R. J.: Approximation Methods in Multidis-ciplinary Analysis and Optimization: A Panel Discussion. Structural and Multidisciplinary Optimization, 27, , 2004 [17] WANG G. G., SHAN S.: Review of Metamodeling Techniques in Support of Engineering Design Optimization. ASME Transactions, Journal of Mechanical design, 2006 [18] CARSTEA, I.: Domain decomposition for coupled fields in electrical engineering. Finite elements. Theory and Advanced Applications. Published by WSEAS Press, ISBN: pp ,