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High Temperatures-High Pressures, Vol. 38, pp. 119–136 Reprints available directly from the publisher Photocopying permitted by license only

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Comparison of inverse methods in the determination of heat flux and temperature in cutting tool during a machining process† Solidônio R. De Carvalho1,∗ , Marcelo R. Dos Santos1 , Priscila F. B. De Souza1 , Gilmar Guimarães1 and Sandro M. M. De Lima E Silva2 1 Federal

University of Uberlândia – UFU, School of Mechanical Engineering – FEMEC, Campus Santa Mônica, Av. João Naves de Ávila, 2121, ZIP code 38408-100, Uberlândia, Minas Gerais, Brazil 2 Federal University of Itajubá – UNIFEI, Institute of Mechanical Engineering – IEM, Campus Prof. José Rodrigues Seabra, 1303 BPS Av. Pinheirinho, ZIP code 37500-903, Box-Postal 50, Itajubá, Minas Gerais, Brazil Received: December 15, 2008. Accepted: March 13, 2009.

During machining, high temperatures are generated in the region of the tool cutting edge. These temperatures have a controlling influence on the wear rate of the cutting tool and on the friction between the chip and the tool. However, direct measurement of temperature using contact type sensors at the tool-work interface is difficult to implement due to the movement of the workpiece and the presence of the chip. Conventional methods such as infrared pyrometer, embedded thermocouple and tool-work thermocouple usually present problems. The use of inverse heat conduction techniques represents a good alternative since these methods uses temperatures measured in accessible positions. This paper proposes a comparison of the inverse techniques Golden Section, Function Specification, SimulatedAnnealing and Dynamics Observers Based on Green’s Function. They are used in an experimental methodology to determine the thermal fields and heat generated at the chip-tool interface during machining. A numerical 3-D transient thermal model was developed to consider both tool and tool holder assembly. The results are validated by controlled experiments and qualitative analysis. Some of the cited procedures are in the computational code INV3D that contain a series of functions for the experimental data acquisition, the three-dimensional mesh generation and the graphical environment analysis. Keywords: Inverse problems, heat conduction, optimization, heat flux identification, machining process. ∗ Corresponding author: E-mail: [email protected] † Paper presented at the 19th International Congress of Mechanical Engineering, November 05–09, 2007 Brasila, DF, Brazil.

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1 INTRODUCTION The inverse heat conduction technique consists of an optimization problem where a functional of the experimental and the theoretical of temperature fields induced by a unknown heat flux is minimized [1]. The advantage of inverse techniques consists in the possibility of indirectly obtaining the solution of a physical problem with partially unavailable information. For example, the determination of a surface temperature without direct access or the diagnosis of diseases using computerized tomography. In both cases, the boundaries are unknown and inaccessible. These problems can be solved using information obtained from sensors located in accessible positions. Several techniques can be found in literature for the solution of Inverse Heat Conduction Problems (IHCP) [2–11] and [12]. This paper presents a comparison of four inverse problems techniques (Golden Section, GS, Function Specification, FS, Simulated Annealing, SA and Dynamics Observers, DO) used to estimate heat flux and then the temperature at the chip-tool interface. The studies of these thermal fields in machining are very important for the development of new technologies aimed to increase tool live and to reduce production costs. Since the direct temperature measurements at the chip-tool interface are very complex this work proposes the estimation of the temperature and the heat flux at the chip-tool interface using the inverse heat conduction problem technique. The thermal model is obtained by the numerical solution of the transient three-dimensional heat diffusion equation that considers both the tool and tool holder assembly. For obtaining the solution equation the finite volume method with a non-uniform grid is used. In this case, the INV3D version 1.0 software was developed. The program INV3D also contains a series of functions that help in the acquisition of experimental data, in the generation of the three-dimensional mesh and in graphical analysis. The methodology proposed here is verified by estimating a heat flux imposed in one controlled experiment. Results of an experimental test are also presented.

2 FUNDAMENTALS 2.1 Thermal model In Figure 1 the schematic model for the thermal problem of machining is presented. The tool and the tool holder are generated from a three-dimensional control volume. The dimensions of the control volume are 86.51 × 81.68 × 31.00 mm. The thermal model of the assembly tool-tool holder, shown in Fig. 1, is described by the transient three-dimensional heat diffusion equation. The equation considering thermal properties temperature dependent can be written as: ∂ ∂T ∂T ∂T ∂T ∂ ∂ λi + λi + λi = (ρCp)i (1) ∂x ∂x ∂y ∂y ∂z ∂z ∂t

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3D Control Volume

Tool holder

Tool Shim

FIGURE 1 Three-dimensional thermal problem: tool, shim and tool holder.

subjected to the boundary conditions −λi

∂T = h(T − T∞ ) ∂η

(2)

in the regions exposed to the environment and −λi

∂T = q (t) ∂η

(3)

at chip-tool-workpiece interface, defined as Ac . The initial condition is T (x, y, z, o) = To

(4)

In Eqs. (1) T is the temperature, ∂T ∂η is the derivative along the outward drawn normal to the surface, k, the thermal conductivity, ρ the density, Cp the heat capacity, h the heat transfer coefficients at the assembly surface, t the time and T∞ the medium temperature. The index i represents the position in the assembly as follows: i = 1 represents the tool cutting; i = 2 represents the tool support and i = 3 represents the tool holder. It should be noted in Fig. (1) that all assembly surfaces are exposed to the environment but the interface area contact, Ac . As mentioned before, this area is subjected to the heat flux q (t) generated due to contact with tool and workpiece. The cutting tool was considered without chip breaker and the material of the tool and tool holder are considered homogeneous. Thermal properties have no temperature dependence and during the experiment the cutting fluid was not used. All sample faces of the tool and tool holder assembly are subjected to heat loss by the heat transfer coefficient, h = 20.0 W/m2 K. If the value of the heat flux, q (t), is known, the Eq. (1) represents the direct problem related to the inverse problem studied.

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FIGURE 2 Irregular finite volume mesh for the tool, shim and tool holder.

2.2 Numerical solution of the direct problem For the solution of Equation (1), a numerical modeling based on the implicit finite volume method was developed [13]. The tool, shim and tool holder assembly was discretized using an irregular three-dimensional mesh as presented in Figure 2. An additional difficulty that appears with the numerical discretization is the need to place the temperature sensor (thermocouple) in the center of a finite volume element. To resolve this problem, a Cartesian grid generator was developed to adapt to any kind of geometry and to allow the generation of the grid from specific points defined by the sensor localization (software INV3D). Since the heat flux determination requires the manipulation of experimental data, the numerical model must not have any limitation on the time duration of the experiment or measurement interval of the experimental temperature. The finite volume method with implicit formulation was used. The direct problem is reduced to the solution of the linear algebraic equations system. For the solution of this system the Successive Over Relaxation method (SOR) is used with a relaxation coefficient W = 1.95 (further details about this method can be found in [13]). 2.3 Inverse problems in heat transfer One way to estimate the heat flux q (t), present in Eq. (3), is to require that the traditional least square function, Smq , defined as the error between the computed temperature Ti and the measured temperature Yi be minimized with

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respect to the unknown q(t). Smq is then defined as Smq =

ntherm

Y (x, y, z, t)i − T (x, y, z, t)i

2

(5)

i=1

where ntherm represents the number of thermocouples used. 2.3.1 Golden Section method (GS) The Golden Section method for estimating the maximum, minimum, or zero of a one-variable function is a popular technique for several reasons. Since, while the function is assumed to be unimodal, it does not need continuous derivatives. Secondly, as opposed to polynomial or other curve fitting, the convergence rate for the Golden Section method is known. Finally, the method can be easily programmed for solution on the digital computer. The Golden Section method is an iterative process where the search interval reduces approximately 62% of the previous iteration interval, until it finds the least value of the objective function [13]. 2.3.2 Simulated Annealing (SA) SimulatedAnnealing is a global stochastic optimization method that originated in the computational reproduction of the thermal process of annealing, where a material is slowly heated and cooled in order to reach a minimum energy state. In the SA method, starting from an initial configuration, a new configuration is generated randomly. If the new configuration has a smaller value of the objective function (in a minimization context), then it will become the current configuration. Otherwise, a stochastic test is applied to indicate whether the new configuration will be accepted or not. This process of movementacceptation is repeated and as the number of analyzed alternatives increases, the acceptance probability of the worse configurations decreases gradually. Due to the possibility of carrying out “wrong way” movements, the search can move from a local optimum toward the global optimum to avoid being trapped in a poor local solution [3, 9]. 2.3.3 Function Specification method (FS) The Function Specification procedure is so called because the explicit underlying nature of the function being estimated is assumed or specified [1]. Typically, a simple form such as piecewise constant or linear function is assumed. In the sequential Function Specification method, the functions components are determined one at a time in sequence rather than all at once in a batch process. 2.3.4 Dynamic Observers Based on Green’s Function (DOBGF) Dynamic Observers [10] are proposed as a solution algorithm for the inverse heat conduction problem (IHCP) of reconstructing a time-dependent surface heat flux at the boundary of a linear heat conductor. The derivation of optimal

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observer equations follows directly from a novel interpretation of the IHCP in the frequency domain: Solving the IHCP is viewed as a filter design problem in which the reconstructed heat flux is obtained by low-pass filtering of the true heat flux.

3 VALIDATION OF THE INVERSE TECHNIQUES AND THE NUMERICAL MODEL A great difficulty in the solution of inverse heat conduction problems is the validation of the technique used. This difficulty is inherent to the problem, since validation of the estimated heat flux requires the previous knowledge of the experimental heat flux. It can be observed that in real inverse problems, such as in machining process, the experimental heat flux is not known. Thus, an alternative for the validation of the inverse technique is to make a controlled experiment, in which, the heat flux and the temperature were measured at the cutting tool. Later, these signals would be compared with the estimated heat flux and the calculated temperature in the Inv3D program for each of the mentioned techniques of inverse problems. Thus, before the analysis of the real process of machining, an experiment with controlled conditions was done. In this case, a cemented carbide tool with the dimensions 0.0127 × 0.0127 × 0.0047 m was used. A heat flux transducer and two thermocouples previously calibrated [13] and a kapton electric heater were used on this tool. The electrical heater was connected to a continuous source current (MCE) that provided the heat generation (Joule effect). The heat flux transducer was located between the heater and the tool, in order to measure the heat supplied to the tool. The temperatures at the tool were measured with two thermocouples. The signals of heat flux and temperatures were acquired using a HP Series 75000 data acquisition system with a E1326B voltmeter controlled by a PC. To guarantee a better thermal contact a thermal paste was used between the transducer and tool. Capacitor discharge was used to attach the thermocouples to the tool surface [14]. Figures 3 and 4 present respectively the heat flux and the temperatures on the tool. The direct problem solution for a three-dimensional non-uniform grid with 13824 points is shown in Figure 5 for the two temperatures. The values of thermal properties used to calculate these temperatures are λ = 43.1 W/m.K and α = 14.8 × 10−06 m2 /s and the minimum distance between the x, y and z grid points are respectively 0.0005, 0.0005 and 0.0002 m. The sample interval, t, was 1.3 s. A good agreement between experimental and calculated temperatures is observed in Figure 5. This fact is more evident when the residuals are analyzed (Figure 6). In this case, the maximum residual obtained was 0.88 ◦ C, which represents an error of approximately 1.46% . The heat flux generated at the tool is estimated from the temperatures shown in Figure 4, for the four techniques of inverse problem. The results of estimated

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FIGURE 3 Experimental heat flux.

FIGURE 4 Experimental temperatures.

heat flux for these techniques are compared with the measured heat flux in Figure 7. It can be seen that the four techniques used in this paper presented good results when compared with the experimental heat flux. Therefore, these results validate each technique and show their efficiency. Figures 8a and 8b show a comparison of the experimental and estimated temperature values at each thermocouple location used in this paper. The residuals of the estimated and experimental values of temperature for each thermocouple location are presented in Figures 9a and 9b.

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FIGURE 5 Comparison of the experimental and calculated temperatures.

FIGURE 6 Residuals of temperature.

In these figures it is possible to see that the maximum residual was obtained by the Golden Section technique and it was 0.88 ◦ C, which represents an error of 1.46%. The performance ranking for the four algorithms shown in Figures 9a and 9b are: the Simulated Annealing (mean residual 0.03521 ◦ C), the Function Specification (mean residual 0.35705 ◦ C), the Golden Section (mean residual 0.3744 ◦ C) and the Dynamic Observer Based on Green’s Function (mean residual 0.43922 ◦ C). Table 1 presents the computational time to estimate the heat flux for the four inverse techniques.

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E xp GS SA FS DOBGF

FIGURE 7 Experimental and estimated heat flux.

Ex p GS SA FS DOBGF

(a)

Exp GS SA FS D O BG F

(b)

FIGURE 8 Comparison of the estimated and experimental temperatures (a) thermocouple 1 and (b) thermocouple 2.

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GS AS FS D O BG F

(a)

GS AS FS D O BG F

(b)

FIGURE 9 Temperature residuals at the locations of : (a) thermocouple 1 and; (b) thermocouple 2.

Inverse Technique Golden Section Simulated Annealing Function Specification Dynamic Observers Based on Green’s Function

Computational time (minutes) 170 300 10 10

TABLE 1 Computational time to estimate the heat flux.

The PC used in the tests had the following configuration: Pentium IV, 2.6 Ghz and memory ram 512 Mb. In this table, it is possible to verify the quickness of the Function Specification and the Dynamic Observer technique when compared to the Golden Section and the Simulated Annealing techniques.

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After validated the numerical thermal model and the inverse techniques using a controlled experiment, the next step is to use the inverse techniques to estimate the heat flux and the temperatures in an experimental machining process.

4 EXPERIMENTAL SETUP OF A REAL MACHINING PROCESS The machining test was carried out on a conventional lathe (IMOR MAXI–II– 520–6CV) without coolant. The material used in the experimental test was gray cast iron (FC 20 EB 126 ABNT) in the form of a cylindrical bar with an external diameter of 77 mm. A Cemented ISO SNUN12040408 K20/Brassinter insert and a ISO CSBNR 20K12/SANDVIK COROMAT tool holder were used. The gray cast iron workpiece has initial diameter of 77 mm. The temperatures were measured at accessible locations of the insert, the shim and the tool holder using K type thermocouple and a HP 75000 Series B data acquisition system with a E1326B voltmeter controlled by a PC. Table 2 presents the locations of the thermocouples. The chip-tool contact area determination represents one of most important error source in the inverse problem solution. Some methods to identify this area can be found in the literature as, for example, the use of a software image analyzer [15] or the application of coatings [16]. In both process, the area is

Position/ Thermocouple X [mm] Y [mm] Z [mm]

1

2

3

4

5

6

7

8

0.000 0.000 0.000 4.490 6.528 7.222 9.512 5.300 6.450 7.250 3.950 4.116 6.579 4.740 1.715 14.55 15.95 21.05 11.52 14.23 14.23 9.400 9.400 4.000

TABLE 2 Thermocouple location shown in Figure 10.

tool holder unknown heat flux tool

thermocouple

FIGURE 10 Location of the thermocouples T1 to T8.

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(a)

(b) FIGURE 11 Chip-tool interface contact area: (a) image as acquired by video camera (b) Image after signal treatment.

measured after machining. This procedure is also used here. However, in this paper, the average interface contact area is obtained from the three tests carried out with the same cutting condition. In order to measure the contact area an image system program with a Hitach CCD, KP-110 model video camera, a AMD K6 450 MHz PC and the GLOBAL LAB Image software were used. A typical contact area is presented in Figure 11.

5 RESULTS OF THE MACHINING PROCESS 5.1 Experimental results In the present paper, the thermal properties of the tool supplied from Sandvik do Brasil S.A. are λ = 43.1 Wm/K and α = 14.8 m2 /s. The material of the tool holder is AISI 1045 steel and the thermal properties are λ = 49.8 Wm/K and α = 14.8 m2 /s [17]. The same properties were used for the tool support. The heat flux was estimated using three inverse techniques for the following

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cutting condition: feed rate of 0.138 mm/rev, cutting speed of 217.72 m/min and depth of cut of 1.5 mm. Figure 12 presents the heat flux estimated by three inverse techniques. In this figure, the heat flux was estimated using the following techniques: GS, FS and DOBGF. It can be mentioned that it was not possible to estimate the heat flux using SA. Although the SA technique is a robust optimization technique, it requires many evaluations of the objective function. It can also be seen in the figure that the heat flux estimated by GS technique tends to reach a steady state condition, while for the other techniques it decreases after a few seconds of machining. This fact can be attributed to the uncertainties of the theoretical model. Some of these uncertainties are the geometric simplifications, the errors in dimensions measured, the values adopted for the thermal properties, the thermal contact resistance, the heat convection between the tool assembly and the air; the measured temperatures and the measurement of the chip-tool interface contact area. In this case, as the experimental heat flux is unknown, only a qualitative analysis of the results is presented. The first step of this analysis consists of comparing the experimental temperature measured at an accessible region of the tool with the calculated temperature from the heat flux shown in Figure 12. The second step is to determine the average temperature at the cutting interface for each instant of time and for each inverse technique and to compare the results with the literature. In Table 3, the computational time to estimate the heat flux is presented. In this table, it is also possible to note that the FS and the DOBGF are quicker. Figure 13 presents the comparison between the experimental temperature measured by thermocouple 3 and the temperature calculated using the heat flux presented in Figure 12. In this case a good agreement between experimental temperatures and temperatures calculated by the GS technique is observed. Nevertheless, in the other

GS FS D O BG F

FIGURE 12 Heat flux estimated for three inverse problems techniques.

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Computational time (minutes)

Inverse Technique Golden Section Simulated Annealing Function Specification Dynamic Observers Based on Green’s Function

5760 – 240 240

TABLE 3 Computational time to estimate the heat flux.

E xp GS FS D O BG F

FIGURE 13 Comparison of experimental and theoretical temperatures for thermocouple 3.

techniques the maximum residue is 100 ◦ C. Figure 14 presents the history of the average temperature at the cutting interface. It is verified in literature that the majority of papers present only the average temperature in steady state at the cutting interface. Few works analyze the transient process. Among the analyzed papers, two presented results for the same materials and cutting conditions [18, 19] as this paper . For a cutting speed of 230 m/min, the average experimental temperature at the cutting interface was 800 ◦ C, whereas the calculated temperature presented variations from 500 ◦ C to 820 ◦ C [18]. In the Ref. [18] a constant feed rate of 0.117 mm/rev was considered. Although the cutting conditions of Ref. [18] are not exactly the same of this work, the experimental and theoretical temperatures coincide with the temperatures presented in Figure 14. In Ref. [19] the methodology based on an ellipsoidal one-dimensional model [20] used the Function Specification method for a cutting speed of 206 m/min, feed hate of 0.176 mm/rev and depth of cut of 1.75 mm. Melo [19] determined a maximum temperature at the cutting

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GS FS DO BG F

FIGURE 14 Temperature at the cutting interface.

interface of 950 ◦ C. This result is very close to the maximum temperature value obtained using the tool-work thermocouple method [21] and with the 900 ◦ C found here, as can be observed in Fig. 13. 5.2 Analysis of transient residues Figure 15 presents the transient residual between experimental temperatures and calculated temperatures using the Function Specification and the Dynamic Observers techniques. Figure 15 shows a tendency of the residues to increase in time. This behavior must be analyzed with care since that it was not observed in the controlled experiment (Figure 9). There are two possible reasons to explain this unexpected behavior: regularization problems or the presence of additional complications, such as bias error in the direct problem. Since both Function Specification and the Dynamic Observers techniques provide regularization solutions for the inverse problem, different from Golden Section and Simulated Annealing, the most probably reason can be the presence of bias error in the model. The main bias error source can be the use of incorrect values of the heat transfer coefficient that vary with temperature. It can be observed that the longer the experimental duration the greater the influence of the heat convection [13]. Another error source can be due to nonlinearity of the thermophysical properties of the insert and tool-holder, such as thermal conductivity and diffusivity which also vary with temperature. Thermal contact resistance between the pieces and the complicated assembly geometry also represent error source. In other words, the better behavior of temperature residues of GS and SA do not guarantee that the real heat flux has been estimated. GS and SA are

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DOBGF FS

FIGURE 15 Transient residual between experimental temperatures and calculated temperatures in cutting tool.

optimization techniques that minimize the objective function for each instant of time without considering the temperature and heat flux history. In this case the heat flux estimated represent the best design variable that is adjusted to minimize the objective function without consideration of the physical problem. In this case, a better behavior just considering the temperature residues was expected. The principal difficulty is that, different from the controlled experiment, we do not have a measured heat flux to compare with the heat flux estimated. The main objective of this paper is to compare four inverse procedures applied to two distinct experiments: i) a controlled experiment and ii) a real machining process.

6 CONCLUSIONS In this paper a comparison of the inverse techniques: Golden Section, Function Specification, Simulated Annealing and Dynamics Observers Based on Green’s Function were presented. For the validation of the proposed methodology, the techniques of inverse problems were tested in controlled experiments in laboratory. All the inverse techniques presented good results in estimating the heat flux. The differences were less than 5% for all the techniques in a comparison with the measured temperature for the controlled experiment. The use of these techniques in an experimental methodology to determine the thermal fields and heat generated at the chip-tool interface during machining process also presented satisfactory results. The temperature field in any region of the tool set (insert, shim and tool-holder) was calculated from the heat flux estimated at the cutting interface. The contributions here are the development

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of a numerical 3-D transient thermal model that takes into account both the tool and the toolholder assembly and the analysis of some inverse techniques applied to the machining thermal problem.

ACKNOWLEDGEMENTS The authors would like to thank the CNPq, FAPEMIG and CAPES Government Agencies for the financial support without which this work would not be possible.

NOMENCLATURE Ac Contact area, Cp Heat capacity, h Heat transfer coefficient q (x, y, t) Unknown heat flux over the arbitrary area x by y T Calculated temperature T∞ Room temperature Initial temperature To x, y, z, Geometric variables, m Y Experimental temperature Greek Symbols α Thermal diffusivity t Time step λ Thermal conductivity ρ Density

m2 J/(kg K) W/(m2 K) W/m2 ◦C ◦C ◦C ◦C

m2 /s s W/(m K) kg/m3

REFERENCES [1] Beck, J. V., Blackwell, B. and St. Clair, C. Inverse Heat Conduction: Ill-posed Problems, Wiley-Interscience Publication, New York, 1985. [2] Blanc, G., Beck, J. V. and Raynaud, M. Solution of the inverse heat conduction problem with a time-variable number of future temperatures. Numerical Heat Transfer, Part B 32(4) (1997), 437–451. [3] Carvalho, S. R., Ong, Tiong H. and Guimarães, G. A mathematical and computational model of furnaces for continuous steel strip processing. Journal of Materials Processing Technology 178 (2006), 379–387. [4] Flach, G. P. and Ozisik, M. N. An adaptive inverse heat conduction method with automatic control. Journal of Heat Transfer 114 (1992), 5–13. [5] Lima F. R. S., Machado A. R., Guimarães G. and Guths S. Numerical and experimental simulation for heat flux and cutting temperature estimation using three-dimensional inverse heat conduction technique. Inverse Problems in Engineering 8(6) (2000), 553–577.

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[6] Lima e Silva, S. M. M., Vilarinho, L. O., Scotti, A., Ong, T. H. e Guimarães, G. Heat flux determination in the gas-tungsten-arc welding process by using a three-dimensional model in inverse heat conduction problem. High Temperatures-High Pressures 35/36 (2003), 117–126. [7] Reinhardt, H. J. Analysis of sequential methods of solving the inverse heat conduction problem. Numerical Heat Transfer, Part B 24(4) (1993), 455–474. [8] Scott, E. P. and Beck, J. V. Analysis order of the sequential regularization solutions of inverse heat conduction problems. J. Heat Transfer 111 (1989), 218–224. [9] Gonçalves, V. C., Scotti, A. and Guimarães, G. Simulated annealing inverse technique applied in welding: A theoretical and experimental approach. Proceedings of the 4th International Conference on Inverse Problems in Engineering, Angra dos Reis, Rio de Janeiro, Brazil, (2002). [10] Sousa, P. F. B., Carvalho, S. R. and Guimarães, G. Dynamic observers based on green’s functions applied to 3d inverse thermal models. Inverse Problems in Science & Engineering 16(6) (2008), 743–761. [11] Woodbury, K. A., Duvvuri, S., Chou, Y. K. and Liu, J. Use of evolutionary algorithms to determine tool heat fluxes in a machining operation, Inverse Problems Design and Optimization Symposium – IPDO2007, Miami Beach, Florida, USA, 2007. [12] Huang, C. H., Jan, L. C., Li, R. and Shih, A. J. A three-dimensional inverse problem in estimating the applied heat flux of a titanium drilling – theoretical and experimental studies. International Journal of Heat and Mass Transfer 50 (2007), 3265–3277. [13] Carvalho, S. R. Determinação do Campo de Temperatura em Ferramentas de Corte Durante um Processo de Usinagem, PhD thesis, Federal University of Uberlândia, MG, Brazil, 2005. [14] Vilarinho, L. O. Avaliação de Gases de Proteção em Soldagem Através de Técnicas Experimentais e Numéricas, Relatório de Qualificação, LAPROSOLDA No. 10/2001, Federal University of Uberlândia, Brazil, 2001. [15] Jen, T. C. and Gutierrez, G. Numerical Heat Transfer Analysis in Transient Cutting Tool Temperatures, Proceedings of the 34th National Heat Transfer Conference, Pittsburgh, Pennsylvania, 2000, 20–22. [16] Yen, D. W. and Wright, P. K. A remote temperature sensing technique for estimating the cutting interface temperature distribution. Journal of Engineering for Industry 108 (1986), 252–263. [17] Matweb, Material Property Data: http://www.matweb.com/2007. [18] Stephenson, D. A. An inverse method for investigation deformation zone temperatures in metal cutting. Journal of Engineering for Industry 133 (1991), 129–136. [19] Melo, A. C. Estimação da Temperatura de Corte Utilizando Problemas Inversos em Condução de Calor, Master degree, Federal University of Uberlândia, MG, Brazil, 1998. [20] Lin, J., Lee, S. L. and Weng, C. I. Estimation of cutting temperature in high speed machining. Journal of Engineering Materials and Technology 114 (1992), 289–296. [21] Fernandes, J. R. S. Projeto e Fabricação de um Dispositivo para Medir Temperatura de Corte no Processo de Torneamento, PIBIC – DIRPE N◦ 40–91, DEEME – UFU, 1992. [22] Machado, A. R. and Silva, M. B. Usinagem dos Metais, 8a ed., UFU, Brazil, 2004.

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