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Finite element analysis of laser tube bending process
Applied Surface Science 208±209 (2003) 437±441
Finite element analysis of laser tube bending process N. Haoa,b, L. Lia,* a
Department of Mechanical, Aerospace and Manufacturing Engineering, Laser Processing Research Centre, University of Manchester Institute of Science and Technology (UMIST), P.O. Box 88, Manchester M60 1QD, UK b Taiyuan Heavy Machinery Institute, Taiyuan 030024, PR China
Abstract Laser bending of tubes is a process in which laser-induced non-uniform thermal stress is used to deform tubes in a controlled way without hard tooling or external forces. For understanding the mechanism of laser tube bending, a thermal±mechanical ®nite element transient analysis is conducted to investigate the developments of stress and strain during laser bending process. The simulation of the moving laser beam during bending process is realized by using individual load arriving time and a special de®ned load time function. The mechanism of the laser tube bending is discussed based on the simulation result. # 2002 Elsevier Science B.V. All rights reserved. PACS: 42.62; 81.20.H; 02.60 Keywords: Laser; Bending; Tube; Deformation mechanism; Computer simulation
1. Introduction Tube bending is important in the manufacturing of boilers, engines, heat exchangers and air conditioners. In industrial practice, mechanical bending has been widely used. In mechanical bending, hard tooling is adopted to exert forces to the tube surface. The material at extrados becomes thinner as it subjected to tensile stress. This usually causes some kind of tensile failures such as neck or fracture occurring at extrados. As a spring back-free and non-contact forming technique, laser bending is achieved by plastic deformation induced by thermal stresses resulted from rapid laser local heating and cooling. Compared to mechanical bending, neither a hard tool nor external forces are required. This makes laser bending much *
Corresponding author. Tel.: 44-161-200-3816; fax: 44-161-200-3806. E-mail address: [email protected] (L. Li).
suitable for small-batch production and rapid prototyping as well as for ®t-up alignments. In addition, the laser based nature of this technique means that it could be employed to bend tubes in locations where it would otherwise be impossible to use mechanical bending, such as in outer space. Silve et al. [1] investigated procedures for laser bending of square cross-section tubes of mild steel. Different scanning sequences were compared experimentally. Kraus [2] conducted a ®nite element analysis of laser bending of square cross-section tubes and investigated the proper heating sequence. Li and Yao [3] explained the mechanism of laser tube bending by stress analysis. The in¯uence of process parameters on the geometry of bending component was investigated by experiment and numerical simulation. A closedform expression for bending angle was also proposed. Strain development during laser tube bending, which is critically important in understanding the mechanism of this process is not reported in the literature. The
0169-4332/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0169-4332(02)01429-0
438
N. Hao, L. Li / Applied Surface Science 208±209 (2003) 437±441
mechanisms of laser tube bending are not yet fully understood. In the present work, the mechanism of circular tube laser bending is studied in detail through thermal± mechanical ®nite element analysis. The traverse of the laser beam during bending has been taken into account in analysis. Temporal and spatial distributions of stress, strain and temperature obtained from simulation are used to explain the mechanism of laser tube bending. 2. Laser tube bending process The laser tube bending process is shown schematically in Fig. 1. One end of the tube is clamped while another is free. During laser bending, the tube rotates along its axis with different speeds according to process requirement while a laser, usually defocused, irradiates on the tube outer surface without causing it to melt. The scanned region of the tube undergoes rapid heating and cooling. This results in uneven thermal stress in the tube and the subsequent bending. To obtain an acceptable bending angle, the tube rotates typically 1808 or more and the laser beam size is usually (5±10) of the tube thickness. Usually the tube is scanned several passes with time intervals for cooling to obtain required bending angle. The tube for simulation is of 20 mm for its outer diameter. The thickness of the tube is 1 mm and length 100 mm. The material of the tube is common engineering low-carbon mild steel (07 M20). Material properties used in simulation are taken from [4±6]. Thermal conductivity, speci®c heat, Young's modulus and yield stress are all temperature dependent.
Fig. 1. Schematic of laser tube bending process.
The laser used for bending is a 400 W Electrox Scorpion Nd-YAG laser emitting at 1064 nm. The defocused laser beam (5 mm diameter) was ®red directly onto the tube surface with power of 400 W. The tube rotated 1808 with a speed of 98/s. The laser irradiating time per pass was 20 s. 3. Numerical simulation A three-dimensional thermal±mechanical ®nite element analysis was carried out to simulate the laser tube bending process. Commercial available general-purpose ®nite element package ADINA was used for the numerical simulation. Eight node hexahedral isoparametric elements are used in the simulation. As shown in Fig. 2, a ®ner mesh of elements is placed under the laser irradiated surface. To conduct thermal±mechanical coupling analysis, the same mesh is created for both thermal and mechanical analysis. Some assumptions are made for the numerical modelling: (a) heat generated by plastic deformation is small compared with heat input by laser beam so that it can be neglected; (b) no melting of the material occurs during laser bending; and (c) the total deformation consists of elastic strain, plastic strain and thermal strain. In thermal analysis, equilibrium of heat ¯ow in the interior of the workpiece is rcT_ r krT qab
(1)
where r is the density, c the speci®c heat, k the thermal conductivity of the material, T the temperature and qab the rate of heat generated per unit volume.
Fig. 2. Finite element mesh.
N. Hao, L. Li / Applied Surface Science 208±209 (2003) 437±441
Boundary heat transfer is modelled by convection and radiation. Convection follows Newton's law. The rate of the heat loss per unit area in Wm-2 due to convection is qc hc Ts
Ta
(2)
where hc is the coef®cient of convection heat transfer; Ts and Ta are the temperatures of the tube surface and air, respectively. The rate of the heat loss per area in Wm-2 due to radiation [7]: qr 5:67 10 8 e Ts4
Ta4
(3)
where e is the surface emissivity. The heat input by laser beam irradiation is taken as the load of heat ¯ux applied to tube surface. The simulation of moving laser beam is realized by using individual load arriving time with a special de®ned load time function. After the temperature ®eld is obtained, the stress and displacement ®eld can be calculated as a response to the thermal loading. Neglecting the dynamic effect, the quasi-static mechanical problem to be solved includes the strain±displacement relations and force equilibrium equations [7]: ui;j uj;i 2 sij;j fi 0 eij
(4) (5)
where e is the strain, u the displacement, s the stress and f the external force. A general Hooke's law is used for elastic deformation and Von Mises criterion is used as the yield criterion. They can be written as 1 1 2n s_ ij 0 dij (6) s_ m e_ eij 2G E r 3 sij 0 sij 0 Y (7) 2
439
The thermal strain rate can be written as _ e_ th ij aT
(9)
The total strain rate is the summation of the elastic, plastic and thermal components e_ ij e_ eij e_ pij e_ th ij
(10)
4. Results and discussion The centre point of intrados is chosen to study the stress and strain variation during laser bending. The developments of stress, thermal strain, plastic strain and total strain of that point in axial direction are shown in Fig. 3. It should be noted that the point of study is at the centre of the laser moving trace. The laser beam reaches this point at time of 10 s and the irradiation lasts for 5 s. It is obvious that the temperature at this point reaches its peak value at time 12 s. The variation of the temperature can be found from the thermal strain curve because they should have the same shape but different scales. The thermal strain represents the material expansion due to temperature rise. The thermal strain increases from zero to its peak value 0.03 then decreases to zero ®nally. This change is quite reasonable because the laser beam ®rst moves in then moves away, causing the temperature go up ®rst then down.
where G, E, n and Y are shear modulus, Young's modulus, Poisson's ratio and yield stress of the material, respectively. The plastic deformation of the material follows the ¯ow rule, which can be written as e_ pij sij 0 l_ where l_ is a positive proportionality constant.
(8)
Fig. 3. Variations of strain and stress in axial direction. (a) strain and (b) stress.
440
N. Hao, L. Li / Applied Surface Science 208±209 (2003) 437±441
The area of laser heating is a small portion of total tube surface. This area with high temperature tends to expand quickly, while other areas still in lower temperature tend to expand in a small scale. The difference of expansion and the integrality of the material cause a thermal stress in the material. The heated area exerts pressure to its surrounding area and the surrounding area exerts pressure on the heated area inversely. That is, the material in heated area is in thermal expansion and mechanical elastic compression. At a certain temperature, which is dependent upon the material and the geometry, the thermal strains reach the maximum elastic strain that the material can endure. Further increase in the temperature would result in a conversion of the thermal expansion into plastic compressive strains. The conversion of thermal expansion into plastic compressive strain could be found in strain curves. At the beginning of laser heating, the material expands due to thermal expansion. When the heating continues, the thermal expansion begins to convert into plastic compression until the cooling period begins. Cooling is mainly due to self-quenching, with the heat ¯owing into the surrounding material resulting in cooling of the heated area within 10±30 s. During cooling, material not only recovers to its original dimension but also shrinkages further, resulting in a shorter dimension. This is due to the fact that the material is plastically compressed already during heating. It is interesting to see that the compressive plastic strain recovers a little during cooling. This is due to the fact that ¯ow stress of heated material is much lower than its surrounding material. For enhancing the bending ef®ciency, the plastic strain recover should be avoided by carefully chosen the laser power and tube rotating speed to reach a suitable maximum heating temperature. The shrinkage of the heated material results in a strain of 0.03 in axial direction. This means the material in that area is shortened. Due to the different lengths of the intrados and the extrados of the tube, a bending angle develops towards the laser beam. Compared to strain variation, the stress variation is more complex. Basically, there are two factors in¯uencing the stress state. One is the thermal expansion and shrinkage of materials due to temperature change both inside and outside of the area for studied. Another is the variation of ¯ow stress of the material due to
Fig. 4. Comparison of experiment and simulation.
temperature variation. At the beginning, the laser heats another area, the thermal expansion of that area exerts tensile stress to this point. When the laser beam moves in, temperature of this point is raised. The negative thermal stress increases rapidly and the stress state changes from tensile to compressive. When temperature increases further, the compressive stress decreases due to the decreasing of ¯ow stress. When cooling begins, the shrinkage of the material makes the stress back to in tensile. Towards the end of laser scanning, the stress in axial direction becomes negative again. This is quite different from the mechanical bending. It is because the temperature at the end of scanning is higher than that at beginning. Higher temperature produces more shrinkage of material. As a result, the material in the centre of laser scanning trace becomes compression. To validate the simulation result, experiments with the same parameters of simulation were conducted (see Section 2). Multiple pass experiments were carried out. Due to the complexity of laser bending, the measurement of dynamic the variation of stress and strain during laser bending was not feasible. Only the ®nal bending angle was compared between simulation and analysis. Fig. 4 shows the comparison of ®nal bending angle with different scanning passes. It can be seen that the trends are in good agreement. 5. Conclusions The development of stress and strain during laser tube bending had been investigated by thermal± mechanical ®nite element transient analysis and the mechanism of laser tube bending was discussed based
N. Hao, L. Li / Applied Surface Science 208±209 (2003) 437±441
on simulation result. During laser local heating, thermal expansion of material in heated area is converted into plastic compressive strain due to its surrounding restriction. This makes the material of intrados of the tube become shorter than that of extrados after cooling and the difference in length enables the tube bend. Plastic strain recovery during cooling would emerge if the process parameters are unsuitable. The negative stress in intrados of ®nal component is caused by uneven temperature distribution during laser scanning. Acknowledgements One of the authors, N Hao, gratefully acknowledges the Shanxi Scholarship Council of China for supporting him conducting research in UMIST as an academic visitor.
441
References [1] S. Silve, W.M. Steen, B. Podschies, Laser forming tubes: a discussion of principles, in: Proceedings of ICALEO, Section E, 1998, pp. 151±160. [2] J. Kraus, Basic process in laser bending of extrusion using the upsetting mechanism, Proc. LANE 2 (1997) 431±438. [3] W. Li, Y.L. Yao, Laser bending of tubes: mechanism, analysis, and prediction, ASME J. Manuf. Sci. Eng. 123 (2001) 674± 681. [4] ASM Handbook Committee, Metal Handbook: Properties and SelectionÐIrons, Steels, and High Performance Alloys, vol. 1, ASM International, Ohio, 1990. [5] H. Arnet, F. Vollertsen, Extending laser bending for the generation of convex shapes, ImechE Part B J. Eng. Manuf. 209 (1995) 433±442. [6] J. Lawrence, M.J.J. Schmidt, L. Li, The forming of mild steel plates with a 2.5 kW high power diode laser, Int. J. Mach. Tool Manuf. 41 (2001) 967±977. [7] K.J. Bathe, Finite Element Procedures, Prentice-Hall, Englewood Cliffs, 1996.
Finite element analysis of laser tube bending process N. Haoa,b, L. Lia,* a
Department of Mechanical, Aerospace and Manufacturing Engineering, Laser Processing Research Centre, University of Manchester Institute of Science and Technology (UMIST), P.O. Box 88, Manchester M60 1QD, UK b Taiyuan Heavy Machinery Institute, Taiyuan 030024, PR China
Abstract Laser bending of tubes is a process in which laser-induced non-uniform thermal stress is used to deform tubes in a controlled way without hard tooling or external forces. For understanding the mechanism of laser tube bending, a thermal±mechanical ®nite element transient analysis is conducted to investigate the developments of stress and strain during laser bending process. The simulation of the moving laser beam during bending process is realized by using individual load arriving time and a special de®ned load time function. The mechanism of the laser tube bending is discussed based on the simulation result. # 2002 Elsevier Science B.V. All rights reserved. PACS: 42.62; 81.20.H; 02.60 Keywords: Laser; Bending; Tube; Deformation mechanism; Computer simulation
1. Introduction Tube bending is important in the manufacturing of boilers, engines, heat exchangers and air conditioners. In industrial practice, mechanical bending has been widely used. In mechanical bending, hard tooling is adopted to exert forces to the tube surface. The material at extrados becomes thinner as it subjected to tensile stress. This usually causes some kind of tensile failures such as neck or fracture occurring at extrados. As a spring back-free and non-contact forming technique, laser bending is achieved by plastic deformation induced by thermal stresses resulted from rapid laser local heating and cooling. Compared to mechanical bending, neither a hard tool nor external forces are required. This makes laser bending much *
Corresponding author. Tel.: 44-161-200-3816; fax: 44-161-200-3806. E-mail address: [email protected] (L. Li).
suitable for small-batch production and rapid prototyping as well as for ®t-up alignments. In addition, the laser based nature of this technique means that it could be employed to bend tubes in locations where it would otherwise be impossible to use mechanical bending, such as in outer space. Silve et al. [1] investigated procedures for laser bending of square cross-section tubes of mild steel. Different scanning sequences were compared experimentally. Kraus [2] conducted a ®nite element analysis of laser bending of square cross-section tubes and investigated the proper heating sequence. Li and Yao [3] explained the mechanism of laser tube bending by stress analysis. The in¯uence of process parameters on the geometry of bending component was investigated by experiment and numerical simulation. A closedform expression for bending angle was also proposed. Strain development during laser tube bending, which is critically important in understanding the mechanism of this process is not reported in the literature. The
0169-4332/02/$ ± see front matter # 2002 Elsevier Science B.V. All rights reserved. doi:10.1016/S0169-4332(02)01429-0
438
N. Hao, L. Li / Applied Surface Science 208±209 (2003) 437±441
mechanisms of laser tube bending are not yet fully understood. In the present work, the mechanism of circular tube laser bending is studied in detail through thermal± mechanical ®nite element analysis. The traverse of the laser beam during bending has been taken into account in analysis. Temporal and spatial distributions of stress, strain and temperature obtained from simulation are used to explain the mechanism of laser tube bending. 2. Laser tube bending process The laser tube bending process is shown schematically in Fig. 1. One end of the tube is clamped while another is free. During laser bending, the tube rotates along its axis with different speeds according to process requirement while a laser, usually defocused, irradiates on the tube outer surface without causing it to melt. The scanned region of the tube undergoes rapid heating and cooling. This results in uneven thermal stress in the tube and the subsequent bending. To obtain an acceptable bending angle, the tube rotates typically 1808 or more and the laser beam size is usually (5±10) of the tube thickness. Usually the tube is scanned several passes with time intervals for cooling to obtain required bending angle. The tube for simulation is of 20 mm for its outer diameter. The thickness of the tube is 1 mm and length 100 mm. The material of the tube is common engineering low-carbon mild steel (07 M20). Material properties used in simulation are taken from [4±6]. Thermal conductivity, speci®c heat, Young's modulus and yield stress are all temperature dependent.
Fig. 1. Schematic of laser tube bending process.
The laser used for bending is a 400 W Electrox Scorpion Nd-YAG laser emitting at 1064 nm. The defocused laser beam (5 mm diameter) was ®red directly onto the tube surface with power of 400 W. The tube rotated 1808 with a speed of 98/s. The laser irradiating time per pass was 20 s. 3. Numerical simulation A three-dimensional thermal±mechanical ®nite element analysis was carried out to simulate the laser tube bending process. Commercial available general-purpose ®nite element package ADINA was used for the numerical simulation. Eight node hexahedral isoparametric elements are used in the simulation. As shown in Fig. 2, a ®ner mesh of elements is placed under the laser irradiated surface. To conduct thermal±mechanical coupling analysis, the same mesh is created for both thermal and mechanical analysis. Some assumptions are made for the numerical modelling: (a) heat generated by plastic deformation is small compared with heat input by laser beam so that it can be neglected; (b) no melting of the material occurs during laser bending; and (c) the total deformation consists of elastic strain, plastic strain and thermal strain. In thermal analysis, equilibrium of heat ¯ow in the interior of the workpiece is rcT_ r krT qab
(1)
where r is the density, c the speci®c heat, k the thermal conductivity of the material, T the temperature and qab the rate of heat generated per unit volume.
Fig. 2. Finite element mesh.
N. Hao, L. Li / Applied Surface Science 208±209 (2003) 437±441
Boundary heat transfer is modelled by convection and radiation. Convection follows Newton's law. The rate of the heat loss per unit area in Wm-2 due to convection is qc hc Ts
Ta
(2)
where hc is the coef®cient of convection heat transfer; Ts and Ta are the temperatures of the tube surface and air, respectively. The rate of the heat loss per area in Wm-2 due to radiation [7]: qr 5:67 10 8 e Ts4
Ta4
(3)
where e is the surface emissivity. The heat input by laser beam irradiation is taken as the load of heat ¯ux applied to tube surface. The simulation of moving laser beam is realized by using individual load arriving time with a special de®ned load time function. After the temperature ®eld is obtained, the stress and displacement ®eld can be calculated as a response to the thermal loading. Neglecting the dynamic effect, the quasi-static mechanical problem to be solved includes the strain±displacement relations and force equilibrium equations [7]: ui;j uj;i 2 sij;j fi 0 eij
(4) (5)
where e is the strain, u the displacement, s the stress and f the external force. A general Hooke's law is used for elastic deformation and Von Mises criterion is used as the yield criterion. They can be written as 1 1 2n s_ ij 0 dij (6) s_ m e_ eij 2G E r 3 sij 0 sij 0 Y (7) 2
439
The thermal strain rate can be written as _ e_ th ij aT
(9)
The total strain rate is the summation of the elastic, plastic and thermal components e_ ij e_ eij e_ pij e_ th ij
(10)
4. Results and discussion The centre point of intrados is chosen to study the stress and strain variation during laser bending. The developments of stress, thermal strain, plastic strain and total strain of that point in axial direction are shown in Fig. 3. It should be noted that the point of study is at the centre of the laser moving trace. The laser beam reaches this point at time of 10 s and the irradiation lasts for 5 s. It is obvious that the temperature at this point reaches its peak value at time 12 s. The variation of the temperature can be found from the thermal strain curve because they should have the same shape but different scales. The thermal strain represents the material expansion due to temperature rise. The thermal strain increases from zero to its peak value 0.03 then decreases to zero ®nally. This change is quite reasonable because the laser beam ®rst moves in then moves away, causing the temperature go up ®rst then down.
where G, E, n and Y are shear modulus, Young's modulus, Poisson's ratio and yield stress of the material, respectively. The plastic deformation of the material follows the ¯ow rule, which can be written as e_ pij sij 0 l_ where l_ is a positive proportionality constant.
(8)
Fig. 3. Variations of strain and stress in axial direction. (a) strain and (b) stress.
440
N. Hao, L. Li / Applied Surface Science 208±209 (2003) 437±441
The area of laser heating is a small portion of total tube surface. This area with high temperature tends to expand quickly, while other areas still in lower temperature tend to expand in a small scale. The difference of expansion and the integrality of the material cause a thermal stress in the material. The heated area exerts pressure to its surrounding area and the surrounding area exerts pressure on the heated area inversely. That is, the material in heated area is in thermal expansion and mechanical elastic compression. At a certain temperature, which is dependent upon the material and the geometry, the thermal strains reach the maximum elastic strain that the material can endure. Further increase in the temperature would result in a conversion of the thermal expansion into plastic compressive strains. The conversion of thermal expansion into plastic compressive strain could be found in strain curves. At the beginning of laser heating, the material expands due to thermal expansion. When the heating continues, the thermal expansion begins to convert into plastic compression until the cooling period begins. Cooling is mainly due to self-quenching, with the heat ¯owing into the surrounding material resulting in cooling of the heated area within 10±30 s. During cooling, material not only recovers to its original dimension but also shrinkages further, resulting in a shorter dimension. This is due to the fact that the material is plastically compressed already during heating. It is interesting to see that the compressive plastic strain recovers a little during cooling. This is due to the fact that ¯ow stress of heated material is much lower than its surrounding material. For enhancing the bending ef®ciency, the plastic strain recover should be avoided by carefully chosen the laser power and tube rotating speed to reach a suitable maximum heating temperature. The shrinkage of the heated material results in a strain of 0.03 in axial direction. This means the material in that area is shortened. Due to the different lengths of the intrados and the extrados of the tube, a bending angle develops towards the laser beam. Compared to strain variation, the stress variation is more complex. Basically, there are two factors in¯uencing the stress state. One is the thermal expansion and shrinkage of materials due to temperature change both inside and outside of the area for studied. Another is the variation of ¯ow stress of the material due to
Fig. 4. Comparison of experiment and simulation.
temperature variation. At the beginning, the laser heats another area, the thermal expansion of that area exerts tensile stress to this point. When the laser beam moves in, temperature of this point is raised. The negative thermal stress increases rapidly and the stress state changes from tensile to compressive. When temperature increases further, the compressive stress decreases due to the decreasing of ¯ow stress. When cooling begins, the shrinkage of the material makes the stress back to in tensile. Towards the end of laser scanning, the stress in axial direction becomes negative again. This is quite different from the mechanical bending. It is because the temperature at the end of scanning is higher than that at beginning. Higher temperature produces more shrinkage of material. As a result, the material in the centre of laser scanning trace becomes compression. To validate the simulation result, experiments with the same parameters of simulation were conducted (see Section 2). Multiple pass experiments were carried out. Due to the complexity of laser bending, the measurement of dynamic the variation of stress and strain during laser bending was not feasible. Only the ®nal bending angle was compared between simulation and analysis. Fig. 4 shows the comparison of ®nal bending angle with different scanning passes. It can be seen that the trends are in good agreement. 5. Conclusions The development of stress and strain during laser tube bending had been investigated by thermal± mechanical ®nite element transient analysis and the mechanism of laser tube bending was discussed based
N. Hao, L. Li / Applied Surface Science 208±209 (2003) 437±441
on simulation result. During laser local heating, thermal expansion of material in heated area is converted into plastic compressive strain due to its surrounding restriction. This makes the material of intrados of the tube become shorter than that of extrados after cooling and the difference in length enables the tube bend. Plastic strain recovery during cooling would emerge if the process parameters are unsuitable. The negative stress in intrados of ®nal component is caused by uneven temperature distribution during laser scanning. Acknowledgements One of the authors, N Hao, gratefully acknowledges the Shanxi Scholarship Council of China for supporting him conducting research in UMIST as an academic visitor.
441
References [1] S. Silve, W.M. Steen, B. Podschies, Laser forming tubes: a discussion of principles, in: Proceedings of ICALEO, Section E, 1998, pp. 151±160. [2] J. Kraus, Basic process in laser bending of extrusion using the upsetting mechanism, Proc. LANE 2 (1997) 431±438. [3] W. Li, Y.L. Yao, Laser bending of tubes: mechanism, analysis, and prediction, ASME J. Manuf. Sci. Eng. 123 (2001) 674± 681. [4] ASM Handbook Committee, Metal Handbook: Properties and SelectionÐIrons, Steels, and High Performance Alloys, vol. 1, ASM International, Ohio, 1990. [5] H. Arnet, F. Vollertsen, Extending laser bending for the generation of convex shapes, ImechE Part B J. Eng. Manuf. 209 (1995) 433±442. [6] J. Lawrence, M.J.J. Schmidt, L. Li, The forming of mild steel plates with a 2.5 kW high power diode laser, Int. J. Mach. Tool Manuf. 41 (2001) 967±977. [7] K.J. Bathe, Finite Element Procedures, Prentice-Hall, Englewood Cliffs, 1996.