Laser Cutting Process | Torch Path Planning

A study on torch path planning in laser cutting processes Part 1: Calculation of heat flow in contour laser beam cutting

Han, Guk-Chan

Abstract

Conductive heat transfer in contour laser beam cutting is analyzed by using a transient, two-dimensional finite difference model, and the result is combined with a simple analytic model. From the calculation results, the correlation is derived between workpiece temperature and opening angles at a corner in contour cutting. As a result, a modified analytic solution is developed to predict if excessive workpiece heating occurs for given cutting contours in a nested plate. The main objective is to use the computation results in the optimization of torch path planning to provide fully automated CNC programming software for laser cutting. To efficiently apply the analytic model in torch path planning, the critical temperature that should be avoided during the cutting sequence is considered. This leads to an improvement of the cutting quality in the automatic cutting process.

Keywords: Contour Laser Beam Cutting, Finite Difference Method, Nonstationary State Heat Conduction, Opening Angle, Critical Temperature, Torch Path Planning

Introduction

Recently, the major objective of laser processing in a production line is the precision or shape-cutting of metallic and nonmetallic thin sheets.' The cutting process is carried out by melting and/or evaporating a material with a focused Gaussian laser beam as a heat source. For a detailed description of the physical mechanism, the theory of laser cutting is given in the references.2-4 Compared with conventional cutting tools, the advantages of using the laser beam are small kerf width and high cutting velocity combined with a narrow heat-affected zone. In combination with a computer-controlled CNC machine, the laser beam can easily produce complex part geometries; however, in the case of complex workpiece geometries, undesirable reduction in cutting quality may result. One cause of this is workpiece preheating as a result of heat conduction. With useful tools or guidelines for determining optimum cutting parameters and sequences in the production line, improvement in cut quality and automation of cutting processes can be expected. For this purpose, numerical calculation methods such as the finite element and finite difference methods can be adopted; however, the simulation of total cutting sequences using these numerical methods is ineffective because of the exhaustive computing time.

Most of the theoretical works on heat transfer in laser cutting have centered on the solution of the classical heat conduction equation for the stationary or quasi-stationary state. The most significant previous work on the analytic solution of heat flow in welding/cutting was done by Rosenthal for a point heat source in three-dimensional analysis or a line heat source in two-dimensional analysis.5,6 The modeling of temperature distributions induced by a distributed heat source was pioneered by Eagar and Tsai7 for a traveling Gaussian intensity distribution and by Ashby and Shercliff8 for a traveling hypersurface line heat source. With the growing interest in laser processing, these models have been investigated further. Most of the theoretical descriptions of the heat conduction are often connected to the Rosenthal solution, and even today some of the solutions obtained by Rosenthal are widely used; however, analytic solutions of the heat conduction equation are inadequate for the temperature profiles within a finite plate or for the cutting of particular contours. Therefore, the temperature field occurring during the contour cutting with the laser beam should be calculated by using a numerical model.

In investigations of the numerical model of the laser cutting process, a three-dimensional heat transfer model was developed by Mazumder and Steen9 for a laser striking the surface of an opaque substrate moving with uniform velocity. The model was solved by the finite difference method, and its results were presented for the temperature distribution and melt depth. Kim and Majumdar10 developed a two-dimensional finite element model based on the transient heat conduction to analyze the material removal process using the laser beam. The numerical experimentation was carried out for mesh refinements and the rate of convergence in terms of groove shape and temperature. Hillebrand, Decker, and Wohlfahr11 solved the workpiece heating during contour laser cutting by using the finite element method. The calculation results showed a possibility of the cutting sequence optimization for complex part geometries and also of the quality assurance. Although these computational models are considerably precise, investigations to optimize the cutting path problem concerning the heat effect have not been presented in the literature.

When the main goal is to examine the temperature distribution along a simple cutting contour, it is convenient to use a simple analytic model. Unfortunately, analytic models of heat conduction are unsuitable for solving the cutting problem of complex contours because the heat-affected zone and/or kerf previously produced affects the current heat conduction. Therefore, the objective of this paper is to develop a modified analytic model for laser beam cutting of complex contours based on the finite difference model. The modified analytic solution should provide a reasonable temperature prediction having a margin of safety for a given cutting contour. This mathematical model can be applied as a tool for avoiding the critical temperature that causes workpiece overheating during the laser cutting process, and then can be applied for carrying out torch path optimization incorporating the heat effect.

Numerical Approach

Physical Definition of Laser Cutting Process

For the cutting contour illustrated in Figure 1, the temperature field is calculated using the finite difference method, and the results are combined with the analytic solution. To develop the mathematical model, the laser cutting process is physically defined as follows:

1. The cross section of a laser beam, moving with uniform velocity, is assumed to be square, having a constant power distribution striking the surface of the workpiece. The laser beam contributes the energy to the workpiece.

2. Heat conduction in the direction of depth is negligible because the workpiece is a thin sheet having a finite width and length.

3. Constant material characteristics, such as thermal conductivity, specific heat, and density, are adopted during the change in temperature.

4. Absence of convective and radiative heat flow is assumed in the formulation.

5. After removal of the molten material along the cutting line, the side walls of the kerf produced are treated as an adiabatic boundary.

6. Influence of the material cooling caused by the cutting gas jet is neglected in calculations.

The advantages of laser cutting can be described by a very simple calculation based on the geometry shown in Figure 1 and the physical definitions given above.

Finite Difference Formulation

The transient heat flow in a two-dimensional workpiece with internal heat generation is governed by the energy conservation equation in the Cartesian coordinate system, as follows:

Analytic Solution

Most of the analytic solutions conducted so far on the heat flow in the workpiece have been on the temperature distribution in the quasi-stationary state. In the quasi-stationary state, in which the temperature distribution is stationary in a coordinate system that moves with the heat source, the mathematical analysis is simple because the problem can be treated as a steady heat flow problem for the moving coordinate. In the area near the start and end of a cut, however, the heat flow is in the nonstationary state. When the cutting is performed over a short length, therefore, the quasi-stationary state is never reached.

In this paper, the nonstationary state heat conduction equation is used as the basic principle. Under the assumptions that the cutting velocity is fast enough, the laser power is high enough, and the heat transfer is only in directions perpendicular to the cutting direction, a useful analytic solution can be derived from the solution presented by Carslaw and Jaeger.12 Figure 4 shows the comparison between the heat flow model in a thin rod and the typical laser cutting model with the laser beam moving at a speed of V. The solution of the temperature distribution at time t, T (xi, t), produced by an instantaneous heat source in a thin rod is expressed as follows (Figure 4a):

where T^sub o^ is the initial temperature, Q' is the net energy input per unit area, xi is the shortest distance between the heat source and point selected to be calculated, and h is the heat flow factor by convection effect on the surface. Because the traveling line heat source can be treated as an instantaneous plane source (Figure 4b), the incident energy density Q' is written as q/Vd in the thin plate cutting, where q is the net input energy per unit time. Also, based on the assumption that there is no convective heat flow through the upper and lower surface, the nonstationary state solution for the temperature distribution in laser beam cutting can be expressed as follows:

The surface temperature distribution around the laser beam obtained by Eq. (3) is presented in Figure 5. The assumptions used in this analysis include the moving line heat source and the constant average thermal properties.

Result of Calculations

Type 304 stainless steel with 1.0 mm thickness is considered as the material for simulations, with the cutting speed of 2 m/min and the laser output power of 250 W The material properties are listed in Table 1. To demonstrate the cutting simulation, opening angles of beta = 180 deg, beta = 135 deg, beta = 90 deg, beta = 63 deg, and beta= 45 deg are selected. Each cutting model for a given opening angle adopts the same cutting condition and cutting length.

The temperature profiles around the laser beam obtained by the finite difference model for beta = 180 deg are compared with the solution of the nonstationary state heat conduction equation [Eq. (3)] in Figure 6. The simulation results obtained by the numerical and analytical model showed very similar temperature gradients for the same cutting condition, except at the side regions of the plate. This difference of temperature gradient may result from the fact that the finite difference model is applied to a finite plate, while the analytic model is applied to an infinite plate. The effect of these differences may decrease as the size of the solution domain increases.

Figure 7 shows the calculation results of isothermal lines for different opening angles. When the laser beam changes its cutting direction at a corner, two types of heat flow are expected. At small opening angles, the burnoff of the corner may take place. This is due to the reduced dissipation of heat caused by two adiabatic sides of the kerf. After the advance of the heat source and a change in the cutting direction at the corner, the fusion zone moves through the workpiece region at higher temperatures. Consequently, the applied heat is dispersed less in the right-hand side than in the left-hand side because two adiabatic boundaries of the kerf have the blocking effect against the heat flow. As a result, the heat accumulation occurs in the side of small opening angles, and a steep temperature gradient is formed. Consequently, a reduction in the cutting quality must be expected at the corner with small opening angles. At the region of large opening angles, however, there is no reduction in the cutting quality along the kerf because the heat applied by laser beam can be easily dispersed in the plate. Simulation results (Figures 7a-7d are in good agreement with the theoretical background mentioned above.

Application of Calculations

On the basis of the finite difference and analytic analysis, a modified analytic model can be introduced to predict the temperature field in laser cutting of complex part geometries. A possible use of this tool is the determination of optimized cutting sequence plans. Figure 8 shows the temperature profiles calculated from the numerical and analytical model for the identical cutting path and cutting parameters. It can be used also as a typical example of illustrating the modification procedure of the analytic model. The temperature profiles plotted by solid lines are simulation results obtained by the finite difference model for an opening angle of beta = 90 deg, while the dashed lines denote the results by the simple analytic model for beta = 180 deg. The black point in the figure is located on the 400 deg C isothermal line for the numerical solution, while it is on the 300 deg C isothermal line for the analytic solution. In this case, the temperature determined by the numerical model is 1.33 times as high as that determined from the analytic model. One of the methods of efficiently considering the heat accumulation effect in the contour cutting process is to use the modified temperature obtained by multiplying a factor to the calculation result of the nonstationary state heat conduction equation [Eq.(3)]. The analytic temperature distribution can be modified for various opening angles as follows:

Critical Temperature

A great number of variables affect the results of laser cutting. They can be divided into materialrelated, laser-related, and process-related variables. The other cutting variable that affects cutting quality is the initial plate temperature. In general, workpiece preheating can result in several advantages for laser cutting. It can improve the cutting operation by an increased cutting speed and reduced temperature gradient in the plate; however, when many contours with complicated shapes are nested and cut with a constant cutting speed and laser output power, workpiece preheating may cause cutting quality deterioration. This is due to the fact that the contour to be cut in the complicated nested plate can be excessively preheated by the heat accumulation from the previous cutting contours.

In this paper, the allowable increase of kerf width is considered as a criterion to judge cutting quality because kerf width is one of the most important factors in precision cutting of small parts. To determine the critical temperature for a given material and cutting condition, a simple lumped heat capacity equation [Eq. (4)] based on the heat balance of the material removed in fusion cutting is used as shown in Figure 10, for which the heat balance equation can be written as follows:

Concluding Remarks

By comparing the results of the numerical and analytical model of heat conduction, a useful tool was developed to predict the temperature rise for various opening angles in contour laser beam cutting. To calculate the temperature field in the cutting of contours, the two-dimensional finite difference model based on the transient heat conduction was introduced. As a basis of the analytic solution, the nonstationary state heat conduction equation was adopted to develop the modification factor. The modified analytic solution was then used for temperature predictions in contour cutting, especially for small opening angles. The critical temperature was defined as an allowable kerf width that increases with the increasing initial temperature of the plate. This approach is used in Part 2 of this paper to optimize the cutting sequence planning, which can be further used in efficiently developing an automated CNC program for the laser cutting process.

References

1. A. Matsunawa, "The State of the Arts of Laser Materials Processing and Their Future Subjects," KWS Spring Conf. (Apr. 1995), pp3-12.

2. D. Schuocker, "Laser Cutting," The Industrial Laser Annual Handbook (1986 ed.), pp87-107.

3. D. Schuocker, "The Physical Mechanism and Theory of Laser Cutting," The Industrial Laser Annual Handbook (1987), pp65-79.

4. WM. Steen, Laser Material Processing (London: Springer-Verlag, 1991).

5. D. Rosenthal, "Mathematical Theory of Heat Distribution During Welding and Cutting," Welding Research Supplement (May 1941), pp220-234.

6. D. Rosenthal, "The Theory of Moving Sources of Heat and Its Application to Metal Treatments," Trans. of SSME (Nov.1946), pp849-864.

7. T.W. Eagar and N.S. Tsai, "Temperature Fields Produced by Traveling Distributed Heat Sources," Welding Research Supplement (Dec. 1983), pp346-355.

8. H.R. Shercliff and M.F Ashby,"Master Plots for Predicting the CaseDepth in Laser Surface Treatments," CUED/C-Mat. (TR134, Oct. 1986).

9. J. Mazumder and WM. Steen, "Heat Transfer Model for CW Laser Material Processing," Journal of Applied Physics (v51, n2, Feb. 1980), pp941-947.

10. M.J. Kim and P Majumdar, "Computational Model for High Energy Laser Cutting Process," Numerical Heat Transfer (Part A, v27, 1995), pp717-733.

11. A. Hillebrand, 1. Decker, and H. Wohlfahrt, "Calculation of Workpiece Heating During Contour Cutting by Laser Beam with the Finite Element Method," Schweiben und Schneiden (v45, 1993), pp487-490.

12. H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids (Oxford, UK: Oxford Univ. Press, 1959).

Guk-chan Han, Production Engineering Center, Samsung Electronics Co., Suwon, Kyungki-do, South Korea E-mail: This e-mail address is being protected from spam bots, you need JavaScript enabled to view it

Suck-joo Na, Dept. of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Taejon, South Korea. E-mail: This e-mail address is being protected from spam bots, you need JavaScript enabled to view it

Guk-Chan Han received his BS in mechanical design from Sung-KyunKwan University (Korea) in 1989 and his MS in precision engineering and mechatronics in 1992 and PhD in mechanical engineering in 1996 from the Korea Advanced Institute of Science and Technology (KAIST) In 1996, he joined Samsung Electronics as a senior researcher He is currently doing research in computerized numerical control systems, especially the development of CNC control software and its applications. His research interests include laser materials processing, combinatorial and stochastic optimization techniques, simulated annealing algorithms and their application, part nesting problems, open CNC architecture, and high-speed motion control algorithms.

Suck-Joo Na received his BS in mechanical engineering from Seoul National University (Korea) in 1975, his MS in mechanical engineering from the Korea Advanced Institute of Science and Technology (KAIST) in 1977, and his Dr.Ing. in welding engineering from TU Braunschweig (Germany) in 1983. In 1983, he joined KAIST to research and lecture on welding and other thermal processes. He is now working mostly in the field of heat and mass flow in the welding process, laser materials processing, vacuum brazing of metals and ceramics, and automation of welding and cutting processes for development of seam tracking sensors and automatic nesting systems. Dr.1a is a member of the Korean Society of Mechanical Engineers, the Korean Welding Society, the American Welding Society, and the German Welding Society.