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3D finite element analysis of evaporative laser cutting

Meung Jung Kim *

Department of Mechanical Engineering, Northern Illinois University, DeKalb, IL 60115, USA

Received 1 June 2004; received in revised form 1 December 2004; accepted 8 February 2005

Available online 17 May 2005

Abstract

A three-dimensional computational model of evaporative laser-cutting process has been developed usinga finite element method. Steady heat transfer equation is used to model the laser-cutting process with amoving laser. The laser is assumed continuous wave Gaussian beam. The finite element surfaces on eva-poration side are nonplanar and approximated by bilinear polynomial surfaces. Semi-infinite elementsare introduced to approximate the semi-infinite domain. An iterative scheme is used to handle the geometric

nonlinearity due to the unknown groove shape. The convergence studies are performed for various meshes.Numerical results about groove shapes and temperature distributions are presented and also compared withthose by semi-analytical methods. 2005 Elsevier Inc. All rights reserved.

Keywords: Finite element method; Evaporative laser cutting; Geometric nonlinearity; Groove shapes; Semi-infiniteelements

1. Introduction

The laser that was invented in 60s has found applications in many manufacturing processes pri-marily due to its precision process and high intensity [13]. The quality of the laser cut is of theutmost importance in laser processing because it would lead to an elimination of post-machiningoperations. Any improvement in laser cut quality would be of considerable significance.

0307-904X/$ - see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.apm.2005.02.015

* Tel.: +1 815 753 9965/9979; fax: +1 815 753 0416.E-mail address: [email protected]

www.elsevier.com/locate/apm

Applied Mathematical Modelling 29 (2005) 938954
mailto:[email protected]:[email protected]

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Nomenclature

Bi Biot numberc specific heathig heat of sublimationh convection heat transfer coefficienti; k unit vectors in the X- and Z-directions, respectivelyI0 laser power density at the center of the beamk thermal conductivityn unit outward surface normalNe evaporation parameterNk conduction parameterqk conduction heat fluxqh convection heat fluxqig heat flux due to material evaporationqL heat flux due to laser radiationRo laser beam radius at the focal pointS(X, Y) groove depths(x, y) non-dimensional groove depthS1

final groove depthT temperatureT1

ambient temperatureTevap evaporation temperature

t non-dimensional timeu non-dimensional laser moving velocityU moving specimen or laser velocityx, y, z dimensionless spatial coordinatesX, Y, Z spatial coordinatesx1, x2, z1, z2 nodal coordinates of an elementxF, yF, zF half the x, y, z-dimensions of the specimenxmin, xmax starting and ending x-coordinates of melting region on the specimenzi surface nodal z-coordinate at ith positionznewi new surface nodal z-coordinate at ith positionznew

i;zold

i! zactual

iactual nodal z-coordinate for iterative computation

Greek letters

ao absorptivitya thermal diffusivitye convergence limit for temperature and position1, g, n dimensionless spatial coordinates of a field pointq densityh dimensionless temperature

M.J. Kim / Applied Mathematical Modelling 29 (2005) 938954 939

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High intensity laser beam can be directed to a narrow region in order to instantly evaporatematerial with very narrow heat affected zone. This ability to cut instantly with extremely narrowlaser beam distinguishes it from other cutting methods. The key to success in precision cut by laserdepends on many factors such as laser characteristics, material properties of the specimen, andmanufacturing parameters. In precision manufacturing the quality of the cut is often measuredbased on the shape of the groove and amount of material removal. Therefore, better understand-ing of the process and thereby the roles of various parameters are essential to successful applica-tions of laser-cutting process.

There have been numerous investigations on laser applications. Some [48] investigated statesof stresses in fracture, chemical compositions and properties, and heat transfer on different typesof materials such as metals, composites, ceramics, and metallic glasses. Others [913] studied heattreatment effects on the material by laser irradiation. Considerable researches [1418] have beendone with heat transfer models on the effects of laser characteristics and material properties forthe quality of laser processes. Other researches [1,1921] are also found about pulsed lasers, melt-ing of thin films, and reflections.

As many applications need to take the melting into consideration, there are also many applica-tions that rely on material evaporation such as cutting plastics and organic materials in medical

operations. In addition, the current method for evaporative cutting can be extended to includemelting pool in the future. This paper primarily focuses on the implementation of a three-dimen-sional finite element method for the first time in order to predict the groove shapes in evaporativelaser cutting as an extension of the previous works by Kim et al. [2225].

2. Mathematical formulation

A typical laser cutting installation is shown in Fig. 1. The typical processes involved in eva-aporative laser cutting are thermal in nature. When a laser beam strikes a material surface, several

effects take place: reflection and absorption of the beam; conduction of heat into the material andloss of heat by convection and/or radiation from the material surface. The amount of energyabsorbed and utilized in removing the material depends on the optical and thermo-physical pro-perties of the material. The mathematical model describing the process of material removal fromthe surface subjected to high intensity laser beam can be found in Modest and Abakians[17,20]and Kim et al. [2225].

It is assumed that there are three different regions on the surface subjected to laser beam asshown in Fig. 2. Region I is too far from the laser to have reached evaporation temperature, re-gion II is the area in which evaporation takes place, and region III is the region in which evapo-ration has already taken place.

hevap dimensionless evaporation temperature (hevap = 1)hi,m, hi,m+1 dimensionless temperatures at nodes zi,m and zi,m+1

k relaxation factors time

940 M.J. Kim / Applied Mathematical Modelling 29 (2005) 938954

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Following assumptions are also made in deriving the model:

(1) laser beam is of Gaussian type in a continuous mode,(2) material moves at a constant relative velocity,(3) material is isotropic and opaque with constant thermal and optical properties,(4) material removal is a surface phenomenon and phase change from solid to vapor occurs in

one step,(5) evaporated material is transparent and does not interfere with incident laser beam,(6) heat losses by convection and radiation from the surfaces to the environment can be approxi-

mated by using a single constant convection coefficient.

Based on these assumptions, the mathematical statement of the problem can be written asfollows:

qcUoT

oX k

o2T

oX2o

2T

oY2o

2T

oZ2

1

subjected to the boundary conditions at edges

qk qh at X XF and Y YF and T T0 at Z ZF for a finite model 2a

Fig. 1. Typical laser installation.

Fig. 2. Energy balance on the surface subject to laser.


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or

T T0 at X 1; Y 1 and Z 1 for a semi-infinite model 2b

and the boundary condition on the surface subject to laser beam is obtained from the balance ofheat transfer on the surface as

qL qk qh qig 3

where qL aoIok neX2Y2=R2 ; qh hT T1; qk kn rT; qig qhigUi n; i and kare

unit vectors in the X- and Z-direction, respectively, and n is the normal outward unit vector tosurface. Here, qUi n represents the rate of material removal when the specimen moves in neg-ative X-direction with the speed U.

With the introduction of the dimensionless variables as follows

x X

Ro; y

Y

Ro; z

Z

Ro; sx;y

SX; Y

Ro; h

T T1

Tevap T1;

Ne qUhig

aoIo; Nk

kTevap T1

RoaoIo; Bi

hRo

k; u

URo

a; a

k

qc

4

Eq. (1) and the boundary conditions (2) and (3) can be rewritten as:

uoh

ox

o2h

ox2o

2h

oy2o

2h

oz25

subjected to

qk qh at x xF and y yF and h h0 at z zF for a finite model 6a

or

h h1 at x 1; y 1 and z 1 6b

and on the surface subject to laser

Region I: z = 0, xF < x < xmin

Nk Bih oh

oz

ex

2y2 7

Region II: z = s(x, y), xmin < x < xmax

h 1; Neosox

ex2y2 Nk Bih

ohoz

1 os

ox

2

osoy

2

" #1=28

Region III: z = s1

, xmax < x < xF

Nk Bih oh

oz

1

os1

ox

2" #1=2 ex

2y2 9

Here Bi is the Biot number representing the ratio of convection to conduction heat losses. u rep-resents the ratio of relative speed of the work specimen to the thermal diffusivity of the material.


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Ne is the ratio of energy utilized in evaporation of material and the absorbed laser energy whileNkrepresents the approximate ratio of conduction losses to the absorbed laser energy.

The convection boundary condition in Eq. (2a) is now expressed in dimensionless form as

qk Bih h1 10

The conduction term in Eq. (3) can now be expressed in terms of others in dimensionless form

qk ex

2

Nkk n Bih h1

Ne

Nki n 11

Note that the regions I and III are subsets of region II and all regions can be handled by one typeof region II in actual analysis.

3. Finite element formulation

The variational formulation of the governing equation by weighted residual method leads to anintegral form

0

ZX

uoh

oxo

2h

ox2o

2h

oz2

dhdX

ZX

uoh

oxdh

oh

ox

odh

oxoh

oz

odh

oz

dX

IC

nxoh

ox nz

oh

oz

dhdC

ZX

uoh

ox

dh oh

ox

odh

ox

oh

oz

odh

oz dX I

C

qkdhdC 12

where X and C represent the domain and the boundary, respectively.The finite element formulation is obtained from this weak form of the variational formulation

by introducing the shape functions /j(x, z) with nodal values hj on an elemental domain as

Keij

h ifhjg fF

ei g 13

where

Keij

ZXe

U/io/j

oxo/iox

o/j

oxo/ioz

o/j

oz

dX

ICe

Bi/i/j ds; i;j 1; 2; . . . ;N

Fei

ICe

qk/i ds

Here Xe and Ce represent elemental domain and boundary. The dimensionless temperatureh(x, z)is approximated by nodal values of temperature and shape functions as

hx;z XNj1

hj/jx;z 14

and N is the number of node per element.


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By assembling the elemental finite element formulation the global finite element model of theproblem can now be written as

Kfhg fFg 15

where [K] and {F} are known matrices and {h} is the unknown column vector.

4. Computational methods

The governing equation is easier to express with moving specimen for the fixed frame to laser,but it is also easier to numerically implement with the moving laser for the fixed specimen. Theyare equivalent except the viewpoint and thus, in the present analysis the laser is handled as a mov-ing source in positive X-direction with fixed specimen for easiness of numerical implementationbelow.

Since the geometry (i.e., groove shape) is not known beforehand, computation begins with anassumed domain, which is the original shape of the specimen at the beginning. Once the temper-ature is calculated for the given domain, the nodal values of the surface temperatures in region IIare examined if the boundary conditions in Eq. (8) are satisfied. If the temperature at any node isgreater than the evaporation temperature, then the material at that node should have melted.

In this paper a simple but effective linear interpolation as described in [24] and repeated below isused for the new position of a node. This simple scheme substantially reduces the computationaltime

znewi zi zi zbottomhi hbottom

hi hevap 16

Once znewi is computed at node i, actual new value for next iteration is relaxed by

zactuali 1 kzoldi kz

newi 17

where k is a relaxation factor used to suppress oscillation in iteration. This new value, zactuali , isused to obtain a new domain.

Since the nodes are moved in z-direction independently to simulate the material removal whenthe temperatures are greater than the melting temperature, the four surface nodes may not be pla-nar and the surface integral due to the laser irradiation cannot be evaluated based on the planarassumption. Thus, the elemental surface with four nodes is approximated by a bilinear polynomialfunction as

zx;y a0 a1x a2y a3xy 18

where the coefficients ais can be obtained by imposing continuity conditions at four nodes as

z1

z2

z3

z4

8>>>>>>>:

9>>>>=>>>>;

1 x1 y1 x1y1

1 x2 y2 x2y2

1 x3 y3 x3y3

1 x4 y4 x4y4

266664

377775

a0

a1

a2

a3

8>>>>>>>:

9>>>>=>>>>;

19

Here, zis are nodal z-coordinates given by zi = z(xi, yi).


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The region III where material has been removed is implemented by extending the maximumgroove depths in the region II.

Once the new shape of the domain is computed, the iteration continues until the relative sums

of squared errors for both nodal temperatures and positions fall below a prescribed tolerance,e.

Etemp XNj1

hnewj holdj

2XNj1

hnewj 2

," #1=220

Epos XNj1

znewj zoldj

2XNj1

znewj 2

," #1=221

Here N is the number of nodes in the domain.Even after the converged solution is obtained by iteration, it is possible that the temperature at

a node has a value below the evaporation temperature but moved to a new position during iter-ation. If this happens, the node should be moved back to the original position and the iterationresumes. This causes numerical difficulty of unstable oscillation of errors during iteration.

5. Numerical results and discussion

First, the three-dimensional results have been compared to two-dimensional results of previousworks for various cases in Fig. 3. The dimensionless parameters for these cases with variousspeeds of laser are

Bi 0.0001; Ne 0.001; Nk 0.4 22

and the dimensions of the specimen are

xF 8; yF 8; zF 2.5 23

These values are chosen to roughly represent the specimen made of typical Aluminum cut by laserpower of 1 kW with the beam focal radius of 0.1 mm that is subjected to natural convection byair. In this case the speed of laser beam in 1 m/s is converted to the dimensionless speed of 1 [25].

Fig. 3. Maximum groove depths of various cases in two- and three-dimensional cases.


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The error criteria are

e 0.01% for position error 24a

e 0.1% for temperature error 24b

The boundary conditions are given by Eq. (6a) and the laser is positioned at the center of thespecimen.

Fig. 3 shows the maximum groove depths against the laser velocity. Both two- and three-dimen-sional results converge with refined meshes, but the three-dimensional cases converge much fasterthan two-dimensional cases. The two-dimensional cases 40 5 and 80 5 or three-dimensionalcases 40 40 5 and 80 80 5 meshes are very close and indistinguishable in the figure. It isnoted that the values of the maximum groove depths of three-dimensional cases are approxi-mately half of those in two-dimensional cases. This may be expected from the heat transfer tothe third direction (positive and negative y directions) in three-dimensional cases that is absent

in two-dimensional cases as illustrated in Fig. 4 (not to show actual direction of heat transfer thatis normal to the surface).

Also, the effect of number of Gauss integration points on maximum groove depths has beeninvestigated for domain and surface integrals in (13). It was found (not shown here) that fine meshof 40 40 5 and the minimum number of integration points, two, are good enough for numer-ically converged results.

The second case considered deals with a semi-infinite body with the following parameters.

U 1; Bi 0.0001; Ne 0.01; Nk 0.10.005 25

The semi-infinite elements used in this analysis are given in Appendix A. The boundary conditions

are given by Eq. (6b). The effect of domain size and the mesh on maximum groove depths hasbeen investigated. In computation, symmetry about xz plane has been utilized in three-dimen-sional analyses.

Table 1 shows two-dimensional results on the maximum groove depths as the mesh is refinedand the domain is increased proportionally. The mesh of 20 15 elements in 2D with the domainsize of 32 50 can be taken as converged results. Further, the result for domain sizes of 16 50(not shown here) has been computed that is very close to the result for the case of 32 50. Thissuggests that the domain size Lx = 16 is good enough for accurate results.

x

yz

Fig. 4. Heat transfer in three directions at the cutting front.

http://-/?-http://-/?-

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Table 2 shows three-dimensional maximum groove depths as the mesh is doubled from left toright and the domain is increased proportionally to check the convergence with location of semi-infinite elements. Also, the number of elements in x- and y-directions from top to bottom showsthe convergence. The mesh of 20 10 15 elements with the domain size of 32 16 50 yields

reasonably converged results for this case. The values with * show very close results to precedingvalues indicating that domain size ofLx Ly = 32 16 is good enough for semi-infinite dimen-sions in x- and y-directions. Modest et al. [17] predicted the max groove depth of roughly 10 intheir semi-analytical analysis that is higher than the current numerical results.

In Fig. 5 the max groove depths and temperature errors are shown during the iteration as wellas the convergence in the error domain. In Fig. 6 the groove shape and temperature distributionwith the half domain for the case 40 20 15 mesh in Table 2 are shown. Figs. 5(e) and 6(c) showthat the temperature is more closely distributed in front of the moving laser showing the Dopplereffect due to the moving source. It is interesting to see the caved-in contour plot at the bottom ofthe groove on the laser receding side. This suggests that the groove bottom cools down faster than

the side surfaces.Further, the effect of number of integration points on maximum groove depths was also inves-tigated for semi-infinite elements (not shown here). The results suggest that the specimen size of 16and 8 in x- and y-directions is good for semi-infinite elements with two integration points. Andconsequently these values are used in the current analyses.

Table 3 shows the case with Nk = 0.01. The convergence can be observed with 40 20 15mesh. The present numerical results for maximum groove depth predict smaller value than thevalue 80 by Modest et al. [17].

Figs. 7 and 8 show typical changes of maximum groove depths, temperature and positionerrors, the temperature distribution, and the groove shape for the case of mesh 40 20 60 in

Table 2Effects of mesh and size of specimen on maximum groove depths with U= 1, Bi= 104, Ne = 10

2, Nk = 0.1

3D Lx Ly Lz

16 8 25 32 16 50 64 32 100

Starting Nx Ny Nz 10 5 15 7.61915 7.38836 7.300457.38839* 7.30045*

20 10 15 6.01683 5.765111 5.74619*

40 20 15 5.80948 5.765111*

* The values are computed only increasing z-dimension from the preceding cases.


2, Nk = 0.1

2D Lx Lz

16 25 32 50 64 100 128 200

Nx Nz 10 15 24.75272 21.52166 20.55286 20.2261921.52356*

20 15 14.55105 14.43881 14.3996740 15 14.69527 14.43894


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Fig. 5. (a) Maximum groove depth with iteration, (b) temperature error with iteration, (c) convergence behavior inerror domain, (d) mid-plane groove shapes with iteration, and (e) mid-plane surface temperatures with iteration for thecase of mesh 40 20 15 in Table 2.

Fig. 6. (a) 3D Groove shape and (b) 3D temperature distribution, (c) temperature contour plot on the groove surface(top view), and (d) heat flux in the mid-plane (at y = 0) for half domain for the case of mesh 40 20 15 in Table 2.


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Table 3. It is observed that the heat affected zone stretches far down from the laser with smallerconductivity of the material.

In the following the sectional shapes of the groove are also presented. These shapes also com-pare well with those by Modest et al. [17] except the maximum groove depth. Here it is noted thatFig. 9(b) shows the backward groove depth for laser motion than forward groove for the movingspecimen. A preliminary study with unsteady model (not shown here) shows forward groove withshallower depth in both cases of moving laser and moving specimen.

Final case with Nk = 0.005 has been also studied and presented in Table 4 and Fig. 10. The con-verged results can be taken for the mesh 20 10 15 with domain size 32 16 400. The present


2, Nk = 0.01

3D Lx Ly Lz

16 8 200 32 16 400 64 32 800

Nx Ny Nz 10 5 15 69.19937 66.09498 64.9744766.09512* 64.97447*

20 10 15 47.52265 47.2509740 20 15 47.67044

* The values in the table are computed only increasing z-dimension from the preceding cases.

Fig. 7. (a) Maximum groove depth with iteration, (b) temperature error with iteration, and (c) convergence behavior inerror domain, (d) mid-plane groove shapes with iteration, and (e) mid-plane surface temperatures with iteration for thecase of mesh 40 20 15 in Table 3.


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numerical result for maximum groove depth predicts much smaller value that of 120 by Modestet al. [17] consistently. With the decrease of the conductivity most heat flux occurs along the lasermotion horizontally.

Fig. 8. (a) Groove shape and (b) temperature distribution, (c) temperature contour plot on the groove surface (topview), and (d) heat flux in the mid-plane (at y = 0) for half domain in the case of mesh 40 20 15 in Table 3.

Fig. 9. Groove section shapes (a) perpendicular to laser motion and (b) in the direction of laser motion for the case40 20 15 in Table 3.


2, Nk = 0.005

3D Lx Ly Lz

16 8 300 32 16 400 64 32 800

Nx N

y N

z10 5 15 125.77696 123.57431 118.57616

123.57510* 118.57482*

20 0 15 74.76116 74.9019774.66565*

40 20 15 75.49031

* The values are computed with same dimensions except doubled z-dimension.


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Fig. 10. (a) Maximum groove depth with iteration, (b) temperature error with iteration, (c) error behavior in errordomain, (d) groove shapes during iteration at mid-section (y = 0), (e) mid-section surface temperatures during iteration,(f) 3D groove shape, (g) 3D surface temperature, (h) temperature contour plot on the groove surface (top view), (i) mid-section heat fluxes, (j) groove section shape in the direction of laser motion, and (k) groove section perpendicular to thelaser motion for the case of mesh 40 20 15 in Table 4.


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Fig. 10 shows similar behaviors of max groove depth with iteration as well as the groove shapeand temperature distribution to the previous cases.

6. Conclusions

A three-dimensional finite element model has been developed to analyze the evaporative laser-cutting process based on steady heat conduction equation with constant laser velocity. The laserintensity is assumed to be sufficiently high to cause direct evaporation of the material from thesurface of the medium. The laser side elemental surface is approximated by semi-quadratic poly-nomial function. Parametric study shows that the numerical results converge with the mesh refine-ment. The predicted groove shapes for a semi-infinite domain well compare with semi-analyticalresults by others except the max groove depths. The present analyses without any limiting

assumptions predict smaller maximum groove depths than semi-analytical results. The tempera-ture distributions show the heat-affected zone is not really limited close to the laser position.The Doppler effect is observed for a moving laser. The steady-state analyses show the backwardgroove shapes than forward ones in real laser-cutting process. The geometric nonlinearity due tothe unknown groove shape has led to an iterative scheme that sometimes resulted in unstableoscillations during iterations.

Appendix A

For eight node linear element (Fig. A.1), assuming that the element extends to infinity along z-

direction the coordinates can be expressed as

x X4j1

xjMj; yX4j1

yjMj; zX4j1

zjMj A:1

where the mapping functions for coordinates are

M1 1 n

2

1 g

2

2

1 1; M2

1 n

2

1 g

2

2

1 1;

M3 1 n

2

1 g

2

2

1 1; M4

1 n

2

1 g

2

2

1 1

A:2

and the shape functions for field variable are standard linear shape functions.

4

5

2

6

18

3

7

Fig. A.1. Natural coordinates of a semiinfinite element.


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The Jacobian J in finite element formulation then becomes

J

ox

on

oy

on

oz

onox

og

oy

og

oz

og

ox

o1

oy

o1

oz

o1

266666664

377777775

P4

j1

oMj

onxj P

4

j1

oMj

onyj P

4

j1

oMj

onzj

P4j1

oMj

ogxj

P4j1

oMj

ogyj

P4j1

oMj

ogzj

P4j1

oMj

o1xj

P4j1

oMj

o1yj

P4j1

oMj

o1zj

2666666664

3777777775A:3

References

[1] W.E. Lawson, Laser cutting of composites, in: Conference on Composites in Manufacturing, Technical PaperEM86-114, Los Angeles, 1986.[2] D.M. Roessler, Detroit looks to lasers, Mech. Eng. 112 (April 1990) (1986) 3845.[3] J. Uhlenbusch, U. Bielesch, S. Klein, M. Napp, J.H. Schafer, Recent developments in metal processing with

pulsed laser technology, Appl. Surf. Sci. 106 (1996) 228234.[4] D.L. Carrol, J.A. Rothenflue, Experimental study of cutting thick aluminum and steel with a chemical oxygen

iodine laser using an N2 or O2 gas assist, J. Laser Appl. 9 (1997) 119128.[5] S. Ghosh, B.P. Badgujar, G.L. Goswami, Parametric studies of cutting zircaloy-2 sheets with a laser beam, J. Laser

Appl. 8 (1996) 143148.[6] J.M. Glass et al., Heat transfer in metallic glasses during laser cutting, ASME Heat Transfer in Manufacturing and

Materials Processing, New York, HTD Vol. 113, 1989, pp. 3138.[7] J. Kusinski, Microstructure, chemical composition and properties of the surface layer of M2 steel after melting

under different conditions, Appl. Surf. Sci. 86 (1995) 317322.[8] K. Li, P. Sheng, Plane stress model for fracture of ceramics during laser cutting, Int. J. Mach. Tools Manufact. 35

(11) (1995) 14931506.[9] K. Brugger, Exact solutions for the temperature rise in a laser heated slab, J. Appl. Phys. 43 (1972) 577583.

[10] J.I. Masters, Problem of intense surface heating of a slab accompanied by change of phase, J. Appl. Phys. 27 (1956)477484.

[11] A. Minardi, P.J. Bishop, Temperature distribution within a metal subjected to irradiation by a laser of spatiallyvarying intensity, ASME Heat Transfer in Manufacturing and Materials Processing, New York, HTD Vol. 113,1989, pp. 3944.

[12] N.N. Rykalin, A.A. Uglov, N.I. Makarov, Effects of peak frequency in a laser pulse on the heating of metal sheets,Sov. Phys.-Doklad. 12 (1967) 644646.

[13] R.E. Warren, M. Sparks, Laser heating of a slab having temperature-dependent surface absorptance, J. Appl.Phys. 50 (1979) 79527957.

[14] B. Basu, J. Srinivasan, Numerical study of steady-state laser melting problem, Int. J. Heat Mass Transfer 31 (1988)23312338.

[15] K.A. Bunting, G. Cornfield, Toward a general theory of cutting: A relationship between the incident power densityand the cut speed, ASME J. Heat Transfer 97 (1975) 116121.

[16] F.W. Dabby, U.C. Paek, High intensity laser-induced vaporization and explosion of solid material, IEEE J. Quant.Electron. QE-8 (1972) 106111.

[17] M.F. Modest, H. Abakian, Evaporative cutting of a semi-infinite body with a moving CW laser, ASME J. HeatTransfer 108 (1986) 597601.

[18] J.F. Reddy, Effects of High Power Laser Radiation, Academic Press, NY, 1971.[19] D. Maydan, Micromachining and image recording on thin films by laser beams, Bell Syst. Tech. J. 50 (1971)

17611789.


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[20] M.F. Modest, H. Abakian, Heat conduction in a moving semi-infinite solid subjected to pulsed laser irradiation,ASME J. Heat Transfer 108 (1986) 602607.

[21] A.P. Zhuravel, A.G. Sivakov, O.G. Turutanov, I.M. Dmitrenko, A low temperature system with a pulse UV laser

for scribing HTSC films and single crystals, Appl. Surf. Sci. 106 (1996) 321325.[22] M.J. Kim, Z.H. Chen, P. Majumdar, Finite element modeling of the laser cutting process, J. Comput. Struct. 49(1993) 231241.

[23] M.J. Kim, P. Majumdar, A computational model for high energy laser cutting process, Numer. Heat Transfer,Part A 27 (1995) 717733.

[24] M.J. Kim, Transient evaporative laser-cutting with boundary element method, J. Appl. Math. Modell. 25 (2000)2539.

[25] M.J. Kim, Finite element analysis of evaporative cutting with a moving high energy pulsed laser, J. Appl. Math.Modell. 25 (2001) 203220.


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