Non-disturbing Boundary Conditions for Modeling of ... - Science Direct

Available online at www.sciencedirect.com

ScienceDirect Physics Procedia 56 (2014) 421 – 428

8th International Conference on Photonic Technologies LANE 2014

Non-disturbing boundary conditions for modeling of laser material processing I.O. Kovalevaa, S.N. Grigorievb, A.V. Gusarova,* b

a Ecole Nationale d’Ingénieurs de Saint-Etienne, 58 rue Jean Parot,42023 Saint-Etienne, France Moscow State University of Technology “STANKIN”, Vadkovsky pereulok 3a, 127055 Moscow, Russia

Abstract Surface laser treatment of a massive body is the typical geometry for various laser-assisted processes. The classical mathematical formulation is a heat source moving over the surface of a half-space target. Generally, such problems are numerically solved in a finite calculation domain. The adiabatic or isothermal boundary conditions are often applied at the boundaries of the calculation domain. Such an approach becomes rigorous when the linear size of the calculation domain is much greater than the size of the melt pool. It is time consuming. Economic non-disturbing differential boundary conditions proposed here are derived from the well-known analytical asymptotics for the steady-state temperature distributions around a moving heat source in 2D and 3D. Finite-difference boundary conditions approximating these differential conditions are tested for modeling of additive manufacturing of massive metallic parts and walls by selective laser melting. It is shown that the linear size of the calculation domain can be as small as double the size of the melt pool. © by Elsevier B.V. by This is an open access article under the CC BY-NC-ND license © 2014 2014Published The Authors. Published Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and blind-review under responsibility of the Bayerisches Laserzentrum GmbH. Peer-review under responsibility of the Bayerisches Laserzentrum GmbH Keywords: selective laser melting; melt pool; evaporation; semi-infinite body; thermal diffusion

1. Introduction Currently, laser treatment of solids is widely used in the technology and varies in objectives, physical and chemical processes involved, and the shape of the treated body. The common is the far heat affected zone (HAZ) where the energy transfer is controlled by heat diffusion. Often HAZ is considerably smaller than the size of the

* Corresponding author. Tel.: +33-477-43-7585 ; fax: +33-477-43-7585 . E-mail address: [email protected]

1875-3892 © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Bayerisches Laserzentrum GmbH doi:10.1016/j.phpro.2014.08.145

422

I.O. Kovaleva et al. / Physics Procedia 56 (2014) 421 – 428

body in all the three (slabs) or two (sheets) dimensions. In this case, including the whole body in the calculation domain at numerical modeling becomes unnecessary and time consuming. The calculation domain should be cut at a reasonable distance from the zone of the principal process of laser-material interaction not to influence this zone. The commonly used boundary conditions for the temperature field at these cuts are the fixed temperature or the adiabatic condition. The latter means that the normal component of the heat flux is zero. Shuja and Yilbas (2011) used the boundary condition of the fixed ambient temperature for modeling the melt pool on the surface of a massive body while Kovalev and Gurin (2014) successfully applied the adiabatic condition for the similar problem. Xiao and Zhang (2008) combined the fixed temperature condition at the forward boundary and the adiabatic condition at the boundary backward relative to the moving continuous laser beam in their modeling of selective laser sintering. Gusarov et al. (2009) also used the fixed temperature at the forward boundary and the adiabatic condition at the lateral and backward boundaries for modelling the selective laser melting. Yilbas and Akhtar (2013) used the condition of fixed temperature for modeling the laser welding of sheets. The objective of this work is to find better conditions at the boundaries of the computation domain cutting the modeled body in the dimensions where HAZ is smaller than the body. The conditions should less disturb the temperature distribution in the far HAZ. This would make it possible to reduce the computation domain and to increase the efficiency of numerical calculations, without influencing the principal process of laser-material interaction. Section 2 presents construction of such boundary conditions based on the well-known analytical steadystate solutions for a moving point source. Section 3 demonstrates implementation examples of the obtained nondisturbing boundary conditions. 2. Differential boundary conditions The laser source is assumed to be localized and to move with a constant velocity u. If the releasing heat is steady then the temperature field T in the far HAZ is steady in the coordinate system moving with the source and is described by the heat diffusion equation a'T  u ˜ ’T

0,

(1)

where a is the thermal diffusivity, ' the Laplace operator, and ’ the nabla operator. The temperature far from the source is assumed to tend to its ambient value Ta. Slabs are considered in Section 2.1. Sheets are considered in Section 2.2. Section 2.3 continues to discuss the domain of applicability of the obtained equations. 2.1. Three-dimensional (3D) case Consider a half-space scanned with a laser beam as shown in Fig. 1 (a). The top plane is supposed to be adiabatic. Carslaw and Jaeger (1976) report point-source solution of Eq. (1) in 3D with thermal power P T  Ta

P § ux uR · exp¨  ¸, 2SOR © 2a 2a ¹

(2)

where O is the thermal conductivity, and R2 = x2 +y2 + z2. The derivatives of the logarithm of Eq. (2) are 

1 wT T  Ta wx

R



1 wT T  Ta wy

R2



1 wT T  Ta wz

x 2

y

z R

2



u §x · ¨  1¸ , 2a © R ¹

(3)



u y , 2a R

(4)



u z . 2a R

(5)

423

I.O. Kovaleva et al. / Physics Procedia 56 (2014) 421 – 428

Laser beam

Scanning direction

Laser beam

Scanning direction

Treated zone

Treated zone

Laser beam

Scanning direction

Treated zone

Z

Z

X X

X

Y

Y

Wall Slab

G

G (a)

(b)

Sheet

(c)

Fig. 1. Geometries of laser treatment with the corresponding coordinate systems: (a) 3D slab; (b) 2D vertical wall; (c) 2D horizontal sheet.

Let the computation domain be bounded by coordinate planes x r X r , y = 0, Y+, and z = 0, Z+ with positive values X+, X-, Y+, and Z+. Then Eqs. (3)-(5) result in the following non-disturbing 3D boundary conditions: X



1 wT T  Ta wx



1 wT T  Ta wx

X



1 wT T  Ta wy

Y



1 wT T  Ta wz

Z



R

R2

2





· u §¨ X   1¸ , at x = -X- , ¸ 2a ¨© R ¹

(6)

· u §¨ X   1¸ , at x = X+ , ¸ ¨ 2a © R ¹

(7)

u Y , at y = Y+ ,  2 a R 2 R

R2



(8)

u Z , at z = Z+ . 2a R

(9)

Face z = 0 is the physical one where the boundary conditions depend on the physical model. Face y = 0 is the mirror plane with the obvious boundary conditions. 2.2. Two-dimensional (2D) case Consider a vertical thin wall of thickness G with a horizontal top as shown in Fig. 1 (b). The top and lateral faces are supposed to be adiabatic. Carslaw and Jaeger (1976) report point-source solution of Eq. (1) in 2D T  Ta

P § ux · § ur · exp¨ ¸K 0 ¨ ¸ , SOG © 2a ¹ © 2a ¹

(10)

where K0 is the Bessel function of the second kind and r2 = x2 + z2. The derivatives of the logarithm of Eq. (10) are

1 wT  T  Ta wx

· § § ur · ¸ ¨ K1 ¨ ¸ u ¨x © 2a ¹  1¸ , 2a ¨ r § ur · ¸ ¨ K0¨ ¸ ¸ © 2a ¹ ¹ ©

1 wT  T  Ta wz

u z 2a r

§ ur · K1 ¨ ¸ © 2a ¹ , § ur · K0¨ ¸ © 2a ¹

(11)

424

I.O. Kovaleva et al. / Physics Procedia 56 (2014) 421 – 428

Equations (11) result in the following non-disturbing 2D boundary conditions: · § § ur · ¸ ¨  K1 ¨ ¸ u ¨X © 2a ¹  1¸ , at x = -X-,  2a ¨ r § ur · ¸ K0 ¨ ¸ ¸ ¨ © 2a ¹ ¹ ©

(12)

1 wT  T  Ta wx

§ § ur · · ¨ K ¨ ¸ ¸ u ¨ X  1 © 2a ¹ ¸  1 , at x = X+, 2a ¨ r § ur · ¸ K0 ¨ ¸ ¸ ¨ © 2a ¹ ¹ ©

(13)

1 wT  T  Ta wz

u Z 2a r

§ ur · K1 ¨ ¸ © 2a ¹ , at z = Z+. § ur · K0 ¨ ¸ © 2a ¹

(14)



1 wT T  Ta wx

A thin sheet shown in Fig. 1 (c) dissipating thermal power 2P is characterized by the same temperature distribution (10) where coordinate z should be replaced with y. This is why Eqs. (12)-(14) are applicable to sheets too. 2.3. Applicability remarks The non-disturbing boundary conditions (6)-(9) in 3D and (12)-(14) in 2D are derived from the well-known moving point-source solutions (2) and (10), respectively, of the linear heat diffusion equation (1). The point-source solutions give the asymptotics of the temperature field in the far HAZ. They depend on the conductive thermal power P dissipated in the treated body. However, the obtained boundary conditions are independent of P. This is important for the modeling of laser treatment because the fractions of incident laser energy redirected to reflected radiation, radiative and convective exchange with the ambience, evaporation, irreversible phase transitions in the condensed phase, chemical decomposition, and other possible processes can be considerable, so that the value of P is unknown at the stage of mathematical formulation of the modeling problem where the boundary conditions should be written. An implicit assumption used to obtain the boundary conditions, is the essentially steady character of the temperature field in the scale of the calculation domain, which is of the order of the far HAZ size d. The simplest case satisfying this assumption is the laser source generating in the continuous mode giving rise a thermally stable process. However, these boundary conditions can be compatible with a pulse periodic laser source used by Yilbas and Akhtar (2013) if its period is much less than the relaxation time of the far HAZ d2/a and no auto-oscillations are generated with the period of the order of d2/a. In this case, P is the mean power of the cumulated heat. Of course, slow variations of the laser power with the period much greater than d2/a can be neglected too. The basic Eq. (1) is linear with no thermal dependence of thermal diffusivity a. This does not mean that the obtained boundary conditions can be only applied to models with linear heat diffusion in the solid phase. The weaker restriction is a low variation of a in the temperature range near the boundaries. This assumption can be better than that of a fixed temperature at the boundaries. Finally, the laser treatment can modify the microstructure of the material. This may significantly change thermal diffusivity a in the treated cylinder of the material behind the laser beam, which is marked as “Treated zone” in Fig. 1. If this is the case, the volume of the calculation domain can be chosen much greater than the treated volume to neglect the modification of thermal diffusivity.

425

I.O. Kovaleva et al. / Physics Procedia 56 (2014) 421 – 428

3. Testing and examples of application A non-linear heat diffusion model with melting/solidification is considered in the moving coordinate system: ’ ˜ (O ’T )  u

wH wx

0,

(15)

where volumetric enthalpy H is related with temperature T by the thermal equation of state

T

H / Cs H d C sTm , ­ ° Tm , C sTm  H  C sTm  Qm , ® °T  ( H  C T  Q ) / C , H t C sTm  Qm s m m l ¯ m

(16)

Cs and Cl are the specific heats in solid and liquid phases, respectively, Tm is the melting point, Qm the latent heat of melting, and O the thermal conductivity approximated by a stepwise function of temperature, O

­O s , T d Tm . ® ¯ O l , T ! Tm

(17)

The obtained non-disturbing boundary conditions are tested in details for a 2D problem in Section 3.1. 2D and 3D examples for a more complicated model including strong evaporation are given in Section 3.2. 3.1. Testing in 2D A thin wall configuration is accepted as shown in Fig. 1 (b). The boundary conditions for Eq. (15) are T o Ta at x o rf and z o f ,

O

wT wz

qa at z

0,

(18)

where the absorbed energy flux density qa

P Sr0 G





exp  x 2 / r02 ,

(19)

with the tentative laser beam radius r0. Equation (15) is numerically solved by a second-order finite volume method in the rectangular computation domain {x = -X- .. X+; z = 0 .. Z+}, which is divided into Nx u Nz rectangular cells with the sizes of 'x = (X- + X+)/Nx and 'z = Z+/Nz and the centers at xi = -X- + 'x/2 + i'x and zl = 'z/2 + l'z, with integers i = 0 .. Nx – 1, l = 0 .. Nz – 1. The non-disturbing boundary conditions (12)-(14) are tested by varying the size of the computation domain. Two computations are compared. The first one is made in a small computation domain, and the second one is made in a larger computation domain. The cell is of the same size 20 x 20 Pm in the both cases. The other parameters of the two domains are listed in Table 1. The laser beam parameters and the physical properties of the medium are listed in Table 2. Figure 2 shows the results of numerical calculation. No visible differences in the shape and the size of the melt pool calculated in the small computation domain and in the large one are observed in Fig. 2 (a). Figure 2 (b) presents the temperature distribution in the small domain. The temperature profiles along two characteristic lines are taken from this 2D distribution and shown in Fig. 2 (c) and (d) by crosses. The profiles along the same lines obtained in the large computation domain are shown by circles. The corresponding profiles coinside and tend to asymptotics (10) shown by full lines in Fig. 2 (c) and (d). Thus, the results do not depend on the size of the tested

426

I.O. Kovaleva et al. / Physics Procedia 56 (2014) 421 – 428

computation domains. For precise numerical calculation with the proposed non-disturbing boundary conditions, it is sufficient to take the linear size of the computation domain about twice the linear size of the melt pool in each dimension. Table 1. Parameters of the tested computation domains. Parameter

X-, Pm

X+, Pm

Z+, Pm

Nx

Nz

Small domain

250

550

200

40

10

Large domain

250

750

400

50

20

Table 2. Parameters accepted to test the non-disturbing boundary conditions. Quantity

Value

Quantity

Value

Radius of the laser beam, r0

40 Pm

Latent heat of melting, Qm

1.46 GJ/m3

Scanning velocity, u

1 cm/s

Specific heat in solid phase, Cs

3.44 MJ/(K m3)

Absorbed specific power, P/G

30 kW/m

Specific heat in liquid phase, Cl

4.68 MJ/(K m3)

Ambient temperature, Ta

298 K

Thermal conductivity in solid phase, Os

7 W/(m K)

Melting point, Tm

1944 K

Thermal conductivity in liquid phase, Ol

20 W/(m K)

(a)

z (Pm)

(b) 0

200

2500

-200

0

200 x (Pm)

400

Fig. 2. Comparison of 2D calculations is the small and the large computation domains with the non-disturbing boundary conditions: (a) Relative sizes of the domains, positions of the laser beam axis shown by vertical hatches, and the calculated melt pools (black); (b) Temperature distribution in the small domain; (c) Temperature profiles along the top of the wall, z = 0; (d) Temperature distributions along the axis of the laser beam continued into the target, x = 0.

3.2. Application to selective laser melting (SLM) Laser-matter interaction is modeled in 3D according to the scheme of Fig. 1 (a) for SLM of massive parts and in 2D shown in Fig. 1 (b) for SLM of thin walls. The strong evaporation is included in the model as described by Khmyrov et al. (2014). Physical boundary conditions are T o Ta at r o f ,

O

wT wz

qa  qe at z

0,

(20)

where r2 = x2 + z2 in 2D and r2 = x2 + y2 + z2 in 3D and qe the evaporation loss estimated from the temperature T at z = 0 by formulas given by Khmyrov et al. (2014). The absorbed flux qa is calculated in 2D by Eq. (19) and in 3D as

427

I.O. Kovaleva et al. / Physics Procedia 56 (2014) 421 – 428

qa

§ x2  y2 · ¨ ¸, exp 2 ¨ ¸ Sr02 r 0 © ¹ P

(21)

where P is the absorbed power of the laser beam. The used parameters of the model are listed in Table 3. The nondisturbing boundary conditions are used at numerical calculation. The results for scanning velocities u equal to 100 and 300 mm/s are shown in Fig. 3. The 3D asymptotics is given in these plots by Eq. (2). In 2D case, the temperature behind the laser beam is considerably higher. This means that cumulation of thermal energy in the target is more important at SLS of walls than at SLS of massive parts. Approaching of the calculated curves to their asymptotics indicates the non-disturbing boundary conditions works for this problem. Table 3. Parameters accepted for modeling SLM with the non-disturbing boundary conditions. Quantity

Value

Quantity

Value

Radius of the laser beam, r0

30 Pm

Latent heat of evaporation, Qb

39.61 GJ/m3

Absorbed laser-beam power, P

18 W

Specific heat in solid phase, Cs

3.44 MJ/(K m3)

Wall thickness, G

60 Pm

Specific heat in liquid phase, Cl

4.68 MJ/(K m3)

Ambient temperature, Ta

298 K

Thermal conductivity in solid phase, Os

28 W/(m K)

Melting point, Tm

1944 K

Thermal conductivity in liquid phase, Ol

28 W/(m K)

Boiling point, Tb

3562 K

Vapor molecular mass, m

47.88 a.m.u.

Latent heat of melting, Qm

1.46 GJ/m3

u = 10 cm/s

u = 30 cm/s

Fig. 3. Comparison of thermal profiles along the scanning axis (OX) for SLM of 3D parts and 2D thin walls at scanning velocity u marked on the top. The asymptotics is given by dashed lines.

4. Minimizing the size of the computation domain The two tested computation domains for the 2D nonlinear problem are shown in Fig. 2 (a). The small domain is a bit greater than the melt pool filled with black in this figure. The large domain has linear dimensions about twice the corresponding dimensions of the melt pool. Temperature profiles calculated in the small domain (crosses in Figs. 2 (c) and (d)) and in the large domain (circles) coincide in the melt pool. This indicates that the variation of the computation domain size doesn’t influence the calculation results within the small domain. Moreover, the asymptotic solution (line) is very close to the numerical points (crosses and circles) near the boundary of the

428

I.O. Kovaleva et al. / Physics Procedia 56 (2014) 421 – 428

computation domain as shown in Figs. 2 (c) and (d). This suggests that the numerical solutions obtained in the small and the large computation domains are correct. Figure 3 shows numerical solutions of another test problem in 2D and 3D. The size of the computation domain is not varied. However, the comparison of the longitudinal temperature profile for the 3D problem with the corresponding asymptotics (broken lines) indicates that the obtained numerical solution can be approximated by the asymptotics at |x| > 10-4 m. Thus, the computation domain could be cut at x = -10-4 m and x = 10-4 m without influencing the numerical solution in the interval -10-4 m < x < 10-4 m. The size of the melt pool in X-direction is about 10-4 m for the 3D problem, while the proposed size of the computation domain is 2.10-4 m. In conclusion, the tested 2D and 3D problems indicate that the linear size of the computation domain can be as small as twice the corresponding linear size of the melt pool. 5. Conclusions For numerical modeling of 2D and 3D problems of laser treatment, economic non-disturbing differential boundary conditions are derived from the well-known analytical asymptotics for the steady-state temperature distributions around a moving heat source. Finite-difference boundary conditions approximating these differential conditions are tested for modeling of additive manufacturing of massive metallic parts and walls by selective laser melting. For precise numerical calculation with the proposed boundary conditions, it is sufficient to take the linear size of the computation domain about twice the linear size of the melt pool in each dimension. References Clark, T., Woodley, R., De Halas, D., 1962. Gas-Graphite Systems, in “Nuclear Graphite”. In: Nightingale, R. (Ed.). Academic Press, New York, pp. 387. Deal, B., Grove, A., 1965. General Relationship for the Thermal Oxidation of Silicon. Journal of Applied Physics 36, 37–70. Deep-Burn Project: Annual Report for 2009, Idaho National Laboratory, Sept. 2009. Fachinger, J., den Exter, M., Grambow, B., Holgerson, S., Landesmann, C., Titov, M., Podruhzina, T., 2004. Behavior of spent HTR fuel elements in aquatic phases of repository host rock formations, 2nd International Topical Meeting on High Temperature Reactor Technology. Beijing, China, paper #B08. Fachinger, J., 2006. Behavior of HTR Fuel Elements in Aquatic Phases of Repository Host Rock Formations. Nuclear Engineering & Design 236, 54. Carslaw, H.S., Jaeger, J.C.,1976. Conduction of heat in solids, Calendon Press, Oxford. Gusarov, A.V., Yadroitsev, I., Bertrand, Ph., Smurov, I., 2009. Model of radiation and heat transfer in laser-powder interaction zone at selective laser melting. J. Heat Transfer 131, 072101. Khmyrov, R.S., Grigoriev, S.N., Okunkova, A.A., Gusarov, A.V., 2014. On the possibility of selective laser melting of quartz glass. Phys. Procedia (submitted to LANE 2014, Contribution ID: 155). Kovalev, O.B., Gurin, A.M., 2014. Multivortex convection of metal in molten pool with dispersed impurity induced by laser radiation. Int. J. Heat Mass Transfer 68, 269-77. Shuja, S.Z., Yilbas, B.S., 2011. Laser produced melt pool: Influence of laser intensity parameter on flow field in melt pool. Optics Laser Technol. 43, 767-75. Xiao, B., Zhang, Y., 2008. Numerical simulation of direct metal laser sintering of single-component powder on top of sintered layers. J. Manufacturing Sci. Eng. 130, 0041002. Yilbas, B.S., Akhtar, S., 2013. Laser welding of AISI316L steel: Microstructural and stress analysis. J. Manufacturing Sci. Eng. 135, 031018.