6.3 Three-dimensional thermo-physical modelling of LIPAA using Finite Element Method (FEM)
6.3.3 Solution methodology
where Io is the peak laser pulse intensity and R is the laser spot radius. The peak laser pulse intensity (Io) can be further computed by using:
peako 1 t P
I R
A
(6.9)
where Rt is the reflectivity of the transparent material and Ppeak is the peak laser power. Since the plasma is thermally coupled with the laser beam, the in situ measurement of the plasma energy is difficult. However, as reported by Saxena et al.
(2014), beyond the plasma generation threshold, a linear increase in the input laser irradiance leads to a linear increase in the energy transfer to the plasma. It is considered that the plasma generation threshold is obtained when the density of the electrons (Ne) reaches a value of about 1018/cm3 (Dahotre and Harimkar, 2008). The plasma energy density can thus be estimated from the electron density by the following equation (Saxena et al., 2014):
plasma e
2 E N E
(6.10)
The value of Eplasma has been determined to be 2.72 × 106 J/m3, taking 34 eV as the ionisation energy (∆E) for air. The plasma power density (q1 ) can thus be estimated from the following expression:
1
plasma on
q E
t (6.11)
where ton is the pulse duration.
(A) Finite element formulation
After formulating the problem in terms of governing equations and boundary conditions, finite element method (FEM) was used to solve the problem. FEM is a numerical technique to solve the engineering problems in which the unknown function is approximated by piecewise defined functions. In this technique, the worksheet is discretized into elements. The collection of elements is called finite element mesh. Each element has nodes which are used to represent values of the field variables (for example, temperature) over the element by an interpolation function. In thermal analysis, the field variable is temperature, which is approximated within the element using its nodal temperatures,
eT N T
(6.12)and
e
eT B T N T
x
(6.13)where [N] is the interpolation or shape function matrix, {Te} is the element nodal temperature matrix and [B] is the general geometric matrix. Galerkin method is used to solve the above equations, and the resulting equations are expressed in the matrix form as
C T K
T T Q
(6.14) where [C] and [KT] are the global heat capacity matrix and global conductivity matrix respectively, which are given as
0
TV
C c N N dV
(6.15)
T 0
T V
K k B B dV
(6.16)and {Q} is the heat flux vector.
(B) Mesh model and mesh sensitivity analysis
The process continuum was discretized with thermal solid 35 with triangular-6 nodes initially for the two dimensional geometrical model and later, with thermal solid 70 with brick-8 nodes for the three-dimensional geometrical model. The entire model was
discretised into coarse mesh and the mesh was further refined at the zone of laser irradiation at the interface of the two materials as shown in figure 6.4.
Figure 6.4 Meshed geometric model (a) 3D view and (b) 2D view
Both the PC and the aluminium were modelled using two different sets of nodes that correspond to two separate materials. A distinct interface of these materials has duly been defined. There was no external pressure applied on the top PC sheet during the experiments, therefore, in numerical simulations no pressure or contact resistance effect was considered at the interface of the two materials. Accordingly, the incoming laser heat flux was applied directly at the interface of the two materials. Mesh size is an important factor that needs to be controlled during the FE analysis. Larger mesh size leads to inconvergent simulation results, whereas smaller mesh size take long computational time. Thus, selection of an optimized mesh size is obtained by carrying out the mesh sensitivity analysis as shown in table 6.1. Mesh sensitive analysis was performed for the process condition of 3.06 MW/cm2 pulse power density, 40 Hz pulse repetition rate and 2 ms pulse duration. From the mesh sensitive analysis, it was found that with a finer mesh size of 20 µm, the simulation gave the best results with computational time of 1.53 hours. The effect of a moving Gaussian heat flux was employed in the numerical model of the LIPAA process. The spatial and temporal thermo-physical investigation of the LIPAA process was carried out by implementing the laser heat source at the interface of the two materials and assuming an automatic time step increment. The spatial investigation lent a hand in predicting the width and depth of the microchannel. In the model, the material removal criterion to achieve ablation in aluminium and PC is such that, the material exceeding its melting point temperature is considered to be ablated or removed from the base material. However, the current study
focuses upon the PC sheet for the determination of the channel width and the channel depth fabricated on its rear side. As such, melting point temperature of the PC sheet i.e.
503 K was taken into consideration to ablate the PC sheet. The temporal temperature distribution for pulse-on time (heating) and pulse-off time (cooling) were also explored.
Figure 6.5 shows the outline of the developed numerical modelling approach for prediction of the dimensions of the microchannel and the machining rate during LIPAA process.
Table 6.1 Analysis of mesh sensitivity for refined mesh zone Mesh Size (µm) Maximum Temperature
(K)
Change in Temperature
(∆T)
CPU time (hours)
35 666.7 - 0.23
30 672.9 6.2 0.43
25 674.4 1.5 0.72
20 674.7 0.3 1.53
15 676.5 1.8 3.17
10 677 0.5 4.69
Figure 6.5 Outline of numerical modelling approach for predicting the width and depth of the microchannel and the machining rate of the LIPAA process