Page 82 - MSAM-4-3

Materials Science in Additive Manufacturing Interpretable GP melt track prediction

t
T
t −1
 RH−  matrix X =[ x , xx, t +1 ] is formed as the input, while
2
−
A track = R arccos   ( RH) 2 RH H− 2 (VIII) t m m m
−
 R 
the hidden layer is defined by the melt track features
Where W and H denote the width and height of the melt Y t = [W t, D t, H t]. The process flow is displayed in Figure 8.
track; R denotes the radius of the bowls; A track denotes the Due to the general complexity of the experimental data in
area of the upper half of the melt track; ρ powder and ρ denote this study, a two-layer DGP model was used, containing
the powder material density and single-pass material the following core layers:
density, respectively; L track denotes the layer thickness; and (i) Melt pool periodic feature extraction layer (first layer):
L, W, and H denote the length, width, and height of the For the melt pool data of the same test set, 10
upper partial melt track. independent GPs – each using the Gaussian cosine
In this study, the powder’s stacking density was assumed composite kernel function – were used two-by-two to
to be 50%; that is, ρ powder = 50% ρ, where ρ is the material learn the characteristic fluctuations of different
3
density, which is taken as 7,800 kg/m . frequencies. For the input of melt pool time series data
1 ()
X t, the output F ( X ) of the ith GP obeys the
t
i
By substituting Equation VII into Equation VI and following distribution (Equation XII).
associating it with Equation VIII, Equation IX is obtained.
,
F 1 () ~ GP µ i ( 1 () X ( ), K ( X X )) , i 1=…,, 10 (XII)
1 ()
’
2
2
 W + 4H   8H 2  i t t t
2
05. WL track =   arccos  1− 2 
2
 8H   W + 4H  The work of Salimbeni et al. also indicated that the
37
 W + 4H 2 W linear mean function predicts better than the constant
2
−  − − H  (IX) mean function, and the computational overhead is less
 8H  2
than that of the non-linear mean function; hence, each GP
j ()
X
Using Halley’s iterative method, the melt track height mean function µ () is in linear form (Equation XIII).
i
H can be solved. However, for the melt track width W, it 1 () T
x
has been indicated that the effect of laser radius on the µ ( ) = a x , i 1= …, , 10 (XIII)
i
m
m
i
16
melt track width is negligible. The width of the melt track Where K ( ’
()1
based on the formation of the melt track by the steady-state X X, ) is expressed as in Equation XIV.
t
t
melt pool can be expressed as Equation X. ’2 ’

t
() 1
’
t
ξP = 2πk(T f–T )W + 0.1•eπρC (T–T ) VW 2 (X) K ( X X, ) =σ i 2 exp − X − l 2 2 X t   ⋅ cos    2π X − X  ,
t




t
0
0
f
t
T
Where the powder absorption rate ξ is taken as  i   i  
0.35; P is the laser power; π is the circumference; the i =…,,1 10 # (XIV)
thermal conductivity k is taken as 14 W/m K; the initial Where σ is used to control the amplitude of the output of
2
i
temperature T is taken as 300 K; the melting temperature the kernel function, l is used to control the smoothness
2
0
T f of the powder is taken as 1700 K; e is a natural constant; of the kernel, and T i defines the periodicity of the
i
the specific heat capacity C is taken as 683 J/kg K; and V is function.
the laser scanning speed. (ii) Melt track geometric feature mapping layer (second
Similarly, the theoretical volume of the melt track Vol is layer): This layer maps the periodic feature F output
(1)
expressed in Equation XI. from the first layer to the melt track geometric feature
F through six GPs. Each GP uses the Matérn kernel
(2)
 π W 2 1 
Vol =  − WH L (XI) employing physical constraints, and for the jth GP, the

2 ()
 4 2  output F () obeys the following distribution
x
j
(Equations XV and XVI).
2.5. DGP-p-based melt track size prediction and
morphology classification F 2 () ~ GP µ j ( 2 () F ( ) , K ( F , F )) , j 1=…,, 6 (XV)
1 ()’
1 ()
2 ()
1 ()
DGPs have a powerful non-linear learning capability, with j
a network structure that consists of an input layer, a hidden K ( F , ) =σ exp (− 1 F ( 1 () − F ) £ − 1 F ( 1 () − F ))
’ ()
’ ()
T
2
() 1
1
() 2
1
()’1
F
layer, and an output layer (Figure 8). m 2
(XVI)
For the melt pool, the geometric feature vector 2
2
2
x m = [s m, ar m, c m, t m, y] at time t is normalized based on the Where σ is the magnitude, Σ = diag ([W , δ, H ), and
m
melt pool features of the neighboring moments. The feature δ is a small perturbation term to prevent the theory from
deviating from 0 and causing numerical instability.
Volume 4 Issue 3 (2025) 8 doi: 10.36922/MSAM025200030