3.3 Free-Surface Tracking Method
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3.3.1 Level Set Method
The Level Set method uses an implicit scalar function to describe the free-surface
position. The scalar function is normally a signed distance field function. It is defined
as the shortest distance from the keyhole interface in respect of laser welding. For
the Level Set function φ( x) in this document, R
3
→ R is defined as follows.
ϕ( x) =
⎧
⎨
⎩
−d( x, ,)
x ∈ in
0
x ∈
−d( x, ,)
x ∈
R
3
− − in
(3.14)
where:
Keyhole surface;
in
Area occupied by molten metal or workpiece;
R
3
− − in
Area of metal vapor/plasma.
Based on Level Set method, the kinematic equation describing the transient
keyhole surface in laser welding can be expressed as follows.
∂ϕ
∂t
+
− →
U · ∇ϕ = 0
(3.15)
where:
− →
U —Kinematic velocity of keyhole surface. In this study,
− →
U is determined
as the kinematic velocity of the molten metal in the weld pool closest to the current
position.
If the Level Set method is used to track the moving surface, the current Level Set
function should be re-initialized after solving of the Eq. (3.15), so as to ensure that
the Level Set function in each time step is a signed distance field. The re-initialization
equation can be described as follows.
|∇ϕ| = 1
(3.16)
Using the Level Set method to describe the free surface of the keyhole can facilitate
the computation of the exact normal vector and curvature value at any position on
the keyhole surface. Based on the distance field function, the calculation formulas
of normal vector and curvature k are as follows.
→
n =
∇ϕ
|∇ϕ|
(3.17)
κ = ∇ ·
∇ϕ
|∇ϕ|
(3.18)