3.5 Numerical Method
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absorption calculation of the initial and multiple laser reflections on the wall of the
keyhole. When the metal vapor/plasma absorption coefficient and its distribution are
given, the laser energy density absorbed by metal vapor/plasma can also be calculated
reasonably.
It should be pointed out that the energy density distribution on the keyhole wall
in the actual laser welding process is closely related to the shape and size of the
keyhole, and therefore the actual energy density distribution on the keyhole wall
may be significantly different from the results of the above two calculation examples.
The above two calculation examples only discuss whether the ray tracing method
proposed can be combined with the method described in the Level Set interface
for calculation purely from the algorithmic perspective, and whether the calculation
result is reasonable, which lays a foundation for further evolution calculation of the
laser welding transient keyhole and moving weld pool.
3.5.2 Keyhole Free Surface Tracking Method
In the laser deep welding process, the morphology of the transient keyhole may be
extremely complex, and its topological shape may also change dramatically due to
the formation of bubbles and spattering. Meanwhile, using the ray tracing method
to calculate the energy density on the keyhole wall and to accurately calculate the
influence of surface tension on the behavior of the keyhole requires high precision
requirements for the calculation of the normal vector and curvature of the transient
keyhole wall. In this section, the realization process of describing the tracking keyhole
interface with the Level Set method is introduced in detail. Additionally, to overcome
the numerical dissipation of Level Set, the high-precision Level Set/Particle Level Set
interface tracking technology has important applications. This paper expounds the
technology and discusses the effectiveness of the high-precision numerical interface
tracking technique by using numerical examples.
3.5.2.1 High-Precision Level Set Interface Tracking and Solving
Technology
The Level Set Eq. (3.14) is a classical Hamilton–Jacobi equation. To reduce the
dissipation of the Level Set method itself, usually, a high-precision discrete scheme
is needed to perform a discrete solution for it. Generally, high precision means
that the truncation error of the difference scheme is at least second order to the
time and spatial steps. In recent years, high-precision schemes based on the idea of
Total Variation Diminishing (TVD), such as Essentially Non-Oscillatory (ENO) and
Weighted Essentially Non-Oscillatory (WENO), have been widely used in the Level
Set method. The idea of the TVD scheme is self-adaptively adjusting the numerical
dissipation of the calculation scheme. When the discrete approximate solution of an
unknown function has a large gradient or amplitude trend, using the TVD scheme can
89
absorption calculation of the initial and multiple laser reflections on the wall of the
keyhole. When the metal vapor/plasma absorption coefficient and its distribution are
given, the laser energy density absorbed by metal vapor/plasma can also be calculated
reasonably.
It should be pointed out that the energy density distribution on the keyhole wall
in the actual laser welding process is closely related to the shape and size of the
keyhole, and therefore the actual energy density distribution on the keyhole wall
may be significantly different from the results of the above two calculation examples.
The above two calculation examples only discuss whether the ray tracing method
proposed can be combined with the method described in the Level Set interface
for calculation purely from the algorithmic perspective, and whether the calculation
result is reasonable, which lays a foundation for further evolution calculation of the
laser welding transient keyhole and moving weld pool.
3.5.2 Keyhole Free Surface Tracking Method
In the laser deep welding process, the morphology of the transient keyhole may be
extremely complex, and its topological shape may also change dramatically due to
the formation of bubbles and spattering. Meanwhile, using the ray tracing method
to calculate the energy density on the keyhole wall and to accurately calculate the
influence of surface tension on the behavior of the keyhole requires high precision
requirements for the calculation of the normal vector and curvature of the transient
keyhole wall. In this section, the realization process of describing the tracking keyhole
interface with the Level Set method is introduced in detail. Additionally, to overcome
the numerical dissipation of Level Set, the high-precision Level Set/Particle Level Set
interface tracking technology has important applications. This paper expounds the
technology and discusses the effectiveness of the high-precision numerical interface
tracking technique by using numerical examples.
3.5.2.1 High-Precision Level Set Interface Tracking and Solving
Technology
The Level Set Eq. (3.14) is a classical Hamilton–Jacobi equation. To reduce the
dissipation of the Level Set method itself, usually, a high-precision discrete scheme
is needed to perform a discrete solution for it. Generally, high precision means
that the truncation error of the difference scheme is at least second order to the
time and spatial steps. In recent years, high-precision schemes based on the idea of
Total Variation Diminishing (TVD), such as Essentially Non-Oscillatory (ENO) and
Weighted Essentially Non-Oscillatory (WENO), have been widely used in the Level
Set method. The idea of the TVD scheme is self-adaptively adjusting the numerical
dissipation of the calculation scheme. When the discrete approximate solution of an
unknown function has a large gradient or amplitude trend, using the TVD scheme can