3.5 Numerical Method
101
during the back substitution process, in order to take into account of the effect of
physical conditions, such as recoil pressure and surface tension, on the motion of the
weld pool.
3.5.3.2 Solution for Temperature Field and Latent Heat Treatment
Method in the Calculation Area
Based on the above discussion, the semi-implicit method is adopted for solving the
temperature field equation (i.e., Eq. (3.3)) in this study. The semi-implicit scheme
means that the explicit scheme is adopted for discretization of all the terms except
the diffusion term. The purpose is to theoretically ensure that the time step of the
temperature field is not limited and that the system of linear equations obtained after
the discretization is symmetric positive definite. Like the discrete flow field equation
(i.e., Eq. (3.2)), the convection term in the discrete temperature field equation of
the fifth-order WENO scheme (i.e., Eq. (3.3)) and the discrete diffusion term of the
implicit central difference scheme are used in this paper, so the semi-discrete form
of Eq. (3.3) can be expressed as:
ρ
n C
n
p
T
n+1
− T
n
t
+ C
(T
n
) = D
(k
n+1 T
n+1
)
∇
− →
U ext · ∇φ = 0
(3.103)
where: C
(·) and D
(·)—Convection and diffusion operators in the discrete energy
equation (i.e., Eq. (3.3)).
When the Eq. (3.103) is discretized, the free boundary condition Eq. (3.60) and the
boundary condition equation on other boundary surfaces of the calculation area (3.61)
must be considered. After this operation, a typical symmetric, positively definite large
sparse matrix can be obtained. Here, the ICCG iterative method is again used to solve
the obtained temperature field matrix, and the new temperature field distribution on
the workpiece at time n + 1 is obtained.
In the laser welding process, the absorption of latent heat of fusion and the release
of latent heat of solidification occur all the time. In practical research, it is found that
both release and absorption of the latent heat affect the temperature at the interface
of the pores and the size of the weld pool all the time. If the latent heat is not
considered, the dynamic behavior of transient pores and weld pool during welding
cannot be accurately simulated.
In general, for latent heat phenomena such as fusion and solidification, studies have
shown that simulation can be well realized by using the temperature recovery/fall
method, the equivalent specific heat method, and the heat enthalpy method. In this
study, the temperature recovery method is used to simulate how to treat the physical
phenomenon of fusion and solidification.
101
during the back substitution process, in order to take into account of the effect of
physical conditions, such as recoil pressure and surface tension, on the motion of the
weld pool.
3.5.3.2 Solution for Temperature Field and Latent Heat Treatment
Method in the Calculation Area
Based on the above discussion, the semi-implicit method is adopted for solving the
temperature field equation (i.e., Eq. (3.3)) in this study. The semi-implicit scheme
means that the explicit scheme is adopted for discretization of all the terms except
the diffusion term. The purpose is to theoretically ensure that the time step of the
temperature field is not limited and that the system of linear equations obtained after
the discretization is symmetric positive definite. Like the discrete flow field equation
(i.e., Eq. (3.2)), the convection term in the discrete temperature field equation of
the fifth-order WENO scheme (i.e., Eq. (3.3)) and the discrete diffusion term of the
implicit central difference scheme are used in this paper, so the semi-discrete form
of Eq. (3.3) can be expressed as:
ρ
n C
n
p
T
n+1
− T
n
t
+ C
(T
n
) = D
(k
n+1 T
n+1
)
∇
− →
U ext · ∇φ = 0
(3.103)
where: C
(·) and D
(·)—Convection and diffusion operators in the discrete energy
equation (i.e., Eq. (3.3)).
When the Eq. (3.103) is discretized, the free boundary condition Eq. (3.60) and the
boundary condition equation on other boundary surfaces of the calculation area (3.61)
must be considered. After this operation, a typical symmetric, positively definite large
sparse matrix can be obtained. Here, the ICCG iterative method is again used to solve
the obtained temperature field matrix, and the new temperature field distribution on
the workpiece at time n + 1 is obtained.
In the laser welding process, the absorption of latent heat of fusion and the release
of latent heat of solidification occur all the time. In practical research, it is found that
both release and absorption of the latent heat affect the temperature at the interface
of the pores and the size of the weld pool all the time. If the latent heat is not
considered, the dynamic behavior of transient pores and weld pool during welding
cannot be accurately simulated.
In general, for latent heat phenomena such as fusion and solidification, studies have
shown that simulation can be well realized by using the temperature recovery/fall
method, the equivalent specific heat method, and the heat enthalpy method. In this
study, the temperature recovery method is used to simulate how to treat the physical
phenomenon of fusion and solidification.