A dynamic welding heat source model in pulsed current gas tungsten arc welding

Journal of Materials Processing Technology 213 (2013) 2329–2338
Contents lists available at ScienceDirect
Journal of Materials Processing Technology
journal homepage: www.elsevier.com/locate/jmatprotec
A dynamic welding heat source model in pulsed current gas tungsten
arc welding
Zhang Tong, Zheng Zhentai∗
, Zhao Rui
School of Materials Science and Engineering, Hebei University of Technology, No. 8, Duangrongdao Road, Hongqiao District, Tianjin 300130, PR China
a r t i c l e i n f o
Article history:
Received 25 January 2013
Received in revised form 9 July 2013
Accepted 11 July 2013
Keywords:
Numerical simulation
Welding temperature field
Heat source model
Pulsed current gas tungsten arc welding
a b s t r a c t
A time-dependent welding heat source model, which is defined as the dynamic model, was established
according to the characteristic of PCGTAW. The parabolic model was proposed to describe the heat flux
distribution at the background times. The recommended Gaussian model was used at the peak times due
to the bell-shaped temperature contour. The dynamic welding heat source was composed of these two
models with a function of time.
To assess the validity of the dynamic model, an experiment was conducted in which the pulsed current
gas tungsten arc deposits on the plate. From the comparison of the experimental and the simulated
values, it can be concluded that the dynamic heat source model, which uses the parabolic model at the
background time, is more realistic and accurate under the same welding conditions.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
With the development of the computer and numerical anal-
ysis technologies, the FEM has become a powerful and reliable
technique for prediction in the welding processing industry. The
temperature field contains sufficient information about the quality
and properties of the welded joint, and determines the distortion,
residual stresses, and reduced strength of a structure in and near
the welded joint. The temperature field is also the foundation of
the metallurgical analysis and phase change analysis. To obtain an
accurate welding temperature field, Goldak et al. (1984) reported
that the importance of a good welding heat source model has been
emphasized by many investigators.
Many welding heat source models have been developed up to
now, and the Gaussian model and the double ellipsoidal model are
the most popular models among them. Some good welding heat
source models can accurately predict the temperature field. How-
ever, most of these models were developed on the assumption that
the heat sources are static and not varied with time in the welding
processes. These models are no longer realistic for some dynamic
welding processes, such as the pulsed current gas tungsten arc
welding (PCGTAW). The objective of this paper is to develop a more
realistic and accurate welding heat source model for PCGTAW.
PCGTAW was developed in 1950s and is widely used in the
manufacturing industry today. In PCGTAW, the welding current is
∗ Corresponding author. Tel.: +86 13512499764; fax: +86 13512499764.
E-mail addresses: zhangtong06@hotmail.com
(Z. Tong), zzt@hebut.edu.cn (Z. Zhentai).
varied periodically from the peak current to the background cur-
rent. Balasubramanian et al. (2008) indicated that the heat energy
to melt the base metal is provided mainly by the peak current, while
the background current is set at a low level to maintain a stable arc.
Therefore, the background time can be seen as brief intervals dur-
ing heating, which allow the heat to conduct and diffuse in the base
metal.
PCGTAW is a widely utilized welding process. Traidia et al.
(2010) and Balasubramanian et al. (2008) pointed out that PCGTAW
has the following advantages over the constant current gas tung-
sten arc welding (CCGTAW): (a) lower heat input; (b) narrower heat
affected zone; (c) finer grain size; (d) less residual stresses and dis-
tortion; (e) improved mechanical properties; and (f) enhanced arc
stability to avoid weld cracks and reduce porosity, etc.
However, the welding parameters of PCGTAW are more com-
plex to define than CCGTAW, and the choice of parameters with
PCGTAW remains empirical. The parameters of PCGTAW were
depicted by Madadi et al. (2012) in Fig. 1. A great deal of work has
been conducted on the numerical simulation of PCGTAW. Fan et al.
(1997) developed a two-dimensional model using the boundary fit-
ted coordinate system to simulate the PCGTAW process. Kim and Na
(1998) computed the fluid flow and heat transfer in partially pen-
etrated weld pool under PCGTAW by the finite difference method.
Traidia and Roger (2011) used the unified time-dependent model
to describe the fluid flow, heat transfer and electromagnetic fields
in the three regions respectively. Many investigations have been
conducted, but far less work has been done on the development of
the welding heat source model under PCGTAW.
Several heat source models have been developed. They are clas-
sified in Table 1. Most of the current heat source models have been
0924-0136/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.jmatprotec.2013.07.007

2330 Z. Tong et al. / Journal of Materials Processing Technology 213 (2013) 2329–2338
Table 1
The classification of current welding heat source models.
One-dimension Two-dimension Three-dimension
Uniform distribution mode
Point heat source Plane heat source Columnar heat source
Line heat source Circular mode –
– Tripped heat source –
– Square heat source –
Gaussian mode
– Circular mode Circular disk heat source
– Oval-shaped heat source Columnar heat source
– Double oval-shaped heat source Cuboid heat source
– Tripped heat source Rotary body heat source
– – Conic heat source
– – Hemispherical heat source
– – Semi-ellipsoidal heat source
– – Ellipsoidal heat source
– – Double ellipsoidal heat source
Exponential decay mode – – Exponential decay heat source
developed on the geometrical shape and distribution in space, but
time as an important factor, which has rarely been considered, in
the model design. In fact, the heat source is varied with time in some
dynamic process, e.g. in the PCGTAW. Therefore, a time-dependent
heat source model, which is available for the dynamic process, is
necessary to be developed.
In this paper, a dynamic finite element model of welding heat
source under PCGTAW is established. Then the moving, time-
dependent heat source was attempted to load onto the structure,
and the FEM was used to compute the temperature field through
the software ANSYS.
2. Theoretical formulations
2.1. Model consideration
With the help of high speed CCD, Traidia and Roger (2011) used
an infra-red camera to capture the characteristic of a welding arc
under PCGTAW, and some good images were obtained which at the
background and peak times (see Fig. 2).
It is easy to see that there is significant difference between the
peak time and the background time, and the arc is bell-shaped
during the peak duration, but not during the background duration.
Fig. 1. Pulsed current GTAW process parameters (Madadi et al., 2012).
In contrast to constant current welding, the heat input in
PCGTAW is supplied mainly during the peak times, and the heating
is halted periodically during the background times. Xu et al. (2009)
pointed out that the characteristic of discontinuity during heating
under PCGTAW is more obvious when the frequency is low. So, two
heat source models must be proposed which will be available in
the peak times and background times. Considering the bell-shaped
temperature contour, the recommended Gaussian model was used
during the peak times; the big problem at present is to propose a
good heat source model which is available during the background
times.
Some good experience can be obtained from the proposed
process of the Gaussian heat source model. The design of the exper-
iment was made to investigate the heat and current distribution of
GTAW, which consists of splitting a water cooled copper anode.
Measure the heat flux to one of the sections as a function of the arc
position relative to the splitting plane. The radial heat distribution
can then be derived by an Abel transformation of the measured heat
flux on the anode. The distribution of heat on the anode is a result
of a series of collisions of electrons with ionized atoms as electrons
travel from the cathode to the anode. The energy released on the
anode surface carried by the electrons constitutes most of the heat,
and Tsai and Eagar (1985) considered that the distribution of the
heat flux on the water cooled anodes should closely approximate
to the distribution across the weld pool.
Similarly, regarding the PCGTAW in this paper, it can be also con-
sidered that the anodic heat flux distribution is closely approximate
to the heat distribution across the weld pool.
2.2. Mathematical model
Traidia and Roger (2011) obtained the numerical simulation
result of the radial heat flux distribution at the anode between the
pulsed current – background time and peak time – and the mean
current, which are shown in Fig. 3a. The third curve which the arrow
points to is the radial heat flux distribution during the background
time.
To simplify the problem, it can be assumed that the radial
heat flux at the background time is parabolic shape, which passes
through three points (0, q(0)), (Rb, 0), (−Rb, 0) in the coordinate
–x plane. The function of radial heat flux distribution at the back-
ground time can be written as:
q(x, ) = q(0) 1 −
x2
R2
b
, − Rb ≤ x ≤ Rb (1)
where q(0) is the maximum value of heat flux and Rb is the radius
of the power density.

Z. Tong et al. / Journal of Materials Processing Technology 213 (2013) 2329–2338 2331
Fig. 2. Infra-red camera images at the background and peak times for both first and last periods.
Substituting q(0) = 43 W/mm2 and R ≈ 2.8 mm which is corre-
sponded with the third curve in Fig. 3a into Eq. (1):
q(x, ) = 43 1 −
x2
2.82
, − Rb ≤ x ≤ Rb (2)
The function image of Eq. (2) is shown in Fig. 3b, which approxi-
mates to the third curve in Fig. 3a that represents the radial heat flux
distribution at the background time, which can be clearly observed
in Fig. 3c which combined Fig. 3a with Fig. 3b in the same scale. So
it can be considered that the radial heat flux distribution at back-
ground time is approximate to parabolic shape, and the welding
heat source is a spinning parabolic shape distribution as shown in
Fig. 4.
The spinning parabolic shape model of welding heat source with
the center at (0, 0, 0) to coordinate axes x, y, can be written as:
q(x, y, ) = q(0) 1 −
x2 + y2
R2
b
(3)
where q(x, y, ) is the power density (W/m2).
For r = x2 + y2 which is the radial distance from the center of
the heat source, then Eq. (3) can be written as:
q(r) = q(0) 1 −
r2
R2
b
, r ≤ Rb (4)
Conservation of energy requires that:
Q = ÁUI = q(r)r dr dÂ =
Rb
0
q(0) 1 −
r2
R2
b
r dr
2
0
dÂ (5)
and produces the following:
Q = ÁUI = q(0)
R2
b
2
(6)
q(0) =
2ÁUI
R2
b
(7)
Substituting q(0) from Eq. (7) into Eq. (4) gives:
q(r) =
2ÁUI
R2
b
1 −
r2
R2
b
, r ≤ Rb (8)
So the dynamic welding heat source model of PCGTAW in one
pulse cycle can be written as:
q(r) =
3ÁpUIp
R2
p
exp −3
r2
R2
p
, t ∈ [0, tp] (at peak times)
or q(r) =
2ÁbUIb
R2
b
1−
r2
R2
b
, t ∈ (tp, tT ] and r ≤ Rb (at background times)
(9)
where q(r) is the power density (W/m2), Áp the heat source effi-
ciency at the peak time, Áb the heat source efficiency at the
background time, U the arc voltage (V), Ip the peak current (A), Ib
the background current (A), r = (x2 + y2)1/2 which is the radial dis-
tance from the center of the heat source (m), Rb the radius of the
heat source at the background time (m), Rp the radius of the heat
source at the peak time (m), tT = 1 pulse cycle time = 1/f (s), f the
pulse frequency, tp the peak time (s), tb the background time (s)
and tp + tb = tT.
3. Evaluation of the dynamic model of welding heat source
in PCGTAW
One experiment was conducted in which the pulsed current
gas tungsten arc was deposited on the plate. The thermocouple
was used to measure the temperature field at the given points,
then the experimental values were compared with the simulated
values to assess the validity of the dynamic welding heat source
model.
Due to the lack of data on material properties, material mod-
eling has always been a critical issue in the welding simulation.

Table 2
The chemical composition of AA7075.
Elements Zn Mg Cu Cr Mn Fe Si Ti Al Impurities
wt.% 5.1–6.1 2.1–2.9 1.2–2.0 0.18–0.28 0.30 0.50 0.40 0.20 Bal. 0.15
Sattari-Far and Javadi (2008) reported that some simplifications
and approximations are usually introduced to deal with this prob-
lem, which are necessary because of the scarcity of material
data and numerical problems when trying to model the actual
high-temperature behaviors of the material. Here we select the
Aluminum Alloy 7075 as the base metal; the chemical composi-
tion is shown in Table 2. The thermal properties of AA7075 shown
Fig. 3. The establishment of the parabolic distribution (a is referred to Traidia and
Roger, 2011).
in Fig. 5 were reported by Guo et al. (2006) which are temperature-
dependent, the emissivity is assumed to be 0.6, and the fusion
temperature range is 477–638 ◦C.
3.1. Experimental procedure
3.1.1. Experiment preparation
The plate of Aluminum Alloy 7075 was cut to the required size of
80 mm × 80 mm × 8 mm. To measure the temperature in the weld-
ing process, the K type NiCr–NiSi thermocouple was used. The
positions of the thermocouples in the plate were shown in Fig. 6.
The thermocouples were glued to a depth of 4 mm, through the
blind holes which were drilled from the bottom of the plate; the hot
end diameter of the thermocouple was 1.5 mm, the cold end was
connected to a multichannel temperature measuring instrument
to acquire the thermal cycle, and the same method was introduced
by Karunakaran and Balasubramanian (2011).
3.1.2. Welding
Bead-on-plate welds were made using the PCGTAW on the sur-
face of the plate along with the center line. The welding parameters
are shown in Table 3.
3.2. FEM calculation
3.2.1. Finite element model
Only half of the plate was selected to analysis for its symmetry.
To reduce the calculation time, the zone near the welding bead has
been modeled with a finer mesh, while the zone further away from
the welding bead has been modeled with a coarser one. Solid70
and Surf152 were used to mesh the model; the surface has been
Fig. 4. Heat source configuration for the spinning parabolic shape.

Fig. 5. Thermal physical properties of AA7075: (a) specific heat and density and (b)
conductivity.
Fig. 6. Schematic diagram of welded plate used in the experiment.
Table 3
Welding parameter.
Process parameter Actual Simulated
Welding current
Peak current 180 A 180 A
Background current 60 A 60 A
Arc voltage 14 V 14 V
Welding speed 1.96–2.03 mm/s 2 mm/s
Pulse frequency 1 Hz 1 Hz
% Pulse on time 50% 50%
Electrode W–2%Th –
Electrode diameter 3.2 mm –
Arc length 2 mm –
Torch angle 60◦
–
Shielding gas Argon 99.9% –
Flow rate 15 L/min –
“coated” with Surf152 to represent the convective heat exchange.
The FEM model is shown in Fig. 7.
3.2.2. Welding heat source
In this research, the APDL programming languages of ANSYS
were applied to realize the moving load of the heat source. A
local coordinate system was established, and the center of the heat
source coincided with the original point of the local coordinate,
then the heat source moved gradually under the control of the loop
command in APDL.
To evaluate the validity of the dynamic heat source model, two
simulation tests were implemented under the same welding con-
ditions, which are described in Table 4. The parameters in the
dynamic welding heat source model are not easy to decide, so a
further study is needed.
3.2.3. Initial condition and boundary conditions
The ambient temperature is 28 ◦C. Considering the moving heat
source, heat losses due to convention and radiation are taken into
account in the finite element models. Heat loss due to convection
(qc) is taken into account using Newton’s law:
qc = hc(Ts − T0)
where hc is the heat transfer coefficient, Ts the surface temperature
of the weldment and T0 is the ambient temperature which is 28 ◦C.
Heat loss due to radiation is modeled using Stefan–Boltzmann’s
law:
qr = −ε · [(Ts + 273)4
− (T0 + 273)4
]
where ε is emissivity which is 0.6 and = 5.67 × 10−8 W/m2 ◦C−4 is
defined as the Stefan–Boltzmann constant.
3.2.4. Latent heat of phase transition
During the welding process, melting and solidifying will occur
in the welding pool, it will absorb or release latent heat in the
phase transition, which is defined as “latent heat of phase tran-
sition”. Lei et al. (2006) use the enthalpy method to deal with the
Fig. 7. FEM model.

Table 4
List of the simulation test.
Simulation test 1 Simulation test 2
Heat source model Dynamic Model 1 Dynamic Model 2
Model description Use Gaussian model at peak times; use parabolic model at
background times
Use Gaussian model both at peak times and background times, but
different values of parameters were used, respectively
Parameters in model
Ip = 180 A, Ib = 60 A, U = 14 V, f = 1 Hz Ip = 180 A, Ib = 60 A, U = 14 V, f = 1 Hz,
Pulse on time = 50%, Rp ≈ 5.0 mm, Rb ≈ 2.8 mm, Áp ≈ 0.68, Áb ≈ 0.62. Pulse on time = 50%, Rp ≈ 5.0 mm, Rb ≈ 2.8 mm, Áp ≈ 0.68, Áb ≈ 0.62.
Notes: The parameters in heat source models are difficult to decide. To simplify the problems, the same parameters in Traidia and Roger (2011) were used for test 1 and test
2 under the same welding condition.
latent heat, and define the material’s enthalpy which varies with
the temperature:
H(T) =
T
0
(T)c(T) dT
where (T) is the density of the material varying with temperature
(kg/m3) and c(T) is the specific heat of the material varying with
temperature (J/(kg K)).
Murugan et al. (2000) reported that the release or absorption of
latent heat can also be considered in the numerical analysis by an
artificial increase in the value of the specific heat over the melting
temperature range.
3.2.5. Others
In the meshed finite element model, the number of the Solid70
element is 848,000, the number of the Surf152 element is 46,640,
and the number of nodes is 887,814 in total.
The heat source defined in a local coordinate system moves with
time, the former load step is deleted when the heat source moves to
the next step. Considering both the calculation time and the com-
puter’s capacity, the minimum size of element is 0.2 mm, and the
cooling time is fixed to 20 s.
Fig. 8. Top view of temperature distribution: (a) at 20.3 s (peak time) and (b) at 20.8 s (background time) computed by the Dynamic Model 1.

Fig. 9. Top view of temperature distribution: (a) at 20.3 s (peak time) and (b) at 20.8 s (background time) computed by the Dynamic Model 2.
Fig. 10. Longitudinal cross-section of temperature distribution: (a) at 20.3 s (peak time) and (b) at 20.8 s (background time) computed by the Dynamic Model 1.

Fig. 11. Longitudinal cross-section of temperature distribution: (a) at 20.3 s (peak time) and (b) at 20.8 s (background time) computed by the Dynamic Model 2.
4. Results and discussion
4.1. Temperature field
4.1.1. Top view of temperature distribution
Figs. 8 and 9 which are in the same scale, show the temperature
field computed by the Dynamic Model 1 and the Dynamic Model
2, respectively, and including the time 20.3 s (peak time) and 20.8 s
(background time) for each. To illustrate the difference of the tem-
perature field between the peak time and the background time in
the welding process, the same area region near the weld pool was
magnified in the same scale.
Comparing the two temperature fields in Fig. 8a and b, it can
be seen that the high temperature region at 20.3 s is larger than
that at 20.8 s. Due to the cyclic variation of the heat input, there is
a thermal fluctuation in the temperature field, which corresponds
to the real dynamic welding process. From Fig. 9a and b, the same
conclusion above can be obtained.
Table 5
Peak temperature comparison of the experimental and simulated results.
Measuring
point
Methodsa
Peak
temperature (◦
C)
Differenceb
(%)
Point Ac
Experimental 402.5 –
FEM (Dynamic Model 1) 397.6 −1.2
Point Bc
Point Cc
a
Experimental: use PCGTAW – welding parameter is shown in Table 3; base metal
– AA7075, chemical composition is shown in Table 2. The description of the Dynamic
Model 1 and Dynamic Model 2 are listed in Table 4.
b .
Difference (%) = (Calculated value − Experimental value)/Experimental value.
c .
The position of the measuring points is depicted in Fig. 6.
In Fig. 8a and b, the maximum temperatures are 892 ◦C and
779 ◦C, respectively. It was found that the maximum temperature
at 20.3 s (peak time) is higher than the value at 20.8 s (background
time). In Fig. 9a and b, the maximum temperatures are 887 ◦C and
849 ◦C, respectively. The maximum temperature appears in the
center of the heat source model for both Figs. 8 and 9.
Comparing Fig. 8a with Fig. 9a, it can be seen that there is small
difference of the maximum temperature between them, which
implies that the maximum temperature is nearly the same at 20.3 s
when using the Dynamic Model 1 and the Dynamic Model 2. How-
ever, the maximum temperature in Fig. 8b is much lower than that
in Fig. 9b, which implies that there is much difference in the max-
imum temperature at 20.8 s (background time) when using the
Dynamic Model 1 and the Dynamic Model 2.
Supplementary Video 1 is available for readers to show the tem-
perature field computed by the Dynamic Model 1 in PCGTAW.
Supplementary data associated with this article can be
found, in the online version, at http://dx.doi.org/10.1016/
j.jmatprotec.2013.07.007.
4.1.2. Longitudinal cross-section of temperature distribution
Along with the midline of the plate in the longitudinal direc-
tion, the cross-sections of temperature distribution which are in
the same scale were obtained, as shown in Figs. 10 and 11. To
demonstrate clearly, the same area region near the heat source was
magnified in the same scale. From Figs. 10 and 11, the same conclu-
sions in Section 4.1.1 can also be obtained. The difference between
the calculated results by the Dynamic Model 1 and the Dynamic
Model 2 is demonstrated in some extent.
4.2. Welding thermal cycles
The comparison of the experimental and simulated welding
thermal cycles at Point A, Point B and Point C are shown in
Fig. 12a–c, respectively.
As can be seen from the figures, the temperatures computed by
the Dynamic Model 1 and the Dynamic Model 2 are slightly lower

Fig. 12. Comparison of the experimental and simulated welding thermal cycles: (a)
Point A, (b) Point B and (c) Point C.
than the experimental values. Table 4 shows the peak temperature
comparison of the experimental and the simulated values. The dif-
ference value listed in Table 5 indicated that the Dynamic Model 1
is more accurate than the Dynamic Model 2, and it implies that the
dynamic model which uses the parabolic model at the background
time is more realistic and accurate.
From Fig. 12a and b, it can be noted that the temperature
is increased slightly during the cooling time while it cannot be
observed in Fig. 12c. The reason for that could be attributed to the
latent heat in the solidifying process. Many experiments show that
Fig. 13. The welding thermal cycle during 5–20 s at point A.
the energy released during solidifying for aluminum alloy is much
bigger than the carbon steel due to the thermal physical properties
of the material in or near the weld pool. However, the latent heat
become less and can be neglected for the areas far away from the
weld pool.
4.3. The characteristic of the pulsed current
Fig. 13 is part of Fig. 12a that magnified with a proper scale.
It is clearly seen that there are some fluctuations in the welding
thermal cycle computed by the Dynamic Model 1 and the Dynamic
Model 2, which can be attributed to the influence of pulsed current.
Wang (2003) used the finite element method to compute the tem-
perature field in molybdenum alloy under PCGTAW, the fluctuation
was observed in the computed welding thermal cycle. Zheng et al.
(1997) developed a three-dimensional model to demonstrate the
transient behavior of temperature field and weld pool in PCGTAW,
and verified that the fluctuations in the thermal cycle curve are
characteristic of the pulsed current welding. Therefore, it can be
concluded that the Dynamic Model 1 and the Dynamic Model 2 can
successfully demonstrate the dynamic process of temperature field
in pulsed current welding.
However, the experiment in this paper failed to capture the
characteristic of the pulsed current. This may be due to the sensi-
tivity of the temperature measuring instrument. The thermocouple
is widely used as temperature sensor for measuring instrument,
which can convert a temperature gradient into electricity. For the
dynamic welding process of PCGTAW, the heat input is varied
periodically in a very short time, which leads to the dynamic char-
acteristic of the process that cannot be obtained easily. This requires
the thermocouples to be sensitive enough to the short-term varia-
tion and the measuring instrument immediately responsive to deal
with the electronic signals from thermocouples at different mea-
sured points. Maybe an improved measuring instrument or a better
measuring method is needed to be developed. Although the exper-
iment failed to capture the temperature fluctuations in PCGTAW,
the temperature values measured by the calibrated instrument are
accurate and convincing.
The peak temperature is obtained when the heat source sur-
passes the measured point. As seen from Fig. 11, the pulsing effect
is more obvious for the pulses closed to the measured point, while it
becomes less for the pulses further away from the measured point.
It can be seen that the region in or near the welded joint has expe-
rienced several heating and cooling processes due to the pulsing
current, and that the soaking time at the high temperature is shorter

compared with CCGTAW. That is why the grain is refined under the
PCGTAW process.
Compared with the welding thermal cycle at different points
in Fig. 12a–c, it can be concluded that the pulsed current has an
significant effect on the points in or near the welded joint, but less
effect on the points far away from the welded joint.
5. Conclusions
(1) Most of the current heat source models are static models that do
not vary with time and cannot represent the heat flux distribu-
tion in some dynamic welding processes; so a good heat source
model for the dynamic welding process must be developed.
(2) The FEM dynamic heat source model was used to simulate the
low frequency PCGTAW, which has successfully demonstrated
the dynamic temperature field in the welding process.
(3) From the comparisons of the experimental and the simulated
values, it can be concluded that the dynamic heat source model
which uses the parabolic model at the background time is more
accurate under the same welding conditions.
(4) In some welding process simulation, especially for those whose
dynamic characteristic is more obvious, the dynamic welding
heat source model has more advantages over the static models.
The static heat source model is the special case of the dynamic
heat source model, which is not varied with time.
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