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Topological optimization of magnetic pulse welding coils with a
connectivity-constrained particle swarm optimization algorithm
Sen Lin
a
, Nengzhuo Chou
a
, Yujia Zhao
a
, Yangfan Qin
a
, Hao Jiang
a
, Junjia Cui
a
, Guangyao Li
b
, Yi Min Xie
c
a
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China
b
Shenzhen Automotive Research Institute, Beijing Institute of Technology, Shenzhen 518118, Guangdong, China
c
Centre for Innovative Structures and Materials, School of Engineering, RMIT University, GPO Box 2476, Melbourne 3001, Australia
highlights
We designed a more efficient MPW
coil by a discrete PSO algorithm.
The continuum problems are
addressed by connectivity constrains
and filtration.
The maximum tensile load of welded
sample (AA5052 and HC420LA) is
improved by 19.88%.
graphical abstract
article info
Article history:
Received 9 June 2022
Revised 21 October 2022
Accepted 29 October 2022
Available online 31 October 2022
Keywords:
Magnetic pulse welding coil
Topological optimization
Connectivity constraints
Particle swarm optimization
abstract
A connectivity-constrained optimization methodology based on a discretized particle swarm optimiza-
tion (PSO) algorithm is proposed for the continuum structural design of a magnetic pulse welding coil.
To address the classical continuum topological challenges of conductors, such as simple connectivity
and checkerboard phenomena in the optimal topology, the PSO algorithm is combined with connectivity
constraints and filtration. The evolutionary history and parametric tests in an electromagnetic structure
study are employed to demonstrate the efficiency and robustness of the novel algorithm. Physical tests
show that the optimized coil can form a larger effective welding area between aluminum alloy
(AA5052) and steel (HC420LA) sheets compared with the joint formed by the original coil. The maximum
tensile load is also significantly improved by 19.88% with a discharge energy of 22 kJ.
Ó2022 The Authors. Published by Elsevier Ltd. This is an open access articleunder the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction
Magnetic pulse welding (MPW) has many excellent characteris-
tics, such as good forming quality, high production efficiency, easy
control, environmental friendliness and energy conservation [1].
Dissimilar materials attached to each other by means of this cold
welding process have been preliminarily applied in new energy
vehicles, rail transit, nuclear power, aerospace, and household
appliances [2]. During MPW, as the conductor transporting current,
the coil plays the role of establishing the magnetic field and driving
the flyer plate [3]. Its structure determines the welding perfor-
mance, and it is one of the key components of MPW equipment
[4]. Therefore, the structural optimization of coils is of great signif-
icance not only for the development of the digital industry but also
research on MPW [5].
Since Isao first took the lead in systematically studying the
structural design of magnetic pulse coils, research in this field
has continued unabated [6]. For different welding processes and
applications, the coil shapes and structures will be different [7].
Taking MPW coils for flat parts as examples, they can be classified
https://doi.org/10.1016/j.matdes.2022.111337
0264-1275/Ó2022 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Materials & Design 224 (2022) 111337
Contents lists available at ScienceDirect
Materials & Design
journal homepage: www.elsevier.com/locate/matdes
into single-sided welding coils, double-sided welding coils and
uniform pressure coils depending on the intended welding task
and action mechanism [8]. Among them, common single-sided
welding coils include I-shaped [9], E-shaped [10] and rectangular
[11] coils. Their common feature is a drastic change in the coil sec-
tion, where a large current density can be generated near the weld-
ing spot [12].
Researchers have employed gradient-based topological opti-
mization algorithms to design various electromagnetic structures
[13]. Dyck proposed the optimized material distribution (OMD)
algorithm and innovatively applied topological optimization to
an electromagnetic field [14]. Since the emergence of the solid iso-
tropic material with penalization (SIMP) method, which shows
excellent performance in traditional mechanical optimization,
electromagnetic topology optimization based on density algo-
rithms has undergone gradual development [15]. Yoo employed
an improved homogenization method for the topological optimiza-
tion design of an H-type electromagnet [16]. The ON/OFF algorithm
proposed by Takahashi [17] and the permeability-based interpola-
tion method proposed by Choi [18] are two typical density algo-
rithms applied for electromagnetic topology optimization. In
addition, level set methods with smooth boundaries are attracting
increasing attention from electromagnetic experts [19].
However, for multifield coupling problems, the sensitivity of
gradient optimization is difficult to ensure [20]. The optimization
process can easily fall into a locally optimal solution [21]. There-
fore, researchers have begun to use evolutionary optimization
algorithms to solve electromagnetic optimization problems [22].
These algorithms simulate optimization phenomena in nature
through numerical methods and establish empirical iterative opti-
mization processes; representative examples include genetic algo-
rithms (GAs) [23], differential evolution (DE) [24], particle swarm
optimization (PSO) [25], and simulated annealing (SA) [26].To
obtain a higher solution quality, it is common to combine these
intelligent optimization algorithms with other deterministic algo-
rithms or mathematical models. For instance, the multiobjective
topology optimization (MOTO) methodology, based on a GA and
DE, has been proposed for the topological optimization of an elec-
tromagnetic actuator [27]. Regarding development for PSO, Men-
des et al. have considered defining evolutionary topologies by
varying the particles’ neighborhood based on certain parameters
[28]. Moreover, intelligent algorithms, which have recently gained
considerable popularity, have been applied to assist PSO, enabling
this evolutionary algorithm to address some optimization prob-
lems that are difficult to express explicitly [29]. However, evolu-
tionary algorithms consume considerable computation time.
Moreover, for electromagnetic topology optimization, they require
sophisticated constraints and termination conditions [30].
The rest of this paper is organized as follows. A discretized PSO
algorithm has been developed for the continuum structural design
of an MPW coil, for which the underlying finite element analysis
and other basic methodologies are introduced in Section 2, includ-
ing associated techniques such as connectivity constraints, muta-
tion and filtration. To demonstrate the efficiency and robustness
of the proposed algorithm, the evolutionary history and corre-
sponding parametric tests have been investigated in an electro-
magnetic structure study. Section 3introduces the details of
mechanical tests and the morphology of the resulting welding
joints, verifying that the optimized coil can form effective connec-
tions and enhance the mechanical properties of the joints between
aluminum alloy and steel sheets. Finally, Section 4concludes the
study.
2. Numerical simulation
2.1. Finite element analysis
The process of MPW is generally treated through cosimulation
involving three physical fields: the electromagnetic field, the struc-
tural field and the thermal field. However, the electric discharge
time is short, while the resistance of the working system is small.
Therefore, the thermal energy generated by the current is rela-
tively small and can be ignored in this case. For such a coupled
electromagnetic–mechanical process, it would be a very compli-
cated project to obtain an analytical solution for the whole welding
process through mathematical means. On the other hand, the
acquisition of test information in the MPW process is costly and
difficult, and the available information is limited. Moreover, weld-
ing outcomes are sensitive to the materials used and the opera-
tions applied. The few and erratic experimental data available are
not sufficient to support accurate theoretical analysis. As an alter-
native, the method of finite element analysis (FEA) can save time
and economic costs. With this method, a large number of accurate
results can be stably obtained through repeatable numerical simu-
lations, which can provide valuable data to be used for optimiza-
tion. In this case, an electromagnetic field model of MPW was
simulated using the commercial software LS-DYNA (Version 12.0,
Ansys Ltd., U.S.). As a high-frequency pulse current is transmitted
through the coil, the aluminum flyer plate (25 120 1 mm)
and the steel base plate (25 120 1 mm) will collide, driven by
a large Lorentz force, as shown in Fig. 1(a). To ensure the accuracy
of the results, the coil (300 170 10 mm) was discretized into
(5 55 mm) hexahedral solid elements, while the flyer and base
plates were discretized into (5 51 mm) hexahedral solid ele-
ments. For these three-dimensional brick elements, the mesh met-
ric indexes were Min: 0.357, Max: 1.000, Average: 0.962, and
Standard Deviation: 0.151, indicating a high element quality. The
properties of each part are listed in Table 1.
The coil and base plate were regarded as rigid parts, while the
Johnson–Cook model was used for the constitutive relation of the
flyer plate. This is because the collision between the flyer plate
and the base plate is a transient process with a high strain rate.
The effective yield stress
r
of each element in the flyer plate can
be derived as
r
¼AþB
n
ðÞ1þCln
ðÞ1T
m
ðÞ ð1Þ
where is the effective plastic strain,
is the normalized effective
plastic strain rate, T
is the homologous temperature, Ais the initial
yield stress, Bis the hardening constant, Cis the strain rate constant,
and nand mare the hardening and thermal softening exponents,
respectively. For the purpose of acceleration, there is a gap
(1.2 mm) between the flyer and base plate, which are separated
by two supporting blocks. These blocks were assumed to be non-
conducting rigid bodies in the simulation. The blocks, base plate
and coil were fixed, while the flyer plate was able to move freely.
Input and output surface currents were deployed at each end face
of the coil. The function describing the 25 kJ discharge current Iin
the coil versus time tcan be defined as
I¼Uffiffiffiffiffiffiffiffiffiffi
C
e
=L
pe
Rt=2L
sin ffiffiffiffiffiffiffiffiffiffiffiffi
t=LC
e
pð2Þ
where the discharge voltage Uand the equivalent capacitance C
e
must be determined experimentally, whereas the equivalent induc-
tance Land the equivalent resistance Rcan be obtained by fitting
the current curve via COMSOL Multiphysics (Version 6.0, COMSOL
Co., Ltd., Sweden). The corresponding current curve is shown in
Fig. 1(b). Since the MPW process is mainly completed within the
first half-wave (0 30
l
s) of the current curve, the peak value
and frequency of the first half-wave determine the deformation of
S. Lin, N. Chou, Y. Zhao et al. Materials & Design 224 (2022) 111337
2
the flyer plate during MPW. Consequently, only the first half-wave
of the current curve was considered in the numerical simulation.
The collision of the two plates occurs at approximately 20
l
s. To
reduce the computational burden, the time step of the structure
field was set to 10 times that of the electromagnetic field, with neg-
ligible influence on the calculation accuracy. Moreover, the nature
of the interaction between the upper surface of the flyer plate and
the bottom surface of the base plate was set to ‘FORMING NODES
TO SURFACE’, the same as the contacts between the flyer plate
and the supporting blocks. The dynamic friction coefficient between
the flyer plate and the supporting block was measured using a fric-
tion coefficient tester (Esm303COF, Mark-10 Corporation, U.S.). In
accordance with the experimental results, the dynamic friction
coefficient was set to 0.15. Regarding the boundary conditions,
the coil and the base plate were specified separately; see Fig. 1(a).
The two supporting blocks were bound to the underside of the base
plate to avoid rigid body displacement. The flyer plate was able to
move freely and was placed on the supporting blocks.
2.2. Evolutionary methodology
For this complex coupled electromagnetic–dynamic optimiza-
tion problem, it is difficult to express the optimization objective
explicitly, which leads to difficulty in applying traditional gradient
optimization algorithms such as SIMP and bidirectional evolution-
ary structural optimization (BESO). As a popular algorithm for evo-
lutionary optimization, PSO is well known for its strong
adaptability and global search ability. However, the binary-
encoded PSO algorithm can still easily fall into local optima. There-
fore, a PSO method based on topology configuration is proposed
here. Assisted by connectivity constraints, it can quickly find the
global optimum that satisfies the requirements of current conduc-
tion. Since each optimization iteration requires an FEA of the opti-
mized coil structure, the collision between the flyer and base plates
consumes considerable computational resources. Therefore, the
coupling analysis of the electromagnetic field and structural field
should be simplified. The main acceleration process of the flyer
plate (0 10
l
s) was selected for the simplified FEA. In the initial
coil design (a flat coil design commonly used for MPW), the current
enters in the middle region and exits from the top side (Fig. 2(a)).
The flyer plate is placed at the long straight section, through which
a concentrated current flows. The welding quality of the joint is
positively correlated with the impact velocity of the flyer plate.
Moreover, the geometry of the joint depends on the velocity distri-
bution of the flyer plate induced by the Lorentz force (Fig. 2(b)).
Therefore, the optimization objective can be chosen to maximize
the velocity in the area around the joint. Simultaneously, the objec-
tive is mainly influenced by the coil elements near the joint, which
can therefore be regarded as the design domain D for optimization.
On the premise that the appropriate collision angle is satisfied, the
two welding plates can be merely ‘‘squeezed” together in the cen-
ter region. Such circumstances are conducive to shear plastic
deformation along the interface of the two plates, where a jet
can be formed. The appropriate combination of the impact angle
and collision velocity is referred to as the ‘‘welding window”.
Accordingly, the objective function for optimization was deter-
mined as follows:
max :V
X
¼R
X
V
X
dX=S
X
s:t:S
W
6S
f
b
low
6b
X
6b
upp
(ð3Þ
where V
X
and V
X
are the average out-of-plane velocity in the objec-
tive area Xand the out-of-plane velocity of each element, respec-
tively. S
X
;S
W
and S
f
are the sizes of the objective area, the
conductor in the design domain and the design domain, respec-
tively. The impact angle in the objective area, b
X
, should also be
in a reasonable range of [b
low
;b
upp
]. Due to the skin effect, the cur-
rent will be conducted along the inner and outer edges of the coil.
Consequently, there is no point in generating more holes in the coil.
Thus, the conductor in the design domain,
W
, should be a simply
connected domain.
In the classical PSO algorithm, the optimal solution for a prob-
lem is determined by comparing a group of particles moving in
the solution domain. For the optimization domain
W
, which is dis-
cretized into finite elements, the position of each particle should be
represented by a matrix X. This matrix is composed of elements
with values in the range [-1, 1], with negative and positive ele-
ments representing insulator behavior and conductor behavior,
respectively. The corresponding movement speed of each particle
is denoted by Vand varies in each iteration of the evolutionary
process. The movement behavior of each individual is influenced
by its best performance (according to the objective function)
Fig. 1. The principle of MPW: (a) a schematic working diagram of the coil; (b) the fitted curve of the current in the coil.
Table 1
The electromagnetic and mechanical properties of the system parts used for
simulation.
Property Coil Base plate Flyer plate
Material CuCrZr HC420LA AA5052
Conductivity (
X
m) 1:75 10
8
1:02 10
7
2:65 10
8
Constitutive model 020-RIGID 020-RIGID Johnson–Cook
Young’s modulus (GPa) 97 210 70
Poisson’s ratio 0.30 0.30 0.33
Density (kg/m
3
) 8900 7850 2730
S. Lin, N. Chou, Y. Zhao et al. Materials & Design 224 (2022) 111337
3
throughout its own evolutionary history as well as by the best per-
formance observed in the entire population. In this way, the topol-
ogy of the design domain can converge to the global optimum as
evaluated by the objective function. Accordingly, the speed and
position of particle iin the kþ1
th
iteration of this finite-element
variant of PSO can be given as
V
kþ1
i
¼
x
V
k
i
þc
1
rand 1ðÞX
k
i;best
X
k
i
þc
2
rand 2ðÞ
X
k
gbest
X
k
i
ð4Þ
where xis a weight controlling the influence of the speed of parti-
cle iin the previous iteration, i.e., an inertia weight applied to the
particle. The influence of this weight value on the convergence of
the PSO algorithm has previously been systematically studied
[31]. It has been proven that a high value of xallows the swarm
to explore the entire design domain to find the global best solution
with considerable efficiency, while a lower value can cause the par-
ticles to concentrate in particular areas to ensure that no locally
optimal solution is missed. X
k
i;best
is the topology with the best per-
formance found by particle iduring its evolutionary history. X
k
gbest
is
the topology found by the global best individual in the swarm. c
1
and c
2
are weight parameters representing the degrees of influence
of the individual performance of particle ithroughout its history
and the performance of the best particle in the entire swarm,
respectively. rand ...ðÞis a random value in the range of [0,1], intro-
duced to improve the robustness of the evolutionary algorithm.
Moreover, the speed of the particles should be constrained to
ensure the convergence of the evolutionary process. To this end, a
maximum speed is defined as V
max
, which depends on the maxi-
mum position of each element, X
max
= 1. In this study, V
max
was
defined as 10%X
max
to avoid premature convergence, as recom-
mended based on previous research on PSO algorithms [32]. Conse-
quently, Vconsisted of elements with values in the range of [-0.1,
0.1] here.
2.3. Connectivity constraints
As mentioned above, the conductor in the design domain
should be a simply connected region. To ensure such geometric
features, connectivity constraints should be considered in the evo-
lutionary algorithm. The evolutionary process may start from an
initial design of a conductor with the form of a simply connected
region, as represented by the blue-gray region in Fig. 3(a). Addi-
tionally,
W
should be connected to the conductor outside the
design domain. The boundary between the conductor and the insu-
lator (empty area) is represented by a black solid line. The ele-
ments along the boundary in the conductor region, f
d
, are
marked in dark gray, while the elements along the boundary in
the insulator region, f
a
, are marked in green. To ensure that the
conductor remains simply connected, the following connectivity
constraints can be imposed:
V
kþ1
i;
j
¼0
j
¼ff
a
ðÞif PV
kþ1
i;f
>0
j
¼ff
d
ðÞif PV
kþ1
i;f
<0
(ð5Þ
where the speeds of the elements in the region jare forced to be
equal to zero. If the sum of the speeds of all elements in the design
domain fin iteration kþ1, as computed via Eq. 4, is positive, then
the scope of jis the external region f
a
. In this circumstance, a con-
ductor element can be added in f
a
in accordance with the speed
matrix. In contrast, a negative value of PV
kþ1
i;f
indicates that those
elements with negative positions in f
d
should be changed from con-
ductor to insulator.
Fig. 2. The discretized electromagnetic model and corresponding FEA: (a) the current density contours of the coil, where the optimization design domain is marked with
black boxes; (b) the out-of-plane velocity distribution of the flyer plate at 10
l
s after the beginning of the MPW procedure, where the objective area is marked.
S. Lin, N. Chou, Y. Zhao et al. Materials & Design 224 (2022) 111337
4
2.4. Mutation and filtration
Although the initial position provides a degree of randomness,
the velocity of a particle can be affected only by the globally and
locally optimal solutions in the current state; consequently, if only
these influences are considered, some potentially excellent solu-
tions may still be missed. Therefore, to increase the randomness
of the geometrical evolution process, we introduce a technique
similar to genetic mutation. To ensure the convergence of the evo-
lutionary process, the probability of such mutation decreases with
an increasing number of iterations. For each element (p,q) in the
addable region f
a
and the deletable region f
d
, the probability of
mutation can be expressed as
V
kþ1
i;j
¼rand 3ðÞj2f
a
ðÞif PV
kþ1
i;f
>0&rand 5ðÞ>k=k
max
ðÞ
a
rand 4ðÞj2f
d
ðÞif PV
kþ1
i;f
<0&rand 5ðÞ>k=k
max
ðÞ
a
(
ð6Þ
where the speed of each element jin f
a
or f
d
will no longer be cal-
culated following Eq. 4if it satisfies the mutation condition. If the
random value rand 5ðÞ¼0;1½is larger than the threshold value,
which is determined in accordance with the current iteration k,
the maximum number of iterations k
max
and an index a, then muta-
tion will occur. The mutation probability decreases sharply as the
iterative process continues. For the case of a¼0:1, the threshold
value is 0.676 at the beginning of the evolution, while mutation
becomes impossible (the threshold value is 1) by the end of evolu-
tion. Moreover, only the elements in the aforementioned modifiable
regions (Fig. 3(a)) can participate in mutation. For instance, if the
evolutionary trend in the current iteration is to remove conductor
elements, then the speed of a mutational element j(Fig. 3(b)) will
be equal to a random value rand 4ðÞ¼V
max
;0½. Conversely, for a
mutation that increases the number of conductor elements, the cor-
responding speed will be rand 3ðÞ¼0;V
max
½. Subsequently, the posi-
tion matrix will be updated as follows:
X
kþ1
i
¼X
k
i
þV
kþ1
i
ð7Þ
As an alternative approach, some researchers have proposed the use
of the sigmoid function to adapt some concepts of GAs and muta-
tion processes for the development of a binary version of PSO
[33]. In contrast, the method proposed here is designed to accom-
plish such a finite-element variant of the PSO algorithm with con-
tinuous position values. Notably, despite the introduction of the
connectivity constraints, the evolutionary process might still gener-
ate discontinuous conductor elements. For example, when remov-
ing some conductor elements along a rectangular boundary, it is
possible to leave a pair of elements that are only diagonally con-
nected. Such unsatisfactory optimization results could cause con-
siderable inconvenience in coil manufacturing. In classical
topological optimization algorithms, filtering techniques are often
used to address similar unfavorable checkerboard phenomena
[34]. Accordingly, to ensure a reasonable current path and a smooth
particle position matrix, a position-based filtering technique can be
applied as follows:
X
kþ1
i;e
¼X
j2
C
H
ej
X
kþ1
i;j
=X
j2
C
H
ej
H
ej
¼max 0;r
min
D
e;jðÞðÞ
8
<
:
ð8Þ
where Cis the set of elements in the modifiable region (f
a
or f
d
) that
lie within a circle of radius r
min
(Fig. 3(b)). The position of element e
is affected by the positions of the other elements C, in accordance
with a weight factor H
ej
.
D
e;jðÞis the distance between elements e
and j. It should be noted that the value of r
min
usually ranges from
1 to 3. Generally, a smaller value produces a more jagged boundary,
while a larger value produces a smoother boundary.
2.5. Optimized results
The flow of the optimization program is shown in Fig. 4. First,
LS-DYNA was employed to perform the coupled analysis of the
electromagnetic and structural fields of the coil and flyer plate.
Thus, the current density contours and the flyer plate velocity were
obtained, and the corresponding script K.file for simulation was
also generated. In accordance with the optimization objective
and design domain introduced above, in combination with the con-
nectivity constraints and mutation and filtration techniques, evo-
lution was carried out. The optimized coil structure was
repeatedly computed via FEA, and the corresponding objective val-
ues were compared until the coil structure converged to the opti-
mal solution. The calculations of the optimization program and
the program called for FEA were executed in MATLAB (MathWorks,
Inc., U.S.). Regarding the number of particles, an excessively small
number could easily result in premature convergence. However,
the computation cost will increase significantly if the population
is too large. To simultaneously maintain sufficient population
diversity and computational efficiency, the number of particles
Fig. 3. Schematic diagrams of the supplementary techniques introduced into the evolutionary algorithm: (a) the addable region f
a
and the deletable region f
d
in the design
domain f; (b) the mutation and filtration techniques.
S. Lin, N. Chou, Y. Zhao et al. Materials & Design 224 (2022) 111337
5
was chosen to be 20 after repeated simulation tests, and the max-
imum number of iterations was k
max
¼50. However, once the his-
torically optimal objectives of all particles had remained stable for
5 iterations, the evolutionary process was considered to have con-
verged and was forced to terminate.
x
was linearly decreased from
1.2 to 0.1 with an increasing number of iterations;
c
1
¼c
2
¼1;
a
¼0:1, and r
min
¼2:3. The objective history and the
corresponding topology of the design domain are shown in Fig. 5
(a&b).
To illustrate the effect of optimization more intuitively, the nor-
malized objective value (the ratio of the objective value of the opti-
mized structure to that of the initial structure) is employed instead
of the absolute value. In the initial stage of evolution (iterations 1
to 13), the objective value increased rapidly. As seen from the geo-
metric features of the design domain, the area of concentrated cur-
rent near the flyer plate (the vertically elongated line on the right)
remained generally unchanged. This is because the alternating cur-
rent in this middle arm directly drives the out-of-plane velocity of
the flyer plate, which is the main contributor to the objective func-
tion. On the other hand, the shape of the conductor on the left was
continually changing. The current here travels along the edge of
the conductor due to the skin effect, and the vertical component
of the corresponding Lorentz force can further enhance the out-
of-plane velocity of the flyer plate. Although it accounts for a rela-
tively small proportion of the objective, this area has a rich diver-
sity of geometric features, thus making it exactly the key area for
the optimization of this coil. In the later stage of evolution, the
growth of the objective value was very slow. In this stage, the pro-
gram was searching for the optimal profile of the left edge of the
conductor, where a peninsula-like structure finally formed. In iter-
ation 38, the evolutionary program achieved the convergence cri-
terion, and the current density contours of the corresponding coil
are plotted in Fig. 5(c).
The evolutionary methodology employed in this study is akin to
the classical PSO algorithm in that it includes several uncertain fac-
tors, such as the initial positions and speed matrices of the parti-
cles as well as the mutation and filtration mechanisms.
Therefore, to test the robustness of the evolutionary algorithm
and explore the appropriateness of the parameter combination, a
statistical study was conducted by performing numerous opti-
mizations with various parameter values, and the corresponding
results are listed in Fig. 6.
To obtain these data, 10 different optimization procedures were
performed for each case, and the means and standard deviations of
Fig. 4. The flow chart of the optimization procedure.
S. Lin, N. Chou, Y. Zhao et al. Materials & Design 224 (2022) 111337
6
the corresponding objective values are shown. The standard devi-
ations were small compared to the mean objective values. This
finding indicates that the algorithm is robust and not affected by
specific instances of random initialization. According to the best
results obtained with different combinations of parameters
(Fig. 6), it was found that a larger value of
a
caused mutations to
occur more frequently, making it more difficult for the evolution-
ary process to converge (consuming more iterations). On the other
hand, a smaller filter radius r
min
could speed up the calculation
(consuming less real time) and result in a more sophisticated coil
design. However, the drawback was that the number of iterations
increased, and the algorithm could more easily fall into locally
optimal solutions (as shown by the topologies in cases 1 and 4).
In contrast, a larger r
min
could make the evolutionary process more
stable and convergent, but the computation speed was reduced
(consuming more real time for each iteration), and many structural
Fig. 5. The results of the optimization procedure: (a) the normalized objective values in some representative iterations, supplemented by the corresponding topologies in the
design domain (black for the conductor region and white for the insulator region); (b) the curve of the normalized objective value versus the number of iterations; (c) the
current density contours of the optimized coil.
Fig. 6. The results of robustness verification and parameter optimization.
S. Lin, N. Chou, Y. Zhao et al. Materials & Design 224 (2022) 111337
7
details of the coil could be lost (as seen from the topology in case
6). Based on these observations, a reasonable combination of
parameter values should be employed. As evaluated in terms of
the mean objective values, the most satisfactory combination of
optimization parameters was that adopted in case 2.
3. Experimental comparison
3.1. Welding procedure
A PS48-16 magnetic pulse generator (PST Products GmbH, Ger-
many), whose maximum discharge energy is 40 kJ, was employed
in this study. Commercial AA5052 aluminum alloy (Southwest Alu-
minum (Group) Co. Ltd., China) and HC420LA steel (HBIS Tangsteel
Co. Ltd, China) were used as the flyer plate and base plate, respec-
tively, during the MPW process. The compositions of the aluminum
alloy and steel materials are listed in Table 2. The collision gap was
controlled by blocks between the flyer plate and the base plate.
When other configuration parameters remain the same, a larger
collision gap provides a larger acceleration distance for the flyer
plate, which can effectively improve the welding quality. However,
a change in the collision gap will also affect the collision angle. As a
kind of percussion welding, MPW has restrictive requirements in
terms of the impingement angle. An improper angle might inhibit
jet formation, resulting in a sharp decline in welding quality or
even welding failure. Therefore, based on previous research expe-
rience concerning MPW [35], an appropriate collision gap of
1.2 mm was deployed in this study. The properties of the welds
formed at discharge energies of 20, 22 and 24 kJ using the opti-
mized and the original coil were tested, with 5 specimens for each
condition.
At the moment of collision, the velocity of the flyer plate could
reach approximately 300–500 m/s. Due to the small collision angle,
metallurgical bonding would not occur in the center area, leading
to the formation of an annular weld configuration at the connec-
tion between the sheets during the MPW process. A larger dis-
charge energy could result in a higher collision speed, resulting
in a stronger jet and more sufficient plastic deformation along
the external edge of the joint. Thus, metallurgical bonding would
occur in areas where no weld could form a weld under a lower dis-
charge energy, leading to expansion of the external edge of the
joint and increasing the effective welding area of the joint. As seen
from the appearance of the joints, there was a concave area on the
flyer plate in the case of a weld formed using the optimized coil
(Fig. 7). This H-like off-plane deformation, distinct from the
rectangle-like deformation induced by the original coil, was caused
by the special current path in the optimized coil, which could be
beneficial to improve tensile strength of the joint. Moreover, the
collision angle around a joint welded with the optimized coil was
also steeper than that of a sample welded with the original coil.
3.2. Mechanical properties of the joints
Quasi-static tensile tests (2 mm/min) were performed using a
universal testing machine (5985, Instron Corporation, Norwood,
U.S.). To eliminate transverse deviation of the joint, two gaskets
were added to the clamping ends on both sides of the plates. The
tension–displacement curves and the maximum tensile capabili-
ties of the joints formed under various discharge energies are
shown in Fig. 8(a&b), respectively. Specifically, the optimized coil
significantly improved the tensile properties and elongation of
the samples; note that the failure mode of all samples was ductile
fracture. For example, for a discharge energy of 22 kJ, the maxi-
mum tensile load of a joint welded with the optimized coil was
increased by 19.88% compared with that of a joint welded with
the original coil. Notably, when the energy was increased to
24 kJ, some of the samples welded with the optimized coil broke
at the aluminum flyer plates rather than at the joints. This finding
indicates that the tensile stiffness of the joint was higher than that
of the aluminum alloy. Conversely, this situation did not occur in
any of the samples welded with the original coil. The correspond-
ing data were eliminated due to this unique behavior; conse-
quently, the strength improvement at 24 kJ appears to be not as
significant as that at 22 kJ. In addition, after the joints were broken,
some detailed features on the flyer plates could be clearly
observed, as shown in Fig. 8(c). Under the same energy level, the
effective welding area produced with the optimized coil was obvi-
ously larger than that of the original coil. This is of interest because
generally, the strength of a welded joint is positively correlated
with the effective welding area. Generally, the effective area also
increased as the discharge energy increased for the joints formed
with the optimized coil (Fig. 8(d)), while the effective welding
areas of the samples welded with the original coil remained almost
the same. Additionally, for a discharge energy above 26 kJ, the joint
failure mode changed from welding interface failure to failure of
the base metal (aluminum plate). This was because when the dis-
charge energy was low, the effective welding area of the joint was
small, and its tensile strength was weak. With an increase in the
discharge energy, the effective welding area of the joint also
increased, until the strength of the joint exceeded that of the alu-
minum plate (Fig. 9).
3.3. Microstructure observations
High-speed collision will lead to severe plastic deformation and
residual stress at the welding interface [36]. To clearly illustrate
the morphology along the welding interface, an optical microscope
(Eclipse Ti2, Nikon instruments (Shanghai) Co., Ltd., China) was
employed to investigate the microstructure (as shown in Fig. 10
(a–d)). Specific shear waves were found along the welding inter-
faces welded with both coils. As seen from the microscopic appear-
ances of the samples, the wavelengths (2728
l
m) and amplitudes
(34
l
m) of the shear waves produced by the optimized coil were
obviously larger than those of the shear waves produced by the
original coil (813
l
m in wavelength and 1.11.6
l
m in ampli-
tude), indicating that the optimized coil could produce a higher
collision speed (Fig. 10(e&f)). Moreover, micron-level transition
zones (marked with white dashed lines) were found beside the
shear waves. Such a transition zone is known to be a mixture of
Al and Fe. Because the impact pressure at the collision point was
much higher than the yield strength of the base metal, adiabatic
shearing and severe plastic deformation were induced at the colli-
sion point during the oblique impact process of MPW. Under these
circumstances, the metal bonds of the Al and Fe atoms each broke
Table 2
Chemical compositions (wt.%) of AA5052 aluminum alloy and HC420LA steel.
Metal Si Cu Mn Mg Zn Al
AA5052 0.25 0.10 0.10 2.50 0.10 Bal.
Metal C Si Mn P S Nb Fe
HC420LA 60.10 60.50 61.60 60.025 60.025 60.09 Bal.
S. Lin, N. Chou, Y. Zhao et al. Materials & Design 224 (2022) 111337
8
separately, and then these two kinds of atoms interpenetrated with
each other, driven by strong kinetic energy. Eventually, a relatively
stable intermetallic phase was formed. The formation of this Fe
i
Al
j
phase (where iand jare positive integers) was determined by the
atomic ratio of the two metals in the transition zone and the cor-
responding change in the Gibbs free energy.
4. Conclusion
In this study, the welding quality of MPW Al–Fe joints was
enhanced through topological optimization of the planar coil used
in the welding process. For this purpose, a connectivity-
constrained optimization methodology based on a discretized
PSO algorithm was proposed. The optimization performance was
validated by comparing the mechanical properties of the MPW
Al–Fe joints produced with the optimized and original coils, whose
welding interfaces were characterized by means of an optical
microscope. The conclusions can be summarized as follows:
(1) A discrete PSO algorithm was innovatively used to optimize
the average particle velocity in the region around the weld-
ing joint of the flyer plate. With the introduction of connec-
tivity constraints, a more efficient MPW coil structure was
successfully obtained. Compared with the original coil, the
value of the optimization objective was enhanced by
13.37% at the 1/4 point of the discharge cycle, which
resulted in a significantly higher impact velocity between
the flyer and base plates.
(2) To enhance the global search capabilities and robustness of
the optimization algorithm, corresponding mutation and fil-
tering techniques were proposed. The enhancement of the
optimization algorithm was verified through a set of numer-
ical analyses, in which the optimal combination of parame-
ters was found to be
a
¼0:10;r
min
¼2:3 and 20 particles in
the swarm.
(3) Quasi-static tensile tests (2 mm/min) showed that for a dis-
charge energy of 22 kJ, the maximum tensile load of an Al–Fe
weld joint produced with the optimized coil was increased
by 19.88% compared with that of a joint produced with the
original coil. Regarding the morphological characteristics of
the joints produced with the optimized coil, they had larger
effective welding areas, as seen from their fracture surfaces,
which was the reason for the improvement in welding
performance.
Declaration of Competing Interest
The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared
to influence the work reported in this paper.
Acknowledgments
The authors gratefully acknowledge financial support from the
National Natural Science Foundation of China (No. 52175315), the
Shenzhen Science and Technology Program
(KQTD20200820113110016), the Natural Science Foundation of
Hunan Province (No. 2020JJ5027), and the State Key Laboratory
of Advanced Design and Manufacturing for Vehicle Body
(51965009).
Appendix A. MATLAB code for FEA & PSO
% Part of the main program
if loop <1 + num % num denotes the number of particles
V(:,:,loop) = rand(size(mapori))*0.2–0.1; % mapori denotes
the topology of the design domain
pos(:,:,loop) = rand(size(mapori))*0.1;
for i = 1:size(mapori,1)
for j = 1:size(mapori,2)
if map(i,j,loop)== 0
pos(i,j,loop)= 1*pos(i,j,loop);
end
(continued on next page)
Fig. 7. The appearances of Al–Fe joints welded using the optimized and original coils with discharge energies of 20, 22 and 24 kJ.
S. Lin, N. Chou, Y. Zhao et al. Materials & Design 224 (2022) 111337
9
end
end
else
[map,V,pos] = pso(map,obj,loop,V,pos,iteration,num); % Call
the PSO subroutine
end
...
% The PSO subroutine
function [map,V,pos] = pso(map,obj,loop,V,pos,iteration,num)
[a,g]=max(obj);
for i = 1:loop
if any(loop-num:-num:1==i)
else
obj(i)=0;
end
end
[b,p]=max(obj);
wst = 1.2;
wed = 0.6;
w = (wst-wed)*(iteration-loop)/iteration + wed;
c1 = 1;
c2 = 1;
Vnew = w*V(:,:,loop-num)+c1*rand*(pos(:,:,p)-pos(:,:,loop-n
um))+c2*rand*(pos(:,:,g)-pos(:,:,loop-num));
Vnew((Vnew>0.1))=0.1;
Vnew((Vnew<-0.1))=-0.1;
X = pos(:,:,loop-num)+Vnew;
X(X>1)=1;
X(X<-1)=-1;
mapnew = map(:,:,loop-num);
% connectivity constraints
se = strel(’diamond’,1);
Ie = imerode(map(:,:,loop-num),se);
dele = map(:,:,loop-num)-Ie;
Ie2 = imerode(1-map(:,:,loop-num),se);
add = 1-map(:,:,loop-num)-Ie2;
% Mutation
for i = 1:size(X,1)
for j = 1:size(X,2)
if add(i,j)+dele(i,j)>0&& rand >wed + 1.2-w
X(i,j)= 1*X(i,j);
end
end
end
Fig. 8. Mechanical validation of the MPW process: (a) the tension–displacement curves of the joints; (b) the tensile properties of the joints; (c) the effective welding areas
(shaded areas) on the flyer plates for the optimized and original coils with a 22 kJ discharge energy; (d) the variations in the welding area with the discharge energy for the
two groups of samples.
S. Lin, N. Chou, Y. Zhao et al. Materials & Design 224 (2022) 111337
10
X = check(size(X,2),size(X,1),2.1,map(:,:,loop-num),X,add,del
e); % Call the filtering subroutine
X(1,5:end)=1;
X(end,:)=1;
X(:,end)=1;
if sum(sum(X >0)) >sum(sum(map(:,:,loop-num)))
mapnew(add==1&>0)=1;
elseif sum(sum(X <0)) <sum(sum(map(:,:,loop-num)))
mapnew(dele==1&X<0)=0;
end
map(:,:,loop)=mapnew;
pos(:,:,loop)=X;
V(:,:,loop)=Vnew;
...
% mesh-independent filter
function [dcn]=check(nelx,nely,rmin,dc,add,dele)
(continued on next page)
Fig. 9. The failure details of the joints in the two groups of samples.
S. Lin, N. Chou, Y. Zhao et al. Materials & Design 224 (2022) 111337
11
dcn = zeros(nely,nelx);
for i = 1:nelx
for j = 1:nely
if add(j,i)+dele(j,i) >0
sum = 0.0;
for k = max(i-floor(rmin),1):min(i + floor(rmin),nelx)
for l = max(j-floor(rmin),1):min(j + floor(rmin),nely)
if add(l,k)+dele(l,k) >0
fac = rmin-sqrt((i-k)
2
+(j-l)
2
);
sum = sum + max(0,fac);
dcn(j,i) = dcn(j,i) + max(0,fac)*(add(l,k)+dele(l,k))*
dc(l,k);
end
end
end
dcn(j,i) = dcn(j,i)/sum;
end
end
end
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