Application of AF-SVM based on the Structure of the Machine
Shengran Meng
1
, Shaohui Su
2
, Lu Lu
3
, Dongyang Zhang
4
, Chang Chen
5
, Guojin Chen
6
1,2,3,4,5,6
Department of mechanical electronics; Hangzhou Dianzi University ;No. 2 Avenue, Hangzhou,China
Key words: Parameter selection, Support vector machine ,Artificial fish swarm algorithm, Grinder bed.
Abstract: The problem of time-consuming in optimization of large complex structures such as grinder, The method of
selecting support vector machine parameters based on artificial fish-swarm algorithm and its application,
The feasibility of replacing time-consuming finite element analysis for structural optimization is validated.
Based on the plane grinder bed, the orthogonal experimental design method was used to select the sample
points in the structure parameter space of the grinder bed; The sample point is simulated by ANSYS, and the
sample set is produced; By using the better parallelism and the strong global optimization ability of artificial
fish swarm algorithm, the optimal parameter combination of SVM is obtained, and the approximate
modeling of the grinder bed model is completed. The results show that compared with the traditional finite
element method, the method not only significantly improves the computation efficiency, but also has good
accuracy.
1 INTRODUTION
The lathe bed is an important supporting part of
CNC machine tools ,which has a great influence on
the performance of the grinder. In the process of
design, it is necessary to have both sufficient
strength and light weight, while also taking into
account its dynamic characteristics. The traditional
design is mainly analyzed and optimized by the
finite element model. However, the finite element
analysis takes a long time. The iterative calculation
of the finite element analysis during the optimization
process will increase the time cost of the whole
optimization process. It is difficult to carry out
multi-scheme analysis and comparison in short time
to meet the requirement of rapid scheme
demonstration in the initial stage of grinder structure
optimization design.
In engineering calculation, in order to save time
cost, the approximate model is often introduced
instead of the simulation model to calculate and
optimize. The approximate model is a mathematical
model based on the experimental design method and
approximate modelling method ,using the finite
input-output parameter pair, the statistical or fitting
method, which is the model after the second
modeling of the original model. The approximate
model can not only reduce the computational time,
but also quickly analyze the complexity of the model
and the sensitivity of the design variables. At present,
there are commonly used response surface methods,
neural networks, support vector machines ,etc.
The traditional approximate model method is
influenced by the number of samples. Increasing the
number of samples can improve the accuracy of the
approximate model calculation. However, in
practical projects, the number of samples is often
limited ,so a more reasonable method is needed to
handle the approximate problem in the case of small
samples. At present, SVM has been well applied in
many fields such as pattern recognition, optimization
design, and data mining.
Therefore, this paper will construct an
approximate model of the grinder bed based on the
support vector machine .on the premise of
guaranteeing the rigidity and natural frequency of
the grinder bed, it can not only improve the
operation speed ,but also improve the overall
multidisciplinary optimization design efficiency of
the bed body, which provide the technical support
for the overall rapid scheme.
2 SVM THEORY
SVM is based on the VC-dimensional theory of
statistical learning theory and the principle of
structural risk minimization.SVM regression is a
generalization of support vector machines in
regression estimation of nonlinear systems, it is the
concrete realization of the idea of nuclear method,
that is, to realize the mapping of input variable to
high dimension feature space implicitly by the
kernel of estimating inner product in feature space,
and then construct linear regression function in high
dimensional feature space, and solve the nonlinear
problem of original space.
Suppose a set of training sets T={(X
i
,Y
i
),i = 1,
2,...,n,},where X
i
is the input vector of the i-th
sample and Y
i
is the output corresponding to the i-th
sample Function,n is the number of
samples.Mapping it to high-dimensional feature
space through nonlinear mapping.Its nonlinear
regression equation:
f(x ) = (x )
ii
b
ϕ
⋅+w
(1)
Where
i
x
ϕ
()
is the nonlinear mapping of input
space Rn to high-dimensional space,
w
is the weight
vector and b is the offset. The SVM regression
equation is based on the structural risk minimization
principle, and the number and generalization ability
of the support vectors are determined by the
insensitive parameter ε, which reflects the sensitivity
of the model to the noise contained in the input
samples. Regression problem transformation
planning problem, consider minimizing, that is:
2
i=1
11
E( ) (x )
2
l
ii
Cyf
l
ε
+⋅
wW=
(2)
In the formula, the first item is the confidence
risk item ,the role is to flatten the regression function,
thereby enhancing the generalization ability ;the
second item is the empirical risk item ,in which C is
the penalty parameter and controls the degree of
penalty over the error ε. When the value of C is
small, the allowable error is larger. On the other
hand, when the value of C is large, the allowable
error is smaller.
After introducing slack variables
i
ξ
and
*
i
ξ
,
formula(2) becomes
*
2
*
,, ,
1
*
*
1
min ( )
2
.. ( (x ) )
y ( (x ) )
, 0
ii
l
ii
b
i
iii
ii i
ii
C
st b y
b
ξξ
ξξ
ϕ
ε
ξ
ϕ
ε
ξ
ξξ
=
+⋅ +
⋅++
−⋅ + +
⎩⎭
w
w
w
w
(3)
By introducing the Lagrange multiplier α and
α*,and according to the duality principle, the
formula (3)is eventually transformed into the
optimization problem:
*
** **
11 ,1
max ,
1
()() ()( )K()
2
nn n
ii iii iijj
ii ij
W
y
εαα αα αααα
== =
++ −−
∑∑
ij
αα
xx
()=
-
(4)
*
11
*
s.t.
0 , ; 1, 2,...,
nn
ii
ii
ii
Ci n
αα
αα
==
=
≤≤=
∑∑
(5)
where K(X
i
,X
j
) is a kernel function satisfying
the mercy condition.
Only part of the α-α* satisfies the non-zero
condition, and the data points connected to the
partial coefficients are called support vectors. If the
number of support vectors isl, the regression
equation for the input vector X can be expressed as
l
*
i=1
f(x) = (x) ( ) ( ) b
ii
bK
ϕαα
⋅+= +
i
wx,x
(6)
The kernel functions commonly used in SVM
include: polynomial kernel function, sigmoid kernel
function and radial basis kernel function.
d
2
(( ) b)
( ) tanh(k( )+v)
2
polynomial kernel function
sigmoid kernel function
radial basis kernel function
K
σ
+
=
i
ii
i
x,x
x,x x,x
xx
()
()
-
exp() ()
(7)
The choice of kernel function is also an
important factor influencing the performance of the
support vector machine, and the radial basis function
can be well adapted. It has good convergence
domain for both low-dimensional spatial data and
high-dimensional space. Influence parameters of the
approximate model performance of the radial basis
kernel function support vector machines The main
parameter has penalty parameters C and σ, they need
to be optimized.
3 ARTIFICIAL FISH SWARM
ALGORITHM
The artificial fish swarm algorithm uses multiple
artificial fishes to perform optimization at the same
time. The optimal value is chosen as the result of
this optimization to achieve parallel operation, so
that the artificial fish swarm algorithm can quickly
converge to the optimal value, and is insensitive to
the given initial value and has the ability of global
optimization. Therefore, it is applied to the optimal
selection of support vector machine parameters, the
optimization objective is to determine the optimal
parameter combination (C,σ)to maximize the
classification accuracy of SVM.
Artificial fish is the virtual entity of a fish, which
encapsulates its own parameters and a series of
behaviors, and the effect of these parameters on the
final result, can accept the environment stimulus
information, and make corresponding activities, Its
environment consists of the solution space of the
problem and the state of other artificial fish, its
behavior depends on its state and the state of the
environment in the next moment, and it also affects
the environment through its own activities, which
affects the activities of other artificial fishes. The
artificial fish swarm algorithm has the following
fourbasicbehaviors:(1)foraging-behavior,(2)clusterin
g-behavior,(3)rearend-behavior(4)random-behavior.
(1)Foraging-behavior is a kind of activity that
imitates fish tending to food, and can be considered
to be the direction of action by perceiving the
amount of food or food concentration in the water by
sight or taste. In the design algorithm, it can be
described as:
Artificial Fish X
i
randomly selects a state X
j
in
its field of vision.
j
*()
i isual
XV Rand=+
(8)
Calculated the objective function value of
X
i
and X
j
respectively Y
i
and Y
j
,if found to be better
Y
j
, then X
i
to X
j
direction move one step.
1
** ()
t
ji
tt
ii
t
ji
XX
X X Step Rand
XX
+
=+
(9)
Otherwise, X
i
continues to select the state X
j
in its field of vision to determine whether to meet
the forward conditions ,and after trying Trynumber
times repeatedly, it still does not meet the forward
conditions, then execute random behavior.
The artificial fish individual state
X=(X
1
,X
2
,...,X
n
) (whereX
i
(i=1,2,...,n) is the
optimization variable),artificial fish vision is Visual,
Rand() is a random function that is a random number
in the interval(0,1).Step for artificial fish move stride
length.
(2)Clustering-behavior is the simulation of a
large number or a small number of fish aggregations,
collective foraging and avoidance of predators,
which is a form of survival that they formed during
the evolution process. There are two rules to follow
in the fish cluster: One is to move as close to the
center of the neighboring partner as possible, and the
other is to avoid overcrowding. In the design
algorithm, it can be described as
Artificial Fish X
i
searches for the number of
partners in the current field (d
ij
<V
isual
) N
f
and
central location X
c
, If Y
c
/N
f
>δY
i
,it indicates that the
position of the partner center is better and less
congested ,then X
i
moves one step toward the center
of the partner.
1
** ()
t
tt
ci
ii
t
ci
XX
X
XStepRand
XX
+
=+
(10)
Otherwise, foraging behaviour .
The total number of artificial fish is N,
congestion factor is δ, the distance between artificial
fish individual i,j,d
ij
=|X
i
X
j
|.
(3)Rearend-behavior is to imitate when a certain
fish or a few fishes finds food, the fish in the vicinity
of them will follow, resulting in a behavior that the
fish in the more distant place will follow. In the
design algorithm, it can be described as
Artificial Fish X
i
searches for the best partner
X
j
of function Y
j
in the partner of current field of
Vision (d
ij
<V
isual
).If Y
j
/N
f
>δY
i
,indicating that the
optimal partner's surroundings are less crowded,X
i
moves one step towards partner X
j
:
1
** ()
t
ji
tt
ii
t
ji
XX
X
X Step Rand
XX
+
=+
(11)
Otherwise, foraging behavior.
(4)Random-behavior is a default behavior of
simulating fish foraging behavior, which refers to
the artificial fish moving randomly in the field of
vision. When food is found ,it moves fast in the
direction that food is gradually increasing. In the
design algorithm, it can be described as
Artificial fish X
i
randomly move one step to
reach a new state
1
*()
tt
iiisual
X
XV Rand
+
=+
(12)
4 APPROXIMATE MODEL
VA L I D AT I O N
Determine The approximate model flow of the
grinder bed is established by the artificial
fish-swarm algorithm. See Figure 1.
The whole process can be roughly divided into
the following steps:
(1)using orthogonal design to generate
parameters, then through finite element analysis, the
design variables and structural responses of each
analysis are recorded, respectively, as experimental
samples.
(2)Setting and initializing parameters of AF and
SVM, respectively.
(3)Through the sample data, SVM training based
on artificial fish swarm algorithm and establish an
approximate model to find the optimal target value,
determine the optimal parameter combination of
SVM.
(4)Contrastive analysis of results with traditional
finite element analysis.
Figure 1: A flow chart of the approximate model
established by AF-SVM.
4.1 Experimental Data
In this paper, the approximate model of the
maximum deformation displacement of the bed is
developed based on the design dimension of a
grinder bed structure. According to the design and
practical experience, the thickness of the bed wall
and fascia as design variables, see table 1.
Table 1:Design variables and symbols.
The selected design variables are shown in
Figure 2.
Figure2:schematic diagram of design variable of grinder
bed body.
In order to establish an approximate model,
sample collection is the first step. Orthogonal test
method has the characteristics of neat comparability
and balanced dispersion. It can use a sample number
as small as possible to obtain more comprehensive
sample points and improve modeling efficiency.4
variable parameters can be taken 4 values in their
respective feasible fields. Use the orthogonal table to
design the sample table 2,select the variables as 4
factors, select the orthogonal table L
25
(4
5
),a total of
25 groups of training.
In order to improve the efficiency of the analysis
and reduce the calculation time, the model is
simplified without affecting the calculation precision
of the model, ignoring the load-hole, the line hole,
the threaded hole, the keyway, the retract groove and
so on, and the partial transition arc is reduced to the
right angle. At the same time, set aside 5mm
distance between the joint surface, establish the
small convex platform, conveniently in the ANSYS,
on the bolt, the Guide slider and so on the joint part
to add the parameter, simulates the combination
parameter the stiffness and the damping unit. The
use of Pro/E software to create a simplified bed solid
model shown in Figure 3.
Figure 3:simplified rinder bed solid model.
The grinder bed material is set to HT200,elastic
modulus E=120gpa,Poisson's ratio v=0.28,density
ρ=7000kg/m3,strength limit σ≥300mpa.The contact
surface between the bottom of the bed and the
foundation is added with the displacement full
restraint to realize the fixing of the bed. The contact
surface of the slider is divided in the guide plane,
and the load pressure is applied 0.689MPa.The
pressure between the bed and the column is
0.12MPa.Through the ANSYS mesh division formed
19,742 nodes, 9,358 units. The meshed model is
shown in Figure 4.
Figure 4: meshed model in ANSYS.
The maximum deformation U (μm) of the sample
data is obtained by ANSYS .See Table 2.
Table 2Orthogonal test results.
4.2 Experimental Simulation
First of all, in order to avoid a certain dimension
feature value being too large and influence the final
result. the sample data is normalized and the
convergence speed of the program is ensured.
The value range of the penalty parameter C of
the radial kernel correlation parameter of support
vector machine is set to (0,10),The range of σ is
(0,10),Population evolutionary algebra is
50,Population size is 5,Try-number is 5,congestion
factor δ is 0.618.Visual is 0.5.Step is 0.1.
Optimized SVM parameter model,C=4.6721
σ=0.0136.In order to verify the validity of the
approximate model,5 output data are randomly
selected as test samples, and compared with the
results of finite element analysis, see table 3.
Table 3Test samples.
No.
Design variable
Analysis
results
X
1
X
2
X
3
X
4
U(μm)
1 65 47 25 44 1.661
2 63 55 16 36 1.719
3 57 54 24 37 1.677
4 58 48 22 43 1.649
5 67 45 19 41 1.664
Compare the output value of the approximate
model on the test sample with the output value of the
finite element ,as shown in Figure 5,where the
diamond point represents the approximate model
and the triangle represents the finite element
solution.
Figure 5Compare the output value of the approximate
model with the output value of the finite element
The actual relative error of the finite element
solution and approximate model value is 3.605%,as
shown in table 4.
Table 4 Simulation results of the test sample SVM
approximation model.
No.
Finite element
solution
Test
solution
Relative
erro
r
1 1.661 1.706 2.709%
2 1.719 1.751 1.861%
3 1.677 1.735 3.458%
4 1.649 1.695 2.789%
5 1.664 1.724 3.605%
From Figure 5 and Table 4,we can see that
compared with the exact verification value of the
finite element analysis, the SVM with optimized
parameters has a very high precision, and the
relative error of the finite element analysis and the
approximate model is less than 5%.Moreover,the
running time of AF-SVM algorithm is very fast ,and
the results obtained by SVM with parameter
optimization have good engineering practicability.
5 CONCLUSIONS
In this paper, in order to calculate the deformation
value of the grinder bed, the problem of time
consuming is too long for finite element analysis.
Replacing finite element analysis model with
approximate model based on AF-SVM. Under the
premise of guaranteeing the accuracy of calculation,
fewer finite element analysis times are invoked,
which significantly reduces the computational time
cost. The experimental results show that the
proposed method can meet the practical
requirements of engineering and can be extended to
approximate simulation of other mechanical
structures to achieve the rapid demonstration of
general scheme.
ACKNOWLEDGEMENTS
Thank the National Natural Science Foundation of
China (Grant No.51475129,51675148,51405117)for
its strong support for this paper. In the writing of this
paper, I got the careful guidance of my tutor, Mr.
SuShaohui. During the preparation and revision of
the papers, Mr. Su gave me guidance and advice
patiently, which made me greatly improve my
studies and writing papers.
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