Application of AF-SVM based on the Structure of the Machine

Shengran Meng

, Shaohui Su

, Lu Lu

, Dongyang Zhang

, Chang Chen

, Guojin Chen

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 = 1,

2,...,n,},where X

is the input vector of the i-th

sample and Y

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 )

⋅+w

(1)

Where

（）

is the nonlinear mapping of input

space Rn to high-dimensional space,

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:

i=1

E( ) (x )

Cyf

+⋅ −

∑

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

and

formula(2) becomes

：

,, ,

min ( )

.. ( (x ) )

y ( (x ) )

, 0

iii

ii i

st b y

ξξ

⎧

⎫

+⋅ +

⎪

⋅+−≤+

⎨

⎬

⎪

−⋅ + ≤+

⎪

≥

⎩⎭

∑

(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 ,

()() ()( )K()

nn n

ii iii iijj

ii ij

εαα αα αααα

== =

++ −− − − ⋅

∑∑ ∑

αα

（）=

(4)

s.t.

0 , ; 1, 2,...,

Ci n

αα

≤≤=

∑∑

(5)

where K(X

) 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：

i=1

f(x) = (x) ( ) ( ) b

ϕαα

⋅+= − +

∑

wx,x

(6)

The kernel functions commonly used in SVM

include: polynomial kernel function, sigmoid kernel

function and radial basis kernel function.

(( ) b)

( ) tanh(k( )+v)

polynomial kernel function

sigmoid kernel function

radial basis kernel function

⎧

⎪

⎨

⎪

−

⎪

⎩

x,x

x,x x,x

（）

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:

①Artificial Fish X

randomly selects a state X

its field of vision.

*()

i isual

XV Rand=+

(8)

② Calculated the objective function value of

and X

respectively Y

and Y

,if found to be better

, then X

to X

direction move one step.

** ()

X X Step Rand

−

(9)

③Otherwise, X

continues to select the state X

in its field of vision to determine whether to meet

the forward conditions ,and after trying Try−number

times repeatedly, it still does not meet the forward

conditions, then execute random behavior.

The artificial fish individual state

X=(X

,...,X

) (whereX

(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：

①Artificial Fish X

searches for the number of

partners in the current field (d

isual

) N

and

central location X

, If Y

>δY

,it indicates that the

position of the partner center is better and less

congested ,then X

moves one step toward the center

of the partner.

** ()

XStepRand

−

(10)

②Otherwise, foraging behaviour .

The total number of artificial fish is N,

congestion factor is δ, the distance between artificial

fish individual i,j,d

=|X

−X

(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：

①Artificial Fish X

searches for the best partner

of function Y

in the partner of current field of

Vision (d

isual

).If Y

>δY

,indicating that the

optimal partner's surroundings are less crowded,X

moves one step towards partner X

** ()

X Step Rand

−

(11)

②Otherwise, 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：

①Artificial fish X

randomly move one step to

reach a new state：

*()

iiisual

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

),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 2：Orthogonal 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 3：Test samples.

No.

Design variable

Analysis

results

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 5：Compare 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

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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