The Method of Polynomial Approximation to Determine the "Optimal" Coefficients of the Mathematical Models of Corrosive Destruction. Algorithm 1

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The Method of Polynomial Approximation to Determine the

"Optimal" Coefficients of the Mathematical Models of

Corrosive Destruction. Algorithm 1

George Filatov

Professor, Doctor of Techn. Sciences, Dnepropetrovsk State Agrarian-Economic University, Ukraine

Abstract − The use of method of polynomial approximation for the determination of the “optimal” coefficients of influence of SSS of the design on the rate of corrosion at its optimal designing is offered. The proposed method is analytical, allowing to extrapolate the research process and to determine the process parameters with high accuracy.

Keywords — Corrosive destruction, Evolution Theory, Identification, Mathematical Models, Optimal Designing, Polynomial Approximation.

I. INTRODUCTION

In the articles [1]-[4] is given the theoretical basis of evolutionary theory, identification of mathematical models of corrosion destruction of structures under their optimal design, taking into account the influence of the stress-strain state (SSS) of structure on the rate of corrosive process and is demonstrated the results of a numerical experiment across multiple optimization objects to confirm this theory. As a result of the research it was concluded that exists the dependence of the coefficient of the influence of the SSS on the rate of corrosive process from starting stiffness of optimization objects. It has been shown that proper account of this dependence can save up to 12% of the mass of the objects at their optimization.

With the help of an iterative procedure has been set the "optimal" coefficient of the influence of SSS on the rate of corrosive destruction. It is shown that the search for the optimal solution by using such "optimal" coefficient of the influence of SSS on the corrosion rate is independent of the starting point and always lead to the optimal design. It is found that the presence of the above-noted patterns in the behavior of the coefficient of influence of SSS on the corrosion rate allows to manage the procedure of strategy search at the optimization of structures. In the work [3], two empirical methods for determining the "optimal" coefficients of the influence of SSS on the rate of corrosion process have been described.

This paper presents first analytical method for determining the optimal coefficient mathematical model of corrosive destruction of construction.

The method consists in applying the procedure of polynomial approximation of the experimental curve of the dependence of the coefficients of influence SSS on the rate of corrosion from the values of the objective function of optimized object.

The proposed method consists of several steps. At the initial stage is carried a preliminary study of the field of search of optimal solutions. We select any point in the field of parameters to be optimized and execute for this point identification of the mathematical model describing the process of corrosive destruction of construction. Next, from the selected point as from the starting point, the search of optimal design is carried out and the intermediate 5-6 points are memorized. For each of these points are carried out the identifications of mathematical model and is building the "experimental" curve of the dependence of SSS influence on the rate of corrosive destruction from the stiffness of optimized design. The next step the "experimental" curve is approximated by some polynomial. The result we get an analytical function of the relationship between the influence of SSS and stiffness of construction to be optimized. The "optimal" values of the coefficients of influence of SSS on the rate of corrosive destruction with the help of the obtained dependence are calculated and specified by the iterative procedure of the specifying identifications, described in the works [1-3].

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[image:2.595.171.421.148.245.2]

14

Fig.1 The thin-walled shell

II. ILLUSTRATION OF THE METHOD OF POLYNOMIAL APPROXIMATION

Let's use the results of the baseline selection (Table 2) [1]. We choose from the table arbitrarily 6 intermediate points in descending order of the objective function and are listed in Table 1.

We formulate two methods of determining "optimal" coefficient of a mathematical model using the procedure of polynomial approximation .

[image:2.595.55.541.384.529.2]

Method 1. The polynomial approximation of the dependence of stiffness of design upon the coefficient of influence of SSS on the rate of corrosive destruction.

Table 1

Selective identification results when optimally designing a thin-walled shell

№ п/п

Selective arbitrary points The coefficients of model

A

(sm2)



(sm)

R

(sm)



0







1 2309,34 3,643 100,89 0,3258 18,650 10,374 0,1667274 2 1976,50 3,088 101,85 0,3297 18,605 10,223 0,1061461 3 1646,78 2,615 100,21 0,3271 18,605 10,311 0,0844576 4 1307,56 2,075 100,24 0,3226 18,614 10,497 0,0588444 5 1134,58 1,783 101,25 0,3224 18,621 10,510 0,0423870 6 1065,51 1,696 100,00 0,3222 18,621 10,621 0,0396290

Analyzing the graph of the dependence of stiffness of the cross-section of shell's wall upon the coefficient



, built on a core sample (Fig. 3) [1], we note that this relationship can be approximated by a polynomial. Let's

call this relationship as "base" and denote

y





 



. Approximating polynomial can be written as:

 

m

m

a

a

a

a













₀



₁



₂ 2



....



(1)

Let us take as a deviation



 



from

A

 



on the

multitude of points



₁

,



₂

,...,



_m the next square approximation:

   













n

i

i

i

A

S

1

2







(2)

The coefficients

a

₀

,

a

₁

,....,

a

_m are selected so that the value of S was the lowest.

Search coefficients approximating polynomial (1) is performed using the method of random search SGEF [ ]. We consider the approximation of a selective ("experimental") curve using third-order polynomial in the form of:

 

3

3 2 2 1

0













a



a



a



a

, (3)

Introducing the notation

3 4 2 3 1 2 0

1

a

;

x

a

;

x

a

;

x

a

x









, formulate

mathematical programming problem: find a vector



x

1

,

x

2

,

x

3

,

x

4



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 

_







 















n

i

A

x

x

x

x

S

1

2 3

3 2 2 2

1









X

(4)

at the performance of restrictions:

 



x





x



x



;



i



1

,

2

,...,

4

;

j



1

,

2

,...,

8



g

_j

X

_{i i i}

.

(5)

[image:3.595.54.544.264.429.2]

approximation results are shown in Table 2.

Table 2

The coefficients of the approximating polynomial and the designed stiffness of the cross-section the shell

№

The starting points The coefficients of approximating polynomial

The designed values of

parameters The relative error

A

(sm2)

opt

A

(sm2)



(%)

1 2309,34 2310,82 0,064

2 1976,50



0

a

849,7853 1959,50 0,860

3 1646,78



1

a

198,9385 1667,95 1,285

4 1307,56



2

a

182041,9 1311,65 0,312

5 1134,58



3

a

769458,4 1109,82 2,182

6 1065,51 1079,90 1,351

The graphs of selective and approximating dependences on the function of SSS influence on the rate of corrosion are shown in Fig. 2.

The proximity of selective and approximating curves allows replace the selective curve of the dependence of stiffness of shell from SSS influence on the rate of corrosion on the analytic dependence in form of the polynomial (3).

Researching the expression (3) on extremum, we obtain:

 

_{2 3} 2 ₀

3 2

1  













a a

a d

d _{. (6)}

Substituting in equation (3) the values coefficients

3 2 1

,

a

,

a

a

and solving it with respect, we find

0005483

,

0





.

[image:3.595.192.409.558.715.2]

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We calculate the estimated value of the stiffness of cross section of the shell (3):



 





A



a

₀



a

₁





a

₂



2



a

₃



3



849

,

5873







198

,

9385





0

,

0005483





182041

,

9



0

,

0005483

2







796458

,

4





0

,

0005483

3



849

,

53

см

2

.

The thickness of the wall



of the shell at

100



R

sm find from the expression:

352

,

1

100

14

,

3

2

53

,

849

2













R

A



sm.

This solution can be obtained with another method, by the putting the problem of minimizing the function (3) as a problem of mathematical programming.

Using the notation

x

₁





, formulate mathematical programming problem: find a minimum of the function:

 

3

1 3 2 1 2 1 1

0

a

x

a

x

a

x

a









X



(7)

at the restrictions:

 

1 1

0

;

2

 

1 1

0

1



x



x





g



x





x



g

X

X

. (8)

The problem (7)-(8) was solved by random search SGEF at the following initial data:

0

,

0

;

1

,

0

1

1



x





x

. The coefficients of the polynomial (7) were taken from the Table 2. Output solution:





0

,

000546892

. The value of the function

 

X



849

,

53



is close to the minimum of

mathematical functions (7) and is almost identical with the analytic solution given above.

For the point with coordinates





1

,

352

sm and

100



[image:4.595.56.542.456.571.2]

R

sm we carry out some specifying identification and optimization, in order to determine the value of the coefficient of mathematical model of corrosive damage that adequately describe the corrosive process. The calculation results are shown in the Table 3.

Table 3

The results of specifying identification and optimal parameters of the shell

№ The starting stiffness The coefficients of model

The optimal parameters of shell. The depth of corrosive damage

A

(sm2_{) (sm)}



_(sm)

R



₀







A

оpt

(sm2₎



(sm)



(sm)

1 849,53 1,352 100,00 0,3224 18,605 10,509 0,02494828 937,51 1,492 0,3648 2 937,51 1,492 100,00 0,3225 18,613 10,509 0,03035749 937,45 1,492 0,3743

The process has converged in just two iterations. The coefficients adduced in the last row of Table 3 allow to satisfactory describe the experimental data shown in Table 3 in the paper [4].

We can offer another method of determining of "optimal" coefficient of the influence of SSS on the rate of the corrosion by the extrapolation of polynomial dependence. The proposed method can be called the method of successive approximations. Let us illustrate this method as an example of optimum designing discussed above thin membrane shell loaded by internal pressure.

We will extrapolate the analytic relationship (3), by determining the values of stiffness of shell for the different coefficients of the influence of SSS on the rate of corrosion



and then checking the strength of shell.

The graph 2 shows that the approximating curve falls continuously with decreasing the coefficient of the influence of SSS on the rate of corrosion



. We will set

the values of coefficient below





0

,

039629

. Let the

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Then, the stiffness of cross-section of membrane, being simultaneously both the objective function, we

calculate according to the formula (3), using the coefficients given in the Table 2:

 

2 3 2 3 3 2 2 1 0

sm

8795

,

986

03

,

0

4

,

769458

03

,

0

9

,

182041

03

,

0

9385

,

198

7853

,

849



































a

a

a

a

A

Taking into account that for considered shell the lower limit values for the radius of the middle surface and that this value corresponds to the value of the optimal solution, we define the thickness of the shell wall from the expression:

A



2



R



,

Where the shell’s wall thickness is equal:

571

,

1

100

14

,

3

2

8795

,

986

2













R

A



sm.

[image:5.595.51.547.319.423.2]

Then we carry out the checks of shell strength. The results are recorded in Table 4.

Table 4

The results of successive approximations to the "optimal" value of the coefficient of the influence of SSS on the rate of thecorrosion

№

_A

_(sm2

)



(sm)

R

(sm)



(МПа)

_P

_r



_t(sm)



_(sm)



(sm) 1 986,88 1,571 100,00 144,03 0,94564 0,3222 0,0463 0,3684 2 945,39 1,505 100,00 151,82 0,98954 0,3222 0,0429 0,3641 3 912,47 1,452 100,00 158,34 1,02609 0,3222 0,0360 0,3581 4 925,22 1,472 100,00 155,86 1,01220 0,3222 0,0385 0,3607 5 932,15 1,483 100,00 154,49 1,00460 0,3222 0,0397 0,3619 6 939,23 1,494 100,00 153,15 0,99700 0,3222 0,0409 0,3630 7 938,71 1,494 100,00 154,70 1,00000 0,3223 0,0521 0,3744

Analyzing the data in the Table 4, it can be concluded that the shell has a reserve by the measure of damage

1

max

_

r

P

[6] as well as by normal stresses



_max

(



_max



 





162

МPа).

Therefore, once again reduce the coefficient of the influence of SSS on the rate of corrosion and take

025

,

0





. Then we calculate the stiffness of the shell:

 

2 3 2 3 3 2 2 1 0

s

3965

,

945

025

,

0

4

,

769458

025

,

0

9

,

182041

025

,

0

9385

,

198

7853

,

849

m

a

a

a

a

A



































The thickness of the shell wall

505

,

1

100

14

,

3

2

3965

,

945

2













R

A



sm.

The results of second verification of shell strength are shown in the Table 4 (line 2). The strength of the shell is provided. We continue to extrapolate the approximating curve. Accept





0

,

02

and calculate the stiffness of the cross-sections of the shell:

 

2 3 2 3 3 2 2 1 0

sm

467

,

912

02

,

0

4

,

769458

02

,

0

9

,

182041

02

,

0

9385

,

198

7853

,

849



































a

a

a

a

A

452

,

1

100

14

,

3

2

467

,

912

2













R

A



sm.

We check the strength of the shell. The results are recorded in Table 4 (line 3).

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 

2 3 2 3 3 2 2 1 0

sm

32

,

925

022

,

0

4

,

769458

022

,

0

9

,

182041

022

,

0

9385

,

198

7853

,

849



































a

a

a

a

A

472

,

1

100

14

,

3

2

22

,

925

2













R

A



sm.

We produce the verification of shell strength. The results are given in Table4 in row №4.

The shell does not pass by strength, as a measure of damage exceeded the limit value P_rmax 1. Consequently, the "optimal" value of the coefficient of influence of impact of SSS on the rate of corrosion in the range 0,022



0,025. Assume the coefficient

023 , 0





. Perform the necessary calculations:

 

2 3 2 3 3 2 2 1 0

s

148

,

932

023

,

0

4

,

769458

023

,

0

9

,

182041

023

,

0

9385

,

198

7853

,

849

m

a

a

a

a

A



































483

,

1

100

14

,

3

2

148

,

932

2













R

A



см.

We carry out the check the shell strength. The calculation results are shown in Table 4 in row №5. The project does not pass, as the measure of damage exceeded the limit value. We accept





0

,

024

.

 

2 3 2 3 3 2 2 1 0

s

229

,

939

024

,

0

4

,

769458

024

,

0

9

,

182041

024

,

0

9385

,

198

7853

,

849

m

a

a

a

a

A



































494

,

1

100

14

,

3

2

229

,

939

2













R

A



sm.

We produce verification of the shell strength. The results of verification are given in the Table 4 in row №6. This project takes place, as the strength of the shell, as at the plastic and in brittle fracture is not broken. Thus, we are taking as an "optimal" coefficient of influence of SSS on the rate of corrosion process





0

,

024

.

The resulting point is taking as a starting, perform for it identification of the mathematical model of corrosive damage and produce the optimization. The optimization results are recorded in the last term of the Table 4.

This example shows that the procedure of the polynomial approximation, adduced above, is possible only, if all sizes of optimized construction can be determined by use the stiffness of this construction for the further control of the bearing ability of structure.

In the case, if this impossible, it is necessary to use the directly approximation of the curves of the changing of optimized parameters of structure. We show this technique on the example of optimizing the parameters of the shell above, loaded with internal pressure. There are two variable parameters in the shell: the wall thickness



and the radius of the middle surface

R

.

We introduce the polynomial approximation with the help of the coefficient of the influence of SSS on the rate of corrosion



for each of these parameters:

 

3

3 2 2 1

0











a



a



a



a



(9)

 

3

3 2 2 1

0









b

b

b

b

R









(10)

The approximating expression for the cross-sectional area of the shell wall (shell’s stiffness) takes the form:

 

   





3



3 2 2 1 0 3 3 2 2 1 0

2

2





























R



a



a



a



a

b



b



b



b

(11)

Let’s search the roots of polynomials (9)-(10 performed using the method of global random search global SGEF [7]. To do this, we introduce the notation:

0

1

a

x



;

x

₂



a

₁;

x

₃



a

₂;

x

₄



a

₃;

x

₅



b

₀;

1

6

b

x



;

x

₇



b

₂;

x

₈



b

₃ and we formulate two mathematical programming problems: to find two vectors



1 2 3 4



1

x

,

x

,

x

,

x

X

and

X

₂



x

₅

,

x

₆

,

x

₇

,

x

₈



, that deliver the minimal values of functional:

 

_







 

















n i

x

x

x

x

S

1 2 3 4 2 3 2 1

1

min









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19

 

_







 















n

i

R

x

x

x

x

S

1

2 3

8 2 7 6 5

2

min









X

(13)

The Table 5 consists the values of the thickness of shell’s wall and the radius of the middle surface of shell's wall for the 6 sampled starting points from (Table 2) [1], the values of the coefficients of approximating polynomials coefficients (9)-(10) and the calculated values of the shell parameters.

The calculated shell’s stiffness at selected points defined by the expression (13). The coefficients of the approximating polynomials (9)-(10), determined by the solving the mathematical programming problem (12)-(14) by the random search method SGEF under

[image:7.595.50.546.283.451.2]

restrictions:

1



10

6



x

_i



1



10

6

,



i



1

,

2

,...,

6



.

Table 5

The coefficients of the approximating polynomials and calculated values of the shell parameters

№

Starting points The coefficients of approximating polynomials

Calculated values of the shell’s

parameters A



(%)

A

(sm2)



(sm)

R

(sm) opt

A

(sm2)



(sm)

R

(sm)

1 2309,34 3,643 100,89 2329,58 3,654 101,47 -0,88 2 1976,5 3,088 101,9

0

a

= 0,62848

b

₀ = 99,68385 1923,1 3,025 101,18 2,70 3 1646,78 2,615 100,21

1

a

= 28,63446

b

₁ = 19,9794 1683,8 2,654 100,97 -2,24

4 1307,6 2,075 100,2

2

a

= 46,9497

b

₂ = 55,6342 1347,9 2,131 100,67 -3,08

5 1134,6 1,783 101,2

3

a

= 95,6192

b

₃= 98,6153 1194,9 1,751 100,43 2,61 6 1065,51 1,696 100,00 1062,20 1,684 100,39 0,31

We pose the problem of determining the coefficient of influence of SSS on the rate of corrosion as the

mathematical programming problem: find a minimum of function:

 

   





3



1 3 2 1 2 1 1 0 3 1 3 2 1 2 1 1 0

2

2



R



a



a

x



a

x



a

x

b



b

x



b

x



b

x









X

X

X

(

15)

under restrictions:

 

1 1

0

;

2

 

1 1

0

1



x



x





g



x





x



g

X

X

. (16)

 

x

1

X

 vector of control variables (the control variable

x

₁





).

The problem (15)-(16) was solved by the random search method SGEF under the following initial data:

0

,

0

;

05

,

0

1

1



x





x

. The coefficients of the polynomial (15),

a

₀

,

a

₁

,

a

₂

,

a

₃ и

b

₀

,

b

₁

,

b

₂

,

b

₃ were taken from Table 2.

Output solution was:





0

,

0247999

. The value of

the function



 

X



824

,

12

is near the minimum of mathematical functions (7) and is near the analytical solution, adduced above.

The wall thickness of the shell, and the radius of the middle surface we find, respectively, using the equations (9) and (10):















₀

_,

₆₂₈₄₈

₂₈

_,

₆₃₄₄₆

₀

_,

_0247999

₄₆

_,

₉₄₉₇

₀

_,

_0247999

2

₉₅

_,

₆₁₉₂

_

₀

_,

_0247999

3

_

₁

_,

₃₀₈

sm;

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20

The shell of this size does not pass by strength

(





186

,

53

МPа;

P

_r



1

,

18184

).

[image:8.595.209.388.507.693.2]

Therefore, from this point we perform several specifying identifications and optimizations. The results are recorded in Table 5.

Table 6

The results of specifying identification and optimal parameters of the shell

№ п/п

Starting stiffness The coefficients of model

The optimal parameters of the shell,

the thickness of damage

A

(sm2_{) (sm)}



_(sm)

R



₀







A

оpt

(sm 2₎



(sm)



(sm)

1 824,12 1,308 100,14 0,3219 18,595 10,527 0,02347457 935,95 1,490 0,3625 2 935,95 1,490 100,00 0,3224 18,603 10,511 0,03028718 936,19 1,490 0,3743

Let's solve the problem of finding an optimal design by the method of successive approximations described above. By replacing the selective curve by approximating curve, we will gradually reduce the value of the coefficient of influence of SSS on the rate of corrosion to the stabilization of the solution.

The latter value is the coefficient of influence of SSS, given in Table 1, is  0,0396629. Since the approximating curve tends to decrease, on the first step we set  0,035 and calculate using the

approximating polynomials (9) and (10) the values of the wall thickness of shell and the radius of the middle surface:

 



























3 2

3 2 2 1

0







0

,

62848

28

,

63446

0

,

035

46

,

9497

0

,

035



a

a

a

a



95

,

6162



0

,

035

3



1

,

569

sm;

 











3













2



3 2 2 1

0







99

,

68385

19

,

9794

0

,

035

55

,

6342

0

,

035



b

b

b

b

R



98

,

6153



0

,

035

3



100

,

31

sm.

Fig. 3. Approximation of selective curve by a polynomial of the third degree

The value of cross-sectional area of the central part of the shell is equal:

 





2





   



R





2



3

,

1416



1

,

569



100

,

31



988

,

88

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Let's verify the strength of the shell. Test results are listed in table 6.12. The strength of the shell is provided.

We make the next step and is calculated using the polynomials (9), (10) the coordinates of the next starting point and stiffness of the cross section of the shell:

 



























3 2

3 2 2 1

0







0

,

62848

28

,

63446

0

,

03

46

,

9497

0

,

03



a

a

a

a



95

,

6162



0

,

03

3



1

,

442

sm;

 











3













2



3 2 2 1

0







99

,

68385

19

,

9794

0

,

03

55

,

6342

0

,

03



b

b

b

b

R



98

,

6153



0

,

03

3



100

,

23

sm.

The value of cross-sectional area of the central part of the shell is equal:

A

 





2





   



R





2



3

,

1416



1

,

442



100

,

23



908

,

12

sm

2

.

We perform the checking of shell strength at approximated parameters and at the approximated coefficient of influence of SSS on the rate of corrosion

03

,

0





. The results are listed in table 7.

Analyzing the results, we conclude that the condition of the shell strength is not ensured.

Hence, the coefficient of influence of SSS on the rate of corrosion should be selected within

035

,

0

03

,

0







. We set





0

,

032

and perform calculations. The results are listed in table 7.

 



























3 2

3 2 2 1

0







0

,

62848

28

,

63446

0

,

032

46

,

9497

0

,

032



a

a

a

a



95

,

6162



0

,

032

3



1

,

493

sm;

 











3













2



3 2 2 1

0







99

,

68385

19

,

9794

0

,

032

55

,

6342

0

,

032



b

b

b

b

R



98

,

6153



0

,

032

3



100

,

27

sm;

A

 





2





   



R





2



3

,

1416



1

,

493



100

,

27



940

,

97

sm2.

The strength of the shell is not ensured. We accept

033

,

0





and perform the necessary calculations.

[image:9.595.53.546.558.647.2]

The results are given in Table 7 on line №3.

Table 7

The results of successive approximations to the "optimal" value of the coefficient of influence of SSS on the rate of corrosion

№

_

_A

_(sm2

)



( sm)

R

( sm)



(МПа)

P

_r



_t( sm)



_(sm)



(sm) 1 0,035 988,88 1,569 100,31 145,75 0,95538 0,3223 0,0547 0,3770 2 0,030 908,12 1,442 100,23 163,14 1,05284 0,3223 0,0556 0,3779 3 0,032 940,97 1,493 100,27 155,70 1,01131 0,3223 0,0553 0,3776 4 0,033 957,09 1,519 100,28 152,14 0,99133 0,3223 0,0551 0,3773 5 0,0325 948,12 1,506 100,27 153,89 1,00116 0,3223 0,0552 0,3775 6 0,0326 950,09 1,508 100,27 153,62 0,99965 0,3223 0,0552 0,3775

The strength of the shell is not ensured. We accept and perform the necessary calculations.

The results are given in Table 7 on line №4.

 



























3 2

3 2 2 1

0







0

,

62848

28

,

63446

0

,

033

46

,

9497

0

,

033



a

a

a

a



95

,

6162



0

,

033

3



1

,

519

sm;

 











3













2



3 2 2 1

0







99

,

68385

19

,

9794

0

,

033

55

,

6342

0

,

033



b

b

b

b

R

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22

A

 





2





   



R





2



3

,

1416



1

,

493



100

,

27



957

,

09

sm2.

We carry out the checking shell strength. The strength of the shell is not ensured. Obviously, the solution lies in the range

0

,

032







0

,

033

.

Therefore, we accept





0

,

0325

and perform the calculations again.

 



























3 2

3 2 2 1

0







0

,

62848

28

,

63446

0

,

0325

46

,

9497

0

,

0325



a

a

a

a



95

,

6162



0

,

03253

3



1

,

506

sm;

 











3













2



3 2 2 1

0







99

,

68385

19

,

9794

0

,

0325

55

,

6342

0

,

0325



b

b

b

b

R



98

,

6153



0

,

0325

3



100

,

27

sm;

A

 





2





   



R





2



3

,

1416



1

,

493



100

,

27



948

,

12

sm2.

The strength of the shell is not ensured

(

P

_r



1

,

00116

).

Obviously, the solution lies within

033

,

0

0325

,

0







. Accept





0

,

0326

.

 



























3 2

3 2 2 1

0







0

,

62848

28

,

63446

0

,

0326

46

,

9497

0

,

0326



a

a

a

a



95

,

6162



0

,

0326

3



1

,

508

sm;

 











3













2



3 2 2 1

0







99

,

68385

19

,

9794

0

,

0326

55

,

6342

0

,

0326



b

b

b

b

R



98

,

6153



0

,

0326

3



100

,

27

sm;

A

 





2





   



R





2



3

,

1416



1

,

508



100

,

27



950

,

09

sm2.

The strength of the shell is ensured (line 6 in the Table 7). We specify the resulting solution with the help of additional identification of corrosive model and additional optimizations.

[image:10.595.55.541.527.634.2]

The results of the iterative procedure specifying identifications are shown in the Table 8.

Table 8

The results of specifying identification and optimal parameters of the shell

№ п/п

Starting stiffness The coefficients of corrosive model

The optimal parameters of shell, the depths of damage

A

(sm2)



(sm)

R

(см)



0







A

опт

(sm2)



(sm)



(sm) 1 950,09 1,508 100,27 0,3219 18,611 10,526 0,03118262 943,64 1,502 0,3746 2 943,64 1,502 100,01 0,3225 18,620 10,506 0,03078890 943,46 1,502 0,3745

As follows from the adduced calculations, both of the approach to the definition of "optimal" coefficient of the influence of SSS on the rate of corrosion give similar results. However, the second approach using a polynomial approximation of the optimized parameters, is preferable, as it allows to calculate the design parameters by which then can calculate the geometrical characteristics of the design and test its strength, stiffness and stability.

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23

In the next article will be consider an another method of polynomial approximation, which will be able to solve this problem.

REFERENCES

[1] Filatov G.V. The Foundations of the Evolution Theory of Identification of Mathematical Models of Corrosion Destruction at the Optimal Planning of Constructions [Text] // G.V. Filatov / − International Journal of Emerging Technology & Advanced Engineering, Volume 6, Issue 3, March 2016, p.p.166-180. [2] Filatov G.V. The Numerical Experimental Verification of

Evolutional Theory of Identification of Mathematical Models of Corrosive Destruction under Stress. Compressed Shell [Text] // G.V. Filatov / − International Journal of Emerging Technology & Advanced Engineering, Volume 6, Issue 4, April 2016, p.p.1-9. [3] Filatov G.V. Application of Evolutional Theory of Identification

of Mathematical Models of Corrosive Destruction at Optimum Designing of Weld-fabricated I-beam [Text] // G.V. Filatov / − International Journal of Emerging Technology & Advanced Engineering, Volume 6, Issue 5, May 2016, p.p.222-236.

[4] Filatov G.V. Optimal design of structures by the combined use of mathematical models of corrosion destruction [Text] // G.V. Filatov / − International Journal of Emerging Technology & Advanced Engineering, Volume 6, Issue 5, May 2016, p.p.6-15. [5] Petrov, V.V Calculation of structural elements, interacting with

aggressive media [Text]: monograph / I.G.Ovchinnikov, Yu.M.Shihov. − Saratov: Saratov State University. 1987. − 288 p. [6] Novozhilov V.V. Prospects for the Construction of the Strength

Criterion under Complex Loading [Text] // V.V. Novozhilov, O.G. Rybakina / − Proceedings of the Academy of Sciences of the USSR, meh. solid , "Nauka", − 1966, − №5, p.p. 101-111. [7] Filatov G.V. The Stochastic Method of Search of Global Extreme