**MECHANICAL OPTIMIZATION **

**WITH ARTIFICIAL IMMUNE SYSTEM **

**Rafał Robak**

^{1,2a}1Instytut Maszyn i Urządzeń Energetycznych, Politechnika Śląska,^{ }

2AvioPolska Sp. z o.o.

aRafal.Robak@avioaero.it

**Summary **

The article shows implementation of artificial immune system in the mechanical optimization. Optimization pro- cess is mainly used during new product introduction phase and ensures that new design is well balanced and in- cludes all critical to quality items. Artificial immune system algorithm in the mechanical optimization assumes op- timal solution as a pathogen and geometric configurations as a lymphocyte which are matching with optimal de- sign.

**Keywords: topological optimization, artificial immune system, algorithm **

**OPTYMALIZACJA Z WYKORZYSTANIEM **

**SZTUCZNYCH SYSTEMÓW IMMUNOLOGICZNYCH **

**Streszczenie **

Artykuł przedstawia zastosowanie sztucznego systemu immunologicznego w procesie optymalizacji konstrukcji mechanicznych. Proces optymalizacji jest stosowany zwłaszcza w trakcie projektowania nowej kontrukcji bądź systemu i pozwala dostarczyć produkt zbalansowany i uwzględniający wszystkie niezbędne kryteria. Algorytm działający na podstawie sztucznego systemu immunologicznego w optymalizacji mechanicznej traktuje optimum jako patogen, a konfiugracje geometryczne jako limfocyty, które adresują optymalną konstrukcje.

**Słowa kluczowe: optymalizacja topologiczna, sztuczny system immunologiczny, algorytm **

**1. INTRODUCTION**

Currently in Research & Development (R&D) area we can experience more attention to the shortening new product introduction timeline maintaining durability and performance aspects on the same or even higher level with respect to legacy designs. Lead time dedicated for new products has been reduced due to business constrains and competitiveness.

Essential is to provide reliable product at the very first time and address all potential issues in advance. Any significant changes in the product definition shall be introduced in early stage design since later can heavily impact program schedule and business needs.

Design is verified typically along a development process using analytical assessments basing on company proce- dures, templates, design practices and using commercial

as well as in-house software for 2D and 3D simulations [5][6][7].

Optimization phase is very often used to balance design and support engineering decision of down selecting the most preferable solution from certain stand point. Is very clear that requirements need to be well determined because can affect final product shape. Component which is very good from performance perspective (e.g.

sophisticated sealing features) could have an issue with durability. In other words, all Critical to Quality (CTQ) items need to be addressed with the same level of resolu- tion including reliability, performance and costs.

In the mechanical optimization process we can distin- guish topologic and parametric optimization. Both are used to determine shape of the component however with different degree of freedom. Topologic optimization is

the most flexible and can be employed for preliminary definition of the component or a system. Second optim zation method is parametric and can precisely tune a product through explicitly defined features of the geometry.

**2. ARTIFICIAL IMMUNE ** **SYSTEMS **

Immunology [4] is a discipline researching biology and biomechanics rules of immune system on pathogens and other structures classified as enemies of the internal system. Artificial immune systems are characterized following features: detection is not driven by the central unit, system is capable to detect anomaly’s;

tion of pathogens never met in the past

created as a network and is constantly updating “lear ing” basing on the information from pathogens tion of the pathogen based on partial conformity

Fig. 1. Lymphocyte type B

No necessity to define negative set of examples since remaining structures (domain subtracted with own structures) became pathogens (Fig. 2).

Fig. 2. Capability detection of immune system

Artificial immune system in the optimization process is treating optimal solution as a pathogen. System is creating lymphocytes to match as much as possible with pathogen and at the same time meet all defined criterions.

**3. OPTIMIZATION ALGORITHM**

Artificial immune system in the optimization process uses clonal selection algorithm [4]. During initialization phase is creating memory cells (lymphocytes). Memory size during optimization process is unchanged. Subs quently each of the cell is producing defined previously number of clones and then are subjected by mutation routine. For each of the element m which belong to the

**High st ruct ural **
**conformit y **

Domain Own structures

the most flexible and can be employed for preliminary definition of the component or a system. Second optimi- zation method is parametric and can precisely tune a

licitly defined features

**ARTIFICIAL IMMUNE **

] is a discipline researching biology and biomechanics rules of immune system on pathogens and other structures classified as enemies of the internal system. Artificial immune systems are characterized by etection is not driven by the central ystem is capable to detect anomaly’s; (identifica- tion of pathogens never met in the past). Memory is created as a network and is constantly updating “learn-

from pathogens. Detec- tion of the pathogen based on partial conformity (Fig.1).

1. Lymphocyte type B

No necessity to define negative set of examples since remaining structures (domain subtracted with own

2. Capability detection of immune system Artificial immune system in the optimization process is treating optimal solution as a pathogen. System is creating lymphocytes to match as much as possible with pathogen and at the same time meet all defined

**OPTIMIZATION ALGORITHM **

Artificial immune system in the optimization process ]. During initialization phase is creating memory cells (lymphocytes). Memory size during optimization process is unchanged. Subse-

each of the cell is producing defined previously number of clones and then are subjected by mutation For each of the element m which belong to the

generated set, algorithm is searching similar element m*.

From elements pair (m, m*) item which

better to the target is maintained while worse one is replaced by new randomly selected. In the next step algorithm is ordering all memory cells. If in some sub zone of the domain there is a high density of cells, then the best element is maintained while remaining are replaced by new randomly selected. In case of not mee ing requirements by the best cell algorithm is starting next iteration and clone process, mutation begins up to reaching convergence criterion.

Fig. 3. Clonal selection algorithm

Algorithm of the selection is responsible for exploitation (worse items are removed) while suppression phase for exploration (domain screening).

Before solving physical task, algorithm shall be verified using dedicated test function. The aim of the verification is to check convergence of the algorithm to the well known test function extremum. During such verification impact of the parameter on the optimization process can be verified as well.

Test function used for validation

physical phenomenon and optimization nature. If phys cal problem basing on engineering judgment has one solution in the interested domain then similar function with one extremum shall be used. For example, Bohachevsky function [8] with jus

Tab. 1. Function test
**Parameters set ** n=2

**Domain ** [X,Y]=(-100;100)
**Global extrema ** F(X,Y)=0 for X=[0] and

Y=[0]

**Recept or **

Remaining structures Pathogens

generated set, algorithm is searching similar element m*.

From elements pair (m, m*) item which corresponds better to the target is maintained while worse one is replaced by new randomly selected. In the next step algorithm is ordering all memory cells. If in some sub- zone of the domain there is a high density of cells, then

ained while remaining are replaced by new randomly selected. In case of not meet- ing requirements by the best cell algorithm is starting next iteration and clone process, mutation begins up to

Algorithm of the selection is responsible for exploitation (worse items are removed) while suppression phase for

Before solving physical task, algorithm shall be verified unction. The aim of the verification is to check convergence of the algorithm to the well- known test function extremum. During such verification impact of the parameter on the optimization process can

Test function used for validation should represent physical phenomenon and optimization nature. If physi- cal problem basing on engineering judgment has one solution in the interested domain then similar function with one extremum shall be used. For example,

with just one optimum:

100;100) F(X,Y)=0 for X=[0] and

Fig. 4. Test function Bohachevsky

On the contrary if engineer is expecting multiple local solutions can select another appropriate test function like Rastrigin function [8]:

Fig. 5. Test function Rastrigin

Quantity of the memory cells and clones need to be set accordingly to the optimization task. More memory cells are improving accuracy of the solution however comp tational time is increasing as well for each iteration.

**4. TOPOLOGICAL ** **OPTIMIZATION **

Essentially topological optimization [2] is set to reduce a mass with constrain on stress results. The aim for alg rithm is to find minimum of the following function:

Ω

Algorithm is removing a portion of the structure which is outside load path. Each lymphocyte represents pote tial solution (structure shape).

On the contrary if engineer is expecting multiple local solutions can select another appropriate test function

Quantity of the memory cells and clones need to be set optimization task. More memory cells are improving accuracy of the solution however compu- tational time is increasing as well for each iteration.

] is set to reduce a ess results. The aim for algo- rithm is to find minimum of the following function:

(1) Algorithm is removing a portion of the structure which is outside load path. Each lymphocyte represents poten-

Shape of the structure is described by set of control points [3]. Each control point has own value of pseudo density (0 – 1). Pseudo-density value determines if portion of the structure remains in the geometry variant or not.

Fig. 6. Density distribution during optimization

Red range of pseudo-density means that element will be removed while green range maintained. Below figure demonstrates a control point concept:

Fig. 7. Control points concept

This approach reduces number of parameters of each memory cell since resultant shape is defined by density function approximated through control points. For 2D structure density function became R3; d=f(x,y) while for 3D structure density function will be represented by R4;

d=f(x,y,z).

As an application example of topolo

by using artificial immune system in the article has been presented 2D structure (200mm x 100mm x 4mm) Following figure shows optimization domain with su ports and mechanical load as well as control points definition:

Fig. 8. Optimization of the 2D plane model mechanical load

Fig. 9. Distribution of control points; system symmetry structure is described by set of control ]. Each control point has own value of pseudo-

density value determines if portion of the structure remains in the geometry variant

optimization

density means that element will be removed while green range maintained. Below figure demonstrates a control point concept:

This approach reduces number of parameters of each nce resultant shape is defined by density function approximated through control points. For 2D structure density function became R3; d=f(x,y) while for 3D structure density function will be represented by R4;

As an application example of topological optimization by using artificial immune system in the article has been (200mm x 100mm x 4mm).

Following figure shows optimization domain with sup- ports and mechanical load as well as control points

plane model subjected to

. Distribution of control points; system symmetry

Structure defined after 68 iterations. Parameters of the algorithm included in the table 2.

Tab. 2. Algorithm parameters
**Clones quantity **
**Memory cells quantity **
**Parameter of mutation **
**Congestion coefficient **
**Allowable von Mises stress **

Fig. 10. Optimized shape

Initially mass of the optimized structure was equal to 0.63kg while in the optimized shape is equal to 0.19kg (achieved percentage reduction 69%). Maximum von Mises stress is 70.5MPa and is below allowable stress set in the algorithm (von Mises stress below 80MPa).

Another example tested in the topological optimization is following 2D structure constrained on left side and subjected to the vertical load on right side (

Fig. 11. Optimization of the 2D plane model mechanical load

Structure defined after 31 iterations. Parameters of the algorithm included in the table 3.

Tab. 3. Algorithm parameters
**Clones quantity **
**Memory cells quantity **
**Parameter of mutation **
**Congestion coefficient **
**Allowable von Mises stress **

Structure defined after 68 iterations. Parameters of the

8 4 0.18 0.21 80 MPa

Initially mass of the optimized structure was equal to 0.63kg while in the optimized shape is equal to 0.19kg (achieved percentage reduction 69%). Maximum von allowable stress set

w 80MPa).

Another example tested in the topological optimization is following 2D structure constrained on left side and subjected to the vertical load on right side (Fig. 11).

. Optimization of the 2D plane model subjected to

iterations. Parameters of the

8 4 0.01 0.50 200 MPa

Fig. 12. Optimized shape

Initially mass of the optimized structure was equal to 0.63kg while in the optimized shape is equal to 0.17kg (achieved percentage reduction 72%). Maximum von Mises stress is 144MPa and is below allowable stress set in the algorithm (von Mises stress below

For both analyzed structures evaluated s

sarily will be global extremum but for sure shall meet design target in terms of requirements

Parametric optimization can be implemented sub s quentially aimed to adjust thicknesses/radiuses of the 2D structure and prepares detailed description for product definition engineer.

**5. CONCLUSIONS **

Artificial immune systems are used in the optimization process and support down selecting of the most balanced design concept. Both optimization types (topologic and parametric) can provide suitable design which meet defined requirements.

In mechanical perimeter typically

to determine structure with minimum material and capable to sustain defined level of the stres

theless, optimization could be used as well to determine structure with desired natural frequency values; (eige values to avoid any resonances in the operating ranges or to increase margin to resonant point)

is part of further research. Currently

on the optimization of natural frequency of the engine low pressure turbine stator

to low frequency excitation coming from rotor imba ance.

Initially mass of the optimized structure was equal to 0.63kg while in the optimized shape is equal to 0.17kg (achieved percentage reduction 72%). Maximum von Mises stress is 144MPa and is below allowable stress set in the algorithm (von Mises stress below 200MPa).

For both analyzed structures evaluated shape not neces- sarily will be global extremum but for sure shall meet design target in terms of requirements set by engineer.

Parametric optimization can be implemented sub se- knesses/radiuses of the 2D structure and prepares detailed description for

Artificial immune systems are used in the optimization process and support down selecting of the most balanced optimization types (topologic and parametric) can provide suitable design which meet

typically optimization is used to determine structure with minimum material and capable to sustain defined level of the stress [1]. Never- theless, optimization could be used as well to determine structure with desired natural frequency values; (eigen- values to avoid any resonances in the operating ranges or to increase margin to resonant point) and this aspect Currently research is focused natural frequency of the aircraft stator component subjected to low frequency excitation coming from rotor imbal-

**Bibliography**

1. Bąk R., Burczyński T.: Wytrzymałość materiałów z elementami ujęcia komputerowego. Warszawa: WNT, 2001.

2. Stadnicki J.: Teoria i praktyka rozwiązywania zadań optymalizacji z przykładami zastosowań technicznych.

Warszawa: WNT, 2006.

3. Szczepanik M.: Optymalizacja ewolucyjna układów powierzchniowych. Rozprawa doktorska. Gliwice: Politechn ka Śląska, 2003.

4. Wierzchoń S.: Sztuczne systemy immunologiczne. Teoria i zastosowania. Warszawa: Akademicka Oficyna W dawnicza Exit, 2001.

5. Zienkiewicz O.C., Taylor R.L.: The finite element method.

6. https://www.ansys.com/

7. https://www.mscsoftware.com/application/structural 8. https://www.scilab.org/

Artykuł dostępny na podstawie licencji Creative Commons http://creativecommons.org/licenses/by/3.0/pl

Bąk R., Burczyński T.: Wytrzymałość materiałów z elementami ujęcia komputerowego. Warszawa: WNT, 2001.

Stadnicki J.: Teoria i praktyka rozwiązywania zadań optymalizacji z przykładami zastosowań technicznych.

a ewolucyjna układów powierzchniowych. Rozprawa doktorska. Gliwice: Politechn

Wierzchoń S.: Sztuczne systemy immunologiczne. Teoria i zastosowania. Warszawa: Akademicka Oficyna W

O.C., Taylor R.L.: The finite element method. Oxford: Butterworth, 2000.

https://www.mscsoftware.com/application/structural-analysis

Artykuł dostępny na podstawie licencji Creative Commons Uznanie autorstwa 3.0 Polska.

http://creativecommons.org/licenses/by/3.0/pl

Bąk R., Burczyński T.: Wytrzymałość materiałów z elementami ujęcia komputerowego. Warszawa: WNT, 2001.

Stadnicki J.: Teoria i praktyka rozwiązywania zadań optymalizacji z przykładami zastosowań technicznych.

a ewolucyjna układów powierzchniowych. Rozprawa doktorska. Gliwice: Politechni-

Wierzchoń S.: Sztuczne systemy immunologiczne. Teoria i zastosowania. Warszawa: Akademicka Oficyna Wy-

Uznanie autorstwa 3.0 Polska.