Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 2, pp. 341-364, Warsaw 2003

EVOLUTIONARY ALGORITHMS AND BOUNDARY

ELEMENT METHOD IN GENERALIZED SHAPE

OPTIMIZATION

Tadeusz Burczyński
Department for Strength of Materials and Computational Mechanics,

Silesian University of Technology

Institute of Computer Modelling, Cracow University of Technology

e-mail: burczyns@polsl.gliwice.pl

Grzegorz Kokot
Department for Strength of Materials and Computational Mechanics,

Silesian University of Technology; e-mail: gkokot@polsl.gliwice.pl

The coupling ofmodern, alternative optimizationmethods such as evolutio-
nary algorithms with the effective tool for analysis of mechanical structures
– BEM, gives a new optimization method, which allows one to perform the
generalized shape optimization (simultaneous shape and topology optimiza-
tion) for elasticmechanical structures. This new evolutionarymethod is free
from typical limitations connected with classical optimization methods. In
the paper, results of researches on the application of evolutionary methods
in the domain of mechanics are presented. Numerical examples for some
optimization problems are presented, too.

Key words: evolutionary algorithms, genetic algorithms, generalized shape
optimization, topology optimization

1. Introduction

For the last twenty years the optimization of mechanical structures has
been divided into four main directions:

• material property optimization,
• size optimization,
• shape optimization,
• topology optimization.



342 T.Burczyński, G.Kokot

When the optimal solutions for two dimensional problems are searched
for, mainly the shape optimization and topology optimization is applied. The
topology optimization reaches wider and wider applications due to its advan-
tages.

The topology optimization concerns problems, in which the topology of a
structure is changed.Within the last ten years three directions of development
of the topology optimization can be observed. The first steps in this field con-
cerned optimization problems of truss structureswhere the optimal layoutwas
looked for.Most of the first papers presenting the topology optimization con-
cerned just the optimization of truss structures e.g.: Achtziger (1995), Dems
(1996), Kirsch (1994, 1995, 1996), Mróz and Piekarski (1995, 1996), Rozva-
ny (1995, 1996), Rozvany and Birker (1995), Rozvany and Károlyi (1997),
Rozvany et al. (1993).

Design variables in such optimization are parameters describing member
positions in the structure, cross section (zero value is equivalent to removing
themember) and the position of trusses joints.

The second type of topology optimization concerns mechanical structures,
mainly surfaces, where material properties and layout are taken as design
variables. The leading researches in this field are: Allaire (1996a,b), Bendsoe
(1995), Bendsoe and Kikuchi (1988), Olhoff (1989, 1993), Pedersen (1995).

In recent years the third direction of the development of the topology opti-
mization has appeared. It is based on generating a new void inside a domain
on the basis on special criteria and next on conducting simultaneous shape
and topology optimization for outside and inner structure boundaries. The
researches on this field are conducted by Eschenauer et al. (1994), Esche-
nauer and Schumacher (1995), Schumacher (1995), Sokołowski and Żochowski
(1999), Burczyński and Kokot (1997, 1998). From the mathematical point of
view those types of optimization relay on replacing a homogenous domain by
a non-homogenous domain.

One of themain goals of this type of the topology optimization of continu-
ous structures,whereamaterial continuumfills anarea, is toplace thematerial
inside thedomain in suchaway, that the optimization criteriawith constraints
are satisfied. The placing of thematerials inside the domain can be equivalent
to the change of the material density, which eventually leads to generating
subregionswith zero density. This special case, particularly desirable from the
structural point of view, is equivalent with generating interior boundaries in
the form of different shape voids. Further transformations of generated voids
and outer boundary require applying the classical shape optimization, which
leads to connecting topology optimization and shape optimization.



Evolutionary algorithms and boundary element method... 343

Both problems – topology optimization and shape optimization – connec-
ted with each other, create a generalized shape optimization problem, which
has crucial importance in the optimal design of structures.
Regardless of the type of the optimization, the formulated optimization

problem must always be solved. To do this, one of well-known optimization
methods can be used (e.g. the optimality criteria approach or mathematical
programmingmethods).
Most of them, despite their advantage, have some limitations, namely:

• the objective function must be continuous,
• the hessian of the objective function should be positive definite,
• there is a substantial probability of getting a local optimum,
• calculations start from a single point, narrowing the search domain,
• the choice of the starting point influences the method convergence.

The limitations mentioned above cause that for some optimization pro-
blems the optimal solution is either very difficult or quite impossible to obtain.
It leads to significant problems in getting the optimal solution in some cases.
Therefore, new optimization methods, free from the limitations mentioned
above, are still being looked for.
Relatively not long ago, the research on simulating processes occurring

in nature has been undertaken (Goldberg, 1989; Holland, 1975; Michalewicz,
1992). Resulting fromthose experiments various algorithms for searching opti-
mal solutions were created. Those algorithms are known asGAs, evolutionary
programming, evolutionary strategies, neural networks, classifier systems and
simulated annealing. Many of them turn out to be alternative methods of
optimization for classical methods such as e.g. well-known gradient methods.
They have been widely applied in solving the search problems and optimi-
zation problems in many disciplines for many years, but their application to
solve optimization problems in mechanics started relatively not long ago.
Particularly, theGAs are often used in solving optimization problems (Mi-

chalewicz, 1992). In GAs, the search of the optimal solution is based only on
the values of the objective function (the fitness function) and does not require
meeting the limitations listed for classical optimization methods. It gives a
free hand in defining optimization problems.
On the basis of articles presented on various conferences and symposia

one may suppose that evolutionary methods will give also good results when
they are applied to mechanical engineering problems. The first attempts of
applyingGAs to the optimization of mechanical structures was undertaken in
the early 1990s (Nagendra et al., 1993; Ponslet et al., 1993; SakamotoandOda,



344 T.Burczyński, G.Kokot

1993). The existing papers in this field concern: truss structures (the optimal
trusses selection, their layouts or looking for proper cross sections) (Oshaki,
1995; Ponslet et al., 1993; Rajeev and Krishnamoorthy, 1997; Sakamoto and
Oda, 1993), the optimization of laminate (composite) structures (Nagendra
et al., 1993; Okumura et al., 1995; Punch et al., 1994); optimization of two
dimensional structures (Annicchiarico and Cerrolaza, 1998; Burczyński and
Kokot, 1998; Chapman and Jakiela, 1994; Fernandes et al., 1998; Kallassy and
Marcelin, 1997; Kita andTanie, 1997), the optimizaton of vibrating structures
(Burczyński et al., 1999b). The genetic algorithms have been also applied in
crack and void identification (Burczyński et al., 1999a). The main goal of
this paper is to present the results of research on the application of a new
alternative optimization approach – the evolutionary method based on GAs
andBEM, to solve problems in the field of the generalized shape optimization.

2. The generalized shape optimization problem

Consider the following class of optimization problems

min
x
: J0(x) (2.1)

with imposed constraints

Jα(x)=0 α=1,2, . . . ,n

J∗β(x)­ 0 β=1,2, . . . ,m

ximax ­xi ­ximin i=1,2, . . . ,k

(2.2)

where x=(xi) is a vector of design variables.
The functionals J0, Jα and J∗β can have the following forms

J(x)=
∫

Ω

Ψ(σ,ε,u) dΩ+
∫

Γ

Φ(p,u) dΓ or J(x)=
∫

Ω

C dΩ (2.3)

where Ψ is an arbitrary continuous function of stresses σ, strains εanddispla-
cements u in the domain Ω of the structures and Φ is an arbitrary function
of the displacements u and tractions p on the boundary Γ .
Functionals (2.1) and (2.2)1,2 can represent the objectives or constraints

described by the stresses or displacements e.g. in the form of the complemen-
tary energy, vonMises stresses, or the cost of the structure. Constraints (2.2)3
are imposed on the design variables, simply the geometry constrains.



Evolutionary algorithms and boundary element method... 345

In the next paragraphs it will be shown how to solve the optimization pro-
blempresented above. The proposedmethod consists of a few steps: geometry
modelling and choosing the design variables, applying the numerical method
for evaluation of the fitness function, creating the internal voids (if necessary)
and applying the evolutionary process. All steps are described below.

3. Geometry modelling

The choice of the geometrymodellingmethod and the design variables has
great influence on the final solution of the optimization process. There is a
lot of methods for geometry modelling. In the proposed approach NURBS or
Bsplines (Piegl and Tiller, 1995) are used to the modelling of the geometry
of the structures. The Bsplines curves are defined as follows (Piegl and Tiller,
1995)

C(u)=
n∑

i=0

Ni,p(u)P i a¬u¬ b (3.1)

where {Ni,p(u)}are the pth-degreeBsplinebasis function, {Pi}are thecontrol
points.
The NURBS curves are defined as (Piegl and Tiller, 1995)

C(u)=

n∑
i=0

Ni,p(u)wiP i
n∑
i=0
Ni,p(u)wi

a¬u¬ b (3.2)

where {Ni,p(u)} are the pth-degree Bspline basis functions defined on a non-
periodic (and nonuniform) knot vector, {Pi} are the control points, {wi} are
the weigts.
The possibility of an easy control of the shape by only a few control points

and the local changes of shapes without affecting on the rest of the structure
are their main advantages (Fig.1), which are very helpful in the optimization
process.
In the shape optimization process the co-ordinates of the control points

become design variables, which gives a small number of the design variables
and simplicity of data preparation in comparison with other methods (e.g.
when the coordinates of boundary nodes (in BEM) or mesh nodes (in FEM)
are taken as the design variables).



346 T.Burczyński, G.Kokot

Fig. 1. Bspline and NURBS

4. Boundary element method in evaluation of fitness function

In the GAs, a fitness function with constraints plays the role of the envi-
ronment (Michalewicz, 1992). In the cases of the optimization problems of
mechanical structures the fitness function, called the objective function, de-
pends on stresses, strains or displacements. In order to determine those qu-
antities the computational methods of mechanics such as the finite element
method (FEM) or the boundary element method (BEM) are used in practical
engineering applications.

BEM is selected as the method of structure analysis. Particularly, BEM
fits to the sensitivity analysis and the optimization of elastic structures for
the sake of its specific features. The main advantages of BEM are the easy
way of discretization (only the boundary of a structure is discretized) and
the computation precision of boundary values. In a iteratively solved shape
optimization problem the easyway of discretization is very important, because
the rediscretization of the domain Ω any time it changes is very inconvenient
(it occurs using FEM).



Evolutionary algorithms and boundary element method... 347

In the shape optimization the boundary of the structure is subjected to
variation, and the goal of the optimization is to shape the boundary in such a
way, that theobjective functiongets the extremum(satisfying theconstraints).
BEM has turned out to be a convenient and natural numerical technique in
the classical shape optimization, because it enables precise describing of the
design variables on the varying boundary and fast meshing of the structure
during the iterative process of boundary shape evolution (only the boundary
is discretizing).
The list of papers in this field can be found by Burczyński (1993).
In BEM, the boundary value problem is described by the following vector

boundary integral equations (Kleiber, 1998)

c(x)u(x)+
∫

Γ

U
∗(x,y)p(y) dΓ(y)=

(4.1)

=
∫

Γ

P
∗(x,y)u(y) dΓ(y)+

∫

Ω

U
∗(x,y)b(y) dΩ(y)

where c(x) is a coefficientmatrix, u(y) and p(y) are vectors of displacements
and tractions, respectively, on the boundary Γ , b(y) is a vector of the body
forces in the domain Ω, U∗(x,y) and P∗(x,y) are fundamental solutions of
elastostatics.

Fig. 2. Discretization of the boundary using quadratic boundary elements

Discretizing the boundary Γ by means of the boundary elements Γe,
e=1, ...,E

x(ξ)=Nne (ξ)(x)
n
e (4.2)

and approximating the fields of displacements and tractions on each boundary
element Γe, in terms of the nodal values une , p

n
e and the shape functions N

n
e

u(y)=Nne (ξ)u
n
e p(y)=N

n
e (ξ)p

n
e (4.3)



348 T.Burczyński, G.Kokot

one obtains a discrete form of boundary integral equation (4.1)

c(x)u(x)=
E∑

e=1

Ne∑

n=1

(u)ne

∫

Γe

P
∗[x,y,ξ]Nne (ξ)J(ξ) dΓ(ξ)−

(4.4)

−
E∑

e=1

Ne∑

n=1

(p)ne

∫

Γe

U
∗[x,y,ξ]Nne (ξ)J(ξ) dΓ(ξ)+

∫

Ω

U
∗(x,y)b(y) dΩ(y)

where (u)ne , (p)
n
e are thenodal values of thedisplacements and tractions fields,

J(ξ) – jacobian, P∗[x,y,ξ], U∗[x,y,ξ] – fundamental solutions in the local
coordinate system.
Finally, equation (4.4) can be transformed into a systemof linear algebraic

equations
AX=F (4.5)

where the unknown values of the boundary displacements and tractions are
placed in the column matrix X.
Solving equation (4.5), allows one to obtain all the unknown boundary

values of the displacements and tractions. Knowing all boundary displace-
ments and tractions on the boundary Γ , the components of the stress tensor
σ=(σij) can be calculated in selected internal points x∈Ω using the follo-
wing equation

σ(x)=
∫

Γ

D(x,y)p(y) dΓ(y)−
∫

Γ

S(x,y)u(y) dΓ(y)+
∫

Ω

D(x,y)b(y) dΩ(y)

(4.6)
where S(x,y) and D(x,y) are the third-order fundamental solution tensors
obtained from suitable differentiation of U∗(x,y) and P∗(x,y) with respect
to the source point x and application of Hooke’s law.
The application of BEM to the optimization shows the great advantage

over other computational methods for the analysis of mechanical structures,
because the discretization is to be done only along the boundary of the struc-
ture, which reduces the problem size in the analysis process.

5. Generating internal voids – the bubble method

Majority of methods of solving optimization problems are based rather
on finite element procedures. BEM has been applied to shape optimization



Evolutionary algorithms and boundary element method... 349

problems but till now it has not been used to the topology optimization. In
the classical shape optimization one optimizes the shape of the existing bo-
undaries. BEM is a exceptionally natural and convenient numerical analysis
technique for such optimization process. It appears, however, that it is impos-
sible to insert any changes inside the domain. As the discretization is made
only on the boundary, there is no possibility to insert the void during the
optimization of the structure.
In order to eliminate this drawback, the idea of using the so-called topology

derivative, proposed by Sokołowski and Żochowski (1999) has been applied.
The topology derivative is defined as follows

ℑ(x)= lim
ρ→0

J
(
Ω \Bρ(x)

)
−J(Ω)

|Bρ(x)|
x∈Ω (5.1)

or by an equivalent formula

ℑ̃(x)= lim
ρ→0

J
(
Ω \Bρ(x)

)
−J(Ω)

ρN
x∈Ω (5.2)

where J is an arbitrary shape functional, Ω ⊂ RN is a domain in the RN

space, Bρ(x) is a bubble with the radius ρ> 0 such that Bρ(x)= {y∈RN :
‖y−x‖<ρ}, Bρ(x) is the bubble surroundings Bρ(x). The function ℑ(x),
called the topology derivative of the functional J(Ω), gives information about
the infinitesimal variation of the shape functional J if a small void is inserted
into the domain Ω. Applying the topology derivative approach, it is possible
to determine the position of the bubble with any shape in the domain.
When the functional describes the complementary energy, the topology de-

rivative problem can be reduced to a method presented by Eschenauer and
Schumacher (1994, 1995), called ”the bubble method”. It is assumed that in-
serting an infinitesimally small void into the domain will produce only local
stress concentration in the vicinity of the bubble, and the global stress field
remains unchanged. Looking for the best position of such a bubble, an opti-
mization process is carried out, and for some shapes of the bubble (i.e. a
circle, triangle, ellipse) the so-called characteristic functions (Eschenauer and
Schumacher, 1995) can be determined:

• for a circular-shaped void

H(σ1,σ2)=
1
2E

[
(σ1+σ2)

2+2(σ1+σ2)
2
]

(5.3)



350 T.Burczyński, G.Kokot

• for a ellipse-shaped void

H =
1
4πE

[
5.65(σ1+σ2)

2+5.52(σ1 −σ2)
2−

(5.4)

−0.22(σ21 −σ
2
2)cos2α+2.34(σ1−σ2)

2cos2α
]

• for a triangle-shape void

H =
1
2E

[
7.0(σ1 +σ2)

2+14.5(σ1−σ2)
2
]

(5.5)

The coordinates, where the characteristic function gets the minimum, po-
int out the centre of the newly generated void. As the characteristic functions
depend on stresses, it is easy to generate internal voids in the topology opti-
mization process using BEM. The coordinates which are the centre of a new
”bubble” inserted into the domain are generated on the basis of the objective
function. In the problem considered above the objective function is expressed
by the complementary energy which is a measure of the mean compliance of
the structure. The point where the complementary energy gets the minimum
is the point of the centre of the new ”bubble”.

6. Evolutionary optimization

6.1. Genetic algorithms

In general, the GAs simulate a natural evolutionary process. The GAs
are able to find the optimal solution satisfying the constraints without the
calculation of derivatives. Many papers or books contain a lot of benchmark
tests for optimization problems, including oneswhich are very difficult to solve
bymeansof the classicalmethods (Michalewicz, 1992;Michalewicz et al., 1994,
1996).
The GAs map an evolutionary process of nature over a span of time, in

order to adapt an individual to conditions of life as fit as possible. It is just
nothingmore than themain goal of optimization. Those algorithms are proce-
dures for searching the space of solutions. They take advantage of themecha-
nisms of natural selection and genetic inheritance, using the neo-Darwinian
principle of reproduction and the survival of the fittest.
The GAs start with a population of randomly generated candidates from

the feasible solution domain (Fig.3). These candidates, called chromosomes,



Evolutionary algorithms and boundary element method... 351

Fig. 3. Flow chart of a genetic algorithm

evolve towards better solutions by applying genetic operators such as amuta-
tion and crossover, simulated on the basis of heredity principles (genetic) and
a selection simulated on thebasis of thenatural selection (theory of evolution).
Ordinarily, after applying genetic operators the new population has a bet-

ter fitness. The population of individuals (chromosomes) undergoes the evolu-
tion.Theobjective functionwith imposedconstraints plays the roleof the envi-
ronment to distinguish between good and bad solutions (Michalewicz, 1992).

6.2. Chromosomal representation and genetic operators

The first step in a GA is to create chromosomes that describe possible
solutions. There are a few possibilities of chromosomal representation (Micha-
lewicz, 1992). The binary coding, theGray coding, the logarithmic coding and
the floating point coding are the best known ones. The selection of the kind
of coding depends on the optimization problem. The floating point coding is
the best representation of the problem described in this paper.

GAs with population of chromosomes coded by floating point representa-
tion use the modified operators (Michalewicz, 1992; Michalewicz et al., 1994,
1996) which are applied to get good convergence, ”jumping” far from a local
optimum, stability, good fitness, better local tuning.



352 T.Burczyński, G.Kokot

The chromosome is represented by the design vector c

c=(c1,c2, . . . ,ci, . . . ,cN) c
min
i ¬ ci ¬ c

max
i (6.1)

where ci, i=1, ...,N are genes.
The population consists of a randomly generated set of feasible chromo-

somes. The population evolves towards better individuals by applying genetic
operators to find a better fitness. The next ”evolved” population is created in
the following steps:

1. Selection

This operator allows the selection of parents from the population. The
rest of genetic operators produce offspring by acting on the parents. The
roulette wheel method (Michalewicz, 1992) is the best known selection.
In this method parents are selected proportionally to themagnitudes of
their fitness.

Having selected chromosomes from the initial population the genetic
operators ”work on them” in order to create a new population. The six
genetic operators for the floating point representation of chromosomes,
briefly shown below, have been completely described by Michalewicz
(1992), Michalewicz et al. (1994, 1996).

2. Mutations

2.1. Uniform mutation

This operator produces a single offspring c′ from a single parent c. One
gene taken from the parent is randomly selected and changed to a new
onewhosevalue is randomly selected fromthedesign space.For example,
when the gene ci is selected

c=(c1,c2, . . . ,ci, . . . ,cN) c
′ =(c1,c2, . . . ,c

′

i, . . . ,cN) (6.2)

This operator is very important in the early phases of the evolution
process, because children are allowed to move freely within the feasible
domain. It is helpful in the case when the local optimum exists, because
a chromosome can ”escape” from it.

2.2. Boundary mutation

This operator produces a single offspring c′ from a single parent c, too,
but the changed genes of a chromosome can take only boundary values
of the design space cmini or c

max
i . The boundary mutation works very



Evolutionary algorithms and boundary element method... 353

well when the solution lies either on or near the boundary of the feasible
search space.

2.3. Non-uniform mutation

This operator produces a single offspring c′ froma single parent c. If the
gene ci has been chosen, the new mutated gene will have the following
value

c′i =

{
ci+∆[t,right(ci)− ci] if a random digit is 0

ci−∆[t,ci− left(ci)] if a random digit is 1
(6.3)

where t is the generation number. The function ∆(t,y) returns a value
from the range [0,y]. The probability of ∆(t,y) increases when t in-
creases. This operator causes that initially the design space is searched
uniformly, but with the increase of t the space is searched very locally,
which gives fine tuning of the algorithm.

3. Crossovers

Thecrossover operation swaps some chromosomes of the selected parents
in order to create an offspring.

3.1. Simple crossover

This operator needs two parents and produces two children. For a ran-
domlygenerated crossingparameter k (k=1, ..., numberof parameters)
it works as follows (e.g. k=2)

c1 =(c1,c2, |c3) c2 =(s1,s2, |s3)

c′1 =(c1,c2, |s3) c
′

2 =(s1,s2, |c3)
(6.4)

A simple crossover may produce a child outside the design space. To
avoid this, a parameter α∈ [0,1] is applied. New chromosomes have the
following forms (c1, c2 are parents)

c1 =(c1, ...,cq) c2 =(s1, ...,sq)

c
′

1 = [c1, ...,ck, ...,sk+1α+ ck+1(1−α), ...,sqα+ cq(1−α)] (6.5)

c
′

2 = [s1, ...,sk, ...,ck+1α+sk+1(1−α), ...,cqα+sq(1−α)]

and they always lie within the feasible design space.



354 T.Burczyński, G.Kokot

3.2. Arithmetical crossover

This operator produces two children that are a linear combination of two
parents

c
′

1 =αc1+(1−α)c2 c
′

2 =αc2+(1−α)c1 (6.6)

where α is a random parameter from the range [0,1]. Applying this
parameter guarantees that the children are in the feasible domain.

3.3. Heuristic crossover

This operator produces a single offspring from two parents

c
′

3 = r(c2−c1)+c2 (6.7)

where r is a random value from the range [0,1], and the parent c2 is
not worse than c1 (i.e. the fitness for c2 is better than for c1 for a
maximization problem). This operator has a few features: it uses values
of the objective function to determine the direction of the search, secures
fine local tuning and the searching in the promising direction.

Whenanewpopulation is created, thefitness is calculated and thewhole
evolutionary process is repeated to producenext populationswith better
fitness. The process is stopped after n cycles or when the solution can
not attain better solution after k cycles.

6.3. Evolutionary algorithm

A flow chart of the proposed approach of the evolutionary optimization is
presented in Fig.4.
The first step in an optimization process is to find the optimal shape of

the external boundary. Itmeans that the typical shape optimization is carried
out. The genetic algorithm is applied as an optimization module. In order
to obtain the information about the fitness function for each individual in
the population, BEM is used. When the main goal of the optimization is to
find only the optimal shape without any changes inside the domain then the
optimization process is finished and the obtained solution is the optimal one.
When it is possible to make any change inside the domain, the second

step of the optimization – the topology optimization – is carried out. A new
void, which changes the topology class, is inserted into the domain. In order
to generate a new void inside the domain the ”bubble method” is used. It
secures the optimal position for the inserted void. The coordinates, where the
characteristic functions get theminimum, are the co-ordinates of the centre of



Evolutionary algorithms and boundary element method... 355

Fig. 4. Evolutionary optimization

the newvoid.The characteristic functions dependon stresses, so theminimum
values are calculated usingBEMandGAs.After inserting the void, during the
optimization process the best place for the inserted void, the optimal shape of
the external boundary and the internal boundary are searched for. It means
that the shape optimization and the topology optimization are being carried
out simultaneously.

7. Numerical examples

7.1. Example 1

A cantilever beam subjected to a point force is considered (Fig.5a). The
objective function is tominimize the complementary energy with the constra-
int imposed on the volume of the structure (Vend =0.5Vstart). All boundaries
of the structure aremodelledbyNURBS.Thefinal shapeof the external boun-
dary after the shape optimization is presented in Fig.5b. The final formof the



356 T.Burczyński, G.Kokot

structure after the simultaneous shape and topology optimization is presented
in Fig.5c and Fig.5d. Figure 5e shows fitness function values versus the num-
ber of iterations. Curve 1 presents the evolution process for the structure in
Fig.5b. The final solution is found very fast in the early stage of the evolution
process. After inserting the first void in the initial iterations the optimization
process converge nearly to the final solution (curve 2). The next iterations are
connectedwith good tuning to the final solution and possible checking if there
is no other global optimum. Curve 3 represents the optimization process for
the structure with two voids. It can be seen that finding the final solution for
this structure requires muchmore iterations than for the initial structure.

Fig. 5. Evolutionary optimization of a cantilever beam

The problemwas solved on the 120MHz PCPentium, and the total time
of the calculations was 75min.



Evolutionary algorithms and boundary element method... 357

In all presented examples the following genetic algorithm parameters have
been used:

GA parameters Value

population size 70
number of generations 300
selective pressure 0.057∗

nonuniformmutation 0.028∗

crossover 0.143∗

∗ – number of chromosomes undergo the operators to the population size.

7.2. Example 2

The example shows the results of the optimization process of the support
presented in Fig.6a. The optimization goal is to find the optimal shape and
topology of the structure for the minimum area of the structure with a con-
straint for displacements of loaded nodes (u¬ 1.3ustart).All boundaries of the
structure are modelled by NURBS. The final shape of the external boundary
is presented in Fig.6b. The form of the structure after the simultaneous shape
and topology optimization is shown in Fig.6c.
Figure 6d shows the objective function values versus the number of itera-

tions. Curve 1presents the evolution process for the initial structure.Thefinal
solution is found very fast in the early stage of the evolution process. After
inserting a void in the initial iterations the optimization process converge very
fast to the final solution (curve 2). The next iterations are connected with
good tuning to the final solution and possible checking if there is no other
global optimum. The problem was solved on the 120MHz PC Pentium and
the total time of the calculations was 118min.

7.3. Example 3

The example shows the results of the optimization process of the rectan-
gular plate presented in Fig.7a. The objective function is to minimize the
complementary energy with a constraint condition in the form of a volume
constraint (Vend = Vstart). It is assumed that the inserted voids changing
the structure topology are circular and the outer boundary is modelled by
NURBS.The final shape of the external boundary is presented inFig.7b. The
form of the structure after the simultaneous shape and topology optimization
is shown in Fig.7c and Fig.7d. Figure 7e shows the objective function values
versus the number of iterations. Curve 1 shows the optimization process for



358 T.Burczyński, G.Kokot

Fig. 6. Evolutionary optimization of a support

the structurewithout voids. Thefinal solution is found in the first steps of opi-
mization process. Curve 2 presents the optimization process for the structure
with two voids. In this case the evolution goes slower. Finally, curve 3 presents
the evolution process for the structurewith four voids. In the initial iterations
the final solution is found and the next iterations give no improvement. Those
iterations are connected only with good tuning to the final solution and possi-
ble checking if there is no other global optimum. The problem was solved on
the 120MHz PCPentium and the total time of the calculations was 160min.

8. Conclusions

Numerical tests prove that the proposed approach of the evolutionary opit-
mization can be applied in order to solve a large class of generalized shape



Evolutionary algorithms and boundary element method... 359

Fig. 7. Evolutionary optimization of a rectangular plate



360 T.Burczyński, G.Kokot

optimization problems of mechanical structures (the simultaneous shape and
topology optimization). The application of GAs as the optimization module
makes this method free from limitations typical for the classical optimiza-
tionmethods. The coupling of the boundary elementmethodwith the genetic
algorithm gives an effective and efficient alternative optimization tool.
TheapplicationofBEMintheoptimization showsthegreat advantageover

other computational methods used in the analysis of mechanical structures,
because only the boundary of the structure is discretized. Also the simplici-
ty of data preparation and the small number of data should be taken into
consideration.
The proposed approach of the evolutionary optimization allows the perfor-

mance of either a complex optimization process in the form of the generalized
shape optimization or a partial process in the form of the shape optimization
without the change in the topology class, or the topology optimization not
changing the shape of the external boundary of the structure.
Now, themaindirection of the research is todevelop thismethodandapply

it to different fields of mechanics such as identification, dynamic problems,
thermo-elastic problems, etc.

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Algorytmy ewolucyjne i metoda elementów brzegowych w uogólnionej

optymalizacji kształtu

Streszczenie

Połączenie nowoczesnych algorytmów optymalizacji, jakimi są algorytmy ewo-
lucyjne, z metodą elementów brzegowych pozwala opracować alternatywną metodę
optymalizacji sprężystych układów mechanicznych w zakresie uogólnionej optymali-
zacji kształtu (połączenie optymalizacji kształtu z optymalizacją topologiczną).Meto-
da ta jest pozbawionawad związanych z typowymi klasycznymimetodami optymali-
zacji (ciągłość funkcji celu, wyznaczanie gradientu funkcji itp.), co znacznie rozszerza
możliwości jej zastosowań.W artykule przedstawiono proponowanąmetodę optyma-
lizacji wraz z przykładami optymalizacji wybranych układówmechanicznych.

Manuscript received October 25, 2002; accepted for print February 4, 2003