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On the identification of linear elastic mechanical behaviour of
orthotropic materials using Evolutionary Algorithms
N. MAGALHÃES DOURADO
a
, J. CARDOSO XAVIER
b
, J. LOPES MORAIS
c
CETAV/UTAD, Departamento de Engenharias, Quinta de Prados, 5000-911 Vila Real, Portugal
a
ndourado@utad.pt;
b
jmcx@utad.pt;
c
jmorais@utad.pt
Abstract: A numerical tool is presented to identify the linear elastic properties of orthotropic and
homogeneous materials, coupled with the off-axis tensile test, as a part of a hybrid numerical-
experimental model. The method combines whole-field displacement measurement techniques and an
optimisation procedure based on the Finite Element Method and a developed Genetic Algorithm. A
Finite Element analysis of the off-axis tensile test was performed using known in-plane linear elastic
properties of wood pine lodgepole to generate a reference nodal displacement field to calibrate the
numerical method. An objective function was chosen to minimise the mean quadratic difference
between the reference displacement field and the displacement field that is calculated for each
potential solution. A Sigma Truncation Scaling mechanism was chosen and a Genetic Algorithm with
Varying Population Size (GAVaPS) was developed, based on an elitist strategy. Good approximation
was acquired for the in-plane elastic properties.
Key-Words: Genetic Algorithm with Varying Population Size; Heterogeneous fields; Inverse method
1 Introduction
The in-plane elastic mechanical behaviour of
orthotropic and homogeneous materials is
characterized through four independent
engineering properties [1, 2]:
1
E
,
2
E
,
12
ν
and
12
G
, where the subscripts represent the material
axes. Usually, two tensile tests, through
directions 1 and 2, are performed to identify the
Young moduli (
1
E
and
2
E
) and the major
Poisson´s ratio (
12
ν
). Additionally, an in-plane
shear test is carried out to obtain the shear
modulus (
12
G
). Different shear test methods
have been proposed to identify the shear moduli
of an orthotropic material. Among them are the
off-axis tensile test [3, 4], the Arcan test [5] and
the Iosipescu test [6, 7].
The use of whole-field displacement
measurement techniques in combination with a
suitable analytical or numerical tool, have been
brought a new approach for the identification of
the mechanical behaviour of orthotropic
materials [8, 9, 10]. The aim of this approach is
to reach a heterogeneous stress and strain fields
through the specimen in a way that all elastic
properties, that should play a balanced role in
the response of the specimen, can be determined
in a single test.
In this work the off-axis tensile test was chosen
to generate the heterogeneous stress and strain
fields. A numerical identification procedure was
developed to determine the in-plane elastic
properties based on an Evolutionary Algorithm
coupled with the Finite Element Method.
2 Inverse methods
Figure 1 illustrates the general inverse approach
of identification of the linear-elastic mechanical
behaviour of an orthotropic and homogeneous
material, based on Evolutionary Search
techniques and on the Finite Element Method.
This approach is based on the experimental
measurement of the specimen response and the
application of a hybrid numeric-experimental
method. Conceptually, this hybrid method
consists in: a numerical model of the mechanical
test, constructed using the Finite Element
Method, and an iterative procedure used to
search the elastic properties compatible with the
measured displacement field.
Evolutionary Algorithms are nowadays an
optimisation method widely used in inverse
problems employing the above solution
methodology [11, 6]. The inverse problem
resolution enables to find out the set of project
variables (i.e., the material in-plane elastic
properties) of the physical problem (i.e., the off-
axis mechanical test), which corresponds to the
set of state known variables (i.e., heterogeneous
displacement field).
Elastic properties:
E
1
, E
2
, n
12
, G
12
Displacement
field measurement
Nodal
displacement
Mechanical
test
F. E. M.
Evolutionary
Algorithm
Solutions
fitness
evaluation
Population in
generation t
Convergence
Fig. 1 Identification of linear elastic behaviour of an
orthotropic material using an inverse methodology.
3 Finite element method analysis
The off-axis tensile test simulation by Finite
Element Method was carried out for the test
specimen geometry shown in Fig. 2, with
mmL 120= , mmw 30= , mmt 10= and °
=
20
θ
,
using the commercial code ANSYS 6.0
. A
reference displacement field was firstly
generated using the in-plane elastic properties
of wood pine lodgepole [12]:
,1210
1
GPa,E = ,0321
2
GPa,E
=
GPa,G 4960
12
=
and
3160
12
,=
ν
. This displacement field was
used to calibrate the numerical method, playing
the role of the experimental data input, for the
identification algorithm.
O
x
y
w
θ
1
2
L
t
Fig. 2 Off-axis test specimen geometric parameters.
The boundary conditions applied to the Finite
Element Model (see Fig.3) are in agreement
with the rigid and non-rotating testing machine
grips. Left-end nodes were fixed and a nodal
displacement prescription
mm,u
x
50
=
was
applied to right-end nodes.
Fig. 3 Finite element model and boundary conditions
of the off-axis tensile test specimen.
A typical “S” deformed shape was observed for
the off-axis uniaxial tensile test specimen after
FEM analysis (Fig. 4a). The obtained reference
displacement field,
x
u and
y
u
(Fig. 4b), exhibits
a clear heterogeneity.
Fig. 4 Reference displacement field:
(a)
x
u and (b)
y
u
.
4 Genetic Algorithm
4.1 Introduction
Among all evolution-based search algorithms,
the Genetic Algorithms (GAs) are perhaps the
most well known. GAs were developed by John
Holland [13,14] in an attempt to explain the
adaptive processes of natural systems and to
design artificial systems based upon these
natural systems. Although not being the first
algorithms to use principles of natural selection
and genetics within the search process, GAs are
today the most widely used [15]. More
experimental and theoretical studies have been
made on the field of GAs than on any other
Evolutionary Algorithms (EAs).
(
a
)
(
b
)
During the last thirty years there has been a
growing interest in problem resolution strategies
of different kinds of problems based upon the
principles of natural evolution and hereditary
laws. These problem solving strategies favour a
population of potential solutions,
The aim of Genetic Algorithms is to identify the
individual (the set of elastic properties) best
adapted to the environment (according to a
fitness function), among the populations (set of
potential solutions) found in successive
generations (iterations).
One variety of these systems is a class of
Evolution Strategies – algorithms that mimic the
main principles of natural evolution for
parameter optimisation problems [16]. Fogel´s
Evolutionary Programming [17] is a technique
for searching through a space of small finite-
state machines. Glover´s scatter search
techniques [18] hold a population of reference
points and breed offspring based on weighted
linear combinations. Holland´s GAs set up
another type of evolution-based systems.
An EA is a probabilistic algorithm that sustains
a population of solutions
{
}
,
1
t
n
t
x,...,x)t(P = in
each generation t (see Fig. 5). Each individual
t
i
x represents a potential solution to the problem
and is encoded according to a predefined data
structure S. A fitness value is determined
according to how well each solution
t
i
x
fulfils
objective function, y, of the problem. Then, a
new generation is formed
()
1
+
= tt selecting
the set of more fit solutions (selection operator).
Members of this emerging generation
experienced transformations produced by
genetic operators. These transformations may be
arranged into two types: unary transformations
i
m
(mutation type) and higher order
transformations
j
c
(crossover type) [19]. The
former is characterized by a small change in
single individual, introducing some extra
variability into the population, and the later by
new individuals generated combining parts from
several (two or more) individuals
()
SS.....S:c
j
→×× . Population size may
remain constant throughout the algorithm or
vary according to any birth-rate control strategy.
After some number of generations the program
converges. The best solution found till then is
hoped to represent a near-optimum (reasonable)
solution.
1011110
1011110
01001010
01001010
0010100
0010100
11110011
11110011
01010101
01010101
01010101
01010101
10001010
10001010
S e l e c t i o n
C r o s s o v e r
M u t a t i o n
10010100 11101011 10101010
11111011 11011101 11100010
Population
G
eneration
t
Generation
t = t + 1
Population
t
i
x
S
F i t n e s s f
Fig. 5 Illustration of a Genetic Algorithm scheme.
4.2 Developed Genetic Algorithm
As mentioned in section 3 a reference
displacement field of off-axis tensile specimen
(Reference solution in Fig. 6), u
i
, was firstly
generated, using the in-plane elastic properties
of wood pine lodgepole. Compliance matrix
[
]
S was previously determined, using the set of
mentioned reference in-plane elastic properties,
with
°
=
20
θ
. Solutions of the GA were
represented by four-dimensional binary design
variables (
1
E
,
2
E
,
21
G
,
21
ν
). Each variable x
i
(
)
4,...,1=i could take values from a predefined
domain
[
]
Rb,aD
iii
⊆=
. The length ascribed
to each variable m
i
, based upon the precision p
required to determine the objective function, y,
was calculated (Bits number evaluation in Fig.
6) considering the smallest integer such that
[19]:
()
1210 −≤⋅−
i
m
p
ii
ab
(1)
An initial population with a number
)t(PopSize ,
(
)
1
=
t , of potential solutions
{
}
,
11
1
1
k
x,...,x)(P = was randomly generated.
Each variable value (Decoding in Fig. 6) is
given by the Equation
()
12
2
−
−
×+=
i
m
ii
ii
ab
stringdecimalax
(2)
where
()
2
stringdecimal represents the hexa-
decimal value of
()
2
string . The objective
function evaluation
[]
)(
i
t
evaliy v=
, of each
chromosome
()
)t(PopSize,...,i
i
1
=
v , was carried
out using
[]
∑
−=
=
N
i
t
ii
N
iy
1
2
1
uu
(3)
where N represents the total number of nodes in
the front plane of the off-axis test model (Fig.
3), u
i
the reference displacement field and
t
i
u
the displacement field obtained by ANSYS for
solution
i
v , in generation t. Compliance matrix
was up-dated for each
)t(PopSize solution.
4.2.1 Selection operator
Population ranking was then performed
according to the values of the objective function,
y, calculated using Eq. (3), and the PopSize(t)
solutions were structured in two subsets:
T
S and
R
S (Fig. 6). The quantities
()
tn
T
and
(
)
tn
R
represent the cardinal of subsets
T
S and
R
S ,
respectively, for generation t.
. . .
1
t,PopSize( t )
.
1
. . . . . .
. . . . . . . . .
t , n+1
1
t , n
t ,1
. . . . . .
. . . .
1
k
t,PopSize( t )
k
t, n+1
t ,n
k
t ,1
k
( x . . . . . . x )
( x . . . . . . x )
( x . . . x )
( x . . x )
Fig.6 Population after Ranking.
4.2.2 Scaling operator
A scaling mechanism (Scaling in Fig. 7) was
used to improve the sensitivity of the algorithm
to find solutions which exhibit objective
function values very close to each other. Thus, a
previous evaluation of the maturity state of
convergence process was performed, and a
Sigma Truncation scaling mechanism was
chosen [19], according to
[
]
[
]
()
σ
×−+= cAvgObjiyiy
'
(4)
where c is chosen as a small integer,
σ
is the
population’s standard deviation and AvgObj
represent average objective function values, in
the current population. Possible negative
evaluations
[
]
iy
'
are set to zero.
4.2.3 Crossover operator
A probability of crossover p
c
was previously
assumed and an expected number
(
)
tPopSizep
c
×
of chromosomes to undergo the crossover
operation was then determined. An even number
of chromosomes was assured by adding or
subtracting one solution from this mapping pool
– subset of
()
tPopSizep
c
×
solutions from the
current population. The decision to add or
subtract a solution from this mapping pool was
done randomly. Thus, once taken the decision to
add a previously rejected solution from the
population, the chromosome selected to figure in
the mapping pool was also done randomly.
Pairs of solutions were randomly settled from
this group of chromosomes and crossing points
were randomly assorted for each pair, from the
range [1..m-1], with
∑
=
=
k
i
i
mm
1
.
Features’ combination of different chromosomes
to form two similar offspring solutions, each
other, was performed swamping corresponding
segments of the parents, i.e. , for a pair of m-
dimensional vectors
(
)
at
m
atat
xxx
,,
2
,
1
... and
(
)
bt
m
btbt
xxx
,,
2
,
1
... , crossing the
chromosomes after the first gene produce the
offspring
(
)
bt
m
btat
xxx
,,
2
,
1
... and
(
)
at
m
atbt
xxx
,,
2
,
1
... .
A lifetime parameter for an i-th offspring
solution,
[
]
(
)
ilifetime , was then determined by
proportional allocation [19]:
[]
+ MaxLT
iy
AvgObj
MinLTminimum ,
η
(5)
(
)
TT
Stnn →
=
(
)
RR
Stn →
MinLT and MaxLT stand for maximal and
minimal allowable lifetime values, respectively.
AvgObj represent average objective function
values, in the current population (Statistical
parameters in Fig. 7), and
()
MinLTMaxLT −=
2
1
η
(6)
4.2.4 Mutation operator
Mutation operator was performed on a bit-by-bit
basis and a probability of mutation p
m
was
assumed. There were a total number of
()
mtPopSize × bits in the whole population and
an expected (on average) number
()
m
pmtPopSize ××
of mutations per generation.
Every bit had an equal chance to be mutated.
Thus, for every bit in the population
)t(P , a
random (float) number r was generated from the
range
[]
1,...,0
. Then, a bit mutation took place
if r < p
m
.
Objective function
[]
iy and lifetime parameter
[]
()
ilifetime values were also recalculated (Eqs. 4
and 5) for recently born (mutated) solutions. A
similar criterion was used to up-date the
objective function values
[]
iy of the current
population as in 4.2.2, according to the
algorithm maturity (Fig. 7).
4.2.5 Aging and elimination operator
Lifetime parameter
[]
()
ilifetime was up-dated
according to
[] []
1
1
−
−
=
t
ilifetime
t
ilifetime
(7)
Solutions belonging to subset S
R
(Fig. 6) are not
up-dated, in terms of the lifetime parameter -
elitist strategy.
Population size in generation
()
1+t , is also up-
dated, eliminating solutions
(
)
tE which
reached null lifetime parameter values, and
adding the offspring solutions generated in
Crossover (4.2.3),
()
tOffspring , according to
( ) () () ()
tOffspringtEtPopSizetPopSize +−=+1
(8)
A full search is done, eliminating clone
solutions among the current population.
Figure 7 resumes all the steps of the developed
Evolutionary Algorithm, described above.
G.A.
G.A.
Input data
Input data
Reference solution
[S]
[S]
FEM
FEM
Bits number evaluation
Initial
Initial
population
population
Decoding
Fitness evaluation
Statistical
parameters
Maturity
Scaling
y
Crossover
Lifetime
Mutation
n
Fitness
Fitness
t = t + 1
t = 1
Aging and/or elimination
Data
output
y
Selection
n
[S]
[S]
t
t
FEM
FEM
[S]
[S]
t
t
FEM
FEM
Maturity
Scaling
y
Lifetime
n
Statistical
parameters
Convergence
Fig.7 Developed Genetic Algorithm.
5. Results and discussion
In the present work an initial population of 10
individuals was used. The domain considered
for each project variable was:
[
]
()
REPaE ∈×∈
1
,
6
1015070,9020
1
[
]
()
REPaE ∈×∈
2
,
6
101537,867
2
[
]
R∈
−
×∈
12
,
3
10392,280
12
νν
[
]
()
RGPaG ∈×∈
12
,
6
101351,400
12
.
which corresponds to the characteristic range of
elastic properties of wood pine lodgepole [12].
The scaling operator was activated after 20
generations, for
3
=
c in Eq. 4, and a total of 40
generations was achieved.
Lifetime parameter,
[]
ilifetime , (Eq. 5) was
determined considering
1=MinLT
and
7=MaxLT
.
A probability of crossover, 25.0=
c
p , and a
probability of mutation
001.0=
m
p , were chosen.
Table 1 shows the set of values of the in-plane
elastic property found for the best solution after
40 generations. The relative errors were
determined with respect to the reference values
presented in section 3. Three of them are inferior
to the coefficient of variation associated to each
of them, which is less than 22% [12].
Design
variables
Elastic
properties:
(pine
lodgepole)
Elastic
properties
(numerical
method)
Relative
error [%]
[]
GPaE
1
10.120 10.335 2.13
[]
GPaE
2
1.032 1.171 13.50
12
ν
0.316 0.320 1.36
[]
GPaG
12
0.496 0.299 39.72
Table.1 Elastic properties obtained according to the
numerical method.
Fitness function, f, was defined as [20]:
ykf
−
=
(9)
k being an arbitrarily large positive value that
ensures that fitness, f, never becomes negative.
The value of the relative error found for the
shear modulus,
12
G
, is greater than the typical
scatter of experimental values of the elastic
properties of wood. This large error can be
attributed to the fact that the number of
generations was not enough to ensure the
stabilisation of the fitness value of the best
solution (Fig. 8).
Thus, the design variables shown in table 1 are
not a suitable solution of the optimisation
problem, from the point of view of the
experimental scatter of mechanical properties of
wood.
6. Conclusions
The numerical method developed to identify the
elastic properties of orthotropic materials led to
good approximation for the Young moduli:
1
E
and
2
E
, and the Poisson’s ratio,
12
ν
,
determination. A great relative error was found
for the shear modulus,
12
G .
7. Acknowledgements
We would like to thank the Portuguese
Foundation for Science and Technology for the
financial support necessary to the execution of
this work in the ambit of the project
POCTI/1999/EME/36270.
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