Determining the Material Parameters of a Polyurethane Foam Using Numerical Optimisation Algorithms

Determining the material parameters of a polyure‐
thane foam using numerical optimisation algorithms
Jernej Klemenc, Andrej Skrlec, Matija Fajdiga
Ansys Conference & 6. Cadfem Austria Users’ Meeting

Table of contents
• Introduction
• Theoretical background
• Practical application
• Results and discussion
• Conclusion

Introduction
• Complex numerical simulations are daily included into
R&D processes.
• In principle material characteristics can be measured,
but what values should be applied for the parameters
of material models in numerical simulations?
• How to estimate the parameters of material models if
they cannot be directly calculated from experimental
data?
Introduction
Theoretical
background
Practical
application
Results and
discussion
Conclusion

Problem definition
• A certain structural behaviour is described with an
ensemble of the experimental data.
• A finite element model for a numerical evaluation of
the phenomenon under consideration is built.
• For this finite element model the parameters of the
applied material models need to be defined.
• Some parameters of the material models (e.g. elastic
modulus, Poisson’s number etc.) can be directly
calculated from experimental data.
• If the parameters of the material models cannot be
directly calculated from the experimental data, a robust
procedure for their estimation should be set‐up.
Introduction
Theoretical
background
Practical
application
Results and
discussion
Conclusion

Applied approach for problem solving
• Experimental data about the structural behaviour is
obtained by testing laboratory samples or prototypes.
• A finite element model is built that represents the
phenomenon under consideration as well as possible.
• A cost function (CF) is defined that measures a
discrepancy between the experimental data and the
results of the numerical simulations.
• The parameters of the material models, which cannot
be directly calculated from the experimental data, are
obtained by a recursive numerical optimisation
procedure that minimises the CF.
• Numerical optimisation procedures: gradient methods,
genetic algorithms, ant colonisation algorithms.
Introduction
Theoretical
background
Practical
application
Results and
discussion
Conclusion

1x
2x
1 2( , )f x x
0P
v
1P
v
2P
v
( )0f P
v
( )1f P
v
( )1f P
v
f(Pt+1)
f(P)
f(Pt)
f(Pt+2)
Pt+2
Pt+1
Initial
searching
directions
Steepest gradient in
the second iteration
the first iteration
(Search continues
in this direction)
• Find a direction of the
steepest descent of the CF
from the current point.
• Move along this direction
to the next point according
to the current step size.
• Calculate the new CF value
and recalculate the step
size.
• Stop the iteration process
if the CF minimum is
found.
Gradient descent optimisation method
Introduction
Theoretical
background
Practical
application
Results and
discussion
Conclusion

Genetic algorithm (GA)
Introduction
Theoretical
background
Practical
application
Results and
discussion
Conclusion
Parent chromosomes
Child chromosomes
{{{ {{{
}}} }}}
1010 001111 0011010 010
1010 010111 0011001010
Crossover points Crossover points
Child chromosome
Child chromosome
after mutation
10 0010010
10 001
1
0 0010
Mutation point
• Find the most successful chromosomes in a population.
• Select candidates for breeding.
• Form children from parents with a crossover.
• Perform mutations in a child population.
• Calculate the new CF values for the child population.
• Stop the iteration process if the CF minimum is found.

Differential ant‐stigmergy algorithm (DASA)
• For each ant find its path through the differential graph
by considering the current deposition of pheromones.
• Improve the past‐best solution according to the newly
selected paths in the differential graph.
• Calculate the new CF values for the potentially
improved solutions and find the new‐best path.
• Deposit pheromones on the new‐best path if the CF
value is reduced. Otherwise evaporate pheromones.
• Stop the iteration process if the CF minimum is found.
Introduction
Theoretical
background
Practical
application
Results and
discussion
Conclusion

Experimentally determined compressive
characteristics of a polyurethane foam
Introduction
Theoretical
background
Practical
application
Results and
discussion
Conclusion
Specimens: 80x80x40 mm Testing machine
Standard testing procedure
according to DIN 3386‐1:2000

Applied finite element model
Introduction
Theoretical
background
Practical
application
Results and
discussion
Conclusion
[ ]∫ ⋅−=
max
0
2
exp )()(
T
sim dttFtFCF
Cost function:LS‐DYNA Material model:
• MAT_LOW_DENSITY_FOAM
Parameters to be estimated:
• hysteretic unloading factor – HU
• shape factor– SHAPE
• decay constant – BETA

Optimisation procedure summary
Introduction
Theoretical
background
Practical
application
Results and
discussion
Conclusion
Gradient method GA DASA
Ranges of the independent
variables
HU=[0.1 ; 0.9999]
SHAPE=[1.0 ; 20.0]
BETA=[500 ; 1000]
HU=[0.1 ; 0.9999]
SHAPE=[1.0 ; 20.0]
BETA=[500 ; 1000]
HU=[0.1 ; 0.9999]
SHAPE=[1.0 ; 20.0]
BETA=[500 ; 1000]
Parameters of the applied
optimisation procedures
n =8
h0=0.0001
=κ 0.0001
=β 0.99
n=10
mi=10
ncrossover_pts=3
pc=0.2
pm=0.1
n=20
b=10
=iε 0.0001
=ρ 0.2
s+=0.02
s-=0.01
Parameter HU 0.157 0.204 0.336
Parameter SHAPE 4.828 5.440 6.837
Parameter BETA 880.44 890.44 880.19
Cost function value 1.2452·106
1.1319·106
1.1317·106
No. of iterations for opt. sol. 40 60 60
Processing time 1h 1h 2h, 30min

Introduction
Theoretical
background
Practical
application
Results and
discussion
Conclusion
Comparison of simulated and experimental
results for optimal finite element models
experimentally determined F(t) curve
simulated F(t) curve with optimal parameters determined by the gradient‐
descent method
simulated F(t) curve with optimal parameters determined by the GA
simulated F(t) curve with optimal parameters determined by the DASA

Concluding remarks
• Three numerical optimisation methods were tested for
the case of the compressive cyclic loading‐unloading
characteristics of a polyurethane foam.
• The optimisation methods were used to estimate the
parameters HU, SHAPE and BETA of the LS‐DYNA
MAT_LOW_DENSITY_FOAM material model.
• The two evolutionary methods (GA and DASA) have
outperformed the classical numerical optimisation
method, i.e., the gradient‐descent procedure.
• The numerical optimisation procedures were
structured in such a manner that the parallel execution
of the finite element simulations could significantly
reduce the processing time.
Introduction
Theoretical
background
Practical
application
Results and
discussion
Conclusion

Determining the material parameters of a polyurethane
foam using numerical optimisation algorithms
Jernej Klemenc, Andrej Skrlec, Matija Fajdiga
All: University of Ljubljana, Faculty of Mechanical Engineering
Abstract
In the event of a crash involving a motor vehicle, a car seat with its backrest and head support
can increase the level of passenger safety. However, the filling material used in the seat and
the head support should absorb a large proportion of the kinetic energy associated with the
moving passenger during the crash. This filling material is usually made of polyurethane foam.
To simulate the behaviour of the seat assembly during a crash the material characteristics of
the seat-filling foam should be appropriately modelled. For this purpose, a low-density-foam
material model was used in LS-DYNA crash-test simulations. This paper will show how
different numerical optimisation algorithms can be coupled with the LS-DYNA explicit
simulations to estimate the material parameters of the low-density-foam material model. The
performance of two evolutionary optimisation algorithms – a genetic algorithm and a
differential ant-stigmergy algorithm – will be compared to the classic gradient-descent
optimisation algorithm. The engineering applicability of the results will be discussed.
Keywords
Polyurethane foam, car seat, crash test, LS-DYNA, gradient-descent algorithm, genetic
algorithm, differential ant-stigmergy algorithm.
1 Introduction
To ensure that newly developed products are reliable, effective and safe for the user, complex
numerical evaluation procedures are applied during the early phases of the R&D process
before the first prototypes are built. The application of these numerical evaluation procedures
can result in reduced R&D costs, provided the geometry of the product is at least partially
optimised before the prototypes are built. However, in order to apply numerical procedures for
the structural evaluation (e.g., the finite-element method) the parameters of the material
models must be known. Some of these parameters can be determined relatively simply with
elementary material testing, e.g., the elastic modulus and the yield stress are determined from
the results of a destructive tensile test. However, some of the parameters that tend to appear in
more complex material models cannot be so easily measured.
A good example of a very complex structure that is made from metal and plastic parts is a car-
seat assembly. Furthermore, a car seat is one of the key elements that ensure a high level of
passenger safety during extreme operating conditions, like, for example, a vehicle crash. The
filling material of the car seat and the head support should absorb a large proportion of the
kinetic energy of the moving passenger during the crash. This filling material is usually made
of a polyurethane foam, and to simulate the behaviour of the seat assembly during the crash
the material characteristics of the seat-filling foam need to be appropriately modelled.
The polyurethane foam in a car seat is mainly loaded with compressive forces, which is why
its compressive material characteristics should be modelled as accurately as possible. Fig. 1
1

shows the typical behaviour of a polyurethane foam during a compressive loading-unloading
cycle.
Fig. 1: Behaviour of a polyurethane foam under a compressive loading-unloading cycle
The MAT_LOW_DENSITY_FOAM (MAT_57) material model is often used in LS-DYNA
crash-test simulations to describe the behaviour of the polyurethane foam during compressive
loading (Hallquist [1], LS-DYNA [2], Chang [3]). However, the following parameters must be
defined for the application of this material model in LS-DYNA:
• material density,
• tensile elastic modulus,
• Poisson’s number,
• )(εσ characteristic during compressive loading,
• hysteretic unloading factor – HU,
• shape factor for unloading – SHAPE,
• decay constant for modelling creep during unloading – BETA.
The factors HU and SHAPE are applied to model the behaviour of the polyurethane foam
during the compressive unloading. In this case the stresses in the finite element are determined
with the following equation:
)(
)__max(
__
)1()( εσεσ loading
SHAPE
unloading
densityenergystrain
densityenergystrain
HUHU ⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅−+= (1).
where )(εσloading is the tabulated characteristic during the compressive loading and
)(εσunloading is the characteristic of the polyurethane foam during the compressive unloading.
The above-described parameters of the MAT_LOW_DENSITY_FOAM material model are
determined on the basis of the standardised test. The material density, the tensile elastic
modulus, the Poisson’s number and the ( )σ ε curve for the compressive loading regime are
determined directly from the corresponding experimental data. However, the material’s
behaviour in the compressive unloading regime and any further reloading regime(s) is
described by the three parameters HU, SHAPE and BETA. These parameters should be
determined on the basis of the measured ( )σ ε curves from the sequential compressive
2

loading-unloading test. This is a reversed-engineering problem and optimisation methods must
be applied in order to solve it.
We applied three different numerical optimisation algorithms that were coupled with the LS-
DYNA explicit simulations to estimate the optimal values of the parameters HU, SHAPE and
BETA from the experimental data. The first method was a classic gradient-descent
optimisation algorithm. In addition, two evolutionary optimisation algorithms were also
applied: a genetic algorithm and a differential ant-stigmergy algorithm.
In the rest of the paper the theoretical backgrounds of the three optimisation algorithms are
first presented. This is followed by a description of the standard experiment for measuring the
compressive loading-unloading ( )σ ε curves of the polyurethane foam. Then the results of the
application of the three optimisation algorithms are presented for the case of a standard
sequential compressive loading-unloading test. Finally, the three optimisation algorithms are
compared and the engineering applicability of the results is discussed.
2 Theoretical background
2.1 Gradient-descent method
The gradient-descent method is one of the most widely used classical optimisation methods.
With this method the minimum of a cost function f(P) of the independent variable(s)
is searched for by following the function f(P) in the downward
direction with the steepest slope. Basically, there are two variations of the gradient-descent
method: 1) searching for the function minimum by calculating the function derivative, and 2)
searching for the function minimum without calculating the function derivative. We applied
the latter approach. In this case the optimum of the cost function is found iteratively using the
following procedure (see
),...,,...,,( 21 Dl xxxx=P
Fig. 2 and Snyman [4], Mathews and Fink [5], Burachlik et al. [6],
Sun and Yuan [7]):
1. Selecting the initial best solution of the independent variables Pt=0 and calculating
the cost-function value f(Pt=0).
2. Selecting the initial step size ht=0 and the initial searching directions rt=0,i ; i=1,…,n
around the initial point Pt=0.
3. Calculating the initial neighbouring points according to the selected initial searching
directions rt=0,i: ;,0,000,0 ititttit h ===== ⋅+= rrPP i=1,...,n.
4. Calculating the cost-function values for the initial neighbouring points f(Pt=0,i) ;
i=1,...,n.
5. Calculating the vectors Rt,i in (D+1)-dimensional space that connect the point
with its neighbours[ )(, tt f PP ] [ ])(, ,, itit f PP : [ ] [ ];)(,)(, ,,, ttititit ff PPPPR −=
i=1,...,n.
6. Calculating a cost-function gradient Ki(t) for all the neighbouring points Pt,i:
[ ] nifftK ititti ,...,1;)()()( ,, =−= RPP .
7. The new best solution Pt+1 is the neighbouring point Pt,i with the largest gradient
Ki(t): ,{ })(max
,...,1
1 tKi
ni
t
=
+ ⇒P 1);()( +→= ti itKtK P .
3

8. Calculating the new step size ht+1 for the next iteration. The new step size is
determined according to the history of the gradient changes (see also Haykin [8]):
11 ++ Δ+= ttt hhh (2),
0)1()(;
0)1()(;
≤−⋅
>−⋅
tKtK
tKtK
⎩
⎨
⎧
⋅−
=Δ +
t
t
h
h
β
κ
1 (3).
9. Selecting the new searching directions rt+1,i and calculating the new neighbouring
points ;,1,111,1 ititttit h +++++ ⋅+= rrPP i=1,...,n. The new searching directions are
located within a corridor originating from the best searching direction rt,i from the
previous iteration.
10. Calculating the cost-function value for the neighbouring points f(Pt+1,i) ; i=1,...,n.
11. If either the cost-function optimum or the limiting number of iterations is reached
the procedure is terminated. Otherwise, the iteration number is updated 1+← tt
and the procedure is returned to step 5.
1x
2x
1 2( , )f x x
0P
v
1P
v
2P
v
( )0f P
v
( )1f P
v
( )1f P
v
f(Pt+1)
f(P)
f(Pt)
f(Pt+2)
Pt+2
Pt+1
Initial
searching
directions
the second iteration
the first iteration
(Search continues
in this direction)
Fig. 2: Principle of an optimum search with the gradient-descent method
2.2 Genetic algorithm (GA)
Genetic algorithms, which are a form of evolutionary algorithm, are often used to solve
complex optimisation problems. The search for the cost-function optimum with a GA is based
on the principles of natural selection. Many different versions of GAs exist. We applied a GA
with binary encoding in our research. When following this approach the optimum of the cost
4

function f(P) of the independent variable(s) ),...,,...,,( 21 Dl xxxx=P is found iteratively with
the following procedure (see also Michalewicz [9], Schmitt [10], Forrest [11], Holland [12]):
1. Defining a suitable binary encoding for the independent variable(s)
. If the independent variables xl l=1,...,D should have nprec
significant digits, the domain of each independent variable should be divided into
equidistant intervals. uppl and lowl are an upper and a lower limit
of the l-t variable’s range. In addition to this, the following condition must be fulfilled:
),...,,...,,( 21 Dl xxxx=P
precn
ll lowupp 10)( ⋅−
1210)( −<⋅− lprec mn
ll lowupp (4).
where ml is the smallest integer number for which the condition in equation (4) is
fulfilled. In this manner each independent variable xi is encoded with a binary set of
length mi: Dlbbbb li mljllll ,...,1),;...;;...;;( ,,2,1, ==s ; are scalar binary variables. The
decimal value of the binary encoded variable xi is calculated as follows:
ljlb ,
12
)(
−
−
⋅+= lm
ll
lll
lowupp
decimallowx s (5).
2. Defining a chromosome template for the problem under consideration. The
chromosome is combined of the biray sets sl that represent the independent variables xl:
== ),...,...,,( 21 Dl ssssC
);...;;...;;;...;...;;...;;;...;;...;;...;;( ,,2,,,,2,1,,1,12,11,1 11 DDll mDjDDDmljlllmj bbbbbbbbbbbb= (6).
The individual components of the chromosome C are called genes. The value of
each gene is called alel.
ljlb ,
3. Selecting a population size n, which is the number of chromosomes in a population.
4. Selecting the chromosomes in an initial population: ;),...,...,,( ,021,0 itDlit == = ssssC
. This can be done by a systematic selection of the initial population or by a
random selection of the initial chromosome’s alels. We applied the random procedure.
Each chromosome Ct=0,i in the population then represents one possible solution of the
optimisation problem (see equations
ni ,...,1=
itit ,0,0 == → PC (5) and (6)).
5. Calculating the cost-function values for the chromosomes Ct=0,i in the initial
population: f(Pt=0,i) ; i=1,...,n.
6. Calculating the fitness p(Ct,i) of each individual chromosome Ct,i from the population:
ni
f
f
p n
i it
it
itit ,...,1;
)(
)(
)(
1 ,
,
,, ==
∑=
P
P
C (7).
7. Selecting the parent chromosomes for mating. The chromosomes with the better fitness
p(Ct,i) should have a higher probability of mating. For this reason, the cumulative
probabilities are first calculated for each chromosome Ct,i in the current generation:
nipq
i
j
ititit ,...,1;)(
1
,,, == ∑=
C (8).
Then n chromosomes are selected for mating from the current generation with a
roulette rule being applied n-times:
5

a. A random number r is selected from an interval [0,1].
b. If itit qrq ,1, ≤<− (qt,0=0) then the i-th chromosome Ct,i is selected for mating.
With this selection rule the chromosomes with the better fitness are selected more
times than the chromosomes with the poorer fitness. These are called the parent
chromosomes.
8. Mating of the selected chromosomes. The chromosomes are mated using a crossover
procedure. During the mating either one (single-point crossover) or more (multi-point
crossover) subsets of genes is exchanged between the parent chromosomes. The size of
the subsets of genes that are exchanged and the crossover points in the parent
chromosomes are chosen randomly (see Fig. 3). In this manner two child chromosomes
are obtained from two parent chromosomes.
Parent chromosomes
Child chromosomes
{{{ {{{
}}} }}}
1010 001111 0011010 010
1010 010111 0011001010
Crossover points Crossover points
Fig. 3: Multi-point crossover
Each chromosome has a probability of mating pc, which is a predetermined parameter
of the GA. This means that not all the parents are mated (if pc<1). Some of the parents
are transferred into the next generation unchanged. The mating procedure is repeated
until n child chromosomes are obtained.
9. Mutation of the child chromosomes. Mutation changes single genes in the child
chromosomes with some predetermined probability of mutation pm. In this manner the
chromosomes of the next generation are obtained: Ct+1,i. Each of them represents
another possible solution of the optimisation problem .itit ,1,1 ++ → PC
Child chromosome
Child chromosome
after mutation
10 0010010
1 0 0 0 1
1
0 0010
Mutation point
Fig. 4: Mutation
10. Calculating the cost-function values for the next generation of chromosomes: f(Pt+1,i) ;
i=1,...,n.
11. If either the cost-function optimum or the limiting number of iterations is reached the
procedure is terminated. Otherwise the iteration number is updated and the
procedure is returned to step 6.
1+← tt
6

2.3 Differential ant-stigmergy algorithm (DASA)
A basic version of the ant-stigmergy optimisation method was first introduced by Dorigo and
his co-workers (Dorigo et al. [13]). The main idea of the ant-stigmergy optimisation method
was taken from the behaviour of ants when searching for food (Goss et al. [14]). When
searching for food the ants deposit pheromones on their way. In the beginning of the search
they search the space randomly. In this manner the ants find many different paths to the food.
If the shorter paths to the food are chosen the ants return to the nest more quickly. This means
that a larger amount of the pheromones is deposited on the shorter paths in the same time
frame when compared to the other paths. When walking around the ants are prone to following
the paths with the larger amount of deposited pheromones. This means that as time passes the
amount of pheromones on the shorter paths increases. On the other hand, the amount of
pheromones on the longer paths eventually decreases due to the process of evaporation,
because only a very small amount of pheromones is deposited on the longer paths.
For solving complex optimisation problems with many independent, continuous variables a
variation of the ant-stigmergy optimisation method was developed by Korosec and his co-
workers called the differential ant-stigmergy algorithm (DASA) (Korosec et al. [15]).
To apply the DASA procedure a differential graph (see Fig. 5) must be composed that
represents the space of the independent variables ),...,,...,,( 21 Dl xxxx=P and considers their
upper and lower limits [uppl , lowl];l=1,...,D.
Fig. 5: Differential graph
The differential graph is composed using the following procedure:
1. Determination of the number of levels for the differential graph. The number of levels
is equal to the number of independent variables D. Each level of the differential graph
represents one independent variable xl;l=1,...,D.
2. Definition of the nodes vl,j for each level of the differential graph. Each node represents
a fixed magnitude of a variation jl,δ of the corresponding independent variable xl. For
each independent variable xl possible variations jl,δ are defined:
) (9),0,( +−
= lll ΔΔΔ
(10)),...,,...,,( 1,,1,,
−−−
−
−−
= ljldldll ll
δδδδΔ
(11)),,...,,...,( ,1,,1,
++
−
+++
= ll dldljlll δδδδΔ
(12)l
Lj
jl djb l
,...,1;1
, =−= −+−
δ
7

(13).l
Lj
jl djb l
,...,1;1
, =+= −++
δ
For each independent variable xl possible variations jl,δ are in the range between
and . b is a logarithmic base,
lL
b
lU
b ℑ∈= llbl LL );(log ε , [ ] ℑ∈−= lllbl UlowuppU ;log ,
and1+−= lll LUd lε is a selected precision for the independent variable xl. The
variations jl,δ are assigned to the vertices vl,j in each layer l=1,...,D as follows:
←= +⋅++++ ),...,,...,,,,...,,...,( 12,1,2,1,,,1, lllll dljdldldldljlll vvvvvvvV
(14).ldljllljldll djll
,...,1);,...,,...,,0,,...,,...,( ,,1,1,,, ==← +++−−−
δδδδδδΔ
Each vertex from the layer l is connected to every vertex in the next layer l+1. A path
p in the differential graph connects the vertices in consecutive layers. Each path p
starts in one of the vertices in the first layer l=1 and finishes in one of the vertices in
the last layer l=D. Between these two layers the path p visits one vertex in each layer.
The path’s direction is always from the first layer to the last layer. The path p never
turns back to the preceding layers.
3. Initial deposition of pheromones to the vertices of the differential graph according to
the Cauchy probability density function:
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
+⋅⋅
= 2
,,
,
1
1
)(
s
vv
s
v
optljl
jl
π
τ (15).
For each layer l=1,...,D the initial factor of the scatter s is equal to s=sglobal=4 and the
initial optimum vertices vl,opt are set to the middle vertices in the layer that
represent a zero variation of the independent variable:
1,, += idloptl vv
01, =+ldlδ . Its corresponding
Cauchy variable has the value 0. The values of the Cauchy variable for the two most
outer vertices in the layer are the following: 4;4 12,1, =−= +⋅ ldll vv . The mid-side vertices
are evenly distributed between these two extremes according to their corresponding
Cauchy variables.
After the differential graph is composed an initial amount of pheromones is deposited to its
vertices and the search for the optimum of the cost function f(P) with the DASA procedure is
carried out as follows (Korosec et al. [15]):
1. Selecting the initial solution: 021 ),...,,...,,()0( =
== tDl xxxxtP .
2. Selecting n paths (ants) pi;i=1,...,n through the differential graph. The paths are
determined by a random choice of the vertices at the differential-graph layers. Each
vertex vl,j at the l-th level is selected with a probability:
∑
+⋅
=
= 12
1 ,
,
,
)(
)(
)( ld
k kl
jl
jl
v
v
vprob
τ
τ
(16).
Each path pi through the differential graph now represents a set of variations for the
independent variables xl;l=1,...,D: niiDiliii ,...,1);,...,,...,,( ,,,2,1 == δδδδp .
8

3. Calculating a new possible solution xl,i(t+1) of the optimisation problem for each path
pi;i=1,...,n:
niDlrtxtx iloptlil ,...,1;,...,1;)()1( ,,, ==⋅+=+ δ (17).
xl,opt(t) is the past best solution of the optimisation problem and r is a weight factor that
is randomly selected from the interval (1,b-1).
4. Calculating the cost-function values for the new possible solutions xl,i(t+1):
[ ] [ ] nixxxxftf tiDiliii ,...,1;),...,,...,,()1( 1,,,2,1 ==+ +
P .
5. Selecting the current best possible solution: [ ]{ })1(min)1(
,...,1
+⇒+
=
tft i
ni
best PP . The best
solution in the current iteration step should have the minimum value of the cost
function, because we were searching for the function minimum in our case.
6. If [ ] [ ])()1( tftf optbest PP <+ the current best solution Pbest(t+1) becomes the new
possible optimum solution Popt(t+1) and the pheromones are redistributed over the
vertices of the differential according to the new best solution. The pheromone
redistribution is done using the Cauchy probability density function from equation (15)
with the adapted factor of scatter
globallocalglobalgloballocalglobal ssssssss ⋅=⋅+=−= + 5.0,)1(; and changed optimum
vertices vl,opt(t+1) according to the new optimal path popt(t+1) through the differential
graph.
7. If [ ] [ ])()1( tftf optbest PP ≥+ the current best solution Pbest(t+1) is discarded and partial
evaporation of the pheromones is carried out. Pheromone evaporation is done using the
Cauchy probability density function from equation (15) with the adapted factor of
scatter ;localglobal sss −= locallocalglobalglobal sssss ⋅−=⋅−= − )1(,)1( ρ and transferred
optimum vertices vl,opt towards the centres of the differential graph at each layer:1, +ldlv
10);()1()1( ,, <<⋅−=+ ρρ tvtv optloptl . ρ is a pheromone-dispersion factor.
8. If either the cost-function optimum or the limiting number of iterations is reached the
procedure is terminated. Otherwise, the iteration number is updated and the
procedure is returned to step 2.
1+← tt
The convergence of the DASA procedure depends on the selection of the DASA parameters:
• the number of ants n,
• the logarithmic base b,
• the pheromone-dispersion factor ρ,
• a factor of scatter reducing s+
for the Cauchy probability density function; 0 < s+
< 1,
• a factor of scatter enlargement s−
for the Cauchy probability density function; 0 < s−
< s+
,
• the selected precision εi
for the independent variable xl.
3 Material characteristics of the polyurethane foam
The experimental determination of the compressive loading-unloading cyclic characteristic of
the polyurethane foam was carried out in accordance with the DIN 3386-1:2000 standard. The
9

specimen had a brick-like shape with the dimensions 80×80×40 mm (see Fig. 6). The
measurements were performed on a ZWICK test stand. The compressive loading and
unloading was introduced along the shortest dimension of the specimen (40 mm). The loading
cycle was composed of two half-cycles: 1) compressive loading up to 70% strain in the
loading direction and 2) controlled unloading until zero compressive load with the same strain
rate as in the loading half-cycle. The complete loading-unloading cycle was repeated three
times. The experimentally determined dependency between the compressive force and the
displacement of the specimen is presented in Fig. 7. Fig. 8 shows the dependency of the
compressive force on the elapsed time during the experiment.
Fig. 6: Test specimens made from polyurethane foam
Fig. 7: Experimentally determined F(Δx) curves for three consecutive compressive loading-
unloading cycles
10

Fig. 8: Experimentally determined F(t) curves for three consecutive compressive loading-
unloading cycles
The finite-element model that was used to simulate the compressive test of the polyurethane
foam is presented in Fig. 9. The polyurethane block of 80×80×40 mm was modelled with eight
finite elements. The nodes on the bottom side of the block all had the degrees of freedom
(DOF) fixed. The top side of the block was loaded by means of a movable rigid block. The
movable block had all the DOFs fixed, with the exception of the displacements in the z
direction. Between the polyurethane block and the rigid block there was an
AUTOMATIC_SURFACE_TO_SURFACE contact without friction. The frictionless contact
was applied, because the Poisson number of this foam in the compressive regime is equal to
zero: 0=ν . This is because no lateral sliding on the contact surface should occur, since there
was no lateral embossing of the foam block during the experiment. The compressive load was
introduced to the foam block by compressing it in the z direction with the rigid block. The
maximum vertical displacement of the rigid block was 70% of the vertical height of the foam
block. In this manner the foam was loaded up to 70% strain in the z direction. During the
simulation the force at the bottom part of the polyurethane foam, the vertical displacement of
the rigid block and the simulation time were saved for further processing.
Fig. 9: Applied finite-element model
A MAT_LOW_DENSITY_FOAM (MAT 57) was applied to the finite elements that were
used for modelling the foam, as was already mentioned in the introductory section. The
material density, the tensile elastic modulus, the Poisson’s number and the ( )σ ε curve for the
11

compressive-loading regime were determined directly from the corresponding experimental
data. The three parameters – HU, SHAPE and BETA – that describe the material’s behaviour in
the compressive-unloading regime and further reloading regime(s) were introduced as
parameters in the LS-DYNA *.k file. Their values represented independent variables for the
optimisation process and were changed between the consecutive iterations of the optimisation
process.
The cost function for the optimisation process was defined as the integral measure of the
discrepancy between the experimentally determined and simulated dependency between the
compressive force and the experimental or simulation time F(t):
[∫ ⋅−=
max
0
2
exp )()(
T
sim dttFtFCF ] (18).
Tmax is the experimental time, Fsim(t) represents the force-time dependency from simulations
and Fexp(t) represents the force-time dependency from experiments. The optimal values of the
parameters HU, SHAPE and BETA should result in the smallest value of the cost function CF.
The optimisation of the cost function according to the three independent variables HU, SHAPE
and BETA was performed with the three numerical optimisation methods, which are described
in Section 2, on a numerical server with two Intel Xeon 2.33 GHz processors, 16GB of RAM
memory and a Linux operating system. The results of the optimisation procedures are
presented in Table 1. Fig. 10 shows a comparison between the experimentally measured and
the simulated F(t) characteristics for the polyurethane foam under consideration.
Table 1: Results of the optimisation procedures
Gradient method GA DASA
Ranges of the independent
variables
HU=[0.1 ; 0.9999]
SHAPE=[1.0 ; 20.0]
BETA=[500 ; 1000]
HU=[0.1 ; 0.9999]
SHAPE=[1.0 ; 20.0]
BETA=[500 ; 1000]
HU=[0.1 ; 0.9999]
SHAPE=[1.0 ; 20.0]
BETA=[500 ; 1000]
Parameters of the applied
optimisation procedures
n =8
h0=0.0001
=κ 0.0001
=β 0.99
n=10
mi=10
ncrossover_pts=3
pc=0.2
pm=0.1
n=20
b=10
=iε 0.0001
=ρ 0.2
s+=0.02
s-=0.01
Parameter HU 0.157 0.204 0.336
Parameter SHAPE 4.828 5.440 6.837
Parameter BETA 880.44 890.44 880.19
Cost function value 1.2452·106
1.1319·106
1.1317·106
No. of iterations for opt. sol. 40 60 60
Processing time 1h 1h 2h, 30min
12

Fig. 10: Comparison of experimentally determined and simulated F(t) curves
( red – experimentally determined F(t) curve; green – simulated F(t) curve with optimal parameters determined
by the gradient-descent method; blue – simulated F(t) curve with optimal parameters determined by the GA;
black – simulated F(t) curve with optimal parameters determined by the DASA)
We can see in Table 1 that the lowest value of the cost function CF was found with the DASA
method. The performance of the gradient method was the worst. The value of the cost function
CF , which was found with the GA, was similar to that of the DASA method, but the optimal
values of the parameters HU, SHAPE and BETA were somewhat different from the optimal
values found by the DASA method. This is because the cost function CF has a ridge-like
shape and different combinations of the parameters HU, SHAPE and BETA result in a similar
value of the cost function. Moreover, if we look at Fig. 10, we can observe that the solutions
found using the GA and DASA methods can hardly be differentiated. We can also see in this
figure that the solution found by the gradient method is different from the solutions of the AG
and DASA methods.
The agreement between the simulated and experimentally determined F(t) curves is very good
for the first loading cycle in all three cases. However, the agreement between the simulated
and experimental curves got worse during the second and the third loading cycles. This is
because the material model MAT_LOW_DENSITY_FOAM (MAT 57) assumes that all the
compressive loading curves should have the same shape, which is not the case in reality. We
applied the material parameter BETA in our model creep during unloading. In this manner we
introduced a time delay after which the compressive loading characteristic follows the initial
loading curve. With such an approach we prevented the compressive loading characteristics
from strictly following the compressive loading curve.
From the processing time in Table 1 we might conclude that the GA outperforms the DASA
method. However, this is not the case. We used only 10 chromosomes with the GA method,
but we used 20 ants with the DASA method. Most of the processing time was consumed by
the LS.DYNA simulations, so this is one reason for the difference in the processing time.
Furthermore, when using GAs, not all the chromosomes in the generations are evaluated in
each generation. The parent chromosomes that were simply transferred to the new generation
were not evaluated anew. On the other hand, each of the 20 ants was evaluated during each
13

iteration when the DASA method was applied. This is another reason for the longer processing
time of the DASA method. If we used full parallelism for the LS-DYNA simulations when the
number of processing cores would be equal to the number of ants, the DASA method would
consume approximately the same amount of computing time as the GA method. The
processing time for the GA method would also increase significantly if greater accuracy was
required. However, this is not the case if higher precision is applied within the DASA method.
4 Conclusion
The paper presents three numerical optimisation methods that can be used to estimate the
parameters of complex material models for finite-element simulations. The three methods
were tested for the case of the compressive cyclic loading-unloading characteristics of a
polyurethane foam. After the compressive characteristic of the foam was measured a finite-
element model was built in LS-DYNA and three parameters of the MAT_LOW_DENSITY
_FOAM (MAT 57) material model, which cannot be simply calculated on the basis of the
measured compressive F(t) characteristic, were determined using the three numerical
optimisation methods.
It turned out that the two evolutionary methods – the GA and the DASA – outperformed the
classical numerical optimisation method, i.e., the gradient-descent procedure. The optimisation
results of the GA and the DASA methods were comparable, but the GA method consumed less
processing time. However, the GA method used fewer chromosomes in each generation when
compared to the number of ants in the DASA method. There were no parallel LS-DYNA
simulations for the chromosomes and the ants, which also resulted in the increased processing
time for the DASA method. Last, but not least, we applied only 5 digits of precision for the
three parameters HU, SHAPE and BETA in our case. If the precision was increased, the DASA
method would outperform the GA with respect to the computing time.
5 References
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corporation, 1998.
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theory and classical and new gradient-based algorithms. Cambridge, Massachusetts:
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Prentice - Hall Inc., 2004.
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[9] Michalewicz Z. Genetic algorithm + data structure = evolution programs. New York:
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[10] Schmitt LM. Optimization with genetic algorithms in multi-species environments. In
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15

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Determining the Material Parameters of a Polyurethane Foam Using Numerical Optimisation Algorithms

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