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




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




Conclusion

1x

2x

1 2( , )f x x

0Pv

1Pv

2Pv

( )0f Pv

( )1f Pv

( )1f Pv

f(Pt+1)

f(P)

f(Pt)

f(Pt+2)Pt+2

Pt+1

Initial searching directions

Steepest gradient in the second iteration

Steepest gradient in 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




Conclusion

Genetic algorithm (GA)

Introduction




Conclusion

Parent chromosomes

Child chromosomes

{{{ {{{

}}} }}}

1010 001111 0011010 010

1010 010111 0011001010

Crossover points Crossover points

Child chromosome

Child chromosome after mutation

1 0 0 0 1 0 0 1 01 0 0 0 1

10 0 0 1 0

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




Conclusion

Experimentally determined compressive characteristics of a polyurethane foam

Introduction




Conclusion

Specimens: 80x80x40 mm Testing machine

Standard testing procedure according to DIN 3386‐1:2000

Applied finite element model

Introduction




Conclusion

[ ]∫ ⋅−=max

0

2exp )()(

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




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




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_FOAMmaterial 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




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 densityenergystraindensityenergystrainHUHU ⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⋅−+= (1).

where )(εσ loading is the tabulated characteristic during the compressive loading and )(εσ unloading 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

,...,11 tKinit =+ ⇒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()(;

≤−⋅>−⋅

tKtKtKtK

⎩⎨⎧

⋅−=Δ +

tt h

hβ

κ1 (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

0Pv

1Pv

2Pv

( )0f Pv

( )1f Pv

( )1f Pv

f(Pt+1)

f(P)

f(Pt)

f(Pt+2)Pt+2

Pt+1

Initial searching directions

Steepest gradient in the second iteration

Steepest gradient in 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

precnll lowupp 10)( ⋅−

1210)( −<⋅− lprec mnll 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

lllll

lowuppdecimallowx 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:

nif

fp n

i it

ititit ,...,1;

)()(

)(1 ,

,,, ==

∑ =P

PC (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:

nipqi

jititit ,...,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

1 0 0 0 1 0 0 1 01 0 0 0 1

10 0 0 1 0

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) lLj

jl djb l ,...,1;1, =−= −+−δ

7

(13). lLj

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,

lLblUb ℑ∈= llbl LL );(log ε , [ ] ℑ∈−= lllbl UlowuppU ;log ,

and 1+−= 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)(

svv

s

voptljl

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

jljl

v

vvprob

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 inibest 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

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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

2exp )()(

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

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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

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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.

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[9] Michalewicz Z. Genetic algorithm + data structure = evolution programs. New York: Springer - Verlag, 1999.

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