Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

48, 3, pp. 567-578, Warsaw 2010

TWO-STAGE OPTIMIZATION METHOD WITH FATIGUE

CONSTRAINTS FOR THIN-WALLED STRUCTURES

Mirosław Mrzygłód

Cracow University of Technology, Institute of Rail Vehicles, Kraków, Poland

e-mail: mrzyglod@mech.pk.edu.pl

The paper describes a two-stage approach to optimization of thin-walled
structures. The optimization procedure takes into consideration high-
cycle fatigue constraints. Hence, Dang Van’s criterion for fatigue dama-
ge estimation is used. In the first stage of optimization, a parametric
model and evolutionary algorithms are used. The topology optimization
method is applied in the second stage. The work is illustrated by an
example of optimization of a thin-walledmechanical structure subjected
to high-cycle load conditions. In both stages of the optimization, the
same parametric FE model, boundary conditions and constraints were
used. The topology optimization applied in the second stage of optimi-
zation significantly improved the final result. The optimization metho-
dology allowed to effectively lower themass of the structuremaintaining
its durability on an established level.

Key words: thin-walled structures, fatigues constraints, topology optimi-
zation

1. Introduction

In optimization of a thin-walled metal structure, the parametric methods are
easy to apply. However, in the case of great number of decision variables in
complex optimization models, like in vehicles, the simple stochastic methods
are inefficient in terms of the computation time. The evolutionary algorithms
are more efficient, but although they give good results they are also time-
consuming. The efficiency of the optimization process can be significantly
improved by using parallel evolutionary algorithms (Mrzygłód and Osyczka,
2006).
The topology optimizationmethodology can also be applied to thin-walled

metallic structures (BendsoeandSigmund,2003;MrzygłódandOsyczka, 2006;



568 M. Mrzygłód

Mrzygłód, 2008). Topology optimization in large vehicle design has been so
far rarely used. But, for fairly simple vehicle structures some successful tests
have been carried out (Fredricson, 2005). For composites,multi-layer objects a
very interesting proposition is the joined topology-sizing optimizationmethod
(Zhou et al., 2004). But for the one-layer design space, the application of
topology optimization is limited.
On the other hand, for structures subjected to high-cycle load conditions

theoptimization procedure should take intoaccount fatigue constraints (Mrzy-
głód and Zieliński, 2006; 2007a,b).
In the proposed optimization methodology, the author wants to join the

advantages of bothparametric and topology optimization algorithms aswell as
to take into consideration the fatigue constraints. For thin-walled structures,
the parametric model and evolutionary algorithms are to be used in the first
stage.Then, theapplication of the topology optimization algorithm isplanned.

2. Two-stage optimization approach

Let us assume the optimization problem to be formulated as follows

minf(η) (2.1)

the constraints are

gj <gj(η)¬ gj j= [1,2, . . . ,K] (2.2)

where η = [η1,η2, . . . ,ηN] is a vector of N design variables; gj is the
j-th constraint (state parameter); gj, gj are the lower and upper bound of
constraints, respectively; f(η) is the objective function (the volume of the
structure), K is the number of constraints.
The design variables in the first stage of optimization represent the size

related NI parameters, in the second stage of optimization thedesignvariables
representapseudo-densityof each NII FEelement thatvariesbetween0and1.
The optimization process consists of the following two stages:

Stage I. Theoptimization investigation of theobjective function isperformed
using evolutionary algorithms and a parametric FEmodel.

Stage II. After finding a solution, the vector of optimal values of decision
variables η∗ form the design space for the topology optimization al-
gorithm. In this stage the same FE model, under the same boundary



Two-stage optimization method with fatigue... 569

conditions, becomes a subject to topology optimization with the aim of
further improvement of the value of the objective function.

Both stages of the optimization are executed with the same constraints.
As the main tool of the first stage of optimization, the software package

EvolutionaryOptimization System (EOS)was chosen.TheEOS is designed to
solve single and multi-criteria optimization problems for nonlinear program-
ming problems. For the purpose of numerical optimization, the EOS software
was combined with the FEM package ANSYS R© and the parallel computing
procedure (Mrzygłód andOsyczka, 2006).
For the realization of the second stage of optimization, a simple procedu-

re of topology optimization was coded using the ANSYS R© APDL language
(ANSYS Inc., 2007).
The following topology optimization algorithm was incorporated (see

Fig.1) (Mrzygłód, 2008, 2009):

1) calculation and record of stress values of the M loading steps;

2) checking the constraint limits (e.g. von Mises eqv. stress); if the state
parameter crosses its bound limit, a layer of finite elements is added to
the structure boundary;

3) selection and removal of the groups of low stressed finite elements
(σ<σmin); the bound value σmin is increased at every iteration;

4) checking the value of the objective function; if the value f(η) is under
the total volume bound value Vref the optimization process is finished,
otherwise go to 1).

It should be noted that it is also possible to start the optimization pro-
cedure from an unfeasible solution and increase the structure to its optimum
topology.

3. High-cycle fatigue constraints

The structural optimizationwith stress constraints is usually suitable for cases
where complex load conditions must be investigated. Among contemporary
stress criteria in the topology optimization, vonMises hypothesis is commonly
used (Bendsoe and Sigmund, 2003). However, this criterion does not take
into account fatigue characteristics of materials but only their yield limit.
In such cases, a multi-axial high-cycle fatigue (MHCF) hypothesis as well as
cumulative damagematrix should be applied.ThemodernMHCFcriteria like



570 M. Mrzygłód

Fig. 1. Topology optimization algorithm

Crossland, Dang Van or Papadopoulos use at least two fatigue limits (e.g.
for bending and torsion) to characterize endurance of the material (Ballard
et al., 1995; Papadopoulos et al. 1997; Dang Van and Papadopoulos, 1999).
In this work, the modern MHCF hypothesis of Dang Van was chosen as the
damage criterion because of its numerical convenience and high reliability
(Ballard et al., 1995; Papadopoulos et al. 1997). Moreover, the author takes a
few assumptions for themodeling and analysis (Mrzygłód andZieliński, 2006,
2007a,b):

• real time load history is simplified,

• possible short time violation of high-cycle loads has been taken into
consideration by means of a suitable safety factor,

• particular components of loads have the in-phase or reverse character,



Two-stage optimization method with fatigue... 571

• frequency of loads is considerably lower than the first eigen-frequency of
studied structures,

• inertial effects are not taken into account.

The Dang Van criterion takes into consideration the average value of shear
and normal stresses in an elementary volume Vel. According to Dang Van,
the fatigue damage appears in a definite time, when the combination of local
shear stresses τ(t) and hydrostatic stress σH(t) cut the borders of the admis-
sible fatigue area. A numerically convenient form of theDangVan criterion is
presented below (Mrzygłód and Zieliński, 2006, 2007a,b)

max
V
[τ(t)+κσH(t)]¬λ (3.1)

where V is volume of the studied object

τ(t)=
σ1(t)−σ3(t)

2
σH(t)=

1

3

(

σ1(t)+σ2(t)+σ3(t)
)

σ1, σ2, σ3 – are principal stresses.

The material parameters can in principle be expressed by data from two
high-cycle fatigue tests: reversed bending (fatigue limit f

−1) and reversed
torsion (fatigue limit t

−1). For the Dang Van criterion it results in λ = t−1
and κ=3t

−1/f−1−3/2.

Dang Van criterion (3.1) proposes an analysis with respect to the time
variable. For the optimization, the following pattern of reduction of the load
time history was proposed:

• real load timehistory (seeFig.2a) is simplifiedtoanequivalent sinusoidal
formwith an equivalent amplitude andmean load values,

• moreover, five load steps: min, 0.5min, 0, 0.5max, max are assumed
(see Fig.2b) and checked in every single fatigue analysis.

Fig. 2. Example of a real load time history (a), transformed to the sinusoidal form
with five load steps: min, 0.5min, 0, 0.5max,max (b)



572 M. Mrzygłód

Fig. 3. Graphical representation of the ”compare and savemaximum” algorithm of
assembling the cumulative damagematrix; L1, . . . ,L5 – are fatigue equivalent stress
matrices for load cases 1 to 5; and D1, . . . ,D5 – are damagematrices 1 to 5. As it is
shown above, the contourmaps of damage fatigue equivalent stresses change their

areas from step 1 to 5



Two-stage optimization method with fatigue... 573

For the topology optimization, the damage matrix should be constructed
for every iteration consisting of a set of load steps. The stress matrix values
for load cases 1 to 5 have been added according to Rainflow Cycle Counting
rules (Dowling, 1993). It means that only the maximum values between two
comparedmatrices are transferred to the resulting damagematrix (see Fig.3).
In particular, for the first damage matrix D1 all stresses are transferred from
the equivalent stress matrix L1. For the second load step, a ”compare and
savemaximum” procedure is employed. For every finite element its equivalent
stress value ofmatrix L2 is compared to the previous one stored in the damage
matrix D1, if the present value of L2 is higher than the former value formD1.
The new value is transferred to the new damage matrix D2. Otherwise, the
equivalent stresses from D1 are transferred to D2 without changes. The ope-
ration is repeated for load cases 3, 4 and 5. The finalmatrixD5 represents the
cumulative damagematrix of thewhole loading sequence. InFig.3, it is shown
how the contour maps of damage matrices of load cases 1 to 5 are gradually
changing their areas.

4. Example of optimization

Anexample of a car rear suspension armwas chosen to demonstrate two-stage
methodology (Fig.4a). The arm is built from two thin sheet drawn pieces
assembled by welding. This part is subjected to high-cycle load conditions.
TheMulti-Body simulation of the real car suspension armwas conducted for
determination of the load history. The equivalent loads for fatigue analysis
were prepared according to the previously established rules (see Table 1). The
parametric FE model of the suspension arm is presented in Fig.4b.
For the assumedmaterialmodel f

−1 =260MPa, t−1 =160MPa, equation
(3.1) takes the form

σeqv,DV =0.5σTG(t)+0.346σH(t)¬ 160MPa (4.1)

where σTG is the equivalent normal stress according to the Tresca criterion.
To find the limit value of damage parameter gj, a preliminary fatigue

analysis for the original shape of the suspension arm (Fig.4b) was done . The
maximumvalue of theDangVan equivalent stress was found (Fig.6a) and the
state parameter bound gj was accepted on the level σeqv,DV = 21MPa. The
low level of admissible stresses can be noticed due to a large safety factor in
the real working structure.
For the optimization process, five from the twelve decision variables we-

re selected after the sensitivity investigation. For the evolutionary algorithm,



574 M. Mrzygłód

Fig. 4. Suspension arm (a) and its FEmodel (b)

Table 1.Equivalent loads for fatigue analysis of the real car suspension arm

Loads
Steps

I II III IV V

Fox [N] −300 −200 −100 0 100

Foy [N] −1030 −1197.5 −1365 −1532.5 −1700

Foz [N] 2050 2337.5 2625 2912.5 3200

Mox [Nm] −310 −350 −390 −430 −4700

Moy [Nm] −50 −60 −70 −80 −90

Moz [Nm] −40 −30 −20 −10 0

Fsx [N] −950 −1175 −1400 −1625 −1850

Fsy [N] −240 −275 −310 −345 −380

Fsz [N] −2450 −2612.5 −2775 −2937.5 −3100

Fax [N] 250 157.5 65 −27.5 −120

Fay [N] 5 2.5 0 −2.5 −5

Faz [N] 390 242.5 95 −52.5 −200



Two-stage optimization method with fatigue... 575

Fig. 5. Decision variables initially accepted for the optimization process; variables
selected through sensitivity investigation aremarked with box

Fig. 6. Initial (a) and optimal shape of structure after the first stage of
optimization (b); (a) objective function f(η)=6438.003g; (b) objective function

f(η)=5713.026g



576 M. Mrzygłód

the following starting parameters were accepted: size of population J =350;
crossover parameter pc = 0.7; mutation parameter pm = 0.4; number of ge-
nerations Lg =70.
As a result of the first stage of optimization, ε1 = 11.3% decrease of the

structure mass was obtained. The results of evolutionary optimization of the
first stage are presented in Fig.6b.
In the second stage of optimization, ε2 =21.5% decrease of the mass was

achieved. The second stage results of topology optimization are presented in
Fig.7.

Fig. 7. Results of topology optimization of the second stage of optimization;
objective function f(η)= 4484.54g

In the two-stages optimization, about εΣ =32.8% of mass reduction was
done, the Dang Van equivalent stress value was maintained on a level of the
original shape of the suspension arm (see Fig.6a). The contour-map of the
Dang Van equivalent stresses represents their maximum cumulative value ob-
tained in the five-step load history.
It should be noted that the obtained optimized structure with sharp edge

holes is difficult to manufacture, and in the final design of the suspension
arm the openings must be replaced by equivalent circular or oval holes. The
dimensions of new holes can be defined by the next-step standard parametric
optimization.

5. Conclusions

In the paper, the complex two-stage optimization methodology with fatigue
constraints was presented. In the proposedmethodology, the author combined
the parametric evolutionary method with the topology algorithm and applied
them to thin-walled structures.



Two-stage optimization method with fatigue... 577

As observed in the numerical example of the structure subjected to high-
cycle loads, theproposedapproachallowedone toeffectively reduce themassof
the structure, maintaining its fatigue durability on an established level taken
from the real object. The second stage of topology optimization improved
the result by removing the excessive material which, clearly, could not be
removed by the first method. What is interesting, the topology optimization
method gives very good results for already optimized objects. The two-stage
optimization approach presented in the paper can be easily applied to complex
thin-walled structures.

References

1. ANSYS Inc., 2007, Release 11 Documentation for ANSYS

2. Ballard P., Dang Van K., Deperrois A., Papadopoulos Y.V., 1995,
High cycle fatigue and a finite element analysis,Fatigue and Fracture of Engi-
neering Materials and Structures, 18, 397-411

3. Bendsoe M., Sigmund O., 2003, Topology Optimization. Theory, Methods,
and Applications, Springer Verlag, NewYork

4. Dang Van K., Papadopoulos I.V. (Eds.), 1999, High Cycle Metal Fati-
gue, From Theory to Applications, C.I.S.M. Courses and Lectures No. 392, A
Springer-Verlag Company

5. Dowling N.E., 1993, Mechanical Behaviour of Materials, Prentice-Hall Int.
Editors Inc., Engelwood Cliffs

6. FredricsonH., 2005,Structural topologyoptimisation: anapplication review,
International Journal Vehicle Design, 37, 1, 67-80,

7. Mrzygłód M., 2008, Topology optimization of structures subjected to high-
cycle load conditions,Proceedings of the 12th AIAA/ISSMO Multidisciplinary
Analysis and Optimization Conference, AIAA, AIAA-2008-5839

8. Mrzygłód M., 2009, Using layer expansion algorithm in topology optimiza-
tionwith stress constraints,Proceedings of CMM-2009 – ComputerMethods in
Mechanics, The University of Zielona Góra Press, 319-320

9. Mrzygłód M., Michalik M., 2008, Topology optimization with stress con-
straint of vehicle structure,Pomiary, Automatyka, Kontrola, 7, 429-432

10. MrzygłódM.,OsyczkaA., 2006,Optimization of railwayvehicle structures
using evolutionary algorithms and parallel computing techniques, Proceedings
of the 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Confe-
rence, 1036-1041



578 M. Mrzygłód

11. Mrzygłód M., Osyczka A., 2007, Two-stage optimizationmethodology for
largestructures,Proceedings of the 7thWorldCongress onStructural andMulti-
disciplinary Optimization, International Society for Structural and Multidisci-
plinary Optimization, 1051-1056

12. MrzygłódM.,ZielińskiA.P., 2006,Numerical implementationofmultiaxial
high-cycle fatigue criterion to structural optimization, Journal of Theoretical
and Applied Mechanics, 44, 3, 691-712

13. Mrzygłód M., Zieliński A.P., 2007a, Multiaxial high-cycle fatigue
constraints in structural optimization, International Journal of Fatigue,
DOI:10.1016/j.ijfatigue.2007.01.032,29, 9-11, 1920-1926

14. Mrzygłód M., Zielinski A.P., 2007b, Parametric structural optimization
with respect to the multiaxial high-cycle fatigue criterion, Structural and Mul-
tidisciplinary Optimization, 33, 161-171

15. Papadopoulos I.V.,DavoliP.,GorlaC., FilippiniM.,BernasconiA.,
1997, A comparative study of multiaxial high-cycle fatigue criteria for metals,
International Journal of Fatigue, 19, 219-235

16. Zhou M., Pagaldipti N., Thomas H.L., Shyy Y.K., 2004, An integrated
approach to topology, sizing and shape optimization, Structural and Multidi-
sciplinary Optimization, 26, 308-317

Dwuetapowa metoda optymalizacji konstrukcji cienkościennych

z ograniczeniami zmęczeniowym

Streszczenie

Artykuł opisuje dwuetapowe podejście do optymalizacji konstrukcji cienkościen-
nych uwzględniające wysoko-cyklowe ograniczenia zmęczeniowe. Do oszacowania
uszkodzenia zmęczeniowych w procedurze optymalizacyjnej użyto hipotezy Dang
Van’a. W pierwszym etapie optymalizacji został zastosowany model parametryczny
oraz algorytmy ewolucyjne.W etapie drugimwykorzystanometodę optymalizacji to-
pologicznej. Działanie algorytmu optymalizacyjnej zilustrowana na przykładzie cien-
kościennej konstrukcjimechanicznej poddanej wysoko-cyklowemureżimowi obciążeń.
Dla obu etapów optymalizacji użyto tego samegomodel parametrycznegoMES oraz
takich samychwarunkówbrzegowych i ograniczeń.Zauważono, że zastosowanawdru-
gim etapie optymalizacja topologiczna znacząco poprawiła końcową wartość funkcji
celu. Zaprezentowanaw pracymetodyka optymalizacyjna pozwoliła skutecznie obni-
żyć masę konstrukcji, utrzymując jej trwałość zmęczeniową na założonym poziomie.

Manuscript received October 16, 2010; accepted for print November 25, 2009