topology optimization of aeronautical structures with

7/25/2019 Topology Optimization of Aeronautical Structures With

1/12

Topology optimization of aeronautical structures withstress constraints: general methodology andapplicationsJ Pars*, S Martnez, F Navarrina, I Colominas, and M Casteleiro

Department of Applied Mathematics, Universidade da Coruna, A Coruna, Spain

The manuscript was received on 22 March 2010 and was accepted after revision for publication on 6 May 2011.

DOI: 10.1177/0954410011411632

Abstract: Structural optimization is a well known and frequently used discipline in aerospaceapplications since most of the optimization problems were stated in order to solve aeronauticstructures. Since first works about structural topology optimization were published, differentformulations have been proposed to obtain the most adequate design. Topology optimizationof structures is the most recent branch of structural optimization. The first works about this topic

were proposed by Bendsoe and Kikuchi in 1988 (Bendsoe, M. P. and Kikuchi, N. Generatingoptimal topologies in structural design using a homogenization method. Comput. Methods Appl.Mech. Engng., 1988,71, 197224). In this field of structural optimum design, the aim is to obtainthe optimal material distribution of a given amount of material in a predefined domain. The mostusual formulation tries to maximize the stiffness of the structure by using a given amount ofmaterial. In this paper, we present a minimum weight formulation with stress constraints thatallows to avoid most of the drawbacks associated to maximum stiffness approaches. The pro-posed formulation is applied to the optimization of aeronautical structures. Some examples have

been studied in order to validate the formulation proposed.

Keywords: topology optimization, aeronautic structures, minimum weight, stress constraints,finite-element modelling

1 INTRODUCTION

The optimization of the topology of structures is a

relatively recent discipline in the field of structural

optimization. Since the first model was introduced

two decades ago [1] a lot of effort has been dedi-

cated to deal with this problem. However, most of

the effort in this area has been focused on the cre-

ation of maximum stiffness formulations due to

computational reasons, among other considerations

[15]. Recently, approaches that use stress (and/or

displacement) constraints have been proposed

since they possess the appealing characteristics of

avoiding checkerboard solutions and guaranteeing

the feasibility of the solution [612]. However,

these formulations require significant computing

resources since the underlying optimization prob-

lem is much more complicated. This is a crucial

aspect since practical topology optimization prob-

lems usually involve a very large number of design

variables and a very large number of non-linear

stress constraints. However, the use of constraints

(stress, displacements, buckling, etc.) in topology

optimization problems produces more realistic

and reliable formulations, and the solutions

obtained satisfy the requirements imposed in engi-

neering projects (minimum cost, bearing capacity,

safety, etc.).

The maingoal of thispaper is to present a model for

the topology optimization problem in aeronautical

applications that is based on the minimization

of the weight with stress constraints. The mainadvantages and drawbacks of this formulation are

explained and some modifications are proposed in

*Corresponding author: Group of Numerical Methods in

Engineering, GMNI, Department of Applied Mathematics,

School of Civil Engineering, Universidade da Coruna, Campus

de Elvina, 15071, A Coruna, Spain.

email: [email protected]

589

Proc. IMechE Vol. 226 Part G: J. Aerospace Engineering

at NANYANG TECH UNIV LIBRARY on June 30, 20 16pig.sagepub.comDownloaded from
http://pig.sagepub.com/http://pig.sagepub.com/http://pig.sagepub.com/http://pig.sagepub.com/


2/12

order to reduce the most critical aspects from numer-

ical and computational points of view (computing

cost, feasibility, robustness). Three different appro-

aches for structural topology optimization that incor-

porate stress constraints are proposed in orderto analyse the obtained improvements. Special

effort is devoted to the analysis of techniques that

reduce the computational requirements. In addition,

a modified minimum-weight objective function is

proposed in order to obtain an essentially binary opti-

mum solution by using continuum design variables.

Finally, some aeronautical structures are optimized

in order to verify the feasibility of the proposed

models.

2 TOPOLOGY OPTIMIZATION PROBLEM

Minimum-weight formulations for the topology

optimization of continuum structures with stress

constraints try to obtain the best distribution of

material in a predefined domain such that the

weight (or the cost) is minimized and stress (and/

or displacement) constraints are satisfied. Thus, the

main goal of this problem is to decide the optimal

distribution of material. According to this idea, the

material distribution is binary since each point of

the domain has two possible states: solid material

or void. This binary approach corresponds to theoriginal formulation of the structural topology opti-

mization problem. In practice, discrete optimization

with a large number of design variables and con-

straints is a challenging problem that has yet to be

correctly solved. Consequently, most of the formu-

lations proposed in the literature state the problem

by means of continuum approaches to the material

distribution. These continuum approaches require

the use of a continuum design variable that defines

the material state: the so-called relative density ().

Thus, the relative density takes the values [0,1]. The

value 0 means no material and 1 meanssolid material. In addition, microstructure models

are required to perform the structural analysis for

intermediate values of the relative density (e.g. solid

isotropic material with penalization (SIMP), hole-in-

cell, rank-layered, etc. [14]). The most usual

microstructure model in topology optimization of

structures is SIMP. A minimum-weight approach

with continuum design variables is presented in

this paper. These variables correspond to the rela-

tive density of each one of the elements used in a

finite-element modelling (FEM) discretization. The

relative density is assumed uniform for each ele-ment of the mesh. The microstructure model used

in this paper is based on a SIMP approach.

According to these statements, the minimum-

weight topology optimization problem with stress

constraints can be written as

find

feg e1,. . .

, Nethat minimizes F verifying gj 40 j1, . . . , m

05min4e41 e1, . . . , Ne

1

where the design variableeis the relative density of

elemente(assumed uniform within the element) and

Ne is the total number of elements in the mesh. The

lower limit of the relative density,min, usually takes

values slightly higher than zero to avoid the stiffness

matrix becoming singular.

2.1 Objective function

According to the previous explanations, the model of

the microstructure used in the proposed formulation

is based on a SIMP model without any penalization of

the intermediate densities. In this formulation the

penalization of the intermediate densities is included

in the objective function as

F XNe

e1

e1=p

Ze matd 2whereeis the element numbere,matis the density

of the material, andp51 is the penalization param-

eter of the intermediate densities. This penalization

parameter is used to favour a mainly solidvoid dis-

tribution of material [13]. This objective function

can be modified in order to improve the obtai-

ned solutions. A modified objective function that

included a penalization on the perimeter of the struc-

ture was proposed in reference [14]. This perimeter

penalization avoided some unexpected phenomena

and produced optimal solutions with a smallernumber of structural elements (trusses).

Objective function (2) can be complemented with a

modified objective function at the end of the optimi-

zation process that forces the material distribution

to be binary although the design variables are con-

tinuous. The modified objective function can be

written as

F XNee1

Ze

bematd 3

where

be 1=12

e sine expe 4

590 J Pars, S Martnez, F Navarrina, I Colominas, and M Casteleiro




3/12

The parameter must take values in the interval

[0.260, 7.404] in order to avoid positive derivatives of

the objective function when e !1. The optimiza-

tion process is developed in two stages. In the first

stage, the optimization process is developed byusing the objective function proposed in equation

(2) until convergence. The second optimization pro-

cess starts with the solution obtained in the first

stage. The parameter is usually modified during

the optimization process. The initial steps are devel-

oped using a small value of the parameter (for

instance, 1:0) which is increased until the max-

imum value is reached. Finally, the process con-

tinues until convergence by using the maximum

value of the parameter ( 7:404). In practice, a

large number of iterations are required in this

second stage, but this is not overly restrictive froma CPU time point of view. In very specific cases the

value of can temporarily take values over 7.404 but

at the end of the optimization stage the limit values

must be satisfied in order to obtain an essentially

binary solution.

2.2 Stress constraints

The formulation proposed for the structural topology

optimization problem tries to minimize the previ-

ously introduced objective function while taking

into account stress constraints. In this paper, threedifferent formulations are presented and analysed

in order to deal with stress constraints. The first pro-

posed model is based on a local approach to the state

stress constraints. In this formulation one stress con-

straint at the central point of each element is imposed

[68, 13, 14]. Thus, each local stress constraint can be

introduced as

ge he

maxe

e

q40 5

and

e

1

" "

e6

where geis the stress constraint of the element eand

is the reference stress (usually the von Mises crite-

rion) obtained through the calculated stress tensor

heat the central point of the element. This local con-

straint is relaxed using the function eproposed in

equation (6) in order to avoid singularity phenom-

enon when the relative density tends to zero [69,

13, 14]. The relaxation parameter " usually takes a

value in the interval [0.001,0.1]. In addition, the expo-

nent qallows one to deal with the real stress (when

q 0) or the effective stress (whenq 1). The use of

effective stresses creates important advantages sinceit avoids discontinuities when the relative density

tends to zero (see [13,14,15] for more details).

The local approach to stress constraints usually

requires a large number of constraints to be imposed

due to the number of elements (and design variables)

involved in the structural analysis (by means of FEM).

Consequently, this approach requires significantcomputing resources when thousands of design vari-

ables are used. Due to this fact, alternative formula-

tions have been derived in order to reduce the

computational cost. In this paper two additional for-

mulations of the stress constraints are introduced: a

global approach that aggregates the effect of all the

local constraints [13] and a block aggregation tech-

nique that aggregates the local constraints of a pre-

defined set of elements of the mesh and then uses

them to state a global constraint [14]. Thus, the

number of constraints is equal to the number of

defined groups of elements. Each group of elementsis usually called a block and, consequently, the pro-

posed formulation is called the block aggregated

approach to stress constraints.

The global approach aggregates all the local

constraints by using a global function. The global

function used in this paper is the Kreisselmeier

Steinhauser function (also used, for example, in refer-

ence [16]) that can be written as

GKS 1

ln

XNe

e1

ee1

!

1

lnNe

" #40 7

with

e he

maxe8

being the normalized reference stress. The coefficient

proposed in equation (8) is the aggregation param-

eter and usually takes values 2 20, 40(see [13,14,

17] for more details). Values of520 allow excessive

violation of the local constraints. Values of 440

excessively increase the non-linearity of the global

functions.

This formulation reduces the required computingcost since the number of stress constraints decreases.

The local approach imposes as many constraints as

design variables multiplied by the number of load

cases. In contrast, the global approach imposes only

one stress constraint per load case.

However, when the number of design variables

(and, consequently, the number of elements)

increases, the global approach aggregates thousands

of local constraints. Thus, the aggregation of a large

number of constraints means a loss of information

about local contributions. Consequently, the sensitiv-

ity analysis becomes useless in practice for the opti-mization algorithms when the global approach is

used.

Topology optimization of aeronautical structures with stress constraints 591




4/12

Due to this fact, a different formulation of the stress

constraints is proposed that combines the advantages

of the global and local approaches: the previously

introduced block aggregated approach of the stress

constraints. This formulation defines groups of ele-ments called blocks. Each block contains approxi-

mately the same number of elements.

The main goal of this approach is to impose a global

constraint that aggregates the contributions of the

elements contained in a block. Thus, the number of

constraints imposed is equal to the number of blocks

defined. In addition, the number of local constraints

aggregated in a global function is much smaller than

the number of design variables. The global function

obtained by aggregating the elements of each block is

defined according to equation (7) as

GbKS 1

ln

Xe2Bb

e e1

!

1

ln Nbe " #

40 9

where eis the normalized reference stress presented

in equation (8),Nbe is the number of elements aggre-

gated in blockb, and Bbis the set of elements stated in

blockb.

This formulation is more general than the local and

global approaches and includes them as special cases

[13, 14]. When the number of blocks is equal to the

number of elements, the block aggregated approach

is equivalent to the local constraints approach. If thenumber of blocks is equal to one, this formulation is

equivalent to the global approach.

The number of blocks used and the way of defining

the geometry of the blocks are the most important

features of this formulation. However, it has been

observed by the authors that the geometry of the

blocks does not significantly influence the final solu-

tion. The number of blocks and the aggregation

parameter are much more critical [13,14].

3 OPTIMIZATION ALGORITHM

Based on the approaches introduced in the previous

section, the topology optimization of structures using

stress constraints leads to mathematical program-

ming problems of the type stated in equation (1)

with highly non-linear constraints of the type stated

in equations (5), (7), or (9) and a non-linear objective

function like the one proposed in equation (2) or in

equation (3). In practical applications, FEM meshes

may produce thousands of design variables and

thousands of stress constraints in two-dimensional

(2D) problems (when the local approach is used).

In three-dimensional problems the number of vari-ables dramatically increases. Thus, the optimization

algorithms used to solve this problem must be

appropriate and specific. Otherwise, the computa-

tional costs will become unaffordable in practice. In

addition, non-specific algorithms may not reach the

optimal solution.

In this paper we propose an improved SLPalgorithm with quadratic line search to solve the

optimization problem [13,14,15,17]. This algorithm

analyses a linearized approach to problem (1) in all

iterations of the optimization process. In addition,

side constraints of the design variables need to be

imposed. The search direction is obtained by means

Fig. 1 Scheme of the geometry (in millimetres), the

prescribed displacements, the loads applied,and the FEM mesh with 4014 quadrilateraleight-node elements used for the structuraloptimum design of an undercarriage





5/12

of the Simplex algorithm [18]. Then, the advance

factor that multiplies the search direction is obtained

by using a directional second-order Taylor expansion

[13,15].

This algorithm has been demonstrated to workproperly for this optimization problem. In fact the

algorithm has an adequate performance in both

opposite situations: when thousands of local stress

constraints are considered (local approach) and

when a reduced number of global stress constraints

is considered (block aggregated approach).

4 SENSITIVITY ANALYSIS

The required sensitivity analysis is obtained analyti-

cally since the resulting algorithms are mathemati-

cally exact and efficient and they produce adequatecalculations of the derivatives.

First and second-order derivatives of the objective

function are obtained by applying a direct differenti-

ation algorithm. This algorithm is directly obtained

from equation (2) or equation (3), respectively, for

each formulation of the objective function by using

the differentiation rules.

Fig. 3 Normalized stress state defined according toequation (8) of the optimum structure of anundercarriage obtained using the localapproach of stress constraints

Fig. 2 Optimum structure for an undercarriageobtained using the local approach of stressconstraints





6/12

First-order derivatives of the stress constraints are

obtained by using an adjoint variable approach [13,

14, 15, 19]. This formulation allows to obtain first-

order derivatives of a set of active (almost violated)

stress constraints and it requires less computationaleffort than a direct differentiation approach [13, 14,

15,19]. In practice, only first-order derivatives of the

active stress constraints are computed in order to

reduce the required computing time. First and

second-order directional derivatives of all the stress

constraints are obtained by using a direct differenti-

ation technique since it requires less computational

effort than the adjoint variable approach. In addition,

this technique allows one to compute the first

and second-order directional derivatives of all stress

constraints even though the whole set of first-

Fig. 4 Optimum structure of an undercarriageobtained using the local approach of stressconstraints and the modified objectivefunction

Fig. 5 Normalized stress state of the optimum struc-ture of an undercarriage obtained using thelocal approach of stress constraints and themodified objective function

Table 1 Summary of the most important parameters

and results for the structural topology optimi-

zation of an undercarriage

Undercarriage m(constraints) " p, Final weightInitial weight %

Local approach 4014 0.01 4.00 8.30Modified

objective4014 0.01 7.00 18.72





7/12

Fig. 6 Scheme of the geometry (in millimetres), the prescribed displacements, the applied loads,and the FEM mesh with 5200 eight-node quadrilateral elements used for the optimumdesign of a structural section of a VTP/HTP

Fig. 9 Optimal solution obtained using the local approach of the stress constraints and the mod-ified objective function proposed in equation (3) for a structural section of a VTP/HTP

Fig. 10 Normalized stress state, obtained based on equation (8), of the optimal solution presentedin Fig. 9

Fig. 8 Normalized stress state, obtained based on equation (8), of the optimal solution presented inFig. 7

Fig. 7 Optimal solution obtained using the local approach of the stress constraints for a structuralsection of a VTP/HTP





8/12

order derivatives has not been previously calculated.

The complete sensitivity analysis procedure was

developed by the current authors in a previous pub-

lication [19].

5 APPLICATION EXAMPLES

The application examples presented correspond to

2D structures in plane stress. Three different exam-

ples have been analysed. The first problem corre-

sponds to the structural optimum design of the

undercarriage of a small plane. The second example

corresponds to the optimization of a section of a ver-

tical/horizontal tail plane (VTP/HTP).The third

example corresponds to the structural optimum

design of an arc section in the rear part of a plane

that bears the VTP. The computations were per-formed on computing nodes with two Quad-Core

processors (2.83 GHz, 1333 MHz FSB) and 32 Gb of

RAM.

5.1 Structural optimum design of an

undercarriage

The first example corresponds to the optimum design

of the undercarriage of a small plane. Null displace-

ments of the structure are imposed in the striped part

of the upper edge. A vertical distributed force of 5550

kN/m is applied on the lower vertical edge.Figure 1 shows the dimensions of the domain and

the position of the external load. The domain of the

structure is discretized by using 4014 eight-node

quadrilateral elements. The material being used is

steel alloy AISI-4340 with density mat7850 kg/

m3, Youngs modulus E 200 GPa, Poissons ratio

0:3, and elastic limit max1790 MPa. The thick-

ness of the structure is 0:1 m. Figure 2 shows the

optimal solution for the undercarriage problem

obtained using the local approach of stress con-

straints. Figure 3 shows the normalized stress state

according to equation (8). Figure 4 shows the optimalsolution for the undercarriage problem obtained by

using the local approach of stress constraints. This

solution includes a second optimization stage that

considers the modified objective function proposed

in equation (3). Figure 5 shows the normalized stress

state for this problem according to equation (8).

The most relevant parameters and results obtained

for this problem can be observed in Table 1. From this

table it can be observed that the weight of the opti-

mum solution obtained with the modified objective

function is higher than the one obtained with the min-

imum weight with penalization approach. This factcan be easily explained since the modified objective

function forces binary solutions and this is a very

important design limitation. However, the solution

obtained by using the modified objective function

has multiple advantages since the number of elements

with intermediate relative density has been decreased.

5.2 Optimum design of a structural section in

the VTP/HTP of a plane

The second example corresponds to the topology

optimization of a structural section in the VTP/HTP

of a plane. Null displacements are imposed on the

edge of the left circumference. A distributed force of

12 kN/m is applied orthogonally on the lower edge of

the section (along 76.6 cm). The dimensions of the

domain and the position of the external loads can

be observed in Fig. 6. The domain of the structure is

discretized using 5200 eight-node quadrilateral ele-ments. Only the lower half of the structure needs to

be analysed due to the symmetry of the geometry and

the applied loads. The material being used is an

aluminium alloy with density mat2770 kg/m3,

Youngs modulus E 73 GPa, Poissons ratio

0:3, and elastic limit max430 MPa. The thick-

ness of the structure is 0:01 m.

Figure 7 shows the optimum solution for a struc-

tural section in the VTP/HTP obtained using the local

approach of stress constraints. The normalized stress

state (represented using equation (8)) can be

observed in Fig. 8. Figure 9 shows the optimum solu-tion of a structural section in the VTP/HTP obtained

using the local approach of stress constraints. This

solution includes a second optimization stage that

considers the modified objective function pro-

posed in equation (3). The normalized stress state

(obtained according to equation (8)) can be observed

in Fig. 10.

Table 2 shows the most important parameters for

this formulation and the most representative results

for the optimization of the VTP/HTP. It can be

observed that the optimal solution obtained with

the modified objective function presents an almostbinary solution. However, the weight of the solution

obtained with the modified objective function is

higher than the one obtained with the minimum-

weight approach with penalization since the require-

ment of a binary solution is a strong design limitation.


and results for the topology optimization of a

structural section in a VTP/HTP

VTP / HTP m(constraints) " p, Final weightInitial weight

%

Local approach 5200 0.015 4.00 9.37Modified

objective5200 0.010 7.00 19.98





9/12


10/12

Fig. 14 Optimal solution obtained using the block aggregated approach of the stress constraintsand the modified objective function proposed in equation (3) for the topology optimizationof an arc section located in the rear part of a plane

Fig. 15 Normalized stress state, obtained according to equation (8), of the optimal solution pre-sented in Fig. 14

Fig. 13 Normalized stress state, computed according to equation (8), of the optimal solutionobtained for an arc section located in the rear part of a plane





11/12

force binary material distributions as it has been pre-

viously explained. On the other hand, the normalized

stress states obtained from the two formulations are

different. The normalized stress state when the local

approach is used (Fig. 13) is strictly satisfied from a

local point of view. On the other hand, the normalizedstress state for the block aggregated approach solu-

tion (Fig. 15) does not strictly satisfy the stress con-

straints from a local point of view. Thus, the local

approach is more reliable than the block aggregated

approach. The main advantage of the block aggre-

gated approach is the obtained reduction in comput-

ing time. This is a result of the small number of

constraints involved, which are reduced from 4650

(local approach) to 186 (block aggregated approach).

The CPU time can be usually decreased one order of

magnitude with this kind of formulation in practical

applications.Table 3 shows the most important parameters and

the most representative results for the optimum

design of a structural arc located in the rear part of

a plane.

The application examples clearly show that the pro-

posed formulations are able to deal with structural

problems in aeronautic applications. Structural prob-

lems with thousands of design variables and thou-

sands of non-linear stress constraints are solved and

appropriate solutions are obtained.

6 CONCLUSIONS

A minimum-weight approach for the topology opti-

mization of structures with stress constraints has

been proposed in this paper. This model is based on

a continuum approach to the design variables by

using a SIMP approach without penalization of the

intermediate relative densities.

The penalization of the intermediate relative den-

sities is included in a modified minimum-weight

objective function that allows 0-1 binary distributions

of material to be favoured. The presented formula-

tions also include stress constraints in order toverify the structural feasibility since material failure

is checked and avoided. Three different formulations

of stress constraints are presented in order to study

the advantages that they offer (computing cost,

robustness, reliability, etc.): the local approach, the

global approach, and the block aggregated approach.

The proposed formulations do not require the use ofany image filtering technique in order to avoid check-

erboard solutions. The optimal solutions are directly

obtained from the analysis and do not include any

smoothing or filtering treatment.

Finally, some application examples related to aero-

nautic and astronautic industries are solved in order

to verify the proposed formulation and the applied

optimization methodology. These examples contain

a large number of design variables and constraints,

which indicates the robustness of the proposed for-

mulation to solve structural topology optimization

problems in aeronautical applications.

ACKNOWLEDGEMENTS

This work was partially supported by the Ministerio

de Educacion y Ciencia of the Spanish Government

(grants DPI2009-14546-C02-01 and DPI2010-16496),

by FEDER funds from the Autonomous Government

of the Xunta de Galicia (grants PGDIT09MDS00718PR

and PGDIT09REM005118PR), by the Universidade da

Coruna, and by the Fundacion de la Ingeniera Civil

de Galicia.

Authors 2011

REFERENCES

1 Bendsoe, M. P.and Kikuchi, N. Generating optimaltopologies in structural design using a homogeniza-tion method. Comput. Methods Appl. Mech. Engng,1988,71, 197224.

2 Bendsoe, M. P. Optimal shape design as a material dis-tribution problem.Struct. Optim., 1989,1, 193202.

3 Bendsoe, M. P. Optimization of structural topology,shape, and material, 1995 (Springer-Verlag,Heidelberg, Germany).

4 Eschenauer, H. A.andOlhoff, N.Topology optimiza-tion of continuum structures: A review. Appl. Mech.Rev., 2001,54(4), 331390.

5 Rozvany, G. I. N.A critical review of established meth-ods of structural topology optimization. Struct.Multidiscip. Optim., 2009,37, 217237.

6 Duysinx, P.and Bendsoe, M. P. Topology optimiza-tion of continuum structures with local stressconstraints. Int. J. Numer. Methods Engng, 1998, 43,14531478.

7 Duysinx, P. Topology optimization with differentstress limits in tension and compression, report:Robotics and automation, 1998 (Institute of Mechanics, University of Liege, Liege, Belgium).


and results for the structural topology optimi-

zation of an arc section located in the rear part

of a plane

Arc m(constraints) " p Final Weight

Initial Weight %

Local approach 4650 0.02 4.00 15.80Block aggregated

approach186 0.02 4.00 14.34





12/12

8 Cheng, G. andJiang, Z.Study on topology optimiza-tion with stress constraints.Engng Optim., 1992,20,129148.

9 Cheng, G. D. and Guo, X. e-relaxed approach instructural topology optimization. Struct. Optim.,

1997,13, 258266.10 Liang, Q. Q., Xie, Y. M., and Steve, G. P. Optimal

selection of topologies for the minimum-weightdesign of continuum structures with stress con-straints. Proc. IMechE, Part C: J. MechanicalEngineering Science, 1999,213(8), 755762.

11 Pereira, J. T., Fancello, E. A., and Barcellos, C. S.Topology optimization of continuum structureswith material failure constraints.Struct. Multidiscip.Optim., 2004,26(1-2), 5066.

12 Yang, R. J. and Chen, C. J. Stress-based topologyoptimization.Struct. Optim., 1996,12, 98105.

13 Pars, J., Navarrina, F., Colominas, I., andCasteleiro, M. Topoloy optimization of continuumstructures with local and global stress constraints.Struct. Multidiscip. Optim., 2009,39(4), 419437.

14 Pars, J., Navarrina, F., Colominas, I., and Cateleiro,M. Improvements in the treatment of stress

constraints in structural topology optimizationproblems.J. Comput. Appl. Math. 2009.

15 Navarrina, F., Muinos, I., Colominas, I., andCasteleiro, M. Topology optimization of structures:A minimum weight approach with stress constraints.

Adv. Engng Softw., 2004,36, 599606.16 Martins, J. R. R. A. and Poon, N. M. K. On struc-

tural optimization using constraint aggregation.In Proceedings of the Sixth World Congress onStructural and multidisciplinary optimization, Riode Janeiro, Brazil, 2006.

17 Pars, J., Navarrina, F., Colominas, I., andCasteleiro, M. Block aggregation of stress con-straints in topology optimization of structures. Adv.Engng Softw., 2010,41, 433441.

18 Dantzig, G. B. and Thapa, M. N. Linear program-ming: 1. Introduction, 1997 (Springer-Verlag, NewYork, New York).

19 Pars, J., Navarrina, F., Colominas, I., andCasteleiro, M.Stress constraints sensitivity analysisin structural topology optimization. Comput.Methods Appl. Mech. Engng, 2010.



at NANYANG TECH UNIV LIBRARY on June 30 20 16pig sagepub comDownloaded from

topology optimization of aeronautical structures with

Documents