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Finite Elements in Analysis and Design 58 (2012) 84–90

Contents lists available at SciVerse ScienceDirect

Finite Elements in Analysis and Design

0168-87

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/finel

Implementation of genetic algorithm for optimum cutting pattern generationof wrinkle free finishing membrane structures

Wonsiri Punurai a,n, Wasan Tongpool b, Jose Hector Morales c

a Department of Civil and Environmental Engineering, Mahidol University, Nakhonpathom 73170, Thailandb Architectural Engineering 49 Limited, Bangkok 10110, Thailandc Centro de Investigacion en Matematicas, A.C., Guanajuato, Gto 36240, Mexico

a r t i c l e i n f o

Article history:

Received 1 August 2011

Received in revised form

10 February 2012

Accepted 15 April 2012Available online 23 May 2012

Keywords:

Optimization

Cutting pattern

Genetic algorithm

Membrane structures

Finite element analysis

4X/$ - see front matter & 2012 Elsevier B.V.

x.doi.org/10.1016/j.finel.2012.04.008

esponding author. Tel.: þ662 889 2138x6391

ail addresses: egwpr@mahidol.ac.th, wonsiri@

a b s t r a c t

The purpose of this paper is to show a practical implementation of a genetic algorithm for minimizing

membrane stresses discrepancies between the actual assembled equilibrium and the specified design

state. The method prevents the surface wrinkle problems in membrane structures under service loading

and determines an optimum cutting pattern, which accounts for the designed stresses of the membrane

structures. Using the displacements of the 3-D surface as the key variables, the proposed method

utilizes a geometrically nonlinear finite element analysis based upon the improved the stress-adapted

numerical form finding of pre-stressed surfaces by the updated reference strategy. The model of genetic

algorithm and the genetic operators are then designed to solve numerically the optimization problem.

The method is validated through examples and compared with the available data. The analysis results

show no significant differences between the assumed designed stresses and the actual stresses in the

membrane.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Membrane cutting pattern generation deals with the problemof defining the subdivision of a large three dimensional surfaceinto subsurfaces. The shape-finding processes must ensure thatthese subsurfaces can be developed in two-dimensions [1]. How-ever, the surfaces which are used in practical membrane struc-tural design possess strong double curvature. Particularly, manyproblems are associated with the cutting pattern generationwhen the high-curvature structural design is unavoidable. Oneof them is in trying to get rid of wrinkles, which could arise instressed membrane surfaces after the construction is completed.The previous studies show that wrinkles are caused by two mainreasons: (i) an extension of the material arises and it is notconsidered during the cutting pattern; (ii) there exists a deviationbetween the actual stress at the construction stage and the designstress given by the designer. When the cutting strips are eval-uated in the light of the morphological parameters of forms,forces and materials, these problems need to be taken intoaccount for minimization.

Many attempts have been made in recent years to developbetter and effective procedures to produce optimum cuttingpatterns. Grundig et al. [2] and Tsubota et al. [3] proposed a

All rights reserved.

; fax: þ662 889 2138x6388.

gmail.com (W. Punurai).

method, which could be used to improve processes of mappingthree dimensional strips with a series of planar triangles. Themethod utilized all the points belonging to the strip edges inthe two dimensional coordinate system, as control variables inthe optimizing cutting patterns. The geometry of edges wascalculated by error minimization techniques which could bringa membrane stress distribution, in the actual assembled equili-brium state, closer to the uniform stress distribution specified inthe designed state. However, this method is considered incon-venient because the process required the separation betweeninner and exterior elements. The method also required the linefitting process on exterior points at each iteration step. Yagi et al.[4] presented a different approach in which the equilibrium state,after deformation, could be simultaneously considered togetherwith the configuration of the cutting pattern as the state of pre-deformation. Shimida et al. [5] took another approach by deter-mining a plane domain consisting of triangular surface elementswhich, once transformed into the 3-D strips, led to minimal strainenergy. The characteristics of the material were included in themechanical formulation. However, parameters related to the pre-stress of the membrane were not considered, thus, it was requiredthat the development of each strip must be followed by anoperation so as to take into account the initial stresses. A bettersolution was eventually proposed. Maurin et al. [6], Ohsaki et al.[7] and Kim et al. [8] proposed a stress composition approachwhere the considerations of forms, forces and materials weresimultaneously taken in the geometrically nonlinear finite

X

Y

Z

X Y

Z

Fig. 1. An evolution process for finding an optimal cutting pattern for a hyperbolic

paraboloid type membrane structure. (a) Initial membrane shape-finding with

initial updated reference geometry. (b) GA-optimized cutting pattern of an

extracted strip.

W. Punurai et al. / Finite Elements in Analysis and Design 58 (2012) 84–90 85

element analysis for cutting the pattern analysis of membranestructures. Their approaches applied an inverse scheme where thedevelopability conditions, based on the displacements and stres-ses for the surface to be reduced to plane sheets, were incorpo-rated in the process of finding equilibrium shape with minimumstress deviation from the target distribution. The technique stillused nodal coordinates of the two dimensional flattening patternas the optimizing control variables, but allowed the constraintpositions of the inner elements to be displaced together with theexternal ones, with no separation required during the iterationprocess. The trade-off between shape and stress deviation couldbe incorporated in the objective function and to improve furtherthe stress distributions of the discretized triangular finiteelements.

These methods mentioned above were able to perform well withstructures with moderate curvature. However, to guarantee a moreexact distribution of stress over all the structure and to overcomeproblems of having non-uniforming stress distributions, over thepart of steeper curvature areas, it is necessary to develop a bettermethod for producing more accurate cutting pattern.

In this paper, a more robust method for computing an optimumcutting pattern is presented. The method is based on an extension ofwork by Bletzinger et al. [9,10] who introduced and improved thestress-adapted numerical form finding of pre-stressed surfaces bythe updated reference strategy. Using the displacements of the 3-Dsurface as the key variables, the proposed method utilizes ageometrically nonlinear finite element analysis combined with themodel of genetic algorithm and the designed genetic operators tosolve numerically the optimization problem. With the presentedapproach, the membrane stress distribution in the actual assembledequilibrium state is brought closer to the uniform stress distributionspecified in the design stage. The paper is organized as follows.Section 2 presents FEM formulations for shape-finding equilibrium.Section 3 describes the cutting pattern procedure in two steps:partitioning the structure into strips and flattening and compensa-tion of the strips. By iteration of these two steps, the optimumcutting pattern can be provided. Section 4 presents the numericalsimulation results which were carried out on the two membranestructural examples with the scheme discussed in Section 3. Resultsof numerical calculations were then verified against the availabledata in the literature. Finally, Section 5 presents conclusions.

Fig. 2. A plot of convergence speed of the optimization process in the GA

optimization procedure.

2. FEM formulations for shape-finding equilibrium

In geometrically nonlinear finite element analysis, a membraneis discretized by using the triangular finite element model withconstant strain. The formulation of Newton–Raphson iterationscheme is discussed in this section to get the solution of the stiffnessnonlinear equations of the membrane, is discussed in this section. Akey issue pertaining to the kinematics of pre-stressed membranes isthat of considerable rigid body rotations along the deformation pathare developed, whereas strains remain in a moderate range. Conse-quently, a hyper-elastic Saint Venant–Kirchhoff constitutive law canbe adopted to describe the behavior of the material in an adequatemanner, as well as a total Lagrangian formulation [11–13].

Following Bletzinger [9], Bathe [14], and Punurai et al. [15], theshape-finding equilibrium equations for the membrane elementused in this study are expressed as

b

ZA

0de : C : 0e dAþb

ZA

0dg :t0S dA¼ tþDtR�b

ZA

0de : t0S dA, ð1Þ

where : is the double contraction; b is the (constant) thickness ofthe membrane; de is the strain tensor corresponding to virtualdisplacements. For hyper-elastic materials, the incremental sec-ond Kirchoff–Piola stresses 0S can be related to the incremental

Green–Lagrange strains 0e using the constitutive tensor

0S ¼ C : 0e, where C is the collective of material constants anddefined in terms of Young’s moduli and Poisson’s ratios. Thesecond Kirchhoff–Piola stress tensor t

0S is related to the Cauchystress r by t

0S ¼ ðdet t0F Þt0F

�1rt

0F�T

, where t0F ¼ @x=@X is the

deformation gradient which provides the relationship (mapping)between a deformed point x and a point of origin X. Bycomponents we have

0eij ¼Xn

I ¼ 1

ðNI,Xj

uIiþNI

,Xiu

IjþNI

,Xiu

IkNI

,Xju

IkþNI

,Xju

IkNI

,Xiu

IkÞ;

0Zij ¼Xn

I ¼ 1

ðNI,Xi

ukNI,Xj

ukÞ;

tþDtR ¼

Iss � ðNduÞ dsþ

Zn

f � ðNduÞ dn,

Fig. 3. Initial shape configuration of the model example I (isometric view).

cable fill1X

warp2X

1

2

3

4

3X

Fig. 4. Discretization of the model example I.

Table 1Membrane material properties of model examples I and II.

Properties Values

Thickness (cm) b¼0.08

Young’s modulus (MPa) E1¼267.05 (fill), E2¼806.05 (wrap)

Shear modulus (MPa) G13¼69.825

Poisson’s ratio n12 ¼ 0:29, n21 ¼ 0:87

initial cutting pattern

optimum cutting pattern

Fig. 5. Initial and optimum cutting pattern for a quarter model example I.

Fig. 6. Stress distributions of an optimum pattern of a quarter model example I.

Fig. 7. Actual stress distributions of a quarter model example I.

W. Punurai et al. / Finite Elements in Analysis and Design 58 (2012) 84–9086

where N is the matrix of the shape functions; u is the vector thatgathers the nodal displacements i,j¼ 1, 2, 3 for the n nodes of asingle finite element; s is the applied surface traction per unitarea; f is the applied force per unit volume; du is the virtualdisplacement evaluated on the surface. All stress componentsnormal to the surface are zero, i.e., S13 ¼ S23 ¼ S33 ¼ 0.

3. Optimum cutting pattern

After the pre-stress and 3-D equilibrium equations are pro-vided the initial membranes shape-finding are determined. Thisprocess is to follow the standard grid method [16] and themethod of updating the reference geometry [10]. The approxi-mate plane sheets can then be obtained by cutting surfaces alongthe cutting lines, and by reducing the stresses at the equilibrium.

At the first stage of the procedure, to optimize a plane sheetfrom a curved surface of a specified shape a quasi-Newtonmethod (to approximate the Hessian matrix from the gradient)is applied to minimize the sum of the squares of the length of

W. Punurai et al. / Finite Elements in Analysis and Design 58 (2012) 84–90 87

differences between 2-D and 3-D configurations. By means ofthese lengths of differences, the objective function to be mini-mized is introduced as

JðxÞ ¼ 12xT x, ð2Þ

where x¼ ðx1,x2,x3ÞT , the superscript T denotes transpose of a

vector, and xi ¼ li�Li; li and Li are the lengths of the element edgesi in 2-D and 3-D configurations, respectively.

In the next step, an actual equilibrium analysis is performedwith these changed 2-D coordinates. By this procedure, the newvalues of displacements and stresses of a 3-D curved surface aredetermined. It is unavoidable that some differences should occurbetween actual and design membrane stresses and that theyshould be minimized or eliminated. Using the displacements ofthe 3-D surface as the key variables from the specified targetstress vector r0, under constraints of the equilibrium equation

Table 2Comparison of membrane stress coefficients for model example I.

Results AVR1 AVR2 AVSD1 AVSD2

Moncrieff et al. [23] 1.080 1.100 0.100 0.200

Kim and Lee [8] 0.992 0.992 0.047 0.113

Present work 1.009 0.992 0.051 0.090

Fig. 8. Initial shape configuration of the

Fig. 9. Discretization of th

(1), the optimization problem for minimizing the stress deviationis stated as

Jðr�r0Þ ¼12ðr�r0Þ

Tðr�r0Þ, ð3Þ

where r is the Cauchy stress of the 3-D element surface. In theoptimization procedure of a nonlinear function in Eq. (2), theimplemented quasi-Newton BFGS method [17–20] avoids muchof the problems in computing a numerical Hessian. However, forthe optimization of Eq. (3), which is highly nonlinear, the solutionby means of a Newton scheme may flow into badly convergencealgorithm. In this study, an alternative approach is taken using agenetic algorithm (GA) [21] to perform the search and optimizingcoordinate choice for expressing geometric elements.

GAs are stochastic and robust search techniques based on theprinciples of evolution and natural genetics [22]. The GA employspopulation of individuals, each of whom represents a possiblesolution to the problem at hand. This population evolves throughthe successive generations according to Darwinian genetic prin-ciples. The members of the population are evaluated by means ofa fitness function. In our case, the fitness function corresponds to J

given by (3), and the population of individuals are the differentvalues that can be adopted by the stress deviation. The fitnessfunction provides a measure of performance of an individual,which is used to bias the selection process in favor of the most fitmembers of the current population. The members undergo theprocess of selection and recombination through probabilistic

model example II (isometric view).

e model example II.

W. Punurai et al. / Finite Elements in Analysis and Design 58 (2012) 84–9088

techniques known as genetic operators, which include reproduc-tion, crossover, and mutation. A new population is created in eachgeneration through replacement of old members with newmembers, and the evolutionary process continues. The used GAtool contains the elitist approach. This means that a solutioncannot degrade from one generation to the next, but the bestindividual of generation is copied to the next generation withoutany changes being made to it.

All the formulations and methods outlined were implemented inMATLAB&. The parameters in GA tool menu in MATLAB& included:Population Type, Double Vector; Population Size, 100; CreationFunction, Uniform; Scaling Function, Rank; Selection Function,Stochastic Uniform; Crossover Fraction, 0.80; Mutation, Gaussian;Crossover Function, Scattered; Algorithm Settings (Initial Penalty:10, Penalty Factor: 100); Hybrid Function, None; Stopping Criteria(Generations: 1000, Time Limit: Inf, Fitness Limit: � Inf; StallGenerations, 1000; Stall Time Limit, 20).

Fig. 1 shows an evolution process for finding an optimal cuttingpattern for a hyperbolic paraboloid type membrane structure. Fig. 2shows the resulting convergence speed of the optimization process.The points at the bottom denote the best fitness values, while thepoints above them denote the averages of the fitness values in eachgeneration. The plot also displays the best and mean values in thecurrent generation numerically at the top. Typically, the best fitnessvalue improves rapidly in the early generation, when the individualsare farther from the optimum. The best fitness values improve moreslowly in later generations (after 300), whose populations are closerto the optimal point. Thus, the GA termination is getting when thecomputation shows no more improvement of the best fitness value

Fig. 10. The cutting pattern strips layout for the model example II.

warpfill

fill

Fig. 11. Initial and optimum cutting p

(500 generations or more). The results from the present study arediscussed in detail through examples in the subsequent section.

4. Numerical examples

In this section, numerical results of structural membrane exam-ples are presented to illustrate the performance of the proposedmethod.

optimum cutting pattern

initial cutting pattern

warp

warp

fill

attern for the model example II.

Fig. 12. Actual stress distributions of a quarter model example II.

Table 3Comparison of membrane stress coefficients for model example.

Strip Results AVR1 AVR2 AVSD1 AVSD2

a Kim and Lee [8] 1.005 1.010 0.043 0.046

Present work 0.990 1.009 0.024 0.022

b Kim and Lee [8] 0.999 0.999 0.043 0.044

Present work 0.981 1.020 0.021 0.022

c Kim and Lee [8] 0.999 1.001 0.158 0.164

Present work 0.988 1.010 0.056 0.052

W. Punurai et al. / Finite Elements in Analysis and Design 58 (2012) 84–90 89

4.1. Example I

This model example was considered in [8,23]. The modelrepresents a more realistic pre-stressed membrane. It is a hyper-bolic paraboloid (HP) type membrane structure with four fixedcorners composed of a fabric textile reinforced by means of cables

Fig. 13. Stress contour plots for strips (a),

around the perimeter of the pre-stressed membrane as shown inFig. 3. According to Fig. 4, the mesh model is comprised of 400membrane elements and 40 cable elements. It is assumed that thedesign target stress is to be equal to 0.49 MPa in both fill andwarp directions. The compensation ratio of 1% is given to theinitial shape-pattern to introduce the pre-stress on each side. The

(b), and (c) of the model example II.

W. Punurai et al. / Finite Elements in Analysis and Design 58 (2012) 84–9090

properties of all materials are listed in Table 1. The results of aninitial shape-analysis, initial cutting pattern and an optimalcutting pattern of a quarter model are shown in Figs. 5 and 6,respectively.

As it can be observed, the change of actual stresses is expecteddue to the curvature variation. Fig. 7 shows the actual membranestress distributions for the fill and warp directions.

For ease of use in comparison with the data obtained in thisstudy and in the literature, membrane stress coefficients aredefined as

AVR1 ¼

Ps011Ps0

11

; AVSD1 ¼

P9s011�s0

119Ps0

11

;

AVR2 ¼

Ps022Ps0

22

; AVSD2 ¼

P9s022�s0

229Ps0

22

, ð4Þ

where subindices 1 and 2 indicate fill and warp directions;s011,s022 are the actual stresses; s0

11,s022 are the design stresses;

AVR1, AVR2 are the average Cauchy stresses; AVSD1, AVSD2 arethe average deviation of Cauchy stresses. It is known that thecomputation is best when the coefficients are getting closer to1 and the AVSD coefficients are getting closer to 0.

Numerical comparisons of stress coefficients with the valuesreported in references [8] and [23] are listed in Table 2. As can beseen from Table 2, the deviations are considerably smaller.

4.2. Example II

A cone shape membrane model example was considered in [8].The model structure shown in Fig. 8 consists of membrane panel,catenary cables in boundaries and ridge cables between thecenter point and fixed boundaries. The material properties arelisted in Table 1. In the analysis, the initial stresses of 3.14 and490 MPa and the compensation ratio of 1% are assigned to pre-tension the membrane and the cables, respectively. The structureis then divided into 10 strips in order to perform the initial shape-analysis and to obtain the target design stresses of 0.49 MPa asshown in Fig. 9. The three strips (a), (b), and (c) are then selectedfor the optimum cutting pattern as shown in Fig. 10. Fig. 11 showsthe extraction of strips, initial cutting pattern as well as theoptimized cutting pattern by the proposed method.

Fig. 12 shows the actual membrane stress distributions for thefill and warp directions. Stress coefficients are also computed asdefined before (Eq. (4)). Thus, the numerical comparisons of stresscoefficients with [8] are listed in Table 3. It can be seen from thetable that the errors for average stresses in the presented methodare considerably smaller than those reported in Ref. [8]. Fig. 13shows the resulting stress contour plots for strips (a), (b), and (c).It is shown that all stresses are well distributed over most of themembrane area. These results showed that the method presentedin this work is feasible and practical.

5. Conclusions

This study proposes a method to determine the optimumcutting pattern to reduce the differences between the assumedand the actual membrane stress. To validate the technique, twomembrane model structures of moderate curvature are analyzed.Results from the analysis indicate that the stresses on theequilibrated state are more uniformly distributed, and the final

stress distributions are sufficiently closer to the assumed designstresses when compared with those of the existing method. Thisshowed that the method presented in this work is feasible andpractical.

Acknowledgments

The authors wish to acknowledge the support from MahidolUniversity, the Society for Industrial and Applied Mathematics(SIAM), Philadelphia, Pennsylvania, and ‘‘Consejo de Ciencia yTecnologıa del Estado de Guanajuato’’ (CONCYTEG), Mexico forpartially funding the visit of Dr. Wonsiri Punurai to Mexico underthe contract 09-02-K662-073.

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