10
Optimization of Topology Reinforce in Flat Panels Edson Bezerra Leal Da Silva Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal. 23 de Novembro de 2009 Abstract This work deals with the development of a computational model for discrete topology optimization of rein- forcements in plate structures with the goal of maximizing the stiffness of the structure. An automatic methodology was developed to optimize the position of a fixed number 1 of reinforcements and a fixed number 1 of holes in the structure, within N (>1) possible positions, defined in a base structure. The optimization is done using a genetic algorithm that can deal with the discrete design variables and the discontinuities of design space. The finite element method is used for the structural analysis required, for the objective function calculation. Optimum topology is obtained from the design domain representation through codified chromosomes, so that the chromosome’s genes characterize the presence and absence of panels in the structure. The present computational model was tested in three case studies, generating different solutions. In order to validate the results, the model was compared with an reference model, based on a plate with uniform thickness and the same amount of material. Analyzing the results, there was a 60% to 90% stiffness improvement for the studied cases when compared with the reference models. Key-words: Topology optimization, genetic algorithms, structural analysis, finite elements method, Reissner- Mindlin plate. 1 Introduction Reinforced plates are widely used in the automoble, aerospace and naval industry. The use of reinforced plates helps to increase the mechanical performance of plates, example, the static and dynamic characteristics as well as reduces the costs associated with the amount of material used. The definition of the topology of the reinforcement in plate type structures is one of the critical phases in reinforced plate project. This has been made based in experience, creativity and rule of thumb or engineer technical judgment and has been used with some success and allowing the satisfaction of the functional requirements of the project. However, in general, this will not produce an optimum reinforcement of the plate. The integration of structural optimization techniques in the project of structural reinforcement of plates, allow us to evaluate all the possibilities of the project in a fast and efficient way in order to guarantee results compared to conventional methods. When applying these techniques in the discrete structural topology optimization, it will be possible to obtain the number, position and the optimum connectivity of elements that compose a specific structure. The topology optimization of reinforced in plates has been extensively studied in literature (see example [2, 4]). This work focus on the Optimization of reinforced plate’s structures submitted to uniformly distributed pressures in order to maximize the stiffness. The objective is to find the position of a fixed number or reinforcement and ”holes” in a plate structure. The methodology was to use a base structure with the presence of all panels and reinforcement. Then a fixed number of reinforcements is imposed. To mathematically model the panels that compose the plates and reinforces, the Reissener-Mendlin theory has been applied. To optimize the plate, genetic algorithms were chosen to take advantage of its characteristics and deal with discrete variables and discontinuous domain projects. Genetics algorithms work with populations of possible solutions that are submitted to a process 1

Optimization of Topology Reinforce in Flat Panels · PDF fileOptimization of Topology Reinforce in Flat Panels ... helpstoincreasethemechanicalperformanceofplates,example ... Interpretation

Embed Size (px)

Citation preview

Page 1: Optimization of Topology Reinforce in Flat Panels · PDF fileOptimization of Topology Reinforce in Flat Panels ... helpstoincreasethemechanicalperformanceofplates,example ... Interpretation

Optimization of Topology Reinforce in Flat Panels

Edson Bezerra Leal Da SilvaInstituto Superior Técnico,

Av. Rovisco Pais, 1049-001 Lisboa, Portugal.

23 de Novembro de 2009

Abstract

This work deals with the development of a computational model for discrete topology optimization of rein-forcements in plate structures with the goal of maximizing the stiffness of the structure.An automatic methodology was developed to optimize the position of a fixed number 𝑁1 of reinforcements anda fixed number 𝑁1 of holes in the structure, within N (𝑁 > 𝑁1) possible positions, defined in a base structure.The optimization is done using a genetic algorithm that can deal with the discrete design variables and thediscontinuities of design space. The finite element method is used for the structural analysis required, for theobjective function calculation. Optimum topology is obtained from the design domain representation throughcodified chromosomes, so that the chromosome’s genes characterize the presence and absence of panels in thestructure.The present computational model was tested in three case studies, generating different solutions. In order tovalidate the results, the model was compared with an reference model, based on a plate with uniform thicknessand the same amount of material. Analyzing the results, there was a 60% to 90% stiffness improvement for thestudied cases when compared with the reference models.

Key-words: Topology optimization, genetic algorithms, structural analysis, finite elements method, Reissner-Mindlin plate.

1 IntroductionReinforced plates are widely used in the automoble, aerospace and naval industry. The use of reinforced plateshelps to increase the mechanical performance of plates, example, the static and dynamic characteristics as well asreduces the costs associated with the amount of material used. The definition of the topology of the reinforcementin plate type structures is one of the critical phases in reinforced plate project. This has been made based inexperience, creativity and rule of thumb or engineer technical judgment and has been used with some success andallowing the satisfaction of the functional requirements of the project. However, in general, this will not producean optimum reinforcement of the plate.The integration of structural optimization techniques in the project of structural reinforcement of plates, allow usto evaluate all the possibilities of the project in a fast and efficient way in order to guarantee results comparedto conventional methods. When applying these techniques in the discrete structural topology optimization, it willbe possible to obtain the number, position and the optimum connectivity of elements that compose a specificstructure.The topology optimization of reinforced in plates has been extensively studied in literature (see example [2, 4]).This work focus on the Optimization of reinforced plate’s structures submitted to uniformly distributed pressuresin order to maximize the stiffness. The objective is to find the position of a fixed number or reinforcement and”holes” in a plate structure. The methodology was to use a base structure with the presence of all panels andreinforcement. Then a fixed number of reinforcements is imposed. To mathematically model the panels thatcompose the plates and reinforces, the Reissener-Mendlin theory has been applied. To optimize the plate, geneticalgorithms were chosen to take advantage of its characteristics and deal with discrete variables and discontinuousdomain projects. Genetics algorithms work with populations of possible solutions that are submitted to a process

1

Page 2: Optimization of Topology Reinforce in Flat Panels · PDF fileOptimization of Topology Reinforce in Flat Panels ... helpstoincreasethemechanicalperformanceofplates,example ... Interpretation

of evolution based in the survival of the strongest to encounter the optimum solution [9]. The optimization processis done by using a toolbox of genetic algorithm from MATLAB and the structural analyze of plates is done usinga commercial software of finite elements called ANSYS.

2 Theorical Background

2.1 Reissner-Mindlin PlatesPlate is a flat structure, where the thickness is much smaller than all the other dimensions. It is geometricallydefined by a medium surface and by the thickness perpendicular to the medium surface. In this work, Reissner-Mindlin’s theory [6, 8] was used for being much closer to the tridimensional model of the elasticity theory.Reisser-Mindlin’s theory is based in the following hypothesis:

1. The transversal displacement of the plate is small compared to its thickness.

2. Normal stresses to the medium surface are neglected (𝜎𝑧𝑧 = 0).

3. The straight lines originally perpendicular to the medium surface remain straight after the deformationoccurs, but, not necessarily perpendicular to the medium surface.

The approximated displacement field is given by [7]:

𝑢(𝑥, 𝑦, 𝑧) = −𝑧𝜃𝑥

𝑣(𝑥, 𝑦, 𝑧) = −𝑧𝜃𝑦

𝑤(𝑥, 𝑦, 𝑧) = 𝑤(𝑥, 𝑦)(1)

where 𝜃𝑥 and 𝜃𝑦 are plane rotations perpendicular to the surface with respect to the x and y axes after deformation.The static equilibrium equations as a function of transversal displacement and rotations are:

𝐸ℎ3

12(1− 𝜈2)

[︂𝜕2𝜃𝑥

𝜕𝑥2+

(1− 𝜈)2

𝜕2𝜃𝑥

𝜕𝑦2+

(1 + 𝜈)2

𝜕2𝜃𝑦

𝜕𝑥𝜕𝑦

]︂+

𝐸ℎ

2𝛼(1 + 𝜈)(𝜕𝑤

𝜕𝑥− 𝜃𝑥) = 0

𝐸ℎ3

12(1− 𝜈2)

[︂𝜕2𝜃𝑦

𝜕𝑦2+

(1− 𝜈)2

𝜕2𝜃𝑦

𝜕𝑥2+

(1 + 𝜈)2

𝜕2𝜃𝑥

𝜕𝑥𝜕𝑦

]︂+

𝐸ℎ

2𝛼(1 + 𝜈)(𝜕𝑤

𝜕𝑦− 𝜃𝑦) = 0

𝐸ℎ

2𝛼(1 + 𝜈)(𝜕2𝑤

𝜕𝑥2+

𝜕2𝑤

𝜕𝑦2− 𝜕𝜃𝑥

𝜕𝑥− 𝜕𝜃𝑦

𝜕𝑦) + 𝑝𝑧 = 0

(2)

2.2 Structural OptimizationStructural optimization is a tool used to design mechanical structures in an efficient way. In other words, structuraloptimization deals with the problem of finding the optimal structure that withstands more efficiently a certainloading. The general approach to formulate structural optimization problems [1] can be expressed by find thevector of variables of project 𝑋 = (𝑥1, 𝑥2, ..., 𝑥𝑛), which

Minimize 𝑓 (𝑋)Subject to 𝑔𝑗 (𝑋) ≥ 0; 𝑗 = 1, ..., 𝑛𝑑

ℎ𝑘 (𝑋) = 0; 𝑘 = 1, ..., 𝑛𝑖

(3)

where 𝑓 (𝑋) is the objective function, 𝑔𝑗 (𝑋) are the inequality constraints, ℎ𝑘 (𝑋) are the equality constraints, 𝑛𝑑

is the number of inequality constraints and 𝑛𝑖 the number of equality constraints.

2.3 Genetic AlgorithmGenetic algorithms (GA) are algorithms that use stochastic search processes, centered on Charles Darwin´s naturalevolution theory based on ”survival of the strongest”. While conventional optimization methods search the solutionfrom a point in search space, GA search the solution from a group of points, randomly generated, called ”population”.

2

Page 3: Optimization of Topology Reinforce in Flat Panels · PDF fileOptimization of Topology Reinforce in Flat Panels ... helpstoincreasethemechanicalperformanceofplates,example ... Interpretation

Each individual of the population, is represented by a chain of characters, which is analogous to a chromossomewhere each character represents a ”gene”. The optimization process using AG see (reference [9]), consists of threebasic operations that operate in an iterative cycle so that the evolution of the population takes place: a) Evaluationof the chromosome ability, b) Selection of chromosomes able for evolution, c) Application of genetic operators ofreproduction to generate a new generation of chromosomes.The flowchart of a traditional genetic algorithm is presented in fig. 1.

No

Yes

Start

Inicialization of Population

Evaluation

Selection

Crossover

Mutation

Create New Population

Converge

Evaluation

End

Figure 1: Flowchart of tradictional genetic algorithms

The evaluation is a measure of performance of the chromosomes, using an evaluation function that attributes anability index to each chromosome based in the value received from the objective function.The Selection process consists in removing random pairs of chromosomes from the population set based in theirability index to form the new generation individuals.One of the existing traditional methods of selection is the roulette, and consists in attributing to each individualof the population a portion of the roulette proportional to its ability [3]. This is the selection method used here.The crossover is the operator responsible for the exchange of genetic material between progenitors, for that reason,crossover is considered the most important operator in the reproduction process. Also it is the most used operatorin genetic algorithm. The crossover is operated in the following way: after selection of two chromosomes, a randomcut off point is also selected. This point is the reference for the chromosome genetic material exchange. In figure2a, there is an example of crossover by a single point. The mutation consists in altering one or more randomly

1 0 1 0 1 1 0 0 0 1

1 1 0 0 1 0 1 1 0 1

1 0 1 0 1 1 1 1 0 1

1 1 0 0 1 0 0 0 0 1

Father

Mother

Child 1

Child 2

Before Crossover After Crossover

(a) Single point Crossover

1 0 1 0 1 1 0 0 0 1Before Mutation

After Mutation 1 0 1 0 1 0 0 0 0 1

(b) Mutation

Figure 2: Genetic operators

selected alleles in the chromosome’s genes. Mutation helps in the optimum search of explored zones of the searchspace, keeping the population’s diversity and avoiding that the solution stays retained in a single optimum local.In figure 2b, there is an application example of chromosome with binary codification.

3

Page 4: Optimization of Topology Reinforce in Flat Panels · PDF fileOptimization of Topology Reinforce in Flat Panels ... helpstoincreasethemechanicalperformanceofplates,example ... Interpretation

3 Optimization of topology of panels using genetic algorithmsThis work studies the problem of topology optimization of a structure in which one seeks the optimum distributionof material in the interior of a domain. Two plates and reinforcements compose the model base of the structureas it is shown in figure 3a. The top and bottom plates and the reinforcements are modeled by flat square platepanels. Consequently the structure is formed by a finite number of cubes where four panel act like reinforcement,the other two belong to the top and bottom plates as figure 3 shows.

Upper Plate

Reinforcement Lower Plate A

(a) Base structural model

Upper Plate Panel

Reinforcemen on the plate panel edge

Lower Plate Panel

A

(b) Representation of a block which forms the struc-ture with panels and reinforcements

Figure 3: Base structure and representation of panels and reinforcements

The objective function, which is equal to the evaluation function, is the minimization of the work of the appliedforces for for given load, and a fixed volume of material.

Minimize {𝑃}𝑇 {𝑑}X

Subjected to [𝐾 (𝑋)] {𝑑} = {𝑃}𝑉 (𝑋) = 𝑉

(4)

where {𝑑} is the displacement vector, {𝑃} the load vector, [𝐾 (𝑋)] is the finite element matrix of rigidity, 𝑉 is thevolume of the structure and 𝑋 = (𝑥1, 𝑥2, ..., 𝑥𝑛) are the design’s variables.In this case the design variables are the panels defined above.

3.1 Problems in study and Design DomainTo find the optimal topology of the structure shown in figure 3, three cases of design space were considered:

∙ Caso I: Reinforcements panelsIn this case, the objective is to determine the topology of the reinforcement panels. The upper and lowerplates do not belong to the design space.

∙ Caso II: Reinforcement panels and holes in the upper plateIn this case, the objective is to determine the topology of the reinforcement panels and the topology of theholes (in the upper plate). The design domain is composed by the reinforcement panels and the panels ofthe upper plate. The number of holes in the upper plate is equal to the number of reinforcement panels.

∙ Caso III: Reinforcement panels by cutting and bending.In this case, the objective is to determine the topology of the reinforcement panels and the topology of theholes (in the upper plate). Reinforcements are obtained by cutting a ”hole” on the upper plate and bendit along one of it’s edges. The panels of reinforcement and the panels of upper plate compose the project’sdomain. The number of holes in the upper plate is equal to the number of panels in the reinforcements.

4

Page 5: Optimization of Topology Reinforce in Flat Panels · PDF fileOptimization of Topology Reinforce in Flat Panels ... helpstoincreasethemechanicalperformanceofplates,example ... Interpretation

3.2 Used techniqueBoth the reinforcements and the plates are modeled by plane panels, and all problems consist in determiningthe optimum position of ”𝑁1” plane panels in ”𝑁 ” possible positions within the design domain. The topology isfounded by allowing the panels to be presents or absent in ”𝑁 ” positions of the structure, with the constraint thatonly ”𝑁1” panels are present in each topology, while the rest ”𝑁 −𝑁1” are absent (with 𝑁1 < 𝑁).In the solution of a topology optimization problem with the AG, variables of the design domain are mapped ina chromosome. Since plane panels physically compose the design domain, a topology is defined a design variablewith N entries of ”ones” and ”zeros”. ”Ones” means that a panel is present, and ” ’zero” means that a panel is absent.Additionally, for GA, a mapping of the chromosomes to the design variable is necessary.

3.2.1 Conversion of the design variables to topology

The vector of design variables has a number of positions equal to the maximum possible number of panels inthe design domain, N, and 𝑁1 < 𝑁 position with ”one” and 𝑁 − 𝑁1 of ”zero” where each position of the vectorcorresponds to a particular panel in the domain. Figure 4 shows an example of a design variable mapped in upperplate. The mapping of the reinforcements is done in a similar way.Since the structural model used is symmetric, the mapping of the design variables was done only by 1/8 of the

1

62

3

4

5

7

8

9

10

1

2

3

4

1

2

3

4

1

2

3

4

6

5

7

6

5

7

6

5

7

8

9

8

9

8

9

10

1010

122

221

1

1

3

33

3

4

44

4

5

55

5

6

6 6

6

7

7 7

7

8

88

8

9

9 9

9

1010

10 10

(a) Mapping of the posi-tions of the vector in theplate

1

1

1

1

1

0

0

1 0 0

1

1

1

1

1

1

1

1

1

1

1

1

1

0

1

1

0

1

1

0

1 0

0

0

0

0

00

00

11 11

11 11

11 11

1 11 1

1 1 0

1 1 0 110

110

0

0

0

0

0 0

0

00

0

0

0

(b) Mapping of the vec-tor’s values in the plate.(1,1,1,1,1,1,0,0,0,0)

(c) Topology

Figure 4: Positions and topology of panels and holes in the upper plate

design domain, thus diminishing the number of variables in the project represented by the chromosomes.

3.2.2 Conversion of the chromosomes in design variables

The chromosomes are mathematically represented by a one-dimensional vector with real codification of the valuesrandomly determined within the interval of [0,1]. The chromosome has N entries, and the biggest 𝑁1 values willbe used to assign ”one” or ”zero” into design variables. The representation of the design variables by the genes isdone in a way so that: a) Each gene of the chromosome corresponds to a position in the design variable (in thefirst and second cases); b) each gene of the chromosome corresponds to five position in the design variables (in thethird case).

3.3 Computational implementationThe process of optimization was done with MATLAB with addition of ANSYS, a commercial software of finiteelements.In MATLAB, a GA toolbox is used to do the optimization. Conversion of population’s chromosomes into vectors of”ones” and ”zeros” is done in a program written in MATLAB environment. In ANSYS, finite elements analyses aredone for the static analysis. ANSYS is executed through command codes written in APDL (ANSYS parametricdesign language) which contains all necessary information to perform the finite elements analysis, writing andcollection of the results.Interaction between MATLAB and ANSYS is written down in the diagram of fig.5.

5

Page 6: Optimization of Topology Reinforce in Flat Panels · PDF fileOptimization of Topology Reinforce in Flat Panels ... helpstoincreasethemechanicalperformanceofplates,example ... Interpretation

Start

Basic parameters definition

Population sizeChromossome sizeCrossover probabilityMutation probability Number of ones in design variable

Random creation of initial population

Chromosome 1Chromosome 2 . . .Chromosome n

Conversion of chromosomes to vectors of 1s and 0s

1. 10010000101002. 1010000101000 . . .n. 1000011000000

Interpretation of APDL code with ANSYS

Creation of topology for each individual of population Attainment of the totalelastic energy for each individual

Objective function evaluation

Maximum number of generation?

Selection, Crossover and Mutation

Creation of new population

Optimal Topology

Yes

No

Ciclo

Figure 5: Scheme of optimization. ANSYS and MATLAB interface

Table 1: Ga parameters and necessary parameters to solve the problem

Parameters Caso I Caso II Caso IIIMaximum generation number: (𝑁𝑔) 50 50 50Chromosome size (Variable numbers): (𝑁) 30 40 10population size (30 *𝑁): (𝑁𝑝) 900 1200 300Number of 1s by chromosome (reinforcement panels): 𝑁1 5 5 5Number of 0s by chromosome(upper plate): 𝑁1 - 5 5Crossover rate: (𝑃𝑐) 0.8 0.8 0.8Mutation rate: (𝑃𝑚) 0.01 0.01 0.01

3.4 Genetic algorithms parameters

The genetic operators used in this work are the roulette selection, the single-point crossover and the uniformmutation that were all described in section 2.3. In table 1 are presented the values of GA parameters used foreach case study. The table also presents the size of chromosome and the value of the variable 𝑁1 that defines: forthe reinforcements, the number of panels (”ones” that the chromosome represents in the project variables) and, forthe plate, the number of holes (panels taken out from the upper plate = ”zeros” that the chromosome representsin the project variables).

4 Solved Examples

For the 3 case problems presented previously, 3 loading condition ( see fig. 6).The values of the loadings applied are 𝑃1 = 𝑃2 = 𝑃3 = 0.1𝑀𝑝𝑎. They were applied in an area of 16*100×100 𝑚𝑚2

for the loading 1 and, in four areas of 4 * 100 × 100 𝑚𝑚2 with symmetrical disposition for the loading 2 and 3.Panels have length and width respectively equal to 𝑎 = 𝑏 = 100 𝑚𝑚, thickness equal to 𝑒 = 10 𝑚𝑚. Structuralmodel to be optimized has 𝐿 = 𝐶 = 1000 𝑚𝑚, of length and width respectively and a thickness of 𝐻 = 100 𝑚𝑚.

6

Page 7: Optimization of Topology Reinforce in Flat Panels · PDF fileOptimization of Topology Reinforce in Flat Panels ... helpstoincreasethemechanicalperformanceofplates,example ... Interpretation

(a) Loading No1: (𝑃1) (b) Loading No2: (𝑃2) (c) Loading No3: (𝑃3)

Figure 6: Model of the base structure with the boundary conditions and applied loads.

The type of finite element used to do the computer modeling of the panels was the plate of 4 nodes and 6 degrees offreedom SHELL181 (ANSYS) used to analyze finite plates and moderately thick (Reissner-Mindlin) plates. Eachpanel was modeled by one finite element. The material properties used for the element are presented in the table2.

Table 2: Material properties on SHELL181 element

PropertyYoung’s modulus 200 Gpa

Shear modulud 77 Gpa

Poisson’s ratio 0.3

Here only the results for loading No1 will be presented. Other loading case can be found in [5].

4.1 Case IThe design domain is constituted only by the reinforcement panels. The finite elements model contains 240 ele-ments in which 40 are part of the design domain and the others are part of the upper and lower plates.The goal is given 5 reinforcement panel, to find for the 30 possible positions to reinforce, which 5 ones maximizethe stiffness of the structure, considering 1/8 of the domain.The search space is constituted by a combination of C30

5 = 30!5!(30−5)! = 142506 possible representations or points in

the search space.The optimization occurs in an evolutional period of 50 generations and in each generation the objective functionis evaluated 900 times, so, there is a total of 45000 chromosomes evaluation.The results obtained are presented in fig. 7. The distribution of elastic energy in the optimal topology is presentedin the figure 7a. and the topology of the reinforcement is presented in fig. 7b.

.269E-07

.828E-07

.139E-06

.195E-06

.250E-06

.306E-06

.362E-06

.418E-06

.474E-06

.530E-06

(a) Distribution of the elastic energy inthe structure

X

Y

Z

(b) Optimal topology of the reinforce

.917E-07

.140E-06

.189E-06

.238E-06

.286E-06

.335E-06

.384E-06

.433E-06

.481E-06

.530E-06

(c) Distribution of the elastic energy inthe inferior plate

Figure 7: Case I. Tridimensional structure with plates and reinforces with four supported corners, and submittedto a loading pressure in the center of the inferior plate.

7

Page 8: Optimization of Topology Reinforce in Flat Panels · PDF fileOptimization of Topology Reinforce in Flat Panels ... helpstoincreasethemechanicalperformanceofplates,example ... Interpretation

.179E-06

.443E-06

.708E-06

.973E-06

.124E-05

.150E-05

.177E-05 .203E-05 .230E-05

.256E-05

(a) Distribution of the elastic energy inthe structure

(b) Optimal Topology ofthe reinforce

MX

(c) Optimal Topology ofthe upper plate

Figure 8: Case II. Tridimensional structure with reinforcement plates and all four corners supported and submittedto uniform load in the inferior plate.

Since, the distribution of elastic energy is greater in the center and diagonals, as shown in figure 7c, it is expectedthat the reinforcements to be allocated on the diagonals in direction to the center. The results confirm this.The optimal topology for this example was found in 13th iteration, which means, only 11700 evaluations were nec-essary to find the best individual. After this phase the chromosomes of the population will appear repeatedly anda significant decrease in computer effort is necessary, since the value of the objective function counter is kept for allchromosomes evaluated. The minimum value of the objective function is 1.1318𝐸−4 𝑁 ·𝑚. A plate model withoutreinforcement with the same material volume to was analyzed, and the work of applied load is 3.23𝐸 − 03 𝑁 ·𝑚.The optimized model has a reduction of 96% in the objective function.

4.2 Case II

In this case for the same domain as in case I and for 1/8 of the model domain, we consider removing 5 panelsfrom the upper plate and use then as reinforcements. Note that, only 10 possible position for reinforcement arepossible. Consequently the search space has C10

5 × C305 = 3.5911𝐸7 points. Since the search space increased, the

population size should also increase and the objective function is calculated 1200 times in each generation makinga total of 60000 evaluations of the objective function.The results are presented on fig. 8. In this example, reinforcements were expected to be placed in the samestructural region as in case I. Thus, the results confirmed the expectations.

4.3 Case III

In this case for the same domain as in case I and for 1/8 of the model domain, we consider removing 5 panels fromthe upper plate, but each panel can only be used as reinforcement in one of its original edges. Then, the searchspace has C10

5 × 45 = 258048 points, allowing the use of less objective function calculation per generation, 300 inthis case making a total of 15000 evaluations.The obtained results are showed in fig. 9

4.4 Elastic Forces Work and Comparison of Results

In order to have a comparison basis, a plate with uniform thickness with the same amount material as the optimizedmodels of the three study cases was analyzed. In table 3, we have presented all elastic force’s work for the threestudy cases and the reference model. Analyzing the results, the greatest value of elastic force work was obtainedfrom the first case as expected because it has more material. The third case showed the smallest elastic force’swork because the work was done in majority by the four panels located in the structure corners (see figure 9a)and no good reinforcement is possible. The results in table 3 show that for all three cases, there was a gain instructural stiffness from the presented model compared with the reference model. Further, case II showed to bethe most efficient.

8

Page 9: Optimization of Topology Reinforce in Flat Panels · PDF fileOptimization of Topology Reinforce in Flat Panels ... helpstoincreasethemechanicalperformanceofplates,example ... Interpretation

.148E-07

.101E-04

.202E-04

.303E-04

.403E-04

.504E-04

.605E-04

.706E-04

.807E-04

.907E-04

(a) Distribution of the elastic energy inthe structure

(b) Optimal Topology ofthe reinforce

(c) Optima Topology of theupper plate

Figure 9: Case III. Tridimensional structure with reinforcement plates and all four corners supported and submittedto uniform load in the inferior plate.

Table 3: Objective function for reinforced plate and reference model

Thickness (𝑚𝑚) Elastic energy (𝑁 ·𝑚)Reference Model Reinforced plate Reference model Gain (%)

Caso I 24 1.132𝐸 − 04 3.23𝐸 − 03 96.5

Caso II 20 1.262𝐸 − 04 5.580𝐸 − 03 97.7

Caso III 20 1.012𝐸 − 03 5.580𝐸 − 03 81.9

4.5 Influence of Symmetry Restriction in the Results Obtained

In the previous examples the use of symmetry allowed the reduction of the chromosomes and the number of individ-uals in the population. In order to asses its influence case I with loading No1 was again considered, but assumingonly 1/4 symmetry and without symmetry for the design variables. As expected this increased significantly thesearch space, and, consequently, the computational effort.We would expect that once we enlarge the domain a better solution would emerge, however this didn’t happen.One way to circumvent this is to introduce the best solution obtained in 1/8 symmetry in the initial population ofthe 1/4 symmetry and without symmetry situation.The basic genetic parameters used in this example where the same from the previews ones except the populationof individuals equal to 1200 and 6600 for 1/4 and all the project domain respectively. The results are presented intable 4 and fig. 10.

Table 4: Objective function for reinforced plate and reference model in restriction’s cases

Thickness (𝑚𝑚) Elastic energy (𝑁 ·𝑚)Reference model Reinforced plate Reference model Gain (%)

1/8 of symmetry 24 1.132𝐸 − 04 3.23𝐸 − 03 96.5

1/4 or symmetry 24 1.161𝐸 − 04 3.23𝐸 − 03 96.4

Whithout symmetry 24 1.191𝐸 − 04 3.23𝐸 − 03 96.3

The results presented in table 4 show that less stiff solutions appear when the symmetry constrain is relaxed.This is exactly the opposite of what was expected. This may be explained by the fact that the genetic algorithmis a probabilistic algorithm, and the results may depend on the initial genetic parameters. It may also occur thatthe obtained solution with 1/8 symmetry is indeed the best solution. One way to asses this is to introduce this

9

Page 10: Optimization of Topology Reinforce in Flat Panels · PDF fileOptimization of Topology Reinforce in Flat Panels ... helpstoincreasethemechanicalperformanceofplates,example ... Interpretation

X

Y

Z

.269E-07.828E-07

.139E-06.195E-06

.250E-06.306E-06

.362E-06.418E-06

.474E-06.530E-06

(a) Topology with 1/8 of design domainsymmetry

X

Y

Z

.513E-07.105E-06

.159E-06.213E-06

.267E-06.321E-06

.375E-06.429E-06

.483E-06.537E-06

(b) Topology with 1/4 of design domainsymmetry

X

Y

Z

.365E-07.124E-06

.212E-06.299E-06

.387E-06.474E-06

.562E-06.650E-06

.737E-06.825E-06

(c) Topology without design domainsymmetry

Figure 10: Tridimensional structure with reinforcement plates and all four corners supported and submitted touniform load in centre of the inferior plate.

solution in the initial population for the 1/4 symmetry and without symmetry optimization problems and verify ifthe GA can find a better solution.

4.6 ConclusionIn this work, a computational model for the topology optimization of the plate reinforcement type structures withthe objective of maximizing the structure’s stiffness was developed. A genetic algorithm was used as optimizationtechnique taking advantage of its characteristics to deal with discrete variables problems and non continuous projectdomain and, the finite elements method for structural analysis associated to the calculus of the objective function.Several types of reinforcements were considered and the model was able to provide significantly better solutions,although the computacional effort was high.

References[1] Jabir S. Arora. Introduction to Optimum Design. Elsevier Academic Press, 2nd edition.

[2] X. Ding and K. Yamakaki. Adaptive growth technique of stiffener layout pattern for plate and shell structuresto achieve minimum compliance. Engineering Optimization, 37(3):259–276, 2005.

[3] D. Goldberg. Genetic algorithms in search, optimization and machine learning. Addson-Wesley, 1989.

[4] Y.C. Lam and S. Santhikumar. Automated rib location and optimization for plate structures. Struct. Multi-discip. Optimiz., 25:35–45, 2003.

[5] E. B. Leal Da Silva. Optimização de topologia de reforços em painéis planos. Master’s thesis, Instituto SuperiorTécnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal, 2009.

[6] R.D. Mindlin. Influence of rotary inertia and shear on flexural motion of isotropic elastic plates. Journal ofApplied Mechanics, 12:A69–77, 1945.

[7] C.A. Mota Soares. Teorias e análises de placas: Métodos analíticos e aproximados. Technical report, 1982.

[8] E. Reissner. The effect of transverse shear deformation on the bending of elastic plates. Journal of AppliedMechanics, 18:31–38, 1951.

[9] S.N. Sivanandam and S.N. Deepa. Introduction to Genetic Algorithms. Springer, 2008.

10