Topology optimization using an adaptive genetic …...Topology optimization using an adaptive genetic algorithm and a new geometric representation B. Sid, M. Domaszewski & F. Peyraut

Topology optimization using an adaptive genetic algorithm and a new geometric representation

B. Sid, M. Domaszewski & F. Peyraut Department of Mechanical Engineering, Laboratoire M3M, University of Technology of Belfort-Montbéliard, France

Abstract

A new method for structural topology optimization using a genetic algorithm is proposed in this paper. This method uses a topology representation by Bézier curves with varying thickness material distribution in a finite element model. Two groups of variables are considered: control points of Bézier curves and thickness values for each curve. This new technique avoids the formation of disconnected elements and checkerboard patterns in optimal topology design. New adaptive strategies for crossover and mutation operators are also proposed. The numerical program has been developed in MATLAB and tested on a benchmark 2-D problem of topology optimization. The results obtained are compared with those obtained by the homogenization method and ANSYS code. Keywords: structural topology optimization, genetic algorithms.

1 Introduction

Structural topology optimization is used to find a preliminary structural configuration that meets some predefined criteria. An optimal structural topology can be obtained by the modifications of holes and connectivities of the design domain. The aim is to redistribute material from within so-called reference domain in an iterative manner in order to arrive at a structural topology which is in some criterion optimal. Topology optimization methods have been discussed in a large number of publications. They are also very attractive for many industrial applications in the domain of mechanical and civil engineering.

Generally, there are two fundamental approaches to topology optimization. The first one is based on the deterministic methods where the sensitivity analysis

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Computer Aided Optimum Design in Engineering IX 127

is necessary and the second one uses stochastic strategies. In the first class of methods for topology optimization, there is a very well known method developed by Bendsøe and Kikuchi [1] which is based on a homogenization technique. This method tries to find the microstructure parameters of each element in a finite element model. There are many other deterministic methods based on the sensitivity analysis and gradient concept [2]. All these methods cannot perform a global search and thus do not converge to the global optimal solution.

A new emerging class of structural topology optimization methods uses the genetic algorithms (GAs) which are based on the mechanics of natural selection and natural genetics. They represent an abstraction of biological evolution using the Darwinian principle of survival of the fittest and transmission of genetic information to the future populations. GAs have been introduced by Holland [3] and developed by a large number of researchers. The comprehensive study of Goldberg [4] is one of the fundamental works in this domain. GAs are actually recognized as a powerful tool for stochastic global search methods and find many applications in various domains of engineering, operations research, economics, …etc.

GAs have been also employed in structural topology optimization [5]. Problems of topology optimization are often formulated using the bit-array representation. A bit-array with 1 and 0 represent the two-dimensional design domain discretized by finite elements. Each element contains either material (1) or void (0) and is treated as a binary design variable. This method of representation can produce the topologies which depend on the finite element mesh and contain the disconnected elements and the checkerboard patterns. Another representation technique based on Voronoï diagrams has been proposed by Schoenauer [6]. It is independent on the mesh but does not suppress the disconnected elements and the checkerboard patterns.

In this paper a new geometric representation method proposed by Tai and Chee [7] is adapted and extended to 2-D structural topology optimization using genetic algorithms. This method develops a model representation by Bézier curves with varying thickness. Two groups of variables are considered: control points of Bézier curves and thickness values for each curve. This technique avoids the formation of isolated elements and checkerboard patterns in the finite element mesh of the design topologies during the evolution of populations in all the generations. New adaptive strategies for crossover and mutation operators are also proposed in this paper. The numerical program developed in MATLAB has been tested on a 2-D benchmark of topology optimization. The results are compared with those obtained by ANSYS program and by homogenization method [8].

2 Topology optimization by GAs

GAs operate with a population of N chromosomes (individuals or potential solutions). The initial population is randomly generated at the beginning of the algorithm and evolves into future generations by applying the three genetic


128 Computer Aided Optimum Design in Engineering IX

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F

Design domain

(a) (b)

operators: selection, crossover and mutation, in each generation. The selection operator chooses those chromosomes in the population that will be allowed to reproduce, and on average the fitter chromosomes produce more offspring than the less fit ones. The crossover operator exchanges subparts of two chromosomes, roughly mimicking biological recombination between two single-chromosome organisms. The mutation operator randomly changes the value of the bit in some locations in the chromosome. The aim of these genetic operators is exploration and exploitation of the admissible solutions and generation of a new population which will have better performances than the populations in the previous generation. The evaluation of each chromosome (topology solution) is performed by finite element analysis according to the objective function (minimum compliance). The iterative process is stopped when there is no improvement in the successive populations or when the prescribed maximum number of generations (iterations) is attained.

2.1 Mechanical problem

The design domain (initial structure to be optimized) is a continuous 2-D solid composed of elastic, homogeneous, isotropic material. This domain is discretized and the static solicitations and boundary conditions are applied (Fig.1). Topology optimization consists in choosing a redistribution of material in each element according to the objective function.

Figure 1: (a) mechanical problem, (b) finite element mesh.

2.2 Formulation

The principle of GAs is a search of the fittest individual (global optimum solution) in the population. If the optimization problem is to maximize the objective function then the fitness function for evaluation of each individual is the same as the objective function. For the minimization problem, the fitness function is defined as the reciprocal objective function. The optimal solution should satisfy some constraints of equality or inequality type. Finally, the optimization problem is stated in the following form

Maximize: f(x)1

Fitness(x) =



Stress concentration

regions

Sf and Su Support

regions ∩

a) b) c)

subject to: g j (x) ≥ 0 j =1,…, J hk (x) = 0 k =1,…, K. Using the penalty methods [9, 10] this problem is transformed into the following form

where Ω is an admissible domain.

2.3 Encoding the topology model

The way in which candidate solutions are encoded is a central factor in the success of a genetic algorithm. Binary encodings are the most common encodings for a number of reasons. Much of existing GA theory is based on the assumption of fixed-length, fixed-order binary encodings. Much of that theory can be extended to apply to non-binary encodings, but such extensions are not as well developed as the original theory. In addition, heuristics about appropriate parameter settings (crossover and mutation rates) have generally been developed in the context of binary encodings. In this paper, it is most natural to use real numbers to form chromosomes, so a real-valued encoding is used.

Figure 2: (a) stress concentration regions, (b) loading and boundary condition regions, (c) support regions.

Some modifications of the topological model developed by Tai and Chee [7] are proposed in this paper. The support regions including loading, boundary condition and stress concentration zones are defined in the initial design (Fig. 2).

max1

1 ,( )

( ) 1 ,max[0., ( )]

J K

jj

if xf x

Fitness xotherwise

f g x+

=

∈ Ω

=

+ −

∑



0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Curve N°2

Curve N°1

Curve N°3

(b) Topology (a) Skeleton

In the skeleton: endpoints In the topology: elements of skeleton

Thickness elements

Control point

Indirect joining point

The cubic Bézier curves form a skeleton and connect directly or indirectly the support regions. An indirect connection means that one endpoint of a Bézier curve is situated on the other curve and can move along this curve (Fig.3a). Another extension of the method proposed by Tai is that the endpoints of Bézier curves placed in the boundary condition zones are not fixed and can also move. Each Bézier curve has a varying thickness which can be affected to one or two sides of the curve (Fig.3b). All these modifications enlarge the exploration design space of admissible solutions and allow obtaining complex topologies and preventing a premature convergence of the genetic algorithm.

Figure 3: Topology representation.

An example of real-valued encoding of a structural topology model is shown in Table 1.

Table 1: Encoding a topology model.

Endpoint N°1

1st thickness

value

1st control point

1st thickness

type

1st segment

Endpoint N°2

Curve N°1 1 3 200 2 1/4. 999 Curve N°2 999 4 997 3 1/2. 345

Curve N° 3 345 0 321 1 2/3. 0.5

2.4 Selection

The so-called “Stochastic Universal Sampling” method is used in this paper. The probability of selection of each individual is proportional to the performance calculated in function of its rank in the population. According to a linear version of this method proposed by Baker [11] a selection pressure of two is applied in



genetic algorithm. The performance of an individual of rank ri is calculated in the following way

1N

N)i(r2)i(rrFitness

−

−⋅−=

This selection technique is one of the approaches of “Sampling Mechanisms” and ensures the maintaining of the population diversity.

2.5 Adaptive crossover and mutation

The exploitation is confided to the operators of crossover and selection, while the exploration is done by initialization and mutation. An adaptive technique for crossover and mutation operators is proposed in this work. The aim of this technique is to adjust the probability of crossover (Pc) and mutation (Pm) on the basis of statistics in each population. These probabilities depend on the average genetic diversity, the rank of individuals in the population as well as on the gap between maximum and average performances. The principle of uniform crossover is used in the developed genetic algorithm. The classical mutation operator based on the diversity of each variable is applied at the beginning of the algorithm, and then it is replaced by a uniform mutation which consists in small perturbation of the individual concerned. At the beginning of the algorithm, a linear distribution of Pc and Pm is assumed. After some generations this distribution becomes uniform.

3 Example

The developed program is tested using a standard benchmark of structural topology optimization problems [2]. This benchmark involves a clamped deep beam with a single load at the centre of the right edge (Fig. 4a). For fixed values of the applied force and the horizontal length L, the optimal height can be found analytically as H=2L, if the frame is assumed to be pin-jointed (Fig. 4c). The design domain (L=10m, H=24m) is discretized using a mesh of 25x60 four-node square plane stress elements. The elastic material parameters are assumed to be: Young’s modulus E=100 GPa, Poisson’s ratio υ=0.3 and thickness t=1 mm. The load is F=400 N. The objective function is the compliance fTu which should be minimized (f is a vector of applied loads and u is a vector of degrees of freedom). The limitations are concerned with the volume of the optimal structure which should be no grater than 20% of the initial volume. The geometric model of this example is shown in the Fig. 4b. Two cubic Bézier curves connect two regions of the fixed boundary conditions with a loading point. The endpoints of the curves can move along the clamped edge. These two Bézier curves represent a skeleton of the topology. To complete this topology some layers of elements should be placed along each curve. The topology of the structure is controlled by the four points of each Bézier curve and by the type and value of thickness of



F

Support regions

Bézier curve

(b)

L

H F

(c)

L

2.4L

F

(a)

each layer which are encoded in chromosomes for genetic algorithm. The thicknesses are increasing during the generations and are stabilized when the genetic algorithm is converged. This reduces the computational cost significantly in comparison with other methods where the entire initial finite element mesh must be analyzed in each population.

Figure 4: (a) design domain, (b) geometric scheme, (c) reference solution.

For this example, the developed program in MATLAB has been applied. An initial population of 100 individuals is generated by random at the beginning of the genetic algorithm. After less than 100 generations, the algorithm converged and the optimal topology obtained is shown in the Fig. 5a which is the same as the reference solution (Fig. 4c). The obtained solution is also compared with the results given by the homogenization method (Fig. 5b) and by ANSYS code

Figure 5: Solutions by: (a) GAs, (b) homogenization, (c) ANSYS.

4 Conclusions

A geometric model using cubic Bézier curves with varying thickness for 2-D structural topology optimization has been presented in this paper. The proposed

(a) (b)

(c)



(Fig. 5c).

model ensures a correct connectivity of elements in finite element mesh and avoids the formation of isolated elements and checkerboard patterns. The genetic algorithm with adaptive crossover and mutation operators has been developed in MATLAB. The real-valued encoding was used in this work. The developed program has been tested on a well-known benchmark in topology optimization concerning a clamped deep beam. The obtained results are in good concordance with those obtained by homogenization method and by ANSYS code. The presented method for structural topology optimization of 2-D elastic structures in plane stress state is also promising for nonlinear structures. In fact, the classic techniques of topology optimization using the density variables are ineffective because of ill conditioning of the tangent stiffness matrix concerning the elements with small densities. This produces numerical problems in finite element nonlinear solvers. In the proposed topology representation model all the skeleton and thickness elements have a real material density. Another advantage of the proposed method is a reduction of computational cost when comparing with other classical techniques of topology optimization where the elements are rejected during the iterations but the entire finite element mesh should be analyzed. In the presented method only the skeleton and thickness elements constituting a valid topology are analyzed for evaluating the individuals in each population.

References

[1] Bendsøe, M.P and Kikuchi, N., Genereting optimal topologies in structural design using a homogenization method. Comput Meth Appl Mech Eng, 71, 197-224, 1988.

[2] Bendsøe, M.P. and Sigmund O., “Topology Optimization: Theory, Methods and Applications”, Springer-Verlag, New York, 2003.

[3] Holland J., Adaptation in Natural and Artificial Systems, Ann Arbor, University of Michigan Press, 1975.

[4] Goldberg, D.E., Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, 1989.

[5] Jakiela, M., Chapman, C., Duda, J., Adewuya, A. and Saitou, K., “Continuum structural topology design with Genetic Algorithms” Comput. Methods Appl. Mech. Engrg, 186, 339-356, 2000.

[6] Schoenauer, M., “Shape representation for evolutionary optimization and identification in structural mechanisms” in Genetic Algorithms in Engineering and Computer Science, G. Winter, J. Periaux, M. Galain and P. Cuesta, Eds. Chichester, John Wiley, pp. 443-464, 1995.

[7] Tai, K. and Chee, T.H., “Design of structures and compliant mechanisms by evolutionary optimization of morphological representations of topology” ASME Journal of Mechanical Design, 122, 560-566, 2000.

[8] Sigmund, O., “A 99 line topology optimization code written in Matlab” Struct. Multidisc. Optim., 21, 120-127, 2001.

[9] Coello, C.A., “Theoretical and numerical constraint-handling techniques used with Evolutionary Algorithms: A survey of the State of the Art”



Computer Methods in Applied Mechanics and Engineering 191(11-12), 1245-1287, 2002.

[10] Deb, K., “An efficient constraint handling method for genetic algorithms” Computer Methods in Applied Mechanics and Engineering, 186(2-4), 311-338, 2000.

[11] Baker, J.E., “Reducing bias and inefficiency in the selection algorithms” in Proceeding of the Second International Conference on Genetic Algorithms and their Applications, New Jersey, USA, pp. 14-21, 1987.



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Topology optimization using an adaptive genetic …...Topology optimization using an adaptive genetic algorithm and a new geometric representation B. Sid, M. Domaszewski & F. Peyraut