Sequential and Distributed Evolutionary Computations in Structural Optimization Tadeusz Burczy´ nski1,2 , Waclaw Ku´s1 , Adam Dlugosz1 , Arkadiusz Poteralski1 , and Miroslaw Szczepanik1 1
Department for Strength of Materials and Computational Mechanics, Silesian University of Technology, Konarskiego 18a, 44-100 Glwice, Poland
[email protected], 2 Institute of Computer Modelling, Cracow University of Technology Cracow, Poland
Abstract. The aim of the paper is to present the application of the sequential and distributed evolutionary algorithms to selected structural optimization problems. The coupling of evolutionary algorithms with the finite element method and the boundary element method creates a computational intelligence technique that is very suitable in computer aided optimal design. Several numerical examples for shape, topology and material optimization are presented.
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Introduction
Evolutionary methods have found various applications in mechanics, especially in structural optimization [2], [5]. The main feature of such applications is the fact that design process of artificial systems like structural or mechanical components is simulated by biological processes based on heredity principles (genetics) and the natural selection (the theory of evolution). The paper is devoted to structural optimization using sequential and distributed evolutionary algorithms. Solutions of optimization problems are very time consuming when sequential evolutionary algorithms are applied. The long time of computations is due to the fitness function evaluation which requires solution of direct (boundary-value or initial boundary-value) problems. The fitness function is computed with the use of the boundary element method (BEM) or the finite method (FEM) [8]. In order to speed up evolutionary optimization the distributed evolutionary algorithms can be proposed instead of the sequential evolutionary algorithms [4]. This paper is extension of previous papers devoted to optimization using sequential and distributed evolutionary algorithms in thermoelastic problems [3],[7], shape and topology optimization of 2D and 3D structures [6],[10].
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Formulation of the Evolutionary Design
A body, which occupies a domain Ω bounded by a boundary Γ , is considered. Boundary conditions in the form of the displacement and traction fields are prescribed. In the case of dynamical problems initial conditions are also prescribed. L. Rutkowski et al. (Eds.): ICAISC 2004, LNAI 3070, pp. 1069–1074, 2004. c Springer-Verlag Berlin Heidelberg 2004
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One should find optimal shape or topology of the structure by minimizing an objective functional: (1) minJo (Ch) Ch
with imposed constraints: Jα (Ch) = 0, α = 1, 2, .., A; Jβ ≤ 0, β = 1, 2, .., B
(2)
Ch is a vector of design parameters which is represented by a chromosome with floating point representation Ch = [g1 , g2 , .., gi , .., gN ]
(3)
where restrictions on genes are imposed in the form giL ≤ gi ≤ giR , i = 1, 2, .., N
(4)
Genes are responsible for shape, topology and material parameters of the structures. General form of the objective functional Jo and performance functional Jα and Jβ can be expressed in structural optimization as follows J = Ψ (σ,ε,u, T ) dΩ + ϕ(u, p, T, q)dΓ (5) Ω
Γ
where Ψ is an arbitrary function of stress σ, strain ε, displacement u and temperature T fields in the domain Ω, respectively, ϕ is an arbitrary function of displacement u, traction p, temperature T and heat flux q fields on the boundary Γ , respectively. Using the penalty function method the optimization problem (1) and (2) is transformed into non-constrained problem and the fitness function consists of functionals Jo , Jα and Jβ . In order to evaluate the fitness function one should solve the boundary-value problem using FEM or BEM.
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Shape, Topology, and Material Parametrization by Genes
The geometry of the structure is specified by NURBS - Non Uniform Rational B-Spline. Co-ordinates of control points of the NURBS play the role of genes. The distribution of material properties as Young’s modulus E (x) , x ∈ Ω in the structure is describing by a surface W (x) , x ∈ H 2 (for 2-D) or a hyper surface W (x) , x ∈ H 3 (for 3-D). W (x) isstretched under H d ⊂ E d , (d = 2, 3) and the domain Ω is included in H d , i.e. Ω ⊆ H d . The shape of the surface (hyper surface)W (x) is controlled by genes gi , i = 1, 2, .., N, which create the chromosome (3). Genes take values of the function W (x) in interpolation nodes xj , i.e. gj = W (xj ) , j = 1, 2, ..., N . If the structure is discretized by FEM
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the assignation of Young’s moduli to each finite element is performed by the mapping [6]: Ee = W (xe ) , xe ∈ Ωe , e = 1, 2, ..., R (6) It means that each finite element contains different material. When the value of Young’s modulus E for the e-th finite element is included in the interval 0 ≤ Ee < Emin , the finite element is eliminated. Otherwise the finite element remains having the value of the Young’s modulus from this material.
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Distributed Evolutionary Algorithms
Distributed evolutionary algorithms [1],[4],[11] are based on the theory of coevolutionary algorithms. The process of evolution is faster, if isolated subpopulations of small number of interchangeable individuals evolve. In this algorithm a population of individuals is divided into several subpopulations. Each of the subpopulations evolves separately, and from time to time only a migration phase occurs, during which a part of individuals is interchanged between the subpopulations. The distributed evolutionary algorithm works as a few isolated sequential evolutionary algorithms [9] communicating between each other during migration phases. The evolutionary optimization is performed in a few steps (Figure 1).
Fig. 1. The distributed evolutionary algorithm (one subpopulation)
At the beginning the starting population of chromosomes is generated randomly. The floating point representation is used. The population is divided into M subpopulations. Then the fitness function values for every chromosome are computed. The evolutionary algorithm operators as the crossover (simple, arithmetical and heuristic) and the mutation (uniform, boundary, non-uniform and Gaussian) are applied next. When the migration phase occurs, some chromosomes from subpopulations migrate to other subpopulations. The topology of
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the migration decides between which subpopulations the migration occurs. The elementary parameters defining the migration are: the migration frequency, the number of migrating individuals, the cloning or the eliminating an emigrating individual from a subpopulation, the selection of emigrating individuals, the way of introducing immigrating individuals to a subpopulation, the topology of the connections among subpopulations. The migration frequency defines the number of generations after which an individual emigrates. When the migration is too frequent, other subpopulations may be dominated by the solutions achieved in the current subpopulation. If it is too rare, in an extreme situation subpopulations will be isolated and they will evolve independently. The number of migrating individuals defines which part of a population will emigrate. If this number is too big, the algorithm starts to behave like a sequentional algorithm with a single population. The ranking selection creates the offspring subpopulation based on the parent subpopulation modified by evolutionary operators. When selection is performed immigrated chromosomes from other subpopulations are also considered. The next iteration is performed if the stop condition is not fulfilled. The end computing condition can be expressed as the maximum number of iterations or the best chromosome fitness function value. The speedup, which is a measure of increasing of distributed computations, is defined as: ko = ttn1 where tn means the time of performing the computation when n processing units (usually processors) are used, and t1 - the computation time on a single unit.
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Numerical Examples of Evolutionary Design Shape and Topology Evolutionary Design of Thermomechanical Structures
A square plate with a circular void is considered (Figure 2a). For the sake of the symmetry only a quarter of the structure is taken into consideration. The considered quarter of the structure contains a gap bounded by an unknown internal boundary shown in the Figure 2b. The values of the boundary conditions are: T10 = 3000 C, T20 = 200 C, q0 = 0, p0 = 100kN/m, α1 = 1000W/m2 K, α2 = 20W/m2 K, u0 = 0. The model consists of 90 boundary elements. The optimization problem consists in searching an optimal: a) shape of the internal boundary, b) width of the gap, c) distribution of the temperature T 0 ∗ on the internal boundary for minimization of the radial displacements given by the 2n functional (5) with Ψ = 0 and ϕ = (u/uo ) , where u is a field of boundary displacements where tractions p0 are prescribed, u0 is a reference displacement, n is natural number. Shape of the internal boundary was modeled using NURBS curve which consists of 7 control points, whereas width of the gap and temperature T 0 ∗ using NURBS curve consist of 6 control points (Figure 2c). For the sake of the symmetry along line AB (Figure 2c) the total number of design parameters was equal to 13. The range of the variability of each control point for the width of the gap is between 0.2 and 0.8, whereas for the temperature is between 50 C and 800 C. The number of subpopulations was 2 and the number of chromosomes in each subpopulation was 10. Figure 2d shows results of the optimization.
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Fig. 2. Square plate with circular void: a) geometry, b) boundary conditions, c) NURBS curves, d) best result
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Fig. 3. Evolutionary design of a bicycle frame: a) plate and boundary conditions, best result, b) proposed frame; Evolutionary design of 3D structure: c) geometry, d) best result
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Shape, Topology, and Material Evolutionary Optimization
Two numerical examples are considered for this kind of evolutionary design. The first example (Figure 3a,b) refers to evolutionary design of 2-D structures: (a bicycle frame - plain stress). The structures is discretized by triangular finite elements and subjected to the volume constraints. The second example (Figure 3c,d) is the 3-D problem where the body is discretized by hexahedral finite elements and subjected to the stress constraint. Using proposed method, material properties of finite elements are changing evolutionally and some of them are eliminated. As a result the optimal shape, topology and material or thickness of the structures are obtained.
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Conclusions
An effective tool of evolutionary design of structures has been presented. Using this approach the evolutionary process of creating the shape, topology and material is performed simultaneously. The important feature of this approach is its great flexibility for 2-D and 3-D problems and the strong probability of finding the global optimal solutions. Acknowledgements. This research was carried out in the framework of the KBN grant no. 4T11F00822.
References 1. Aleander J.T., An Indexed Bibliography of Distributed Genetic Algorithms, University of Vaasa, Report 94-1-PARA, Vaasa, Finland, (2000). 2. Burczy´ nski T., Beluch W., Dlugosz A., Ku´s W., Nowakowski M. and Orantek P.: Evolutionary computation in optimization and identification. Computer Assisted Mechanics and Engineering Sciences, Vol.9, No. 1 2002, pp. 3-20. 3. Burczy´ nski, T., Dlugosz, A., Evolutionary optimization in thermoelastic problems using the boundary element method. Computational Mechanics, Vol. 28, No. 3-4, Springer 2002. 4. Burczy´ nski, T., Ku´s, W., Distributed evolutionary algorithms in optimization of nonlinear solids. In: Evolutionary Methods in Mechanics (eds. T. Burczy´ nski and A. Osyczka). Kluwer, 2004. 5. Burczy´ nski T., Osyczka A. (eds): Evolutionary Methods in Mechanics, Kluwer, Dordrecht 2004. 6. Burczy´ nski T., Poteralski A., Szczepanik M., Genetic generation of 2D and 3D structures. Computational Fluid and Solid Mechanics (ed. K.J. Bathe), Elsevier, Amsterdam 2003, pp. 2221-2225. 7. Dlugosz, A., Evolutionary computation in thermoelastic problems. In: Evolutionary Methods in Mechanics (eds. Burczynski, T. and Osyczka, A.), Kluwer, 2003. 8. Kleiber, M., (ed.), Handbook of Computational Solid Mechanics, Springer, Berlin, 1998. 9. Michalewicz Z.: Genetic Algorithms + Data Structures = Evolutionary Algorithms, Springer-Verlag, Berlin 1996. 10. Poteralski A., Burczy´ nski T., Szczepanik M., Evolutionary optimisation of 3-D structures. Proc. CMM 2003 15th International Conference on Computer Methods in Mechanics, Wisla, 2003. 11. Tanese R.: Distributed Genetic Algorithms. Proc. 3rd ICGA, pp.434-439, Ed. J.D. Schaffer. San Mateo, USA, (1989).
