Computer Methods in Applied Mechanics and Engineering
A multiobjective and fixed elements based modification of the evolutionary structural optimization method
Introduction
Many elegant methods for structural optimization are presented in the literature. Short surveys of this field can be found in [47], [27], [41], [46]. Structural optimization is often divided into three classes: sizing, geometrical, and topology optimization [36]. The differences between these classes can be explained by a truss example:
In sizing optimization, the cross-sectional dimensions can be chosen as the design variables. The values of the design variables are not allowed to go to zero, in other words, individual structural members cannot be removed.
In geometrical optimization, the coordinates of the joints can be chosen as the design variables.
In topology optimization, the number of structural members and the connectivity of the members is optimized.
In the case of plate-type structures, the term shape or fixed-topology optimization may be used instead of geometrical optimization. In these cases, the boundary line of a structure may be varied during the optimization process. If additional holes are introduced into the structure, the terms variable-topology and generalized shape optimization may be used [17].
It should be noted that there may be cases in which the above classification does not apply. The terminology concerning the structural optimization classes also varies in the literature. For simplicity, the optimization scheme including all three classes of optimization is also called topology optimization in this paper.
It is characteristic for sizing and geometrical optimization that the topology of a structure cannot be altered during the optimization process. When applying these methods, it is not guaranteed that the structure obtained is the best or even a good one: another initial topology might produce a remarkably better solution for the minimization problem. Since topology optimization also looks for the best overall topology satisfying the problem constraints, the optimization process cannot be misled as easily by a poor initial guess. For this reason, only by using topology optimization it is possible to produce the best overall structure.
Topology optimization was pioneered by [26], who studied statically determinate trusses for a number of loading and support conditions. His analytical results, so-called Michell trusses, have an infinite number of members of varying length. In Michell trusses, each bar is subject to a constant strain (stress). It has also been analytically proved that the Michell truss cannot have any greater compliance to the given load than any other truss using the same amount of material (for linearly elastic material compliance equals twice the work done by the external forces or twice the total strain energy of the structure) [40]. Since the Michell trusses have an infinite number of structural components, they are rather impractical in the engineering applications.
In the 1960s, topology optimization was remarkably improved when the so-called ground structure approach was first introduced [12]. In the ground structure approach, the design domain is formed by a finite number of truss members, and each member is a potential part of the optimal truss. By applying numerical optimization methods, the additional bars can be removed from the design domain, and as a result, the remaining truss members represent the optimal topology.
Originally, ground structure problems were solved by using direct optimization methods, i.e the mathematical programming (MP) algorithms. However, they were, and still are, inefficient in solving large optimization problems. On the other hand, the MP algorithms are well suited for handling all kinds of objective functions and constraints. To solve a realistic optimization problem a rather large design domain has to be employed, and consequently, the MP algorithms may limit the use of the ground structure approach. Instead of the MP algorithms, indirect optimization methods, i.e. optimality criteria (OC) algorithms, can be employed to solve structural problems. In the OC optimization, it is necessary to determine an appropriate criterion on which the optimality of the solution is based. The criterion may be related, for instance, to the structural stresses: it is often assumed that for the least-weight truss each bar is subject to the corresponding allowable stress value. The approach based on the above criteria is also called the fully stressed design (FSD) method [14]. In the fully stressed state each structural component is subject to its maximum/minimum allowable stress value. The allowable stress values may be different for each component. The allowable stresses may also be different in tension and compression, if, for instance, buckling is considered. If the limiting stresses are equal for every structural component, the resulting FSD structure is also equally stressed. Most often in the literature, the equally stressed state is also called fully stressed. Typically, the OC algorithms consist of consecutive, iterative redesign loops, in order to produce, for instance, a fully stressed state. Prager’s contribution to the OC-based topology optimization, starting from the late 1960s, should be especially acknowledged [30]. Compared with the MP algorithms, the OC methods are efficient in large optimization problems, but lack generality in various kinds of minimization problems. OC algorithms are also discussed by Save and Prager [40].
Topology optimization of trusses using both MP and OC algorithms is studied, for instance, in [29], [33], [37], [38], [39].
In the literature, topology optimization is most often applied to truss ground structures based design domains. However, optimization procedures have been developed to deal with general layout optimization problems in which continuum design domains are employed. The topology optimization of such problems is discussed in Section 1.2.2.
Instead of a structural universe of truss members, the design domain can be defined as a finite number of volumes, which can have either material or a void. A mathematical model for this type of problems can be formed by using the finite element method (FEM). For continuum-type design problems the design variables have to be chosen so that they affect both element volumes and stiffnesses; for plane (2-D) problems, element thicknesses or densities can be used. Typically, the design variables are subject to continuous values. Either MP or OC algorithms can be employed to solve continuum-type problems. This topic is studied, for instance, in [3], [52], [15], [44].
Considerable attention has recently been paid to the work of Bendsøe and Kikuchi [5] who first introduced the so-called homogenization method. Here the design domain is constructed from a finite number of cells, each of which can have individual microstructure, and furthermore, each cell can have either material or a rectangular void. The orientation angle and the side lengths of the void are typically chosen as the design variables. The main objective of the homogenization is to determine a mechanical relationship between the microstructure and the material properties of a cell, and for this purpose, the FE method is typically employed. By varying the orientation angle and the side lengths of a void the element stiffness as a function of the orientation angle and the element density (function of side lengths) can be determined. Because of the large number of design variables, OC methods are typically utilized in homogenization based problems. For detailed information of homogenization, the reader is referred to the publications of Olhoff et al. [28], Suzuki and Kikuchi [42], Bendsøe et al. [6], [7] and Bendsøe [4].
Topology optimization has also been applied by using so-called evolutionary optimization algorithms. Generally, evolutionary optimization methods do not have a firm theoretical background, and their convergence is so far unproven. The evolutionary methods imitate natural selection, i.e. the survival of the best, and the evolution observed among living organisms, and this is believed to lead towards an optimal solution. Topology optimization has also been applied by some evolution based algorithms: genetic algorithms, a method based on biological growth, and evolutionary structural optimization (ESO).
Genetic algorithms are based on the theory of natural selection [21], [16]. In genetic algorithms, the properties of each point of the design domain (organism) are expressed by so-called chromosomes, which are represented by genes. Each chromosome symbolizes a possible optimal value of a design variable. At each iteration loop, a group of organisms form a population. By applying different evolution strategies (crossover, mutation, etc.) an attempt is made to obtain a new, better population. The genetic algorithm is continued as long as there is any improvement between two consecutive populations. Genetic algorithms can be applied to minimization problems having various kinds of objective functions and constraints. This approach was also applied to topology optimization, see e.g. [18], [8], [9], [31].
The optimization method based on biological growth mimics the growth of a tree or a bone. The first approach has been applied to shape optimization [19], [22], [23], [24]. The latter method, the so-called soft kill option (SKO), has been applied to topology optimization [1], [2], [25]. SKO is explained by the analogy with biological structures, like bones, which attempt to reach an equally (fully) stressed state. This phenomenon is also called adaptive mineralization that can be observed as an increasing stiffness of the highly stressed portions of bones. SKO also utilizes the FE method in the formulation of a mathematical model of the problem. The SKO optimization consists of consecutive FE analyses, and at each FE analysis, Von Mises stress values are determined for the elements. Based on the ratios calculated by Von Mises stresses and an intuitively chosen reference stress, the elastic modulus of each element is either increased or decreased. The optimization procedure is continued until the material, i.e. elements subject to the high values of the elastic modulus, is clearly concentrated into certain parts of the design domain. The SKO method does not employ a distinct objective function or constraints, except the ones concerning the lower and the upper limits of the elastic modulus used in the analysis.
Xie and Steven [48] introduced an approach called evolutionary structural optimization (ESO) in 1993. Again, this method employs a design domain constructed by the FE method, and furthermore, external loads and support conditions are applied to the element model. ESO is based on the simple idea that the optimal structure (maximum stiffness, minimum weight) can be produced by gradually removing the ineffectively used material (elements) from the design domain. It should be noted that a similar idea had been proposed earlier by Rodriguez-Velazquez and Seireg [34]. The material removal can be carried out by assigning the corresponding elements a relatively small elastic modulus or thickness value, for instance, 1/106 times the initial value [20]. The element removal is typically based on the element Von Mises stresses. The element strain energy based criterion has also been utilized [10]. This iterative ESO procedure is to be repeated until the rejection criterion values of all the elements are within a given range. The ESO optimization has been studied in [35], [49], [50], [13], [51]. Error estimations have been carried out by Chu et al. [11]. The evolutionary optimization methods can be applied to both ground structure problems and continuum-type problems. They are more often used to solve the latter rather than the former.
Considering the engineering aspects, ESO seems to have some attractive features: the ESO method is very simple to program via the FEA packages and requires a relatively small amount of FEA time. Additionally, the ESO topologies have been compared with analytical ones, e.g. Michell trusses, and so far the results are quite promising (see pp. 594–596 in [26], p. 246 in [10], and p. 891 in [48]). The theoretical aspects of ESO have been studied in [43] and it seems that ESO also has very distinct theoretical basis. In [43] it was outlined that the ESO method minimizes the compliance-volume (CV) product of a finite element model. The logarithmic form of the problem can be written as follows:where Wext is work done by external forces, and in this paper, this energy term is called the compliance. V denotes to the total volume of a structure. Element thicknesses or element elastic moduli can be chosen as the design variables. Above, the element thicknesses, tj (j = 1, … , m), are employed. m refers to the number of elements in the design domain, and tmax denotes the maximum thickness value of each element. In ESO the above problem is solved by using the SLP-based (Sequential Linear Programming) approximate optimization method followed by the Simplex algorithm. Practically, at each ESO iteration elements are removed from the design space based on the current gradient vector terms of the objective function: the design variable/variables having the greatest positive gradient term will be rejected.
It is often proposed that if topology optimization is utilized to solve a design problem, the optimization should be done in two separate stages: in the first stage, the overall structure is outlined by applying a topology optimization method, and in the second stage, the sizing optimization can be employed, see e.g. [1], [28].
The standard ESO lacks generality, i.e. no specific stress or displacement constraint can be added into the minimization problem. For the reasons discussed in [43], the optimal solution may still be reached by ESO. Besides, if the two-stage procedure is employed, the sizing of the structural components will not take place until the second stage. Consequently, in that case, there is no actual need to enforce the stress and displacement constraints yet. ESO is obviously well suited to solve the first-stage optimization problems.
The second-stage sizing optimization can be performed independently, regardless of the first-stage optimization approach. Additionally, the objective of the minimization can be changed, if necessary. However, if some of the design constraints are not considered until the second stage, the ESO topology may fail in the sizing optimization. This problem will be studied next.
Section snippets
General
For many design (weight minimization) problems having displacement and stress constraints ESO will produce a good topology to start with. There may still exist some additional constraints which the designer has to consider, and which are not taken into account in any way in ESO. As a consequence, it may happen that the ESO topology does not yield the least-weight structure.
Quite often, a design problem includes some geometrical constraints. For instance, the cross-sectional areas may be limited
General
In this section, the performance of MESO is studied by means of two numerical plane truss examples. For that purpose, the ground structure design domains are utilized and geometrical constraints added into the problems. The cross-sectional areas of the bars are chosen as the design variables. The MESO results are evaluated on the basis of the corresponding ESO results. The objective of both examples is the minimum weight of the final structure. The ANSYS 5.4 program has been employed. The
Conclusions
In this paper, the so-called evolutionary structural optimization method (ESO) was examined. On the basis of the previous studies, ESO was known to minimize the compliance-volume (CV) product of an element model. Consequently, ESO was seen to be analogous with an SLP-based optimization method, called the approximate optimization method. It can be stated that ESO is basically a standard MP algorithm, which just minimizes a particular objective function.
The ESO optimization is very easy to apply.
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