Micro-scale truss optimization using genetic algorithm
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DOI: 10.1007/s00158-010-0603-x
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- Prendes-Gero, M.B. & Drouet, J. Struct Multidisc Optim (2011) 43: 647. doi:10.1007/s00158-010-0603-x
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Abstract
This paper describes the development of a genetic algorithm that is capable of optimizing the mass of micro-scale trusses. Belonging to the group of periodic cellular materials, micro-scale trusses are characterized by the creation of a base cell with a pattern that is repeated in space until a global structure is obtained. Investigation in this field has generally been focused on the design of base cells and their resistance once the final structure is obtained. In this project we have attempted to optimize each individual cell and in particular its elements according to the loads and boundary conditions applied to the global structure. With this objective, we defined a dichotomic search algorithm that establishes a set of cross-sectional areas suitable for the micro-scale truss, formulated the penalty coefficient for the over-sized elements, and studied the clones and rebirth process in order to avoid stagnation of the genetic algorithm. The cell elements used in this project were equal to or less than to 1 mm long, with a cross-sectional area in the order of 10^{ − 9} m^{2}.
Keywords
Micro-scale trussGenetic algorithmInitializationPenalty coefficient1 Introduction
The objective of this study is to present an approach based on an elitist genetic algorithm (GA) capable of minimizing the mass of a micro-scale truss. The micro-scale trusses considered in this paper were built from two-force bar members with a typical unit cell size of 1 mm.
The optimization of truss structures using a GA has been studied many times in the past (Deb and Gulati 2001; Galante 1996; Domínguez et al. 2006; Gil and Andreu 2001; Greiner et al. 2004; Prendes-Gero et al. 2005, 2006; Tang et al. 2005). However, because these studies deal with large rather than micro-scale trusses, the approaches described therein cannot be directly applied to micro-scale truss optimization. In all the papers on the optimization of large trusses, the set of cross-sectional areas is known beforehand (Deb and Gulati 2001; Kelesoglu 2007) and in some cases the values in the set are standard and come from a construction code (Domínguez et al. 2006; Galante 1996; Greiner et al. 2004; Prendes-Gero et al. 2005). In micro-scale truss optimization, we cannot choose from among standardized values of cross-sectional areas and are thus confronted with the problem of selecting an adequate set of cross-sectional areas for a given micro-scale truss configuration and boundary conditions.
With micro-scale trusses, the relative difference in mass between two individual parts can be very small (<1%). In order to maintain population diversity, clone studies must be carried out in the chromosome; although this has been employed in some research work in the past (Deb and Gulati 2001; Greiner et al. 2004), there is no description of clone study implementation. In these studies, the binary chains were compared bit to bit, but the genetic information remains to be investigated.
At minimum, a second selection criterion is required to enable selection of the best individual. This criterion is the penalty weight of the oversized structures. There is a dual objective to optimization: to obtain trusses with the lowest possible mass that can support the applied loads, and to obtain highly loaded truss members.
In the case of rebirth, and contrary to the majority of the published studies where with each rebirth the search space is reduced (Galante 1996; Greiner et al. 2004), a single rebirth process must be implemented (Goldberg 1989). In the single rebirth, the best individual obtained from the previous evolution is retained and a new process of optimization is carried out: selection of the set of cross-sectional areas as proposed by the authors makes this set highly suitable for the studied problem; the GA makes full use of the set and it is therefore impossible to reduce the initial search space. In addition, a convergence criterion must be used to take into account the lack of improvement in the results with successive rebirths.
- 1.
Selection of a set of cross-sectional areas based on micro-scale truss configuration and boundary conditions.
- 2.
Use of a new formulation to achieve the penalty coefficients for oversized elements.
- 3.
Analysis of the genetic information from clones, instead of using the result of a modified objective function.
- 4.
Use of a single rebirth without a reduction in search space.
- 5.
Use of a convergence criterion that takes into account stagnation of the algorithm after several rebirths.
2 General formulation of the problem
3 Application to micro-scale trusses
Since 1970 when Holland (1992) first developed genetic algorithms, thousands of examples on structural optimization have been implemented. Nevertheless, in the development of micro-scale trusses, modifications to the optimization process are needed to ensure that they work correctly with these very small structures.
3.1 Initialization
One of the main differences between the optimization of larger-sized trusses and the optimization of micro-scale trusses is the selection of the set of cross-sectional areas. In the first case, the set of cross-sectional areas can be obtained from commercial catalogues (discrete variables), and this is a known entity at the beginning of the optimization process (Domínguez et al. 2006; Galante 1996; Prendes-Gero et al. 2005, 2006). In the case of micro-scale trusses, the cross-sectional area can be considered a continuous variable and the range is not known beforehand.
3.2 Evaluation
In general, the penalty coefficients applied in structural optimization only concern the individuals for which the stress value is higher than the maximum allowable stress, and these coefficients present values that are so high as to rapidly eliminate individuals from the population (Deb and Gulati 2001; Galante 1996; Gil and Andreu 2001; Greiner et al. 2004).
However, in addition to the sub-sized elements participating in the calculation of the structures, oversized elements are also present (Prendes-Gero et al. 2005). Structures with minimum weight must be obtained and the stress level in each element must be close to the maximum allowable stress. Moreover, certain criteria are required to help select the best structure from the final population in the case where many structures in that population have the same weight.
3.3 New populations and convergence criteria
The individuals from new populations, with the exception of the initial one from which the individuals were randomly generated, are created from the best individuals in the previous population. The selector known as “aptitude” is used (Mahfouz et al. 1998).
Selection takes into account population dispersion, and eliminates individuals with low aptitudes to prevent them from having descendants. The survivors are ordered according to their probability of rejection so that the individuals with better aptitude, i.e. a lower weight, will have a smaller probability of rejection.
- 1.
Elite individuals: These are selected from among the best individuals in the current population, with no mutation and whose number, N_{e}, is the result of multiplying the elite probability P_{e} by the population size N_{p}.
- 2.
Crossover individuals: These are selected from the surviving individuals as a function of their probability of rejection, with mutation and whose number, N_{c}, is the result of multiplying the crossover probability P_{c} by the population size N_{p}.
Since each chromosome represents a cross-sectional area defined within the set of cross-sectional areas, the one-point crossover (Domínguez et al. 2006; Galante 1996; Kulkarni et al. 2004) or the two-point crossover (Kelesoglu 2007; Tang et al. 2005), can only be employed when the number of sections in the set is similar to the maximum number of sections defined from the number of bits n_{b}. In the other case, the crossover could lead to a cross-sectional area that does not exist in the set. To avoid this, we have selected the crossover points from among the ones located between phenotypes (Prendes-Gero et al. 2005, 2006).
- 3.
Random individuals: The remaining new individuals are selected and mutated from the best individuals in the current population. This group is empty if the total of the elite probability P_{e} and the crossover probability P_{c} are 100%.
The creation of new populations in which the best individuals survive from one generation to the next allows us to increase the rate of convergence and to keep the best genetic information from the previous populations. However, when the number of elite individuals is very high, rapid stagnation of the algorithm can occur, leading to poor results or a local minimum.
To avoid this situation, the implemented GA works with an elite probability below or equal to 30%, and duplicated individuals are eliminated through clone studies (Deb and Gulati 2001; Galante 1996). When a new individual is generated, it is compared with the individuals that are already present in the population and if it is essentially identical to them, it will be discarded and a new individual will be generated.
In this case, the study of clones compares the discrete elements in the chromosomes of the individuals and not their weight, because different individuals with different genetic information can nonetheless have the same weight.
Rebirth is, along with the study of clones, one of the mechanisms employed to maintain population diversity (Goldberg 1989). Even when multiple variations have been implemented (Galante 1996; Greiner et al. 2004; Whitley et al. 1991), the objective is to reinitiate the optimization process once convergence has taken place and to preserve the best individual from the previous evolution. This individual is allowed to keep the previous knowledge, and the randomly generated remaining individuals are provided with a new space to work and evolve. The search space is not reduced, because the GA has a great capacity to adapt to the set of sections, employing most of the sections in the set, due to the way they are defined.
- 1.In the first case, partial convergence takes place when:
- (a)
The best individual from the optimization process has not been modified in the past 10 generations.
- (b)
The number of generations has reached 150.
- (a)
- 2.
In the second case, global convergence takes place when some of the previous criteria are fulfilled and when the best individual from the optimization process has not been modified in the past five rebirths.
4 Case studies
In the last 10 years, micro-scale truss studies have focused on three aspects: manufacturing processes, material used, and the behavior of the structure when stress is applied to it (Gordon et al. 2008; Jacobsen et al. 2008; Wadley 2006). However, all of these have treated the micro-scale truss as a group of unit cells. The new manufacturing processes allow us to work not only with cells, but also with truss bars and to study the dimensions these elements must present in order to fulfill the normal stress (Jacobsen et al. 2008).
4.1 Analyzed structure
4.2 Penalty coefficients
- 1.
c_{p1}, in this case the penalty coefficient is null according to most of the works published on the application of genetic algorithms in structure optimization.
- 2.
c_{p2}, penalty coefficient is defined by (6) and the power of the divisor p takes the value 8 according to (7).
- 3.
c_{p3}, is similar to the previous one but with a value of 4 for the power p.
Figure 5 shows that the behavior of the genetic algorithm is very similar to the three penalty coefficients. For coefficients of set c_{s} between 0.1 and 0.2, the mass of the structure has a very important reduction. From this point (c_{s} = 0.2), the reduction in mass is continuous but smoother until it reaches the minimum values with a coefficient of set c_{s} of 0.6. Finally, values equal to or greater than 0.8 produce structures that cannot support the stress they are subjected to.
On the other side, the comparison of the three curves shows that for the penalty coefficient c_{p3}, the real mass of the structures is always higher than the real mass obtained with the coefficients c_{p1} and c_{p2}. In this case, the penalty coefficient c_{p2} achieves better results than c_{p1} when it works with coefficients of set c_{s} smaller than 0.3, whereas its behavior is similar for values higher than 0.3.
The subpar behavior of the penalty coefficient c_{p3} is produced by an imbalance between the two terms on the left hand side of (3). The penalty term presents values far superior to the real mass of the structure, and the genetic algorithm focuses on this second term in an attempt to reduce the value of the modified objective function. However, the coefficient c_{p2} presents values of the penalty term with an order of magnitude in the order of the real mass. In this case, the genetic algorithm looks for structures with low mass and low penalties, avoiding the oversized structures.
4.3 Number of bits
One of the most innovative features of this study is the use of a set of cross-sectional areas adapted to the structure. This is contrary to the majority of previous studies on structural optimization where a fixed set of cross-sectional areas is used without prior analysis of suitability for the proposed problem.
The set of cross-sectional areas depends on the truss for optimization, because the truss topology as well as the boundary and loading conditions are used to define the section of reference A_{r}. But it also depends on the number of bits n_{b} since this number determines the number of terms in the set. In this study, 2-, 3- and 4-bit phenotypes were considered with the goal of finding the most appropriate number of bits for the genetic algorithm. The set of cross-sectional areas has four, eight and 16 different terms, respectively. As with the study of the penalty coefficient, we took into account a probability of mutation P_{mut} of 0.02, a probability of crossover P_{c} of 0.70, a probability of elite P_{e} of 0.30, a population size of N_{p} of 20 individuals, and the c_{p2} penalty coefficient for the oversized elements.
Real mass and penalty term for different number of bits n_{b}
Coefficient of set (c_{s}) | Real mass (kg) | Penalty term | ||||
---|---|---|---|---|---|---|
2-bit | 3-bit | 4-bit | 2-bit | 3-bit | 4-bit | |
0.1 | 5.78 × 10^{ − 7} | 1.96 × 10^{ − 6} | 4.25 × 10^{ − 7} | 4.44 × 10^{ − 7} | ||
0.2 | 2.24 × 10^{ − 7} | 2.23 × 10^{ − 7} | 4.23 × 10^{ − 7} | 4.04 × 10^{ − 7} | ||
0.3 | 1.58 × 10^{ − 7} | 1.90 × 10^{ − 7} | 4.04 × 10^{ − 7} | 3.80 × 10^{ − 7} | 3.94 × 10^{ − 7} | 4.83 × 10^{ − 7} |
0.4 | 1.44 × 10^{ − 7} | 1.89 × 10^{ − 7} | 3.95 × 10^{ − 7} | 3.41 × 10^{ − 7} | 3.76 × 10^{ − 7} | 3.98 × 10^{ − 7} |
0.5 | 3.15 × 10^{ − 7} | 1.50 × 10^{ − 7} | 3.73 × 10^{ − 7} | 3.50 × 10^{ − 7} | 3.68 × 10^{ − 7} | 3.96 × 10^{ − 7} |
0.6 | 2.56 × 10^{ − 7} | 1.42 × 10^{ − 7} | 1.26 × 10^{ − 7} | 6.11 × 10^{5} | 3.55 × 10^{ − 7} | 3.51 × 10^{ − 7} |
0.7 | 2.40 × 10^{ − 7} | 1.61 × 10^{ − 7} | 1.60 × 10^{ − 7} | 8.03 × 10^{5} | 3.66 × 10^{ − 7} | 3.56 × 10^{ − 7} |
0.8 | 2.11 × 10^{ − 7} | 2.97 × 10^{ − 7} | 1.63 × 10^{ − 7} | 9.17 × 10^{5} | 1.49 × 10^{5} | 3.57 × 10^{ − 7} |
0.9 | 3.11 × 10^{ − 7} | 2.31 × 10^{ − 7} | 2.68 × 10^{ − 7} | 9.57 × 10^{5} | 8.36 × 10^{5} | 3.60 × 10^{ − 7} |
In Table 1, the values written in cursive script correspond to structures that are not able to support the effort they are subjected to, and the blank spaces correspond to sets of cross-sectional areas with terms so small that it is impossible to perform calculations using the Code Aster solver. From this table, the minimum real mass and penalty term are found for 4-bit binary chains. However, we chose to work with a 3-bit phenotype for two reasons: (1) to reduce computation time (7 h for 3-bit vs 15 h for 4-bit), (2) to keep the size of the set of cross-sectional areas (8 for 3-bit vs 16 for 4-bit) at a realistic level considering that too many different sections increase both the complexity and the cost of the micro-scale truss construction process (Gordon et al. 2008; Jacobsen et al. 2008; Wadley 2006).
4.4 Probability of mutation
One of the most controversial parameters in the optimization of structures using genetic algorithms is the probability of mutation P_{mut}. Most authors consider its influence over the results to be very small, so they work with values below 5% (Domínguez et al. 2006; Kelesoglu 2007). Others prefer working with higher values to increase the search space (Deb and Gulati 2001).
In order to determine the most appropriate probability of mutation P_{mut} on the implemented genetic algorithm, the structure was optimized using probabilities of mutation P_{mut} going from 0.01 to 0.10 with 0.01 increment. This study has been done with a probability of crossover P_{c} of 0.70, a probability of elite P_{e} of 0.30, a population size N_{p} of 20 individuals, phenotypes with 3-bit binary chains, and a coefficient of set c_{s} of 0.6.
Weights and computation time for different probabilities of mutation P_{mut}
Probability of mutation (P_{mut}) | Real mass (kg) | Penalty term | Total weight | Computation time (min) |
---|---|---|---|---|
0.01 | 1.39 × 10^{ − 7} | 3.57 × 10^{ − 7} | 4.96 × 10^{ − 7} | 566 |
0.02 | 1.42 × 10^{ − 7} | 3.55 × 10^{ − 7} | 4.98 × 10^{ − 7} | 463 |
0.03 | 1.40 × 10^{ − 7} | 3.58 × 10^{ − 7} | 4.98 × 10^{ − 7} | 401 |
0.04 | 1.43 × 10^{ − 7} | 3.56 × 10^{ − 7} | 4.98 × 10^{ − 7} | 508 |
0.05 | 1.43 × 10^{ − 7} | 3.55 × 10^{ − 7} | 4.98 × 10^{ − 7} | 693 |
0.06 | 1.48 × 10^{ − 7} | 3.56 × 10^{ − 7} | 5.04 × 10^{ − 7} | 528 |
0.07 | 1.53 × 10^{ − 7} | 3.58 × 10^{ − 7} | 5.11 × 10^{ − 7} | 460 |
0.08 | 1.58 × 10^{ − 7} | 3.59 × 10^{ − 7} | 5.17 × 10^{ − 7} | 434 |
0.09 | 1.86 × 10^{ − 7} | 3.67 × 10^{ − 7} | 5.53 × 10^{ − 7} | 247 |
0.10 | 1.83 × 10^{ − 7} | 3.62 × 10^{ − 7} | 5.45 × 10^{ − 7} | 205 |
From this point (P_{mut} = 0.01), the computation time decreases until it reaches a minimum for a probability of mutation P_{mut} of 0.03 which corresponds to a higher real mass and penalty term. At this point the time of computation again increases until a probability of mutation P_{mut} of 0.05, but in this case only the penalty term is reduced. For probabilities greater than 0.05, the time of computation decreases and both the real mass and penalty term increase. It is therefore preferable to work with low probabilities of mutation because they produce better micro-scale trusses.
4.5 Size of population
Most of the published studies on structure optimization use large populations (Deb and Gulati 2001) to improve the diversity of the search space. However, an increase in population size N_{p} leads necessarily to an increase in computation time. Some studies have analyzed the influence of the population size N_{p} over the results and show that it is possible to define a range of the population size N_{p} where the behavior of the genetic algorithm is better (Greiner et al. 2004; Prendes-Gero et al. 2005; Goel et al. 2010).
To test this point, the structure has been studied with different population sizes N_{p}, a probability of crossover P_{c} of 0.70, a probability of elite P_{e} of 0.30, phenotypes with binary chains of 3 bits, a coefficient of set c_{s} of 0.6 and a probability of mutation P_{mut} of 0.01 according to the previous studies.
This situation can be explained by setting up a clone study. Using this study, the stagnation of the genetic algorithm is avoided and the work space is increased; therefore the influence of population size N_{p} decreases until it becomes nearly null in the abovementioned range.
4.6 Rebirth influence
Results of a typical converged solution for a maximum allowable stress S = 4.20 10^{8} Pa
Element | Cross-sectional area (m^{2}) | Length (m) | Axial load (N) | Normal stress σ (Pa) | σ/S (%) |
---|---|---|---|---|---|
1 | 1.21 × 10^{ − 10} | 1.00 × 10^{ − 3} | 3.55 × 10^{ − 19} | 2.93 × 10^{ − 9} | 0 |
2 | 2.01 × 10^{ − 10} | 1.41 × 10^{ − 3} | 3.70 × 10^{ − 2} | 1.84 × 10^{8} | 43 |
3 | 2.59 × 10^{ − 9} | 1.00 × 10^{ − 3} | 9.87 × 10^{ − 1} | 3.81 × 10^{8} | 90 |
4 | 1.21 × 10^{ − 10} | 1.00 × 10^{ − 3} | − 1.87 × 10^{ − 2} | − 1.55 × 10^{8} | 36 |
5 | 1.21 × 10^{ − 10} | 1.41 × 10^{ − 3} | 1.28 × 10^{ − 2} | 1.06 × 10^{8} | 25 |
6 | 2.59 × 10^{ − 9} | 1.00 × 10^{ − 3} | 9.88 × 10^{ − 1} | 3.81 × 10^{8} | 90 |
7 | 1.21 × 10^{ − 10} | 1.41 × 10^{ − 3} | 1.37 × 10^{ − 2} | 1.13 × 10^{8} | 26 |
8 | 1.21 × 10^{ − 10} | 1.00 × 10^{ − 3} | − 1.24 × 10^{ − 2} | − 1.03 × 10^{8} | 24 |
9 | 1.21 × 10^{ − 10} | 1.41 × 10^{ − 3} | 1.76 × 10^{ − 2} | 1.45 × 10^{8} | 34 |
10 | 1.21 × 10^{ − 10} | 1.00 × 10^{ − 3} | 4.95 × 10^{ − 20} | 4.09 × 10^{ − 10} | 0 |
11 | 1.21 × 10^{ − 10} | 1.41 × 10^{ − 3} | 1.37 × 10^{ − 2} | 1.13 × 10^{8} | 26 |
12 | 2.59 × 10^{ − 9} | 1.00 × 10^{ − 3} | 9.54 × 10^{ − 1} | 3.68 × 10^{8} | 87 |
13 | 1.21 × 10^{ − 10} | 1.00 × 10^{ − 3} | − 1.87 × 10^{ − 2} | − 1.55 × 10^{8} | 36 |
14 | 1.21 × 10^{ − 10} | 1.41 × 10^{ − 3} | 1.76 × 10^{ − 2} | 1.45 × 10^{8} | 34 |
15 | 2.59 × 10^{ − 9} | 1.00 × 10^{ − 3} | 9.82 × 10^{ − 1} | 3.79 × 10^{8} | 90 |
16 | 2.01 × 10^{ − 10} | 1.41 × 10^{ − 3} | 3.70 × 10^{ − 2} | 1.84 × 10^{8} | 43 |
17 | 1.21 × 10^{ − 10} | 1.00 × 10^{ − 3} | − 1.24 × 10^{ − 2} | − 1.03 × 10^{8} | 24 |
18 | 1.21 × 10^{ − 10} | 1.41 × 10^{ − 3} | 1.28 × 10^{ − 2} | 1.06 × 10^{8} | 25 |
19 | 2.59 × 10^{ − 9} | 1.00 × 10^{ − 3} | 9.87 × 10^{ − 1} | 3.81 × 10^{8} | 90 |
20 | 2.59 × 10^{ − 9} | 1.00 × 10^{ − 3} | 9.88 × 10^{ − 1} | 3.81 × 10^{8} | 90 |
5 Conclusion
- 1.
Definition of a reference section from which the set of cross-sectional areas was obtained. The set is specific to the micro-scale truss to be optimized.
- 2.
Definition of penalty coefficients for the oversized bars to facilitate selection of the best micro-scale trusses and obtain final structures with few oversized elements.
- 3.
Use of a bit-to-bit clone study, with the goal of avoiding algorithm stagnation.
- 4.
Use of a single rebirth to extend the search space without reducing the number of set sections because, as mentioned above, this is specific to the micro-scale truss to be optimized.
- 5.
Consideration of a new convergence criterion that analyzes algorithm stagnation after several rebirths.
Finally, we believe that the best solution of each optimization run can be trusted as global optimum because (1) the normal stress levels are very close to the maximum allowable stress and (2) its configuration is similar between runs with the largest cross-sectional associated with vertical elements and the smaller cross-sectional areas associated with horizontal and diagonal elements.
Acknowledgments
The work described in this paper was supported by a grant from the Spanish Ministerio de Educacion y Ciencia through the National Programme of Mobility of Human Resources del Plan Nacional de I-D+I 2008–2011.