Reliability-Based Design Optimization of Structures Combining Genetic Algorithms and Finite Element Reliability Analysis Luis Celorrio
Abstract Structural optimization has undergone a substantial progress. Most of these research efforts deal with deterministic parameters. However, in a realistic structural design, it is necessary to consider inherent uncertainties in geometric variables and material properties to ensure safety and quality. Then, design constraints are formulated in terms such as probability of failure or reliability index. The process of design optimization enhanced by the addition of reliability constraints is referred as Reliability-Based Design Optimization (RBDO). Most of RBDO methods use classical mathematical optimization algorithms and require computing gradients of objective function and constraints. This task sometimes can be cumbersome and very hard because reliability constraints are implicit functions of design variables. However, the increased power of computers has made possible to apply heuristic methods, especially Genetic Algorithms in RBDO problems. In this paper Genetic Algorithms have been combined with Nonlinear Finite Element Reliability Analysis software, named OpenSees, to solve RBDO problems. A numerical example shows the performance of the implementation.
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Keywords Reliability based design optimization Nonlinear finite elements Opensees Genetic algorithms Discrete design variables
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1 Introduction Structural optimization has undergone a substantial progress. Design and analysis software programs based in finite elements have recently added optimization and sensitivity assessment capabilities. Design methods for daily practice deal only with
L. Celorrio (✉) Mechanical Engineering Department, Universidad de La Rioja, Logroño, La Rioja, Spain e-mail:
[email protected] © Springer International Publishing Switzerland 2015 Á. Herrero et al. (eds.), 10th International Conference on Soft Computing Models in Industrial and Environmental Applications, Advances in Intelligent Systems and Computing 368, DOI 10.1007/978-3-319-19719-7_13
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deterministic parameters. However, uncertainties are inherent in design variables and parameters such as material properties, loading and geometry parameters and, therefore, these uncertainties have to be considered in the design of any engineering system to ensure safety and quality. Traditionally, these uncertainties have been considered through partial safety factors in structural optimization methods. These partial safety factors are established in structural design codes for buildings. However, partial safety factors do not provide a quantitative measure of the safety margin in design and are not quantitatively linked to the influence of different design variables and their uncertainties on the overall system performance. Therefore, it is crucial to account for uncertainties explicitly in structural design. Reliability Based Design Optimization (RBDO) is a set of methods to design optimal structures subject to reliability constraints. Each reliability constraint is written in probabilistic terms, such as, probabilities of failures or reliability indexes. These terms are computed using structural reliability methods. In structural engineering, failures are formulated in mathematical form as limit state functions, which are expressed in terms of stresses and displacements. Violations of limit state usually occur when structures support extreme natural actions like large wind loads, earthquakes, wave loads, etc. In these cases linear structural analysis could be very inaccurate and material and geometrical nonlinearities must be considered. In this paper a power software program named OpenSees with excellent capabilities to nonlinear finite element analysis and structural reliability analysis has been used [1, 2]. Classical RBDO methods consist in a double loop procedure. At the outer loop, design optimization is carried out by mathematical methods and requires gradients of the objective functions and reliability constraints. Sequential Quadratic Programming (SQP), Sequential Linear Programming (SLP), and other nonlinear constrained optimization methods are often used. Reliability constraints and their gradients are evaluated at the inner loop. Other types of mathematical methods have been developed to solve RBDO problems, like single loop methods and decoupled methods. Several reviews can be found in the literature about these methods [3, 4]. Recently the increased power of computers makes it possible to develop RBDO methods based in heuristics, like the Evolutionary Computation methods. In this paper Genetic Algorithms are proposed to solve RBDO problems. There is an extensive list of references about the application of GAs in deterministic structural optimization [5, 6]. However, relatively few efforts have been done to apply GA to RBDO problems involving frame and truss structures. Dimou and Koumousis [7] consider multiple-population GA in the RBDO of planar trusses. Multiple populations contend among themselves for computational resources and a meta-GA assigns these resources to the most-fit sub-populations. Shayanfar et al. [8] develop a GA-based RBDO method and apply it to analytical and structural problems. They only consider linear elastic structures. However nonlinear analysis has to be considered when extreme loads are applied and that is the case when reliability analysis is worthy. This paper proposes a GA-based RBDO method applied to nonlinear structures.
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Some Multiobjective Genetic Algorithms (MOGA) have been used to solve multiobjective RBDO problems [9, 10]. In these problems, the weight of the structure and the reliability indexes of the probability constraints are defined as objective functions. Next section explains the formulation of RBDO problems. Section 3 describes same features of GA based optimization. A communication transmission tower truss subjects to random loads is optimized applying the proposed method in Sect. 4. Finally, conclusions are outlined in Sect. 5.
2 Formulation of RBDO Problem An RBDO problem can be formulated as minimization of an objective function subject to reliability and deterministic constraints. Also lateral constraints are considered for the design variables. The objective function is the cost of the structure and can include the cost of construction and costs along its life cycle, such as maintenance cost, reparation cost when failure occurs and demolition cost. Usually the probability of failure is very little and reparation cost can be negligible with respect to the construction cost. Constraints describe design conditions stated in codes of structural design and are written as limit state functions. The most common mathematical formulation of a RBDO problem is: min f ðd, μX , μP Þ d, μX
s.t. Pfi = P½ɡi ðd, X, PÞ ≤ 0 ≤ Ptfi , i = 1, . . . , n
ð1Þ
dL ≤ d ≤ dU , μLX ≤ μX ≤ μU X where d ∈ Rk is the vector of deterministic design variables. dL and dU are the upper and lower bounds of vector, respectively. X ∈ Rm is the vector of random design variables, that is, random variables whose mean values, μX are design variables. μLX q and μU X are the upper and lower bounds of vector μX . P ∈ R is the vector of random parameters. μP is the mean value of P. f ð ⋅ Þ is the objective function, n is the number of reliability constraints, k is the number of deterministic design variables, m is the number or random design variables and q is the number of random parameters. ɡi ðd, X, PÞ is the i-th limit state function. Pfi is the probability of violating the i-th probabilistic constraint and Ptfi is the target probability of failure for the i-th probabilistic constraint. Reliability constraints are written in terms of nodal displacements and in terms of internal forces in any section of a member. If these displacements or internal forces efforts exceed a limit value a failure occurs. Limit state functions are defined in a way that ɡi ð ⋅ Þ ≤ 0 represents the failure domain. Then, the probability of
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failure for the i-th limit state function Pfi could be computed using the multivariate integral: Pfi = ∫ ɡ ðd, X, PÞ ≤ 0 fX ðxÞdx i
ð2Þ
where fX ðxÞ is the joint probability density function of vector of random variables X. The close-form solution of this integral is not usually available. Numerical solutions have been obtained until dimension 4 or 5. Because that, approximate reliability methods and simulation methods are available. Simulation methods as MonteCarlo Simulation (MCS) and Importance Sampling based MCS need a large computing time and generally are not used. The approximate First Order Reliability Methods (FORM) provides accurate reliability approximation. FORM is used in this research to compute Pfi of each reliability constraint. Alternatively the reliability of a limit state function can be written in terms of the reliability index. In FORM, these values are related by this equation: Pfi ≈ Φð − βi Þ
ð3Þ
where Φ is the normal standard CDF. Now, if FORM is used to evaluate the reliability constraints the RBDO problem is expressed with the next equation: min f ðd, μX , μP Þ d, μX
s.t. βi ≥ βti , i = 1, . . . , n
ð4Þ
dL ≤ d ≤ dU , μLX ≤ μX ≤ μU X where βti is the target or admissible reliability index corresponding to the i-th reliability constraint. The first step in FORM is to transform the limit state function from the space of original random variables to the space of uncorrelated standard normal random variables using an isoprobabilistic transformation. Then, each limit state functions is approximate by a first order function using the Taylor series around a determinate point in standard normal random space. FORM iteratively compute the reliability index moving from an initial point to a point named design point or Most Probable Point (MPP). The value of the limit state functions and its gradients with respect to the random variables are computed in each iteration. In structural design, the limit state functions are not explicit functions of random design variables, they rather than depend implicitly of design variables. An iteration in FORM implies a call to finite element software with capabilities to compute the gradients of the structural response with respect to random variables. In this paper, the Finite Element software OpenSees is used to determine the response of the structural analysis. OpenSees is an open source code developed in University of California at Berkeley. OpenSees has capabilities of sensitivity and
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structural reliability analysis. Two methods are implemented to compute gradients: the Forward Finite Difference method (FFD) and the Direct Differentiation Method (DDM). As DDM is more efficient than FFD, it is used in this research.
3 GA Based Optimization GAs are well known algorithms that have been applied extensively in optimization and classification problems. As other Evolutionary Computation procedures, they are inspired in the principles of natural selection and simulate the adaptive behavior of live beings. Basically, GAs start from an initial population of potential solutions that evolves by some genetic operators which act in a very similar manner than natural selection laws [11]. More important operators are: Selection. This operator is used to select individuals for reproduction in the current population. These individuals are named parents and form the mating-pool and generate children to form the next generation. There are several types of selection operators. The GA usually selects the individuals that have better fitness values. This value is the value of the objective function in unconstrained optimization problems or is a modified fitness that results adding to the objective function a penalty function in constrained optimization problems. Selection operators are stochastic, because are based in random numbers generators. This fact explains the capability of GAs to explore very complex search space of design variables. In the RBDO problems formulated here a tournament selection operator has been chosen. Elitism. Genetic Algorithms often maintain the best solution/s from a previous population through a mechanism termed elitism. In simplest terms, elitism is simply taking the best solution/s from the current population and simply carrying it/them over to the next generation. Crossover. This operator takes pairs of parents of the mating-pool and combines or interchanges the genes of these parents generating children. There are various types of crossover operators: mask, single point, two point, heuristic, etc. Mutation. This operator works after crossover and makes small random changes in single parents of the mating-pool. These changes area applied in stochastically selected genes. Mutation operator provides genetic diversity and enables the genetic algorithm to search a complex search space. GAs present some advantages with respect to classical optimization methods. Unlike mathematical optimization methods which perform optimum search considering only a single solution at a time, GAs work with a population of individuals
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in each generation. Each individual is a codified version of the design variables of the structure. Objective and constraints functions are evaluated for all these individuals. GA is a free-gradient optimization method. This feature makes possible to consider objective and constraints functions with mathematical difficulties such as discontinuous or non-derivable functions [11]. Initial population and offspring in next generations are generated stochastically in GAs. This permits to maintain several search points and to explore complex search space. Therefore, the convergence of the optimization to local minima is prevented, although starting point has been poorly chosen. All these aspects result in more chances of finding the optimal solution than gradient based methods, even on problems having hard search spaces with multiple local minima. The main drawback of these methods is the extreme computational effort required. Classical RBDO problems require a large quantity of objective and reliability constraints evaluations. In the case of structural applications, for each reliability constraint assessment some runs of structural analysis software as, for example, a finite element package, have to be carried out. The replacement of mathematical optimization methods by GAs increases even more the computational time because several design solutions are computed at the same time. However, parallel computing resources and High Performance Computing techniques allow dealing with complex GA-based RBDO problems.
4 Numerical Example: Space Truss This example deals with the design of the space truss shown in Fig. 1. This structure has been frequently used as a transmission tower and has been proposed in the literature for evaluate optimization methods. This truss contains 25 bars and 10 nodes. It is subjected to the nodal loads represented also in Fig. 1. The relation stress – strain of the steel is represented with a bilinear model, represented in the Fig. 2. Elastic modulus, E, and yield stress, fy , are considered random variables. The strain hardening ratio b is deterministic. Bar elements are modeled as nonlinear finite elements using the corotational transformation to consider geometric nonlinearity. The 25 bars are grouped in 8 groups. Bars in the same group have the same cross sectional area and these areas are the random design variables of the problem. Also, nodal loads are lognormal random variables. The random variables of the problem are registered in Table 1. This, the RBDO problem is formulated as: 8
nbi
i=1
j=1
min V ðμA , μP Þ = ∑ μAi ∑ li
μA1 ....μA8
s.t. βij ≥ βtij = 3.7 i = 1, 2 j = x, y
ð5Þ
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Table 1 Random variables in the tower truss example Random variable
Description
Dist.
Mean
CoV
A1 , . . . , A8 E fy b P1 P2 P3 P4
Cross section area Elastic modulus Yield stress Strain hardening ratio Lateral load Lateral load Vertical load Lateral load
N LN LN
Design Variable 210E + 6 kPa 250E + 3 kPa Deterministic 60 kN 100 kN 200 kN 60 kN
0.1 0.05 0.05 0.00 0.1 0.1 0.1 0.1
LN LN LN LN
190.5cm
P3
P3
2
1
P1 1
P2
P2 2
254cm
2
3 3
2
2
4
5
3
P4
3
3
4
5
5
6
P4
190.5cm
4
8 8
254cm
8 6 8
6
6 7 7
z
8
7 x
508cm
7
9
y 10
Fig. 1 Transmission Tower Truss
6
7
508cm
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Fig. 2 Nonlinear behavior model for steel
P ɡij ðA, PÞ ≤ 0 < Φ − βti i = 1, 2 j = x, where nbi is the quantity of bars of the i-th group and the reliability constraints are written in terms of the displacements of the nodes 1 and 2 on coordinates x and y. These constraints are: ɡi ðA, PÞ = 1 −
uij ðA, PÞ i = 1, 2 j = x, y umax
ð6Þ
Here the allowed displacement umax = 2 cm has been set. Design variables μAi are discrete design variables and can take 20 different values of the set ½1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30 cm2 . The population size considered is 40 and the vector size of desing variables is 8. Initial population has been created using uniform selection, that is, all individuals have been selected randomly from an initial range. GA does not work directly with the mean values of the cross sectional areas, rather than it works with a codified version consisting in integer values in the range ½1, .., 20. The maximum number of generations used is 300. Elistism is considered taking the two best fitness inviduals of the current population over the next generation. The rest of individuals for the next generation are formed applying crossover and mutation operators to the parents of the current population. A single point crossover operator is chossen whose crossover fraction is 0.8. This is the fraction of individuals of the next generation, deducting elite children, that are created by crossover. The type of mutation operator is gaussian. The probabilistic optimization problem has been solved four times considering different analysis models and various types on material behavior. This shows the flexibility of the GA based method.
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In the first case, geometric and material nonlinearities are considered. The convergence is reached in 76 generations. The set of design variables with best fitness in the last generation is considered as the probabilistic optimum design. The optimum volume of the truss is 52901.8 cm3 , and the mean values of the cross sectional areas are: μA1 , . . . , μA8 = 1, 10, 7, 1, 3, 3, 2, 16 cm2 . In this probabilistic optimum the four reliability constraints are verified and the penalty function is equal to the fitness function. The values of the reliability indexes for the optimum design are: β1x = 6.060, β1y = 3.781, β2x = 6.064, β2y = 4.895. These results are summarized in Table 2. Table 2 Results for the tower truss example Case
μA1 . . . . μA8 [cm2]
1 2 3 4
1, 8, 1, 4,
10, 14, 14, 14,
7, 3, 3, 2,
1, 2, 1, 2,
3, 2, 2, 3,
3, 2, 2, 4,
2, 1, 2, 1,
16 14 16 16
Volume [cm3]
β1x , β1y , β2x , β2y
52901 49129 51967 54136
6.060; 3.781; 6.064; 4.895 14.075; 3.812; 14.294; 4.984 5.246; 3.713; 5.156; 4.165 8.564; 3.752; 8.381; 4.570
It is worth to note that the second constraint that refers to the displacement of node 1 along y axle is more critical than the others constraints. Also, its value 3.781 is not exactly equal to the admissible value for the reliability index. The reason of this different is that optimization problem deal with discrete design variables. Case 2 corresponds to consider geometric nonliearity and linear elastic material. Case 3 corresponds to consider geometric linearity and nonlinear material. And case 4 corresponds to consider geometric and material linearities. Resuts for these three cases are also registered in Table 2. Although this numerical example involves a reduced quantity of random design variables, each probabilistic optimization took approximately an hour to provide the results. This period of time is about four times the computational time needed with gradient based RBDO methods. MonteCarlo Simulation with Importance Sampling has been used to verify the results obtained by the GA-based RBDO method. Reliability indexes obtained with simulation were very close to reliability indexes obtained with the proposed method. The proposed method has been applied to frames. However, due to space limitations a numerical example cannot be provided here. The interested reader is referred to coming works.
5 Conclusions In this work a RBDO method combining GAs with Nonlinear Finite Element Reliability Analysis software OpenSees has been carried out. The proposed method has been applied to engineering structures subjected to extreme random loads
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caused by natural actions. In these cases large strains and displacements occur and geometric nonlinearity must be considered. The implemented method can be used to design structures when traditional methods prescribed in construction codes are not applicable or are imprecise. In addition to the case with geometric and material nonlinearities, other three types of structural analysis have been implemented: geometric nonlinearity with linear elastic material, geometric linearity with nonlinear material behavior and both geometric and material linearity. The designer have to decide what is the adequate analysis and then, to select the values of the cross-sectional areas or the standardized steel profiles. GA can deal with several types of design variables: continuous, integer or mixed. This allows the design of structures using standardized commercial steel sections. Also a near global optimum design can be reached. The numerical example shows that precise results can be obtained in moderate period of time thanks to the increased power computing available even in home computers. Advanced evolutionary algorithms such as distributed GA, MOGA, coevolutionary algorithm and others could be applied to solve complex RBDO problems and research efforts are focused on them.
References 1. OpenSees: open system for earthquake engineering simulation. http://opensees.berkeley.edu 2. Haukaas T, Scott MH (2006) Shape sensitivities in the reliability analysis of nonlinear frame structures. Comput Struct 84(15–16):964–977 3. Celorrio-Barragué L (2010) Metodología eficiente de optimización de diseño basada en fiabilidad aplicada a estructuras. Ph. D Thesis. Universidad de La Rioja, Logroño 4. Celorrio-Barragué L (2012) Development of a reliability-based design optimization toolbox for the FERUM software. In: 6th International Conference Scalable Uncertainty Management, 2012, LNAI, vol 7520, pp 273–286. Springer, Berlin 5. Pezeshk S, Camp CV (2002) State of the art on the use of genetic algorithms in design of steel structures. In: Burns S (ed) Recent Advances in Optimal Structural Design. American Society of Civil Engineers, Reston 6. Foley CM (2007) Structural optimization using evolutionary computation. Optimization of structural and mechanical systems. World Scientific, Singapore Chap. 3, pp 59–120 7. Dimou CK, Koumousis VK (2003) Competitive genetic algorithms with application to reliability optimal design. J Adv Eng Softw 34(11–12):773–785 8. Shayanfar M, Abbasnia R, Khodam A (2014) Development of a GA-based method for reliability-based optimization of structures with discrete and continuous design variables using OpenSees and Tcl. Finite Elements Analy Des 90:61–73 9. Mathakari S, Gardoni P, Agarwal P, Raich A, Haukaas T (2007) Reliability-based optimal design of electrical transmission towers using multi-objective genetic algorithm. Comput-Aided Civil Infrastruct Eng 22(4):282–292 10. Deb K, Gupta S, Daum D, Branke J, Mall AK, Padmanabhan D (2009) Reliability-based optimization using evolutionary algorithms. IEEE Trans Evol Comput 13(5):1054–1074 11. Golberg DE (1989) Genetic algorithm in search, optimization, and machine learning. Addison-Wesley Professional, Reading
