Reliability-Based and Robust Design Optimization with Artificial

6th World Congresses of Structural and Multidisciplinary Optimization Rio de Janeiro, 30 May - 03 June 2005, Brazil

Reliability-Based and Robust Design Optimization with Artificial Neural Network Sangook Jun1,Yong-Hee Jeon1,Joohyun Rho1,JeongHwa Kim2,Juhyun Kim2,and Dong-Ho Lee3 (1) School of Mechanical and Aerospace Engineering, College of Engineering, Seoul National University, Seoul, Republic of Korea [email protected] (2) School of Mechanical and Aerospace Engineering, College of Engineering, Seoul National University, Seoul, Republic of Korea [email protected] (3) School of Mechanical and Aerospace Engineering, College of Engineering, Seoul National University, Seoul, Republic of Korea [email protected] 1. Abstract This research is accomplished for the validation that it is possible to evaluate reliability and robustness simultaneously under uncertainty with an approximated model. For this, Monte Carlo Simulation and an artificial neural network were used as implements to obtain the probabilistic and stochastic properties, such as expectation and standard deviation. Monte Carlo Simulation is suitable to this study because it can evaluate reliability and robustness together. However, as requiring large number of sample, it is very inefficient in that the computational cost is expensive. So, to complement this defect, the surrogated model generated by an artificial neural network was applied. Moreover, since it has remarkable characteristic to capture nonlinearity, the approximation of limit-state function and robust region in the design space is feasible. Drawing on this approach, we applied to the aero-structural optimization problem of transonic wing. Assuming that manufacturing and operating uncertainties are statistically independent and normally distributed, we defined multi-objective as expectation and standard deviation of performance and constraints as limit-state function. Through the probability density function and the cumulative distribution function of optimized wing, we were able to guarantee the improvement of wing performance from the increase of expectation and the robustness from the decrease of standard deviation, and also to perform the optimization design satisfying the desired reliability. From these facts, we could confirm the feasibility that reliability-based and robust design optimization is achieved at once. 2. Keywords: Reliability-Based Design Optimization, Robustness Design Optimization, Artificial Neural Network, Wing Design 3. Introduction Generally the reliability is the ability which a system or product accomplish given capacity for a given time. The failure is a case in which a system cannot do required capacity at intended point of time. From designer’s angle, the reliability means that they carry out the design in consideration of unpredictable matters, an uncertainty. Conventional design optimization has a limit. There is strong possibility that optimal solution exists in a boundary of the upper and the lower bound. In this instance, the optimum solution may disobey a constraint through fluctuation under uncertainty. So the value of an objective function may not be satisfied with a target value. According to the design optimization using probabilistic and stochastic approach, a concept of the reliability and robustness is often defined as follows. The reliability focuses on constraints. The higher probability which optimal solution is satisfied with a constraint, the higher reliability there is. The robustness focuses on an objective function. The smaller standard deviation for value of an objective function, the stronger robustness there is. Established design optimization method is in short consideration of uncertainty for the physical variables and the present conditions. DeLaurentis give a definition for the uncertainty. According to him, an uncertainty is “the incompleteness in knowledge that causes model-based prediction to differ from reality in a manner described by some distribution function.” [1] An uncertainty is classified by two divisions. Firstly, the uncertainty in laboratory includes fidelity uncertainty and approximation uncertainty. And that is controllable. Secondly, the uncertainty out of laboratory is uncontrollable. Designers do not know well about accurate information for the performance and the reliability of realistic system. That is, an uncertainty exists. So, when designers work out a design, they assume the worst case. A disadvantage in economic effect occurs due to assumption mentioned above, according to Mavris. [2] Lee considers an uncertainty in design process. He ensures the reliability of optimal solution by inducing the probabilistic and stochastic theory. [3] And Nah tries the construction of design frame using probabilistic approach about an uncertainty. [4] The cases other than research mentioned above are presented as follows. It is much more expensive to perform reliability-based optimization for large-scale problem than small-scale problem, that is, computational cost is increased dramatically. To solve the problem, Tsompanakis proposes two level approach using evolution strategy. [5] Grooteman presents efficient and stochastic method for probabilistic analysis. [6] To analyze reliability, Choi compares reliability index approach with performance measure approach. [7] In order to work out reliability-based and robust design optimization, an evaluation of the reliability is required. The evaluation of the reliability is next to impossible analytically about complex system. In such a case the reliability is evaluated using Monte-Carlo simulation. Monte-Carlo simulation can implement almost all problems and can evaluate the reliability and the robustness at once. But, owing to Monte-Carlo simulation is evaluated for at least one million samples, computing time is so long. To complement this disadvantage, approximate model is used. Also merit of analytic method, efficiency is obtained using an approximate model. Approximate model have to reproduce well nonlinearity and have to be possible for rapid response to obtain. The line of this study is as follows. The uncertainty is divided into two. The one is controllable factors, and the other is uncontrollable factors. Each factor is assumed independent and probabilistic model. The optimization is performed to be satisfied with reliability and robustness condition and is evaluated using Monte-Carlo simulation and artificial neural network. By applying to aircraft wing problem coupling aerodynamics with structure analysis, we want to tap the possibility of reliability-based and robust design optimization.

4. Theoretical Background 4.1 Reliability-Based and Robust Design Optimization Conventional optimization problem is defined as follows.

Find x minimize f (x) subject to hi (x) = 0 ; i =1,L,k

(1)

gj (x) ≤ 0 ; j =1,L,l xL ≤ x ≤ xU In the above formulas, x is vector of design variables, f(x) is objective function, h are equality constraints and g are inequality constraints. xL is the lower bound of design variables and xU is the upper bound of design variables. If (1) is changed to design optimization problem in consideration for uncertainty, the design variables x become random variables that is independent and has normal or non-normal distribution. The objective function and limit-state function are evaluated using mean value, representative value of random variables. The points mentioned above can be expressed in following formulas. Either the objective function cost(X) has to consist of the deviation that can represent robustness of f(x), or cost(X) has to be composed so as to consider mean value and deviation at once.

minimize cost(X) subject to P[Gi (X) ≤0] ≤pi

(2)

In (2), G(X) is limit-state function and when this function is smaller than 0, a failure is occurred. pi is the failure probability which presents a target. The reliability or the failure probability is expressed by probability which system is satisfied with design requirement. This is important factor in the design process. And it is helpful for designer to consider an uncertainty. Mathematically, failure probability, Pj is expressed as follows.

Pj = P(G(X) ≤ 0) = ∫

G(X) ≤ 0

g(x)d x

(3)

In an evaluation for the robustness of an objective function or a limit-state function, the evaluation can be performed by various methods. For instance, these methods contain sensitivity analyses, first order reliability method and second order reliability method, and so forth. It is difficult that designers evaluate a robustness of an objective function and a limit-state function at once through the methods stated above. A random variable and differentiable failure function required previous information is required by these methods. But only probability density functions of all random variables which known before reliability is analyzed is required by Monte-Carlo simulation. In general, first order reliability method and second order reliability method is very effective in small-scale problems, but as the numbers of random variables are increased and as a problem is complicated, Monte-Carlo simulation is more reliable. [6] In a complex and a large-scale problem, since a great deal of computing time is required by Monte-Carlo simulation, a great difficulty is occurred in case reliability analysis is performed. To solve this difficulty, approximate model which can reproduce given design space well is required. So, in this research, we trace a design space by the approximate approach using artificial neural network and choose the method by which robustness and limit-state function are evaluated using Monte-Carlo simulation. Artificial neural network is introduced in the next section. 4.2. Artificial Neural Network Theory The artificial neural network theory [8] is theory which mimics information processing of human brains. In the brains, neurons, basic units of nerve, get information and process received information. After that, neurons transmit processed information to the next neurons.

Figure 1. Communication among Human Neurons As the brains can judge through such connection of neurons, artificial neural network can prove proper judgment or a value by clustering information processing units called artificial neuron.

Figure 2. Neural processes in artificial neural network The figure 2 shows artificial neuron. An artificial neuron multiplies stimuli entered from outside to proper weight and add these values, then conveys the information to the next neuron through pertinent transfer function. A set which utilizes former information together is called layer, whole artificial neural network consist of many layers. The output received from outside can express target value as a result through proper weight and transfer function. In general way, artificial neural network which is composed of a hidden layer and an output layer is used. A tangent sigmoid and a purelinear transfer function are used as a transfer function.

Figure 3. Layer neural network When the number of layers of artificial neural network and a transfer function of each layer are determined, the number of neurons and the weights are determined. In two-layer neural network which is widely used, the number of output layers can be chosen the same as the number of output variables. On the other hand, there is not a distinct correlation between the number of neurons of hidden layers and the number of the input variables or the output variables. The weights of artificial neural network are determined in regard to number of neurons. A given initial weight is changed by learning. A form of back-propagation is widely used as a learning method. In the form of back-propagation, a change of the weight be propagated from a output layer to a former layer. An objective function of learning is to reduce sum of squared errors between the results of experiment and the output. Sum of squared errors is called performance index. Levenberg-Marquardt method is widely used, the convergence speed of this method is rapid. The nonlinear design space can be reproduced by the artificial neural network through sufficient numbers of a design points and a hidden layer. Although values are beyond the bounds of variables, the artificial neural network is robust. [9] But in the case using limited numbers of design points, a character of the design space may not be expressed by design points. A form of space reproduction can be changed by neuron’s numbers of hidden layer. And according to initial weight, after the weight is learned, this may be local minimum. [10] As has been mentioned, the artificial neural network has several disadvantages. But nonlinearity can be reproduced by this. And because expressed a form of matrix, a rapid response can be given by this. To obtain responses for a large numbers of samples like Monte-Carlo simulation, approximate model which is represented as a form of function like response surface method is good. But, because of ability that reproduce nonlinearity does not come up to different approximate model, it is difficult to evaluate robustness. In case of Kriging model, this model has advantage of excellent reproduction of nonlinearity and the use of stochastic approach. But this do not present designer response efficiently compared with artificial neural network. On this account, the reproduction of design space is performed using not response surface model and Kriging model but the approximate model that consist of artificial neural network.

Polynomial Regression

Kriging model

Neural Network

Figure 4. Comparison among approximate models 5. Observation 5.1 Definition of problem for conventional design optimization The wing of the aircraft is one of the most important components that have it fly in the air and hold out severely structural load. And the planform of the wing has dominant influence on performances of the airplane. So, the design of the wing is not only the kernel of the aircraft design but also the part required much effort. In this research, reliability-based and robust design optimization is applied to the wing design that is the core in the aircraft design, considering aerodynamic and structural disciplines Wing for a commercial aircraft of DC-9 is modeled. We conduct multidisciplinary optimization consisted of aerodynamic and structural disciplines. For disciplinary analyses, vortex lattice method (VLM) is used for aerodynamic analysis and Wing-box modeling for structural analysis. The original code is decomposed along aerodynamic and structural disciplines. The objective is range maximization:

Range =

V L ⎛⎜ Wi ⎞⎟ ⋅ ⋅ln SFC D ⎜⎝Wf ⎟⎠

(4)

In (4), V is cruise speed, SFC is specific fuel consumption, L/D is lift-to-drag ratio, Wi is weight of aircraft before mission is executed and Wf is weight of aircraft after mission is finished. The design problem is consisted of seven design variables represented in Table 1. Table 1. Design space of wing design problem Design Variables Minimum Baseline Span (ft) 41.989 46.655 Sweep angle (deg) 22.050 24.500 Taper ratio at 30% span 0.685 0.761 Taper ratio 0.184 0.204 Incidence angle (deg) -2.000 0.000

Maximum 51.320 26.950 0.837 0.224 2.000

The constraints are supposed as follows. As if you know in (4) because lift-to-drag ratio (L/D) is proportional to range, this ratio is larger than baseline, lift (L) is larger than take-off gross weight, fuel weight (Wfuel) and wing weight(Wwing) are smaller than the baseline, and displacement at the tip of the wing (dtip) is within 5% of the baseline. The initial value of each design variable is determined based on DC-9 specification. For this problem, the flight condition is like following. The aircraft cruises at 25,000 ft above the ground with Mach number 0.75. Angle of attack is considered to be zero and take-off gross weight is set to 108,000 lb. 5.2 Definition of problem for design optimization under uncertainty When the optimization problem that is defined previously is changed into reliability-based and robust design optimization problem, this is showed like Figure 5. To consider an uncertainty factor, two kinds of uncertainty is assumed. The first, five design variables representative of wing planform are uncertainty factors which are occurred in the process of a production, means are assumed the values of baseline to these, and the uncertainty factors is assumed that these are ranged in conformity to normal distribution N(µ,σ2). A standard deviation of normal distribution is two percent of the means. These variables are used directly in the process of design optimization and are changed by designers, so these variables are controllable variables. The second uncertainty factor is an uncontrollable factor which designers are hard to control and aircraft’s cruise speed is chosen as uncertainty factor which is occurred in the process of a operation. This is assumed in the same way with the first factors.

Figure 5. Transform of design problem for reliability-based & robust design optimization When mean and standard deviation is found through Monte-Carlo simulation, the objective function consist of as follows using method of imposing weights, to increase mean of range (E) and decrease standard deviation of range (σ).

⎛ σ (Range ) ⎞ ⎛ E (Rangebase ) ⎞ ⎟⎟ + (1 − w) ⋅ ⎜⎜ ⎟⎟ Objective = w ⋅ ⎜⎜ ( ) Range σ ⎝ E (Range ) ⎠ base ⎠ ⎝

(5)

And all constraints including of side constraints allow five-percent failure probability so as to have reliability of 2σ. 5.3 Construction of Surrogated Model Using Artificial Neural Network In this research, because evaluations for one million samples have to be carried out, approximate model has to respond rapidly and has to capture well the nonlinearity such that robust region. The approximate model using artificial neural network agreed with these conditions. Sixty six design points is chosen using D-optimal method and these points is learned by artificial neural network. Approximate models for an objective function and constraints are constructed, to evaluate fitness root mean square error and R2 is displayed in Table 2.

RMSE R2

Range 0.0136 0.9985

Table 2. Analysis of variance Lift L/D Wfuel 0.0135 0.0095 0.0152 0.9985 0.9993 0.9982

Wwing 0.0139 0.9985

dtip 0.0115 0.9989

5.4 Results of reliability-based & robust design optimization The results of reliability-based and robust design optimization using artificial neural network are showed by the following Figure 6. For the means and the standard deviations of range, Pareto optima are illustrated. We want to consider about a case that standard deviation is the smallest among these and that mean is the smallest among these.

Figure 6. Pareto plot

Compared to the result which is performed by multidisciplinary feasible method, one of multidisciplinary design optimization methodology, this result is displayed in Table 3. “RBDO & RDO (w = 0.0)” is the result of case in which the weight of objective function is zero, only standard deviation is minimized. “RBDO & RDO (w = 1.0)” is the result of case in which the weight of objective function is one, only mean is maximized. And limit-state function expresses the failure probability of all constraints, in the case of range, baseline and MDF express value without a probability distribution, and RBDO & RDO means the means. (E(range), σ(range) is normalized range and this mean that the range is distributed according to normal distribution at which mean is E(range) and standard deviation is σ(range).

Design Variables Span (ft) Sweep angle (deg) Taper ratio at 30% span Taper ratio Incidence angle (deg) Output Values L/D Wfuel (lb) Lift (lbf) Wwing (lb) dtip (ft) Limit-state function, G(x) Range (nm) ( E(range), σ(range) )

Table 3. Comparison of the optimization results RBDO & RDO RBDO & RDO Baseline (w = 0.0) (w = 1.0) 46.655 43.699 47.939 24.500 25.881 22.657 0.761 0.689 0.737 0.204 0.190 0.210 0.000 0.830 1.082 RBDO & RDO RBDO & RDO Baseline (w = 0.0) (w = 1.0) 18.21 21.08 22.88 23,132.4 19,339.2 23,033.4 2.603*106 1.821*106 1.975*106 10,002.3 6,967.9 8,260.6 8.50 8.17 8.89 0.8229 0.0277 0.0214 1,659.3 1,572.2 2,074.6 (1.000, 0.0329) (0.856, 0.0224) (1.212, 0.0284)

MDF 47.927 22.050 0.747 0.199 1.187 MDF 23.14 23,242.7 1.914*106 7,915.2 8.93 0.8827 2,119.4 (1.237, 0.0274)

In MDF which is the result of conventional design optimization, sweep angle is situated in lower bound of design variable. And fuel weight is much the same as value of baseline, that is, this is situated in boundary of constraint. In the result of RBDO & RDO, sweep angle and fuel weight is shifted to safe region. Such aspect is showed in common for all constraint and in Figure 7, probability density function of L/D among constraints of “RBDO & RDO (w = 0.0)” is illustrated. In case of baseline, constraint of L/D has failure probability of 84.5%, but in “RBDO & RDO (w = 0.0)”, as the probability is decreased to 0.02%, the fact that “RBDO & RDO (w = 0.0)” guarantee the reliability above 2σ is verified.

Figure 7. Failure probability in probability density function of L/D In Table 3, in the case of range, objective function, the result of “RBDO & RDO (w = 0.0)” is decreased from 1,659nm of baseline to 1,572nm, that is, range is decreased about 15%. But standard deviation is decreased from 0.0329 to 0.0224, so this is improved by some 32%. Compared with result of MDF, in this result, mean is decreased about 40% and standard deviation is improved by some 18%. Owing to effect of mean is not considered and problem is made to minimize only standard deviation in the process of defining of design optimization problem, mean is not improved and only standard deviation is improved.

Figure 8. Optimized results of RBDO & RDO (w = 0.0) On the other hand, in the case of “RBDO & RDO (w = 1.0)”, range is increased from 1,659nm of baseline to 2,084nm, that is, this is improved by some 20%. Standard deviation is decreased from 0.0329 to 0.0284, so this is improved by some 14%. Compared with result of MDF, mean is decreased about 3% and standard deviation is increased by some 4%. But, looking at the value of limit-state function, constraint, in MDF when an uncertainty is considered, success probability is on 12%, so the reliability of 2σ is not guaranteed. In view of reliability-based design optimization, we think of that the case of “RBDO & RDO (w = 1.0)” is reasonable.

Figure 9. Optimized results of RBDO & RDO (w = 1.0) 6. Conclusions In this research, Monte-Carlo simulation is carried out using approximate model which consists of artificial neural network. By this way through applying reliability-based and robust design optimization to problem of aircraft wing design, the possibility is investigated. As the result, the conclusion runs as follows. Firstly, because of reliability focuses on constraints and robustness focuses on objective function generally, the fact that optimization can be carried out through combining these is verified. Secondly, nonlinear design space is reproduced with approximate model using artificial neural network, by reason of this robust design space can be searched. 7. Postscript This paper is carried out by support of Innovative Design Optimization Technology Research Center designated by the Korea Science and Engineering Foundation. 8. References 1. D. A. DeLaurentis and D. N. Mavris. Uncertainty Modeling and Management in Multidisciplinary Analysis and Synthesis. Proc. of 38th AIAA Aerospace Science Meeting and Exhibit, 2000, Reno 2. D. N. Mavris, D. A. DeLaurentis, O. Bandte and M. A. Hale. A Stochastic Approach to Multi-disciplinary Aircraft Analysis and

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