Robust and Reliability Based Design Optimization

Transcription

1 Robust and Reliability Based Design Optimization Frederico Afonso Luís Amândio André Marta Afzal Suleman CCTAE, IDMEC, LAETA Instituto Superior Técnico Univerdade de Lisboa Lisboa, Portugal May 22, 2015 F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

2 Outline Presentation Table of Contents 1 Outline Presentation 2 Uncertainty Based Design Optimization Definition Uncertainty Models Uncertainty Categorization Optimization Under Uncertainty 3 Statistical Concepts 4 Robust Design Optimization Problem Formulation Numeric Computation of the Mean and Standard Deviation Numeric Methods 5 Reliability Based Design Optimization Problem Formulation Numerical Computation of the Probability of Failure 6 Robust and Reliability Based Design Optimization Problem Formulation Numeric Methods 7 Analytic Problem - Rosenbrock Function with 1 Constraint RDO Approach RBDO approach R 2 BDO Approach 8 Analytic Problem - Problem with 3 Constraints Results 9 Analytic Problem - MDO Benchmark Problem RDO Approach RBDO Appraoch 10 References F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

3 Uncertainty Based Design Optimization Table of Contents 1 Outline Presentation 2 Uncertainty Based Design Optimization Definition Uncertainty Models Uncertainty Categorization Optimization Under Uncertainty 3 Statistical Concepts 4 Robust Design Optimization Problem Formulation Numeric Computation of the Mean and Standard Deviation Numeric Methods 5 Reliability Based Design Optimization Problem Formulation Numerical Computation of the Probability of Failure 6 Robust and Reliability Based Design Optimization Problem Formulation Numeric Methods 7 Analytic Problem - Rosenbrock Function with 1 Constraint RDO Approach RBDO approach R 2 BDO Approach 8 Analytic Problem - Problem with 3 Constraints Results 9 Analytic Problem - MDO Benchmark Problem RDO Approach RBDO Appraoch 10 References F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

4 Definition Uncertainty Based Design Optimization Definition Uncertainty Nikolaidis [1] defined uncertainty as: Certainty, in the context of decision theory, is the condition in which a decision maker knows everything needed to select the most desirable outcome. Uncertainty is the gap of what the decision maker presently knows and certainty. Every real engineering problem has an associated uncertainty, which can arise from many different sources and are present during design, manufacturing and operation [2]. For aeronautic, Yu and Du [3] elaborated the following list: Uncertainties in operations - e.g. aerodynamic loads, flight speed, altitude, angle of attack Uncertainties in material properties - e.g. material tensile strength and Young s modulus Uncertainties in manufacturing processes - e.g. tolerances on dimensions and shapes Modeling uncertainties - e.g. simplifications of computational models, experimental determination of parameters, fidelity appropriate to the design stage F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

5 Uncertainty Based Design Optimization Uncertainty Models Uncertainty Models Uncertainties can be represented in diverse ways and can be used in computational simulations or mathematical models. Several mathematical models have been proposed which can be divided in 3 main categories [4]: Interval bound; Membership function; Probability density function. Historically uncertainty formulation have been done in terms of probability theory, although, recently it application is being questioned since several other distinct mathematical theories are being shown to perform well in designing with uncertainty [5], [6]. Uncertainty Descriptions [4] F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

6 Uncertainty Based Design Optimization Uncertainty Categorization Uncertainty Categorization Uncertainty Categorization According to Wojtkiewicz et al [7] uncertainty can be classified in two categories: Aleatory, stochastic or random - if it is related to the inherent variability in natural phenomena. Hence, it cannot be reduced, short of changing the phenomenon itself. It is irreducible even if more sample data/information is collected. Epistemic or subjective - in which case the shortcomings of the models used to describe physical phenomena come into play. It stems from a lack of knowledge and is therefore reducible through obtaining additional information. It is also usually biased. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

7 Uncertainty Based Design Optimization Optimization Under Uncertainty Optimization Under Uncertainty A deterministic optimization formulation does not account for the uncertainties in the design variables and parameters and simulation models. Deterministic optimum solutions are usually associated to a high probability of failure (e.g. the optima can be very new constraint boundary). Some of the advantages and disadvantages related to the deterministic optimization are summarized below: Pros Cons Easy to implement Relatively good results Industrial has years of experience with this kind of optimization Hard to apply to vehicles with novel configurations Difficulty when accounting for uncertainties Robustness and reliability inconsistency throughout the vehicle Nowadays the markets competitiveness have impelled that the design is at the same time optimum and robust and reliable solutions. Hence, it is of paramount importance that design optimizations take uncertainties into account [8]. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

8 Uncertainty Based Design Optimization Optimization Under Uncertainty Uncertainty Based Design Optimization In order to include uncertainties in design optimization process one has to change the standard deterministic problem appropriately. Two main methodologies that incorporates uncertainty in the design optimization are: RDO - Robust Design Optimization [9] [10]; RBDO - Reliability Based Design Optimization [10] [11] [12]. There have been developments in the way of combining the RDO and RBDO formulations: R 2 BDO - Robust and Reliability Based Design Optimization [10] [13]. Uncertainty-based design domains [14] Reliability versus robustness [4] F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

9 Statistical Concepts Table of Contents 1 Outline Presentation 2 Uncertainty Based Design Optimization Definition Uncertainty Models Uncertainty Categorization Optimization Under Uncertainty 3 Statistical Concepts 4 Robust Design Optimization Problem Formulation Numeric Computation of the Mean and Standard Deviation Numeric Methods 5 Reliability Based Design Optimization Problem Formulation Numerical Computation of the Probability of Failure 6 Robust and Reliability Based Design Optimization Problem Formulation Numeric Methods 7 Analytic Problem - Rosenbrock Function with 1 Constraint RDO Approach RBDO approach R 2 BDO Approach 8 Analytic Problem - Problem with 3 Constraints Results 9 Analytic Problem - MDO Benchmark Problem RDO Approach RBDO Appraoch 10 References F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

10 Statistical Concepts Statistical Concepts Random Variable A random variable X is a variable that can assume any value from a set of possible x values, each associated with a given probability. Random variables can be either discrete or continuous. Probability Density Functions The Probability Density Function (PDF), f X (X ), is the mathematical function that describes the distribution of the possible values x of X and their respective probabilities. Any PDF satisfies the following conditions: f X (X ) 0 for all values of X and + f X (t)dt = 1 and two properties of the Cumulative Density Function (CDF): P( < t < + ) = 1 P(t = a) = 0 a R F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

11 Statistical Concepts Statistical Concepts Expected Value The expected value, E(X ), or mean value µ X of a random value X is defines the centre of its position and for a continuous random variable is given by: E(X ) = µ X = + t f X (t) dt Variance The variance, V (X ), or the second moment σx 2 a distribution and is given by: of a random value X is a measure of dispersion of + V (X ) = σx 2 = E[X µ X ] 2 = E(X 2 ) E(X ) 2 = (t µ X ) 2 f X (t) dt The standard deviation of X, σ X, is the positive square root of the variance. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

12 Statistical Concepts Statistical Concepts High Order Moments For any positive integer n, X s n th order central moment is defined by: E[X µ X ] n = + (t µ X ) n f X (t) dt The Skewness (S(X )) and the Kurtosis (K(X )) are respectively the third and fourth central moments. Coefficient of Variance The coefficient of variance, c.o.v., is the standard deviation divided by the absolute value of the mean: c.o.v. = σ µ F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

13 Statistical Concepts Statistical Concepts Multivariate Distributions For independent random variables, the joint probability function is defined as the product of each individual PDF: f X1,X 2,...,X N (X 1, X 2,..., X N ) = f X1 (X 1) f X2 (X 2 )... f XN (X N ) If Z = g(x 1,..., X N ) is another random variable, its means is given by: E(Z) = E[g(X 1,..., X N )] = + + g(t 1,..., t N ) f X1,...,X N (t 1,..., t N ) dt 1,..., dt N A generic formulation for the multivariate central moments is given by: + + N Mm n 1,...,m N = f X1,...,X N (t 1,..., t N ) (x i µ xi ) m i dt 1,..., dt N i=1 F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

14 Statistical Concepts Statistical Concepts Normal Distribution and Sigma Levels This is an important concept to help understanding the reliability targets. In an uncertainty based optimization and in particularly in a Reliability Based Design Optimization the designer has to ensure that the probability of the design failing is within certain prescribed values. Assuming that both objective and constraint functions have normal distributions, these probabilities are associated with different Sigma levels (or standard deviations σ). Normal distribution, 3σ design [15] Sigma Level Percent Variation ±1σ ±2σ ±3σ ±4σ ±5σ ±6σ Sigma level as percent variation F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

15 Robust Design Optimization Table of Contents 1 Outline Presentation 2 Uncertainty Based Design Optimization Definition Uncertainty Models Uncertainty Categorization Optimization Under Uncertainty 3 Statistical Concepts 4 Robust Design Optimization Problem Formulation Numeric Computation of the Mean and Standard Deviation Numeric Methods 5 Reliability Based Design Optimization Problem Formulation Numerical Computation of the Probability of Failure 6 Robust and Reliability Based Design Optimization Problem Formulation Numeric Methods 7 Analytic Problem - Rosenbrock Function with 1 Constraint RDO Approach RBDO approach R 2 BDO Approach 8 Analytic Problem - Problem with 3 Constraints Results 9 Analytic Problem - MDO Benchmark Problem RDO Approach RBDO Appraoch 10 References F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

16 Robust Design Optimization Robustness Definition Noor [16] defined robustness as: the degree of tolerance to variations (in either the components of a system or its environment). A robust ultra-fault-tolerant design of an engineering system is depicted. The performance of the system is relatively insensitive to variations in both the components and the environment. By contrast, a nonrobust design is sensitive to variations in either or both. Robust Design Optimization (RDO) methods: seek a design whose performance is insensitive to small changes in the uncertainty quantities; deal with everyday fluctuations; focus on the event distribution near the mean value. In this way the impact of the uncertainties design is harder to perceive. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

17 RDO Formulation Robust Design Optimization Problem Formulation minimize x,r F (µ f (x, r), σ f (x, r)) subject to G i (µ gi (x, r), σ gi (x, r)) 0 i = 1,..., n g ( ) P xk LB x k xk UB P bounds k = 1,..., n DV µ mean σ standard deviation x deterministic design variable r stochastic design variable n g number of constraints n DV number of design variables P probability F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

18 Robust Design Optimization Numeric Computation of the Mean and Standard Deviation Numeric Computation of µ f (X ) and σ f (X ) The robust objective and constraints are now functions of the mean and standard deviation of objective and constraints, which in turn depend on the probabilistic distribution of variables. Numeric Computation of the Mean and Standard Deviation The analytic integration of the mean and standard deviation is not possible at the majority of the cases. + + µ f (x) =... f (t)p x,r (t)dt σ f (x) = [f (t) µ f (x, r)] 2 p x,r (t)dt So a numerical technique is required: Monte Carlo Integration (MC); Taylor based Method of Moments (MM) [9]; Sigma Point Method (SP) [17]; Surrogate Approximation of µ f and σ f directly [18]; Gaussian Quadrature [19]. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

19 Robust Design Optimization Monte Carlo (MC) Method Numeric Methods Monte Carlo based methods are classical numerical approaches to solve an integral. Were first introduced in the 1940s and have been ever since widely used in uncertainty analysis. It is a stochastic method based on the probability distribution of the output of a process determined by the stochastic distribution of the inputs. The process is repeated n iterations by building a new sample with the desired characteristics at each iteration. µ f = 1 n f (x i ) N i=1 σf 2 = 1 n (f (x i ) µ f ) 2 n 1 i=1 F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

20 Robust Design Optimization Numeric Methods Taylor based Method of Moments (MM) As the name implies, the Taylor Based Method of Moments employs a Taylor series expansion of the function f of a random vector x about the mean of x: N RV ( f f (x) = f (µ x ) + x i=1 i ( ) N 1 RV N RV ( 2 f + 2 x i=1 j=1 i x j ( ) N 1 RV N RV N RV ( + 3! i=1 j=1 k= ) (x i µ xi ) + ) (x i µ xi ) ( x j µ xj ) + 3 f x i x j x k ) (x i µ xi ) ( x j µ xj ) (xk µ xk ) + F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

21 Robust Design Optimization Numeric Methods Taylor based Method of Moments (MM) N RV σf 2 = ( ) N ( ) 1 RV 2 f µ f = f (µ x ) + 2 x 2 σx 2 i +... i=1 i ( ) f 2 σx 2 x i + i=1 i N ( ) RV ( + 3 ) f f ( κxi x 3 i=1 i x i 3 N RV N [( ) RV ( 3 f f + x 2 i=1 j=1,i j i x j x j +... ( ) 2 f + xi 2 ) + ) 2 ( κxi 1 4 ) σ 4 x i + ( ) ( 1 2 ) 2 ] f + σx 2 2 x i x i σx 2 j + j These estimates are third order accurate. The κ is the kurtosis of standard deviation and N RV is the number of random variables. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

22 Robust Design Optimization Numeric Methods Sigma Point Method (SP) The Sigma Point (SP) method is a derivative of the Taguchi method, used in statistical tolerance estimation since 1978 [17]. The idea behind SP is that it is easier to match an input distribution (typically a normal distribution) than to linearise (or in general, approximate) a non-linear mapping. To compute the integrals in SP employs a procedure similar to Gaussian integration, but where the sample locations and respective weights are optimized to match the first moments of the input probability distribution. N RV ˆµ f (χ 0 ) = W 0 f (χ 0 ) + W i (f (χ +) + f (χ )) i=1 i=1 N 2ˆσ f 2 (χ RV 0) = W i (f (χ +) f (χ )) 2 + N RV + i=1 ( Wi 2W 2 i ) ((χ+) + f (χ ) 2f (χ 0 )) 2 F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

23 Robust Design Optimization Sigma Point Method (SP) Numeric Methods ( ) x i K χ 0 = µ x χ + = µ x + (N RV + K), i = 1,..., N RV x i χ = µ x (N RV + K), i = 1,..., N RV x i K W 0 = N RV + K 1 W i = W i+ = W i = 2 (N RV + K) the i th row in the square root of the covariance matrix real constant that should be so that N RV + K = 3, the kurtosis of the standard norma The number of evaluations required for compute sensitivities in a finite difference estimate for gradient based optimization is a huge problem in terms of computational time, if finite differences are used. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

24 Robust Design Optimization Numeric Methods Gaussian Quadrature Method (GQM) σ 2 f = µ f = N N N W i1 ( W i2 (... ( W in f (x i1,i 2,...,i n )))) i 1 =1 i 2 =1 i n=1 N N N W i1 ( W i2 (... ( W in (f (x i1,i 2,...,i n ) µ f ) 2 ))) i 1 =1 i 2 =1 i n=1 This method provides an approximation of the mean and standard deviation of a function on a domain by a suitably weighted sum, where x i s are called nodes and are suitably selected in the domain. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

25 Reliability Based Design Optimization Table of Contents 1 Outline Presentation 2 Uncertainty Based Design Optimization Definition Uncertainty Models Uncertainty Categorization Optimization Under Uncertainty 3 Statistical Concepts 4 Robust Design Optimization Problem Formulation Numeric Computation of the Mean and Standard Deviation Numeric Methods 5 Reliability Based Design Optimization Problem Formulation Numerical Computation of the Probability of Failure 6 Robust and Reliability Based Design Optimization Problem Formulation Numeric Methods 7 Analytic Problem - Rosenbrock Function with 1 Constraint RDO Approach RBDO approach R 2 BDO Approach 8 Analytic Problem - Problem with 3 Constraints Results 9 Analytic Problem - MDO Benchmark Problem RDO Approach RBDO Appraoch 10 References F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

26 Reliability Based Design Optimization Reliability Based Design Optimization (RBDO) methods: Seeks a design whose probability of failure is less than a certain value Deals with extreme events that may lead to failure Focuses on the event distribution near the tails of the PDF F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

27 Reliability Based Design Optimization Problem Formulation RBDO Formulation minimize x,r subject to f (x, r) g rc i (x, r) 0 i = 1,..., n rc g d j (x) 0 x LB k x k x UB k j = 1,..., n d k = 1,..., n DV Alternatively, minimize x,r subject to P (f (x, r) target 0) or P (target f (x, r) 0) g rc i (x, r) 0 i = 1,..., n rc g d j (x) 0 x LB k x k x UB k j = 1,..., n d k = 1,..., n DV F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

28 Reliability Based Design Optimization Reliability Constraints Problem Formulation gi rc = P fi P allowi = P (g (x, r) 0) P allowi P (g (x, r) 0) = p x,r (t)dt g(x,r) 0 x r gi rc gj d n rc n d n DV P P fi P allowi deterministic design variable stochastic design variable reliability constraints other design constraints number of reliability constraints number of other constraints number of design variables probability probability of failure allowable probability of failure F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

29 Reliability Based Design Optimization Numerical Computation of the Probability of Failure Numerical Computation of the Probability of Failure Determining the probability of failure, P fi, requires either sampling (again, Monte Carlo Method ) or techniques such as the: First Order Reliability Method (FORM) [20] [21] [22]; Second Order Reliability Method (SORM) [20] [21] [22]; Sequential Optimization and Reliability Assessment (SORA) [23] [24]; Reliable Design Space (RDS) [25]. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

30 FORM Reliability Based Design Optimization Numerical Computation of the Probability of Failure In essence, FORM consists of creating a linear approximation to the limit state function g(r) (r now being a generalized set of random variables - which encompasses the uncertainties in both design variables and parameters). According to FORM the probability of failure is evaluated (approximately) as: P fi = Φ ( β), where Φ is the cumulative distribution function of the standard normal distribution and β is the distance from the Most Probable Point (MPP) of failure to the current iterate (also called reliability index), measured in the standard normal space - u. Two approaches to solve Most Probable Point problem was investigated: Reliability Index Approach (RIA) Performance Measure Approach (PMA) A transformation from the r-space to the u-space is needed: Distribution type Transformation, r k = T 1 (u k ) Normal (µ, σ) µ + σu k Log-normal (µ, σ) e µ+σu k ( )) Uniform (a, b) a + (b a) erf (u k 2 ( ) 2 1 Gamma (a, b) ab u k 9a a F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

31 Reliability Based Design Optimization Numerical Computation of the Probability of Failure RIA The MPP problem using RIA can be defined as: minimize u ( ) u T 1 2 u subject to g (r (u)) = 0 The reliability constraint in the RBDO problem may be written in terms of the reliability index β: g rc i = β reqd β i, where β reqd is the specific reliability index. This approach has draw-back, when for a particular set of design variables failure does not occur, or when the limit state surface is far from the origin. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

32 Reliability Based Design Optimization Numerical Computation of the Probability of Failure PMA To suppress the RIA draw-back the PMA was devise [26]. In the PMA approach the inverse problem of that one stated in RIA is solved instead with: minimize u subject to g (u) ( u T u) 1 2 β reqd = 0 This is not only a more robust formulation than RIA, it also immediately returns the required value of the reliability constraint: g rc i = g (r (u)) Another important advantage is that the MPP subproblem may be formulated in a minimax approach, effectively handling multiple. constraints simultaneously, something that is not possible with RIA. There are other alternative approaches to the RIA and PMA presented here. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

33 Reliability Based Design Optimization Numerical Computation of the Probability of Failure SORM SORM has seen little practical use in the main reliability problems since it requires higher order information on the objective function and constraints [27] [12] [22]. Thus FORM is the most widely used. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

34 Reliability Based Design Optimization Numerical Computation of the Probability of Failure SORA Huang [23] define the Sequential Optimization and Reliability Assessment (SORA) method as: a single-loop method containing a serial of cycles of decoupled deterministic optimization and reliability assessment for improving the efficiency of probabilistic optimization. The original SORA approach does not take into account the effect of changing variance in design problems. Currently some efforts are being made in order to improve SORA efficiency to solve problems with changing variance The goal of SORA is to use serial single loops to efficiently optimize the objective function and assess its reliability. The difference between this and other similar methods is the way it uses inverse MPP to establish deterministic constraints that are equivalent to the probabilistic ones. Flowchart of the SORA method [2] F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

35 Reliability Based Design Optimization Numerical Computation of the Probability of Failure RDS The Reliable Design Space (RDS) method was introduced by Shan and Wang [25] and aims to convert the original deterministic constraint into a probabilistic one. By this approach, only one single optimization loop needs to be done to reach the solution. As a consequence, the number of required function evaluations is expected to drastically reduce, thus making this the most efficient method is theory. However, it is importance to notice that this method requies partial derivatives, which are not always obtainable and thus reducing its efficiency on those cases. Single-loop method [25] F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

36 Robust and Reliability Based Design Optimization Table of Contents 1 Outline Presentation 2 Uncertainty Based Design Optimization Definition Uncertainty Models Uncertainty Categorization Optimization Under Uncertainty 3 Statistical Concepts 4 Robust Design Optimization Problem Formulation Numeric Computation of the Mean and Standard Deviation Numeric Methods 5 Reliability Based Design Optimization Problem Formulation Numerical Computation of the Probability of Failure 6 Robust and Reliability Based Design Optimization Problem Formulation Numeric Methods 7 Analytic Problem - Rosenbrock Function with 1 Constraint RDO Approach RBDO approach R 2 BDO Approach 8 Analytic Problem - Problem with 3 Constraints Results 9 Analytic Problem - MDO Benchmark Problem RDO Approach RBDO Appraoch 10 References F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

37 Robust and Reliability Based Design Optimization The Robust and Reliability Based Design Optimization (R 2 BDO) is a method that combine the RDO and RBDO characteristics. R 2 BDO uses the robust objective function and the reliable constraints. It was introduced by Paiva [10]. According to the author this change outperforms the original RBDO formulation, because this last has frequent problems in dealing with the probabilistic objective function targets. A bad choice of the starting point can lead to probabilities which are either too close to zero or to one, varying slowly. What leads to insensitive objective functions and most likely the optimizer stops before reaching a true candidate to local/global minimum Also the RBO formulation is not the best in terms of treating with constraint since the user is left with the choice of weights, from which it is defined how far from the failure surface should the average optimum lie. These can be solve by calibrating weights in order to mimic a probabilistic constraint. The way the RBDO deals with constraints is better suited for optimization with uncertainties. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

38 Robust and Reliability Based Design Optimization Problem Formulation R2BDO Formulation minimize x,r subject to F (µ f (x, r), σ f (x, r)) g rc i (x, r) 0 i = 1,..., n rc g d j (x) 0 x LB k x k x UB k j = 1,..., n d k = 1,..., n DV F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

39 Robust and Reliability Based Design Optimization Numeric Methods Numeric Methods Mean and Standard Deviation Computation The numeric methods to compute the mean and standard deviation of the objective function are the same ones used in RDO: Monte Carlo Integration (MC); Taylor based Method of Moments (MM); Sigma Point Method (SP); Surrogate Approximation of µ f and σ f directly; Gaussian Quadrature. Probability of Failure Computation The reliability constraint is also treat by: First Order Reliability Method (FORM); Second Order Reliability Method (SORM); Sequential optimization and reliability assessment (SORA). F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

40 Analytic Problem - Rosenbrock Function with 1 Constraint Table of Contents 1 Outline Presentation 2 Uncertainty Based Design Optimization Definition Uncertainty Models Uncertainty Categorization Optimization Under Uncertainty 3 Statistical Concepts 4 Robust Design Optimization Problem Formulation Numeric Computation of the Mean and Standard Deviation Numeric Methods 5 Reliability Based Design Optimization Problem Formulation Numerical Computation of the Probability of Failure 6 Robust and Reliability Based Design Optimization Problem Formulation Numeric Methods 7 Analytic Problem - Rosenbrock Function with 1 Constraint RDO Approach RBDO approach R 2 BDO Approach 8 Analytic Problem - Problem with 3 Constraints Results 9 Analytic Problem - MDO Benchmark Problem RDO Approach RBDO Appraoch 10 References F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

41 Analytic Problem - Rosenbrock Function with 1 Constraint The Rosenbrock function is a difficult objective function and in this test its application is mainly focus on assessing how the accuracy of each method is affected by different uncertainty levels and target reliabilities. To the classic Rosenbrock function, it can be added a constraint to study the robustness and reliability of one of the design parameters. The constraint is basically an unitary circle. The standard formulation of the Rosenbrock function is given by: minimize x=(x 1,x 2 ) f (x) = 100(x 2 x 2 1 )2 + (1 x 1 ) 2 subject to g(x) = x x The analytic solution of this optimization problem is (x 1, x 2 ) = (0.7864, ) with the function value: f = For the reported analyses in this presentation it is used x 1 as the parameter that carries uncertainty, while x 2 remain deterministic. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

42 Analytic Problem - Rosenbrock Function with 1 Constraint RDO Approach RDO Formulation The Rosenbrock function constrained to a unit circle can be posed in the RDO Formulation as follows: minimize x=(x 1,x 2 ) F (x) = µ f (x) + σ f (x) subject to G(x) = µ g + 2σ g 0 The factor 2 in the constraint is called Robustness Constraint. To solve this problem, the methods briefly introduced earlier were applied. Monte Carlo Method was only used for post-optimality analysis of the others selected methods. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

43 Analytic Problem - Rosenbrock Function with 1 Constraint RDO Approach RDO Results σ Method x 1 x 2 µ f σ f ɛµ f ɛσ f fcalls dfcalls gcalls MM SP MM SP MM SP One can observe that by decrease the covariance of x 1, the optimum attained is closest to the global minimum. MM method proves to be inaccurate even for low covariances, while the SP is very accurate at the predicting the desired statistical measures, comparing favourably with the very heavy (computational time speaking) Monte Carlo method. SP performs even less function evaluations than the MM in return for a considerably higher accuracy, although it presents more constraint function calls. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

44 Analytic Problem - Rosenbrock Function with 1 Constraint RDO Approach RDO Results f(x) Constraint Minimum evolution SP Minimum obtained SP Minimum evolution MM Minimum obtained MM Analytical Minimum kσg Method x 1 x 2 µ f σ f ɛµ f ɛσ f fcalls dfcalls gcalls dgcalls 1 SP SP SP F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

45 Analytic Problem - Rosenbrock Function with 1 Constraint RBDO approach RBDO Formulation The Rosenbrock function constrained to a unit circle can be posed in the RBDO Formulation as follows: minimize x=(x 1,x 2 ) F (x) = f (µ x ) subject to g r c(x) = 0 Two different methods of FORM were implemented: RIA; PMA. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

46 Analytic Problem - Rosenbrock Function with 1 Constraint RBDO approach RBDO Results σ Method x 1 x 2 f ɛ β fcalls gcalls RIA PMA RIA PMA RIA PMA One can observe that by decrease the covariance of x 1, the optimum attained is closest to the global minimum. The objective function is defined using the mean values of the design variables as input, becoming virtually the same as if using the first order Method of Moments (RDO). The errors (ɛ β ) are obtained by comparison of the real reliability with the target. The accuracy of both formulations in terms of the reliability index is acceptable, especially considering that FORM uses a first order approximation to the real failure hypersurface. From the table above one can notice that the PMA approach requires less constraint function calls than RIA, although presents a little more functions calls. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

47 Analytic Problem - Rosenbrock Function with 1 Constraint RBDO approach RBDO Results f(x) Constraint Minimum evolution RIA Minimum obtained RIA Minimum evolution PMA Minimum obtained PMA Analytical Minimum k βreqd Method x 1 x 2 f ɛ β fcalls gcalls 1 RIA RIA RIA F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

48 Analytic Problem - Rosenbrock Function with 1 Constraint R 2 BDO Approach R 2 BDO Formulation The Rosenbrock function constrained to a unit circle can be posed in the R 2 BDO Formulation as follows: minimize x=(x 1,x 2 ) F (x) = µ f (x) + σ f (x) subject to g r c(x) = 0 It was used the Sigma Point method to compute the mean and standard deviation of the objective function. For the constraints it was selected the FORM with PMA. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

49 Analytic Problem - Rosenbrock Function with 1 Constraint R 2 BDO Results R 2 BDO Approach σ x1 β req x 1 x 2 f ɛ β fcalls gcalls β req σ x1 x 1 x 2 f ɛ β fcalls gcalls As expected the computational time has increased significantly since this hybrid method uses the parts of RDO and RBDO that require more function calls such as the computation of the mean and standard deviation and the MPP problem in the reliability constraint. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

50 Analytic Problem - Rosenbrock Function with 1 Constraint R 2 BDO Results R 2 BDO Approach f(x) Constraint Minimum evolution Minimum obtained Analytical Minimum F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

51 Analytic Problem - Problem with 3 Constraints Table of Contents 1 Outline Presentation 2 Uncertainty Based Design Optimization Definition Uncertainty Models Uncertainty Categorization Optimization Under Uncertainty 3 Statistical Concepts 4 Robust Design Optimization Problem Formulation Numeric Computation of the Mean and Standard Deviation Numeric Methods 5 Reliability Based Design Optimization Problem Formulation Numerical Computation of the Probability of Failure 6 Robust and Reliability Based Design Optimization Problem Formulation Numeric Methods 7 Analytic Problem - Rosenbrock Function with 1 Constraint RDO Approach RBDO approach R 2 BDO Approach 8 Analytic Problem - Problem with 3 Constraints Results 9 Analytic Problem - MDO Benchmark Problem RDO Approach RBDO Appraoch 10 References F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

52 Analytic Problem - Problem with 3 Constraints The scope of this test case is to evaluate not only the SORA and RDS methods not used in the previous problem, but also to assess if all the methods still remain accurate if more than one constraint exists. The problem was introduced by Shan and Wang [25] and is posed as: minimize µ 1,µ 2 f (µ 1, µ 2 ) = µ 1 + µ 2 subject to P(g i (X ) 0) R i, i = 1, 2, 3 g 1 (X ) = X 2 1 X 2/20 1 g 2 (X ) = (X 1 + X 2 5) 2 /30 + (X 1 X 2 12) 2 /120 1 g 3 (X ) = 80/(X X 2 + 5) 1 0 µ j 10, j = 1, 2 σ 1 = σ 2 = 0.3β i = 3i, i = 1, 2, 3 where mu 1, µ 2, σ 1 and σ 2 are the mean values and standard deviations of 2 random variables (X 1 and X 2 ), and R i is the required reliability (which is the same for every constraint). A target reliability index (β reqd = 3) and a standard deviation of the random variables (σ = 0.3) was chosen. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

53 Analytic Problem - Problem with 3 Constraints Results Results This results of this problem were computed by L. Amândio [28] [29] RBDO RDO R 2 BDO Method RIA PMA PMA alt SORA SORA alt RDS MM SP SP + PMA alt Design Variables µ µ Objective function Constraint Reliability ε β1 [%] ε < 2 ε < 2 ε < 2 ε < 2 ε < 2 ε < ε < 2 ε β2 [%] ε < 2 ε < 2 ε < 2 ε < 2 ε < 2 ε < ε < 2 ε β3 [%] Function Calls #Obj. func. eval #Const. func. eval #Total. func. eval the number of function evaluations required to determine partial derivatives, were not accounted for the number of necessary function calls within the main function evaluations, were taken into account F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

54 Analytic Problem - MDO Benchmark Problem Table of Contents 1 Outline Presentation 2 Uncertainty Based Design Optimization Definition Uncertainty Models Uncertainty Categorization Optimization Under Uncertainty 3 Statistical Concepts 4 Robust Design Optimization Problem Formulation Numeric Computation of the Mean and Standard Deviation Numeric Methods 5 Reliability Based Design Optimization Problem Formulation Numerical Computation of the Probability of Failure 6 Robust and Reliability Based Design Optimization Problem Formulation Numeric Methods 7 Analytic Problem - Rosenbrock Function with 1 Constraint RDO Approach RBDO approach R 2 BDO Approach 8 Analytic Problem - Problem with 3 Constraints Results 9 Analytic Problem - MDO Benchmark Problem RDO Approach RBDO Appraoch 10 References F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

55 Analytic Problem - MDO Benchmark Problem Analytic The original problem was first introduced by Sellar et al [30]. In this presented test case it was reduced the number of global design variables by erasing the x 2 variable from discipline 1. This analytic problem was chosen due to its simplicity and despite its low dimensionality, it presents features of larger Multidisciplinary Design Optimization (MDO) problems, which allows each of the MDO architecture implementations to be verified before passing to more complex problems. This problem in the standard formulation is given by: minimize z 1,x 1,x 2 f (z 1, x 1, x 2) = x x2 + y1 + e y 2 subject to 1 y y z x 1, x 2 10 Discipline 1 y 1(z 1, x 1, y 2) = z x1 0.2y2 Discipline 2 y 2(z 1, x 2, y 1) = y 1 + z 1 + x 2 The introduction of the uncertainties in the design optimization was performed using the Multi-Discipline Feasible (MDF) architecture, due to it simplicity and because the MDF solve a single optimization problem, which makes easier the introduction of uncertainties in the MDO architectures study. The local design variables, (x 1, x 2), remain deterministic, while z 1 carries now uncertainty. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

56 RDO Formulation Analytic Problem - MDO Benchmark Problem RDO Approach minimize x=(x 1,x 2 ),r=µ z1 subject to F (x, r) = W f µ f + W g σ f G(x) = K µg µ g K σg σ g le 10 µ z x 1, x 2 10 with f = x1 2 + x 2 + y1 (x 1, y 2, µ z1 ) + e y2(x 2,y 1,µ z1 ) 1 y1(x 1,y 2,µ z1 ) 0 g = 3.16 y2(x 2,y 1,µ z1 ) µ z1 = z 1 σ z1 = c.o.v. z1 µ z1 W f = 1 W g = 1 K µg = 1 K σg = 2 Discipline 1 y1 (x 1, y 2, µ z1 ) = µ 2 z 1 + x 1 0.2y 2 Discipline 2 y2 (x 2, y 1, µ z1 ) = y 1 + µ z1 + x 2 F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

57 Analytic Problem - MDO Benchmark Problem RDO Approach RDO Results Optimum c.o.v. z1 = c.o.v. z1 = c.o.v. z1 = 0.01 c.o.v. z1 = µ f σ f x x z One can note that when the random parameter z 1 covariance increases the objective function becomes farther, as it was expected. With c.o.v. z1 = 0.005, mean and standard deviation errors under 1% in a second were achieved, which is just a little more than using the MDF ( second) without uncertainties. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

58 RBDO Formulation Analytic Problem - MDO Benchmark Problem RBDO Appraoch minimize x=(x 1,x 2 ),r=µ z1 f (x, r) = x1 2 + x 2 + y1 (x 1, y 2, µ z1 ) + e y2(x 2,y 1,µ z1 ) subject to { gi r β reqd β 0 if RIA g (r(u)) 0 if PMA 10 µ z x 1, x 2 10 with µ z1 = z 1 σ z1 = c.o.v. z1 µ z1 Discipline 1 y1 (x 1, y 2, µ z1 ) = µ 2 z 1 + x 1 0.2y 2 Discipline 2 y2 (x 2, y 1, µ z1 ) = y 1 + µ z1 + x 2 F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

59 RBDO Formulation Analytic Problem - MDO Benchmark Problem RBDO Appraoch RIA Formulation: PMA Formulation: minimize u ( ) u T 1 2 u subject to g(r(u)) = 0 { with g(r(u)) = 1 y y 2 24 minimize u subject to g(u) ( u T u) 1 2 β reqd with g(u) = [(3.16 y 1 ) + (y 2 24)] The r-space to u-space transformation is using a Normal Distribution. F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

60 References Table of Contents 1 Outline Presentation 2 Uncertainty Based Design Optimization Definition Uncertainty Models Uncertainty Categorization Optimization Under Uncertainty 3 Statistical Concepts 4 Robust Design Optimization Problem Formulation Numeric Computation of the Mean and Standard Deviation Numeric Methods 5 Reliability Based Design Optimization Problem Formulation Numerical Computation of the Probability of Failure 6 Robust and Reliability Based Design Optimization Problem Formulation Numeric Methods 7 Analytic Problem - Rosenbrock Function with 1 Constraint RDO Approach RBDO approach R 2 BDO Approach 8 Analytic Problem - Problem with 3 Constraints Results 9 Analytic Problem - MDO Benchmark Problem RDO Approach RBDO Appraoch 10 References F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

61 References I References E. Nikolaidis. Engineering Design Reliability Handbook, chapter Types of Uncertainty in Design Decision Making. CRC Press, X. Du and W. Chen. Sequential optimization and reliability assessment method for efficient probabilistic design. In Probabilistic Design, ASME Design Engineering Technical Conferences, pages , X. Yu and X. Du. Reliability-based multidisciplinary optimization for aircraft wing design. Structure and Infrastructure Engineering: Maintenance, Management, Life-Cycle Design and Performance, 2: , F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

62 References II References T. A. Zang, M. J. Hemsch, M. W. Hilburger, S. P. Kenny, J. M. Luckring, P. Maghami, S. L. Padula, and W. J. Stroud. Needs and opportunities for uncertainty - based multidisciplinary design methods for aerospace vehicles. Technical report, National Aeronautics and Space Administration, J. G. Klir and M. J. Wierman. Uncertainty-Based Information. Physica-Verlag, S. Parsons. Qualitative Methods for Reasoning under Uncertainty. The MIT Press, S. F. Wojtkiewicz, M. S. Eldred, R. V. Field, A. Ubrina, and J. R. Red-horse. Uncertainty quantification in large computational engineering models. In 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Seattle, Washinghton, USA, F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

63 References III References H. Agarwal. Reliability based design optimization: formulations and methodologies. PhD thesis, University of Notre Dame, Notre Dame, Indiana, USA, M. Padulo, S. A. Forth, and M. D. Guenov. Robust Aircraft Conceptual Design using Automatic Differentiation in Matlab, pages Springer, R. M. Paiva. A Robust and Reliability-Based Optimization Framework for Conceptual Aircraft Wing Design. PhD thesis, University of Victoria, Victoria, British Columbia, Canada, D. Padmanabhan, H. Agarwal, J. E. Renaud, and S. M. Batill. A study using Monte Carlo simulation for failure probability calculation in reliability-based optimization. Optimization and Engineering, 7: , F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

64 References IV References D. M. Frangopol and K. Maute. Life-cycle reliability-based optimization of civil and aerospace structures. Computers and Structure, 81: , R. M. Paiva, C. Crawford, and A. Suleman. Robust and reliability based design optimization framework for wing design. AIAA Journal, 0(0):0 0, L. Huyse. Solving problems of optimization under uncertainty as statistical decision problems. AIAA Paper No , P. N. Koch, R.-J. Yang, and L. Gu. Design for six sigma through robust optimization. Structural and Multidisciplinary Optimization, 26: , F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

65 References V References A. K. Noor. Engineering Design Reliability Handbook, chapter Perspectives on Nondeterministic Approaches. CRC Press, J. R. D Errico and N. A. Zaino Jr. Statistical tolerancing using a modification of taguchi s method. Technometrics, 30(4): , M. Bonte, A. van den Boogaard, and J. Huétink. Deterministic and robust optimisation strategies for metal forming processes. In Forming Technology Forum 2007 Application of Stochastics and Optimization Methods, Zurich, Switzerland, F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69

66 References VI References M. Padulo, M. S. Campobasso, and M. D. Guenov. Comparative analysis of uncertainty propagation methods for robust engineering design. In Proceedings of International Conference on Engineering Design, ICED 2007, Paris, France, August. X. Du. First order and second reliability methods. Course notes ME Probabilistic Engineering Design, Missouri University of Science and Technology, Chapter 7. A. Kiureghian. Engineering Design Reliability Handbook, chapter First- and Second-Order Reliability Methods. CRC Press, F. Afonso, L. Amândio, A. Marta and A. Suleman Robust and Reliability Based Design Optimization May 22, / 69