CAS-D-15-00814

7/23/2019 CAS-D-15-00814

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Computers and Structures

Manuscript Draft

Manuscript Number: CAS-D-15-00814

Title: Efficient optimization of reliability-constrained structural

design problems including interval uncertainty

Article Type: Research Paper

Keywords: surrogate model; optimization; interval uncertainty;

reliability; composite structures

Abstract: A novel interval uncertainty formulation for exploring the

impact of epistemic uncertainty on reliability-constrained design

performance is proposed. An adaptive surrogate modeling framework is

developed to locate the lowest reliability value within a multi-

dimensional interval. This framework is combined with a multi-objective

optimizer, where the interval width is considered as an objective. The

resulting Pareto front examines how uncertainty reduces performance while

maintaining a reliability threshold. Two case studies are presented: a

cantilever tube under multiple loads and a composite stiffened panel. The

proposed framework is robust and able to resolve the Pareto front with

fewer than 200 reliability evaluations.

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Efficient optimization of reliability-constrained

structural design problems including interval uncertainty

Yan Liua, Han Koo Jeongb, Matthew Collettea,∗

a Department of Naval Architecture and Marine Engineering, University of Michigan,

United States bDepartment of Naval Architecture, Kunsan National University, Korea

Abstract

A novel interval uncertainty formulation for exploring the impact of epis-temic uncertainty on reliability-constrained design performance is proposed.An adaptive surrogate modeling framework is developed to locate the low-est reliability value within a multi-dimensional interval. This framework iscombined with a multi-objective optimizer, where the interval width is con-sidered as an objective. The resulting Pareto front examines how uncertaintyreduces performance while maintaining a reliability threshold. To case stud-ies are presented: a cantilever tube under multiple loads and a compositestiffened panel. The proposed framework is robust and able to resolve thePareto front with fewer than 200 reliability evaluations.

Keywords: surrogate model, optimization, interval uncertainty, reliability,composite structures

1. Introduction

Managing uncertainty, especially in the early stages of structural design,is critical to many novel designs and materials. While robust design andreliability-based optimization frameworks have been successfully developed(see e.g. [1, 2, 3, 4] for recent work in this field), most of these formulationsrequire a precise stochastic definition of the uncertainty involved. However,

early stage structural design with novel materials or applications is marked by

∗Corresponding AuthorEmail address: [email protected] (Matthew Collette)

Preprint submitted to Computers and Structures September 8, 2015

nuscript

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7/23/2019 CAS-D-15-00814


comparatively limited and vague information about the design and associated

uncertainties. Uncertainty associated with limited knowledge is epistemic innature and is normally reducible by investment in engineering investigation.Consequently, traditional reliability-based design tools may not be well suitedto model the design situation as the epistemic uncertainty normally cannotbe stochastically defined.

Reliability analysis with non-probabilistic interval uncertainty [5, 6] hasbeen studied as an approach to modeling epistemic uncertainty. While thisremoves the need to define a precise stochastic distribution, reasonable un-certainty bounds still must be developed. The approach proposed here takesa different route. Reliability-based design optimization and interval uncer-tainty approaches are combined. Then, the width of the uncertainty inter-

vals are treated as an objective in a multi-objective optimization approachwhile maintaining consistent reliablity levels. The resulting Pareto frontsshow the impact of lack of knowledge on design performance and allows theengineering design team to prioritize where to invest time in reducing epis-temic uncertainty. The core contribution required to make such an approachpractical is a novel adaptive surrogate modeling technique for efficient inter-val reliability analysis. This surrogate approach allows the combination of reliability models, interval uncertainty, and multi-objective optimization toremain computationally feasible.

The central concept of the proposed framework is that interval uncer-

tainty is a useful representation of uncertainty in early-stage design knowl-edge. Conventional structural optimization approaches [7] often use stochas-tic form to account for uncertainty in the design. However, in the author’sfield of marine structures, such models are difficult to apply early in the de-sign process as precise stochastic uncertainty information does not yet exist.Any error in assumption of distribution can be harmful later in the design [8].This is especially true for marine structures, where too low early structuralweight estimates can cause extensive design re-work or in-service structuralfailures [9]. It is argued that interval uncertainty can be used to address thisconcern where no assumption of distribution is needed [10]. Among othernon-traditional uncertainty models [11], interval uncertainty was selected as

in most cases the non-deterministic parameters and variables are only knownwithin intervals [12]. This work is the first to explore the coupling betweenstructural optimization with interval uncertainty that is arranged in reliabil-ity analysis. The interval uncertainty range will be treated as design vari-ables, and the optimizer will determine feasible design configurations with

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respect to various uncertainty intervals. The aim is to resolve the trade-off

between interval uncertainties and design performance through the optimiza-tion, thus unveiling the overall impact of early stage design uncertainty onthe design.

A complication in selecting interval uncertainty is that working with in-tervals in optimization can be computationally expensive. In design opti-mization involving interval uncertainty, a max-min optimization [13] is oftenneeded. In min-max approaches, the optimizer searches for a solution thathas the best worst-case performance in the uncertainty interval. In terms of reliability analysis involving interval uncertainty, the worst case of intervalanalysis needs to satisfy the specified reliability constraint. As reliabilitysimulation itself normally involves a search for most probable point of failure

[14], reliability analysis with interval uncertainty becomes a nested optimiza-tion in which the worst case performance needs to be located. Consideringthat such analysis is repeatedly requested in population-based heuristic de-sign optimization approaches, the computation can soon become intractable.This work presents a method that is capable in locating the worst case re-liability result while remaining computationally efficient, therefore allowingthe aforementioned interval uncertainty study to be carried out within amoderate computation budget in reliability-based design optimization.

Many previous authors have studies how to efficiently solve reliability-based design optimization (RBDO) problems. One way is focused on reli-

ability problem formulation, where the performance measure approach [15],sequential RBDO [16] and single-loop RBDO [17] have been proposed. How-ever, they are not ideal as solutions to interval uncertainty problems whereworst case reliability needs to be computed. A more direct way to reducecomputational cost is referring to the research field of surrogate modeling.In recent years, development in this direction has been reported [18]. Amongthe surrogate methods proposed, Kriging [19] shows promise as an approxi-mation tool in reliability simulation. Kriging was first proposed for structuralreliability problem by Kaymaz [20], and recent development can be found inworks of Bichon et al. [21], Echard et al. [22], and Dubourg et al. [23]. Thesestudies on Kriging methods for reliability mainly focused on the approxima-

tion of the limit state function, and then applying Monte Carlo Simulation(MCS) method for reliability analysis. Such methodology has also been ap-plied in solving reliability analysis with interval uncertainty problems [24].While Kriging-assisted MCS methods may be a viable strategy for a singlereliability analysis, adopting this methodology in population-based optimiza-

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tion algorithms, such as evolutionary algorithm, can be problematic, as the

large number of reliability analysis required can quickly render MCS methodextremely costly even with the help of Kriging.This manuscript introduces a new Kriging modeling strategy tailored to

the specific need in design optimization with interval uncertainty. The pro-posed Kriging method is advantageous in that it can give a predicted worst-case performance along with the estimated function response; therefore, theinner-loop search in the interval uncertainty domain can be avoided. Thestrategy is achieved by exploiting the Kriging model information to estab-lish a local quadratic approximation for reliability in the interval variablesdomain. With the quadratic approximation, the worse-case performance canbe directly located to avoid a search operation. Additionally, two refine-

ment criteria for the Kriging are implemented to ensure the quality of worst-case prediction from the proposed surrogate model and the accuracy of thequadratic approximation. It is demonstrated in this manuscript that theproposed method can accurately estimate worst-case reliability performancefor multi-objective evolutionary algorithm with a high level of efficiency.

The rest of the paper is organized as follows. Section 2 reviews the intervaluncertainty and interval reliability analysis. Subsequently, Section 3 intro-duces the proposed surrogate modeling method and the multi-objective op-timization framework for trade-off study. In Section 4, the proposed methodis examined in an interval benchmark problem [5], and then the validated

method is applied to CFRP top-hat stiffened panel design [25]. Discussionand concluding remarks are given in the last section.

2. Review of interval reliability analysis

This work treats uncertainties that are due to lack of information via aninterval formulation. There are two critical components to such an approach:the definition of the interval model and the application of this model inreliability analysis. Each of these components is reviewed in turn in thissection.

2.1. Interval uncertainty

Interval uncertainty provides an appropriate alternative from stochasticuncertainty, as no information regarding stochastic distribution is required.Interval modeling [26] is usually applied in a simple close form:

Y = [Y l, Y h] = Y ∈ I |Y l ≤ Y ≤ Y h (1)

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Interval uncertainty model is concerned with investigating the whole range of

the potential values bounded by a higher bound and a lower bound. There isno assumption of probabilistic distribution within these bounds; specifically,a uniform distribution is not assumed. The interval definition can also beinterpreted in another form:

∀Y ∈ [Y 0 − ∆, Y 0 + ∆] (2)

where ∆ is defined as the maximum deviation of uncertainty from nominalvalue, representing the range of the interval uncertainty. In this form, it isclear that interval uncertainty can reflect the tolerance for error in the design[27]. The impact of interval ranges on the design performance is the aim of

this work. This trade space can be studied by making a version of ∆ one forthe objectives of the optimization problem, as reviewed in Section 3.

2.2. Worst-case reliability index with interval uncertainty

Without interval uncertainty, reliability analysis is concerned with calcu-lating the probability of failure in a limit state function:

P f = P r(g(X) ≤ 0) =

g(X)≤0

f X(X)dX (3)

where g denotes the limit state at which the system is safe if g(X) ≥ 0. X

is a vector of random variables accounting for stochastic uncertainties, and

f X(X) is the joint probability density function. Required and achieved valuesof P f are normally given in terms of the safety index, β :

P f = Φ(−β ) (4)

As direct integration of the expression given in Equation 3 is difficult, manyapproximate methods of estimating β have been proposed. Here, for simplic-ity, the first order reliability method (FORM) [14] is used. In FORM, thereliability index can computed in the following procedure:

β = minU

U

s.t. g(U ) = 0(5)

where U is the transformed standard normal variables from random variables:ui = Φ−1(F xi(xi)). The reliability index can also interpreted as the minimumdistance from origin to limit state in U space.

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Given that the information about stochastic uncertainty models are usu-

ally incomplete in the early stage of innovative marine structure design, thecalculated reliability index may not be accurate enough to ensure sufficientsafety in design [8]. Interval uncertainty can be introduced into reliabilityanalysis to model the epistemic contribution to uncertainty and address thisconcern [28]. Generally, when interval uncertainty parameter is involved inthe limit state function, the interval reliability index output becomes an in-terval: β ∈ [β L, β H ]. In this reliability-based optimization involving intervaluncertainty, the worst-case reliability index β L is used for safety examination.

Here, interval parameter Y is used to illustrate the process in comput-ing β L. After the transformation of random variables X to U space, thelimit state function g(U, Y ) = 0 is defined with both normal variables U and

interval parameter Y . Though there is proposed work [29] conducting mono-tonicity analysis for reliability with interval parameter, most of the times anoptimization search is needed to locate β L in the interval reliability output.The procedure is defined with the equation below:

β L = minU,Y

U (Y )

s.t. g(U, Y ) = 0

Y L ≤ Y ≤ Y U

(6)

The above procedure is a nested optimization process where the outer

loop minimizes the reliability index by locating the worst-case combinationof interval variables Y ∗, while the inner loop is a reliability determinationvia any standard reliability evaluation method. In this work, the FORMprocedure defined in Eq. 5, is used for the reliability index evaluation.

2.3. Impacts of interval reliability analysis

A reliability-based design optimization that accounts for worst-case reli-ability using FORM analysis can be expressed as the following:

min f (µ(X ))

s.t. β L = minY

f FORM (X ,Y ) ≥ β t

Y ∈ [Y 0 − ∆,Y 0 + ∆](7)

where f is a performance function associated with design variables and β t isthe target reliability index constraint. f FORM denotes the process by which

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to evaluate the reliability index. Additional constraints that do not involve

Y or may be deterministic could also be added.While most studies on interval reliability assumed fixed interval domain,this work argues that various range values ∆ of interval uncertainty needto be examined. The range ∆ reflects the epistemic uncertainty present.However, epistemic uncertainty is normally reducible by investing engineer-ing time and expense to generate and analyze additional data. From theproject management perspective, it is interesting to explore what the impactof epistemic uncertainty is on design performance and what performance gaincould be achieved by reducing it. It is proposed that the width of ∆ can beset as an objective in multi-objective optimization along with conventionalobjectives such as weight, cost etc. The resulting Pareto trade space will

show the engineers the impact of epistemic uncertainty on performance.Computational efficiency is the primary issue that needs to be addressed

in order for the proposed interval study. Due to the optimization searchfor worst-case combination of interval variables Y ∗, the number of reliabil-ity analyses needed can increase significantly and makes the computationintractable for design optimization. This manuscript proposes an advancedtechnique that is capable of conducting the worst case search in intervaldomain and clearing the obstacle in proposed trade study on interval uncer-tainty.

The following section will introduce an adaptive surrogate modeling strat-

egy to locate the worst-case reliability and a multi-objective optimizationframework in order to investigate the trade-off between interval uncertaintyand design performance.

3. Efficient trade-off analysis with Adaptive Kriging modeling

3.1. Overview

In this section, an efficient implementation of the interval reliability anal-ysis is put forward in a multi-objective optimization framework. Since thedesign optimization cycle time is closely related with the number of reli-ability simulation in Eq. 7, in this study, FORM reliability simulation is

approximated by a proposed Kriging surrogate model in the optimization.Moreover, the surrogate method proposed here specifically address the prob-lems where the worst-case search is needed in interval variable domain. Ineach surrogate prediction for the reliability performance of the individual, anestimated worst-case reliability index can also be provided simultaneously in

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the proposed method. Thus, a higher level of efficiency is achieved by re-

moving the inner-loop search. The surrogate-assisted RBDO that accountsfor worst-case reliability can be expressed as the following:

min f (µ(X ))

s.t. β L(X ,Y ) = y(X ,Y ∗) ≥ β t(8)

where [X ,Y ∗] is the estimated worst-case scenario of design point [X ,Y ]from the surrogate model. y means that the surrogate model is used toapproximate FORM simulation. The Kriging theory and the derivationof worst-case estimator are presented in sequence. Afterwards, the multi-objective optimization for interval uncertainty trade study is introduced.

3.2. Kriging modeling

Kriging is a powerful surrogate model that has been widely used to ap-proximate computationally expensive simulations. An in-depth Kriging the-ory can be found in works of Sacks [19] and Simpson [30]. As a brief expla-nation, Kriging predicts the function value y at an unobserved point basedon a set of sampling points through a realization of a regression model anda stochastic process:

y(x) = f T β + z (x) (9)

where f is regression basis functions by user’s choice and β are regressional

coefficients. In this work, ordinary Kriging is used; thus, f is a vector of all1.0 with length ns. The stochastic process z is assumed to have zero meansand a covariance of:

E [z (xi)z (x j)] = σ2R(θ, xi, x j) (10)

where σ is the process variance and R(θ, xi, x j) is the correlation model. Acommonly used Gaussian correlation model is adopted here:

R(θ,xi,x j) = exp(−nvk=1

θk(|xik − x j

k|2) (11)

where θ is a correlation parameter vector that is found by optimizing a maxi-mum likelihood function. After that, the Kriging predictor for an unobservedpoint x can be expressed as the following:

y(x) = f T β + rT (x)R −1(Y − Fβ) (12)

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where β is computed by least square regression, the vector r measures the

correlation between the prediction point x and the sampled points [x1...xm].The mean square error s2 of the predictor can also be provided:

s2(x) = σ2

1 + uT F T R −1F

−1u − rT R −1r

(13)

where u = F T R −1r − f . As an indication of prediction quality, s2 will beused later for adaptively improve the modeling.

From Eq. 12 it follows that the gradient of the Kriging predictor can bederived as follows:

J y = J f (x)T β + J r(x)R −1(Y − Fβ) (14)

where J f and J r are the Jacobian of f and r, respectively. J f is [On×1] as f is a vector of constants in ordinary Kriging.

The ith row of the Hessian matrix of the Kriging predictor,

H i,: = [ ∂ 2y

∂xi∂x1,

∂ 2y

∂xi∂x2,...,

∂ 2y

∂xi∂xnv

] (15)

can also be derived as follows:

H i,: = ∂J r(x)T

∂xi

R −1(Y − Fβ) (16)

where: ∂J r(x)

∂xi

mn

= ∂ 2R(θ,x,xm)

∂xi∂xn

, m = 1,...,ns, n = 1,...,nv (17)

These properties will be used to accelerate the worst-case performanceprediction search, as explained in next section.

3.3. Worst-case prediction and adaptive refinement

In this study, the Kriging surrogate model is used as a worst-case esti-

mator to replace the optimization search that is conventionally required inlocating the worst case Y ∗. Y ∗ is expressed as follows:

Y ∗ = arg minY ∈I

f (X ,Y ) (18)

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where I = [Y L,Y U ] is the bounded interval variables domain. In the pro-

posed surrogate method, Y ∗

is estimated in a single Kriging evaluation andtherefore avoids an additional nested optimization search.For every candidate point D that has n design variables and m interval

variables D = [X 1,...,X n, Y 1,...,Y m], the predicted response yD, Jacobianmatrix J D, and Hessian matrix H D, can be provided by the surrogate basedon Section 3.2. Within J D and H D, the components associated with intervalvariables Y will be sorted out and assembled to form local Jacobian andHessian matrices for interval domain:

J Y = [ ∂ y

∂Y 1,...,

∂ y

∂Y m]T

H Y =

∂ 2y

∂Y 21 · · ·

∂ 2y

∂Y 1∂Y m...

. . . ...

∂ 2y

∂Y m∂Y 1· · · ∂ 2y

∂Y 2m

(19)

Afterwards, a quadratic model for the interval domain can be establishedusing J Y , H Y and yD:

y(Y + sY ) ≈ 1

2sY

T H Y sY + J Y sY + yD (20)

where sY is called the Newton step to the minimum of the quadratic, which

is the estimate of worst-case performance. sY can be obtained by:

sY = −H −1Y J Y (21)

Assuming the quadratic approximation implied by this approach matchesthe underlying Krigin model, the worst case can be directly determined asY ∗ = Y + sY . This method is much more efficient for the worst-case searchas it eliminates the inner optimization run originally required in Eq. 18.

However, there are two levels of nested assumptions in this approach.First, it is assumed that the quadratic approximation of the Kriging surfaceis accurate. Second, it is assumed that the Kriging model is a faithful model

of the underlying reliability simulations. To ensure that these assumptionsare both reasonable, two online refinement criteria are used. These criteriaare evaluated every time a worse-case performance prediction is made, andif either is violated, an updated prediction is generated. This scheme willensure that an accurate worst case performance prediction is provided for

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optimization while refining the Kriging model to increase the quality of the

surrogate model as the optimization runs. The refinement criteria are asfollows:

1. Based on the predicted worst-case design, a repeated worst-case predic-tion is made reestablishing the quadratic approximation at the worst-case point found. The two outcomes of these two searches need to besufficiently close;

2. The U metric proposed in [22] for Kriging model accuracy is used here:

U = y(Dworst) − β t

s(Dworst) (22)

where s is the Kriging variation derived in Eq. 13. This metric comparesthe distance from the reliability constraint boundary to the predictederror in the Kriging model. Points close to a constraint boundary orwith large prediction errors would be highlighted for refinement. Aminimum value of 3.0 is required for U metric in this study.

The first criterion ensures that the worst-case prediction is stable by aniterated prediction check. The second criterion establishes a lower confidencebound [31] regarding the target value β t. The U metric utilizes the Krigingvariation information to ensure the accuracy of prediction. A minimum valueof 3.0 suggests that the probability of making a mistake on y ≥ β t is Φ(−3) =

0.135%. In each worst-case estimation, these two criteria will be examined,and any violation will be used as a guidance to update the surrogate model.The procedure is outlined in Algorithm 1.

3.4. Multi-objective optimization coupling interval variables

The goal of the proposed method is to explore the impact of reducing thewidth ∆ of interval uncertainties on design performance. The resulting tradespace will help prioritize areas for engineering investment to reduce epis-temic uncertainty. With the developed worst-case estimation technique, thistrade-off study of interval uncertainty can be resolved in a multi-objective

optimization framework. This work proposes to couple interval uncertaintyinto structural design optimization by treating interval widths ∆ as designvariables, and an interval reduction metric as objective function. Some in-terval reduction measures can be found in the work of Li et al. [32]. In thispaper, a simple inverse metric of interval variable ranges production is used

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Algorithm 1 Worst case prediction and updating scheme.

BEGINInitialize: Train an initial Kriging modelfor Candidate design Di = [X ,Y ] do

Predict the worst case performance D(1)worst = [X ,Y ∗]of Di;

Repeat the process by predicting D(2)worst of D

(1)worst;

Compute the U metric, and compare two predicted worst cases;if |y(D

(2)worst) − y(D

(1)worst)| ≤ 10−5 and U ≥ 3.0 then

Do not update Krigingelse

Evaluate β (Di) using exact reliability analysisUpdate Kriging modelEstimate Dworst again

end if

end for

Set reliability index of Di as y(Dworst)END

as an indication of the cost needed to gain relevant information and reducesthe interval uncertainty:

Cost = 1

I

i=1

∆i

(23)

The interval reduction cost function and structural performance functionwill be optimized within NSGA-II - a multi-objective genetic algorithm op-timizer proposed by Deb et al. [33]. The NSGA-II algorithm is an elitistnon-dominated sorting genetic algorithm that has proven to be powerful incapturing a set of diversified Pareto optimal solutions. In NSGA-II, childrenpopulation is created from parent chromosomes by both crossover and muta-tion. The simulated binary crossover algorithm is adopted here for crossoverwith an exponent of 4.0. The random mutation rate is set at a low probabilityof occurrence of 0.1%.

Relying on the proposed adaptive surrogate modeling strategy to esti-

mate worst-case reliability, the optimizer can quickly evaluate the adequacyof structural design with different uncertainty interval widths, and thus keepsthe interval computation tractable. The surrogate-assisted optimization pro-cess is summarized in Algorithm 2 below.

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Algorithm 2 Adaptive surrogate-assisted MOGA with interval variable.

BEGIN

Initialize: Latin Hypercube Sampling method to collect initial samplepoints, evaluate sampled points using FORM and construct a Kriging sur-rogate model;Initialize NSGA-II.while Termination criteria of GA is not met do

for Individual Di in population do

Evaluate fitness values of Di using objective functions [f,Cost];Estimate the worst case reliability index β L of Di by surrogateCheck updating criteria described in Algorithm 1

if Status is updating Kriging then

Evaluate β (Di) using exact reliability analysisUpdating sampling database

Refine Kriging model and estimate β L againend if

Set constraint violation as max[0, β t − β L]end for

Apply GA operators to create the next generation of populationend while

END

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4. Case studies

In this section, the proposed method is examined with an engineering de-sign problem [5] and a marine composite stiffened panel design problem [25].Through these case studies, the functionality of worst-case performance sur-rogate prediction is well examined, and the ability of proposed algorithm indelivering a closely converged Pareto front compared to high-fidelity simula-tion results is shown.

4.1. Cantilever tube

A cantilever tube problem proposed in Du’s [5] interval reliability studyis used in this work to test the framework. The cantilever tube shown inFig. 1 is subjected to external forces F 1, F 2, P and torsion T . The limitstate function is expressed as follows:

g(X ,Y ) = S y − σmax (24)

where S y is the yield strength, and σmax is the maximum von Mises stressgiven by:

σmax =

σ2x + 3τ 2zx (25)

The calculation of stress is determined from classical structural mechan-ics:

σx = P + F 1 sin θ1 + F 2 sin θ2

A +

M c

I

M = F 1L1 cos θ1 + F 2L2 cos θ2

A = π

4[d2 − (d − 2t)2], c = d/2,

I = π

64[d4 − (d − 2t)4], τ zx =

T d

4I .

(26)

The random variables and interval variables setting are given in Table 1and Table 2. Note that the interval variable ranges are fixed in the origi-nal problem [5] for a single analysis. In this work, the test multi-objectiveoptimization problem with varying interval ranges is formulated as follows:

min V olume(t, d) = π(d − t)tL1

min Cost(∆1, ∆2)

s.t. β L(X ,Y ) = y(X ,Y ∗) ≥ β t

where [t, d] = [µ(X 1), µ(X 2)]

Y 0 ≤ Y ≤ Y 0 + ∆

(27)

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Figure 1: Cantilever tube problem [5].

Table 1: Random variables

Variables Parameter 1 Parameter 2 DistributionX 1(t) 5mm 0.1mm Normal

X 2(d) 42mm 0.5mm NormalX 3(L1) 119.75mm 120.25mm UniformX 4(L2) 59.75mm 60.25mm UniformX 5(F 1) 3.0kN 0.3kN NormalX 6(F 2) 3.0kN 0.3kN NormalX 7(P ) 12.0kN 1.2kN GumbelX 8(T ) 90.0Nm 9.0Nm NormalX 9(S y) 220.0Nm 22.0Nm Normal

Table 2: Interval variables

Variables IntervalsY 1(θ1) [0, 10]Y 2(θ2) [5, 15]

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Table 3: Design variables

Design variables Description Lower bound Upper boundt Mean value of thickness X 1 3mm 6mmd Mean value of diameter X 2 38mm 44mm∆1 Interval variable range of Y 1 0 10∆2 Interval variable range of Y 2 0 10

The mean value of thickness and diameter are set as design variables alongwith two interval variable ranges. The design variable domain is defined inTable 3. The interval uncertainty lengths ∆1, ∆2 range from 0 to 10, wherezero ranges means the interval variables are reduced to the deterministic

values: [Y 1, Y 2] = [0, 5], and the largest ranges mean the interval variablescover the same range shown in Table 2. In conducting the analysis, theinitial surrogate model is constructed from 150 points sampled by the LatinHyper-square Method evenly in the design space. Reliability simulations areconducted using the PyRe (python Reliability) module.

To validate the proposed method, initial, the worst reliability in the in-terval range is located by Algorithm 1 directly (e.g. no optimization). Thiscomputation is also performed by Du [5] which allows for comparison withthe current method. The design point [t,d, ∆1, ∆2] = [ 5.0, 42.0, 10, 10] ischosen as the prediction point, which means the random variables X and

interval variables Y are the same as Tables 1 and 2 in the literature. Thepredicted worst-case interval variable combination is Y ∗ = [3.9289, 7.8074]and the associated reliability index is 3.5933. A comparison of the resultwith Du’s method is shown in Table 4. It is clear that the worst-case pre-diction successfully estimated the worst-case design compared to nonlinearoptimization results from Du’s study. The computational efficiency of thesurrogate model is also promising, the 952 limit state function evaluationsare made for 150 reliability analysis during the sampling stage. Though thisis higher than the 147 function calls used by Du, the surrogate is now readyfor optimization while Du’s work only finds a single value.

Next, the test problem stated in Eq. 27 is optimized using the proposed

multi-objective optimization framework in Algorithm 2. As a reference, thetest problem is also optimized within NSGA-II with all the reliability analysisconducted directly via FORM simulation. The two optimizations both used100 generations and a population size of 100, and a target reliability index

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Table 4: Worst case reliability

Worst case reliability Proposed surrogate prediction Du 2007 pmax

f (= Φ−1(−β L)) 1.63 × 10−4 1.63 × 10−4

function evaluations of g 952* 147

*:Note this surrogate was built for optimization, hence it contains morefunction calls than Du’s single-value computation.

constraint of 3.0. A direct comparison between the two Pareto fronts is shownin Fig. 2. Note that the product of the interval width is plotted directly inFig. 2. This is the inverse of the objective space the problem was solved invia Eq. 23.

Both Pareto fronts show a clear trade-off between interval uncertaintyrange and design performance, though the absolute impact on structuralmaterial volume is small. The increase of interval range multiples ∆1 ∗ ∆2

causes penalty on design performance measured by volume of cantilever tube.The penalty stops after reaching a certain value of interval range. This isdue to the nonlinearity of interval uncertainties in the original problem: if alocal minimum of performance is already contained in the interval, expandingthe interval does not further reduce performance unless a new minimum isincluded. The front found by the surrogate closely follows the front foundby direct FORM simulation. For this problem, this demonstrates that the

proposed approach is efficient and practical.A detailed comparison illustrated in Fig. 3 can disclose the relative im-portance between the two interval uncertainties involved. It can be seen thatPareto optimal points favor large ∆1 values and small ∆2 values, which indi-cates that reducing interval uncertainty Y 2 is more worthwhile in improvingdesign performance compared to Y 1. It also indicates that there is a criticalvalue of ∆2 at which point the design performance worsens rapidly.

The worst case prediction of the surrogate is examined in the optimizationrun by computing the bias of the prediction:

bias = | predicted − actual|

∆max

(28)

where the bias is defined as the normalized difference between predicted worstcase Y

∗and the actual Y ∗ from high-fidelity analysis. An average bias value

of 2.7×10−4 is recorded for the 10100 individual evaluations in the evolution-ary algorithm. This indicates that the proposed surrogate model’s worst-case

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0 2 0 4 0 6 0 8 0 1 0 0

∆

1

∗ ∆

2

. 0 0

. 0 1

. 0 2

. 0 3

. 0 4

H i g h - f i d e l i t y F O R M a n a l y s i s

P r o p o s e d s u r r o g a t e m e t h o d

Figure 2: Pareto fronts comparison.

Figure 3: Impact of interval range on performance.

prediction is sufficiently accurate for interval analysis throughout the opti-mization run. The closely converged Pareto front from surrogate method inFig. 2 also suggests that the proposed surrogate model is capable in locatingthe worst-case performance in an accurate and efficient way. Overall, thequadratic approximation to locate the worse performing point directly from

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the Kriging surrogate, and the coupling of this technique with the NSGA-II

appears to successfully solve this two-interval problem.

4.2. Top-hat stiffened panel structure design

Marine composite structures are often used for high-speed, innovativevessels. Compared to conventional steel vessels, there is larger uncertainty inboth the loading of the structure and the material variability of the compos-ites for such vessels. Marine composite structural design then requires signif-icantly larger loading and material uncertainty factors than steel structures.In this case study, it is attempted to apply interval uncertainties to representlarge variabilities in loading component, while using conventional reliabilitymethods to capture the material variability of the composite. Through the

proposed method, a trade study between interval uncertainty range and de-sign performance of a top-hat stiffened panel structure is performed. Such ananalysis is similar to an early-stage vessel design problem, in which a robustestimation of structural weight is needed while loading remains uncertain.

(a) Picture of a FRP top-hat stiffenedgrillage.

(b) Schematic of a multi-stiffener gril-lage.

Figure 4: Configuration FRP top-hat stiffened grillage plate.

A simplified grillage structure with carbon fiber-reinforced plastic (CFRP)materials [25] is adopted here to demonstrate an application of proposedmethod. The typical configuration of this type of structure used in marine

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Figure 5: A sectional view of the CFRP single skin with top-hat stiffeners.

structures is shown in Fig. 4. The size of this top-hat stiffened grillage plateis 4.2m in length and 3.2m in width, and spacing values for longitudinaland transverse frames are 800mm and 700mm, respectively. This gives thegrillage plate three longitudinal and five transverse top-hat stiffeners.

The sectional view of a top-hat shape stiffener with effective breadth isshown in Fig 5. This top-hat stiffener including base plate is consisted of crown, web, non-structural former. For simplicity, these components areconstructed to have geometrical symmetry, and longitudinal configuration is

of the same as in the transverse direction. The depth of the stiffened plateis fixed at 180.00mm. Furthermore, it is assumed that CFRP laminates aremonolithic; hence no attempt is made to ascribe thickness of ply details,make-up, fiber volume fractions, etc. Such decisions would be taken up ata later stage after gross decisions about the choice of particular structuraltopology and material are made.

Under lateral pressure, both the strength and the deflection of such pan-els are important to structural design. For this work, the maximum paneldeflection is considered as the limit state equation for the structure. Thedeflection limit state function is defined as follows:

g(X ,Y ) = wmax − w(X ,Y ) (29)

where wmax is the allowed maximum deflection, taken as L/150 in this study.The deflection w of the grillage is estimated by using Navier’s energy method[34], where the deflection is determined by equating total strain energy to

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the work done by the load. The equation is given below:

w =∞

m=1

∞n=1

16P l4b/EI r

π6mn[m4(r + 1) + I sI r

l3

b3n4(s + 1)]

sin mπx

l sin

nπy

b (30)

where I r and I s are moments of inertia for the longitudinal stiffeners andtransverse stiffeners, respectively; m and n are wave numbers; E is equivalentYoung’s moduli; and a reasonable pressure load P is estimated by the ABSHigh Speed Craft rules [35]

The descriptions of distributions for each components in Eq. 29 are pre-sented in Table 5. In this work, the geometric design variables are treatedas random variable X , while the interval variable Y is used to change the

COV of the loading estimate in the limit state equation. The mean value of equivalent Young’s modulus µE is taken as 140GPa.

Table 5: Distributions of variables in CFRP problem

Variables Description Mean COV Distributiontw Web thickness µ1 0.03 Normaltcr Crown thickness µ2 0.03 Normalt p Plate thickness µ3 0.03 NormalE Equivalent Young’s modulus µE 0.05 NormalP Pressure Load pd 0.2 ± ∆ Normal

The multi-objective optimization problem for interval trade-off study isformulated in Eq. 31, within which the independent design variables aredefined in Table 6.

min Weight(X )

min Cost(∆)

s.t. β L(X ,Y ) = y(X ,Y ∗) ≥ β t

where X = [µ1, µ2, µ3]T

Y 0 − ∆ ≤ Y ≤ Y 0 + ∆

(31)

The above problem is optimized in both the proposed surrogate methodpresented in Algorithm 2, and the all FORM analysis in terms of differentways in computing β L. The initial surrogate model is constructed using 150sampling points. The NSGA-II parameters are set as 100 generations and a

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Table 6: Design Variables of CFRP problem

Design variables Description Lower bound Upper boundµ1 Mean value of tw 1.0mm 15mmµ2 Mean value of tcr 1.8mm 15mmµ3 Mean value of t p 5mm 45mm∆ Interval variable range 0.01 0.1

population size of 100. For a more rigorous comparison, two different targetreliability criteria β t are investigated at 2.5 and 3.0, respectively. The cor-responding optimized results are shown in Fig. 6. Again, the Cost objective(Eq. 23) is inverted to interval range for a better trade space view.

0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 0 . 1 2

∆

2 1 0

2 2 0

2 3 0

2 4 0

2 5 0

2 6 0

2 7 0

2 8 0

2 9 0

W

e

i

g

h

t

(

k

g

)

β

t

= 3 . 0 F O R M a n a l y s i s

β

t

= 3 . 0 P r o p o s e d m e t h o d

β

t

= 2 . 5 F O R M a n a l y s i s

β

t

= 2 . 5 P r o p o s e d m e t h o d

Figure 6: Pareto fronts comparison under different target reliability index

It is clearly shown in Fig. 6 that at different target reliability index lev-els, the proposed surrogate method successfully demonstrated its ability in

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worst-case estimation and converging very closely to the high-fidelity results

achieved from all FORM analysis. In this composite structure case study,the absolute value of the penalty for information uncertainty is higher thanin the cantilever tube case. The increased interval uncertainty range causesapproximately a 20% weight increase on the design in the worst scenario.The strong linear trend is related to linear relationship between pressure anddeflection in Eq. 30. The potential weight penalty is critical for compositematerial application in naval ship designs where weight requirement is strin-gent. Therefore, this trade-off information can be highly valuable in earlystage structural weight estimation for these innovative designs.

The proposed method is also highly computationally efficient comparedto direct reliability analysis. A comparison of the number of FORM reliabil-

ity simulation required for these optimization runs is presented in Table 7.Without using a surrogate, over 105 FORM reliability analyses are requiredto converge the Pareto fronts. Thus, the direct simulation method can easilybecome inapplicable when reliability simulation becomes more complex andtime-consuming. However, the proposed method used on 150 FORM callsinitially, and then another 7-8 for refinement as the optimization proceeded.On average the proposed method used far less than one FORM call in locat-ing each worst case β L, and about 1.6 FORM calls per final point found onthe Pareto front. Examining the statistics of the iterative quadratic searchand refinement for the worse performance shows that highest number of itera-

tions was 3 Kriging predictions. Directly finding reliability from MCS-basedmethod on a Kriging surrogate can take up to 106 Kriging predictions incomputing β L for a single individual evaluation. Coupled with the resultsof the cantilever tube presented previously, the proposed method has beenshown to be both robust and efficient on these problems.

Table 7: Computation cost comparison

Cases Overall reliability sim-ulation f FORM calls

Average f FORM callsin computing β L

β t = 3.0 FORM analysis 203079 20.10β t = 3.0 Proposed method 158 0.016β t = 2.5 FORM analysis 200101 19.81β t = 2.5 Proposed method 157 0.015

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5. Conclusion

This manuscript proposed and demonstrated a novel interval-uncertaintyoptimization approach to explore the value in reducing epistemic uncertaintyin design. Initially, a combined interval-reliability model was presented. Inthis approach, the limit state equation can contain both stochastic (aleatory)variables and interval-uncertainty variables. Then, a surrogate modelingapproach was demonstrated that extended existing Kriging modeling ap-proaches by directly locating minima on the Kriging surface via the Jacobianand Hessian of the model. This approach was used to remove the innermostloop in the optimization - the location of the worst reliability performancewithin a given interval. Adaptive refinement criteria to ensure the accuracy

of this approximation were also presented. Next, the approach was coupledto a conventional NSGA-II optimizer, and an information cost function de-fined in terms of the product of interval uncertainty widths. By includingthis information cost function as an objective, the impact of epistemic un-certainty on the design performance was revealed in the Pareto trade-spacesthat resulted from the NSGA-II. The proposed approach was demonstratedon a cantilever tube and composite panel design problem. For both prob-lems, the proposed approach was shown to be accurate and efficient, usingfewer than 200 direct reliablity analysis while finding Pareto fronts similarto those found without the use of any surrogate.

With the help of the presented method, the decision makers are better

equipped to explore novel structural designs in early stage. As shown in thecomposite panel case study result, the interval uncertainty study can fullyprepare the designer to design against the worst-case scenario and make struc-tural weight estimations under incomplete information. Additionally, thisinterval uncertainty trade study can quickly inform the designer regardingwhether engineering improvement is critically needed for the design project.

Acknowledgments

The team at the University of Michigan was supported by Ms. KellyCooper, Office of Naval Research, under grant N00014-11-1-0845. Her con-tinued support is greatly appreciated. Additionally, Han Koo Jeong wishesto express his gratitude to Kunsan National University for the financial sup-port received through the Kunsan National University’s Long-term OverseaResearch Program for Faculty Member in the year 2014.

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6. Additional Resources

The code used to implement the optimization approach and a sampleproblem is available under the MIT Open Source License at DeepBlue. Note

for the purposes of peer review, these files are available by request, and will

be assigned a final URL when the article is published

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