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Risk-averse structural topology optimization under random fields using stochastic expansion methods Jes´ us Mart´ ınez-Frutos 1* , David Herrero-P´ erez 1 , Mathieu Kessler 2 , Francisco Periago 1,2 1 Computational Mechanics and Scientific Computing Group, Technical University of Cartagena, Campus Muralla del Mar, 30202 Cartagena, Murcia, Spain 2 Department of Applied Mathematics and Statistics, Technical University of Cartagena, Campus Muralla del Mar, 30202 Cartagena, Murcia, Spain Abstract This work proposes a level-set based approach for solving risk-averse struc- tural topology optimization problems considering random field loading and ma- terial uncertainty. The use of random fields increases the dimensionality of the stochastic domain, which poses several computational challenges related to the minimization of the Excess Probability as a measure of risk awareness. This problem is addressed both from the theoretical and numerical viewpoints. First, an existence result under a typical geometrical constraint on the set of admissible shapes is proved. Second, a level-set continuous approach to find the numerical solution of the problem is proposed. Since the considered cost functional has a discontinuous integrand, the numerical approximation of the functional and its sensitivity combine an adaptive anisotropic Polynomial Chaos (PC) approach with a Monte-Carlo (MC) sampling method for uncertainty propagation. Fur- thermore, to address the increment of dimensionality induced by the random field, an anisotropic sparse grid stochastic collocation method is used for the ef- ficient computation of the PC coefficients. A key point is that the non-intrusive nature of such an approach facilitates the use of High Performance Computing (HPC) to alleviate the computational burden of the problem. Several numer- ical experiments including random field loading and material uncertainty are presented to show the feasibility of the proposal. Keywords: Risk-averse topology optimization; Random fields; Material uncertainty; Stochastic expansion methods; Existence theory; Level-set method. * Corresponding author Email addresses: [email protected] (Jes´ us Mart´ ınez-Frutos 1 ), [email protected] (David Herrero-P´ erez 1 ), [email protected] (Mathieu Kessler 2 ), [email protected] (Francisco Periago 1,2 ) Computer Methods in Applied Mechanics and Engineering

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Risk-averse structural topology optimization underrandom fields using stochastic expansion methods

Jesus Martınez-Frutos1∗, David Herrero-Perez1 , Mathieu Kessler2 , FranciscoPeriago1,2

1Computational Mechanics and Scientific Computing Group, Technical University ofCartagena, Campus Muralla del Mar, 30202 Cartagena, Murcia, Spain

2Department of Applied Mathematics and Statistics, Technical University of Cartagena,Campus Muralla del Mar, 30202 Cartagena, Murcia, Spain

Abstract

This work proposes a level-set based approach for solving risk-averse struc-tural topology optimization problems considering random field loading and ma-terial uncertainty. The use of random fields increases the dimensionality ofthe stochastic domain, which poses several computational challenges related tothe minimization of the Excess Probability as a measure of risk awareness. Thisproblem is addressed both from the theoretical and numerical viewpoints. First,an existence result under a typical geometrical constraint on the set of admissibleshapes is proved. Second, a level-set continuous approach to find the numericalsolution of the problem is proposed. Since the considered cost functional has adiscontinuous integrand, the numerical approximation of the functional and itssensitivity combine an adaptive anisotropic Polynomial Chaos (PC) approachwith a Monte-Carlo (MC) sampling method for uncertainty propagation. Fur-thermore, to address the increment of dimensionality induced by the randomfield, an anisotropic sparse grid stochastic collocation method is used for the ef-ficient computation of the PC coefficients. A key point is that the non-intrusivenature of such an approach facilitates the use of High Performance Computing(HPC) to alleviate the computational burden of the problem. Several numer-ical experiments including random field loading and material uncertainty arepresented to show the feasibility of the proposal.

Keywords: Risk-averse topology optimization; Random fields; Materialuncertainty; Stochastic expansion methods; Existence theory; Level-setmethod.

∗Corresponding authorEmail addresses: [email protected] (Jesus Martınez-Frutos1),

[email protected] (David Herrero-Perez1), [email protected] (MathieuKessler2), [email protected] (Francisco Periago1,2)

Computer Methods in Applied Mechanics and Engineering

1. Introduction

Topology optimization aims at finding the optimal distribution of materialwithin a design domain such that an objective functional is minimized undercertain constraints [1]. Contrary to size and shape optimization methods, topol-ogy optimization permits to obtain a material distribution without assumingany prior structural configuration. This provides engineering designers with apowerful tool to find innovative and high-performance conceptual designs atthe early stages of the design process. Such a technique has been successfullyapplied to improve the design of complex industrial problems, such as aeronauti-cal, aerospace and naval applications [2]. However, deterministic conditions areusually assumed, and thus obviating the different sources of uncertainty whichmay affect not only the safety and reliability of structures but also their perfor-mance. Robustness and reliability have been recognized as a desirable propertyin structural systems and have sparked considerable interest in the scientific andengineering communities. The need for including uncertainty quantification andpropagation stages during the design process has shown to be a key issue forsolving real-world engineering problems in several fields, such as aeronauticaland aerospace [3], civil [4], automotive [5] and mechanical [6] engineering, toname but a few. This fact, together with the development of probabilistic un-certainty propagation methods, has fostered the interest for considering uncer-tainty within the topology optimization problems, giving rise to the formulationof several approaches embraced under the term of Topology Optimization UnderUncertainty (TOUU) methods.

Such methods can be broadly classified, according to the representation andtreatment of uncertainties, into non-probabilistic and probabilistic approaches.The former methods [7] do not require the statistical information about theuncertainty of the phenomenon but a qualitative notion about its magnitude.The worst-case approach [8, 9, 10], taking the form of a min-max optimizationproblem, and fuzzy techniques [11], making use of fuzzy set theory, are methodsincluded in this category. The main drawback of these approaches is that theyare often too conservative, due to overestimation of uncertainty, and may leadto optimal designs with poor structural performance. The latter methods makeuse of the probabilistic characterization of the uncertainty of the phenomenon.Several formulations have been proposed in this context to address the wide con-cept of “structural robustness”. These formulations differ from each other in thedesign objective as well as in the way the uncertainty is incorporated within theoptimization formulation. Robust Topology Optimization (RTO) incorporatesthe first two statistical moments of the cost functional to obtain optimal de-signs which are less sensitive to variations in the input data [12, 13, 14, 15, 16].Reliability-Based Topology Optimization (RBTO) aims at determining the bestdesign solution (with respect to prescribed criteria such as stiffness, weight orconstruction costs) while explicitly considering the unavoidable effects of un-certainty. This is done by posing the constraints in terms of the probabilityof constraint violation (probability of failure) [17, 18, 19]. On the other hand,Risk-Averse Topology Optimization (RATO) [20, 21, 22] does not aims at mini-

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mizing a deterministic prescribed criteria but a risk function that quantifies theexpected loss related with the damages (e.g. excess probability) [23]. WhereasRBTO provides optimal designs in terms of a deterministic prescribed criteriawith enough reliability level, RATO provides the best design from the point ofview of risk-aversion.

One of the main challenges of TOUU methods is the computational burdenof addressing the problem, which still remains even though significant numericaland theoretical advances have been achieved in the last years. A key point ofthe computational challenge is that when the solution of the underlying PartialDifferential Equation (PDE) is expensive, one can only afford to solve a fewhundred samples. This is far from the required number of samples for estimatinga probability. Such a drawback is exacerbated in high-dimensional stochasticdomains, such as those obtained with random fields. This work is concernedwith this issue in the context of RATO problems under random fields.

The RATO problem is introduced by the early work of Conti et al. [21]proposing a risk-averse approach in the finite dimensional context of shape op-timization. The uncertainty is introduced in the applied loads, and hence, thereis a linear dependence of the system response with respect to uncertainty. Mak-ing use of the assumption of linear elasticity and quadratic functional cost, theapplied loads can be expressed as a linear combination of a small number of de-terministic basis loads and a set of uncertain coefficients, which are representedby means of discrete random variables. The main advantage of this approachis that the computational cost scales linearly in the number of linearly inde-pendent applied loads. Allaire and Dapogny [24] addressed a similar problemunder the assumption of small uncertainties using a first-order Taylor expan-sion of the cost functional with respect to the uncertainties. Similar ideas areused in [25] for the case of geometrical uncertainties. These approaches lead todeterministic approximations of the cost functional with similar computationalburden to that of multiple load problems, where the number of loads is similarto the finite number of random variables. Existence of solution for worst-caseoptimization problems has been obtained, e.g. in [26, 27]. However, this issue ispresented in [20] as an open problem for RATO. The early work of Conti et al.[21] is recently extended to introduce a new class of stochastic shape optimiza-tion problems by means of the concept of stochastic dominance [28]. Ratherthan handling risk aversion in the functional cost, dominance constrains areconsidered for bounding the Excess Probability of the design shape with thatof a given reference shape. This issue, is the main difference with respect toother formulations which consider probabilistic constraints such as RBTO. Werefer the interested reader to [28] for more details on this passage. To the bestknowledge of the authors, these are the only works dealing with RATO problemfor the system of linear elasticity. Such articles are numerical in nature and noexistence results of optimal shapes have been proved so far.

In this paper, the topology optimization problem of risk awareness for con-tinuous elastic structures with random field uncertainties in material propertiesand applied loads is analyzed. In contrast to previous works in RATO, theevaluation of Excess Probability functional under random field material and/or

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loading uncertainties requires integrations over failure regions, which representsa key difficulty. This is not the case of the RTO problem considered in our pre-vious work [13], where the integrands in the cost functionals are smooth withrespect to the random variable. This permits an efficient use of stochastic col-location methods to approximate the integrals in the random domain that arisein such a problem. In this work, the problem is addressed both theoreticallyand numerically. From a theoretical point of view, existence of a solution undervolume and geometrical constraints on the set of admissible designs is proved.The proof of this result is obtained by adapting the one in [13] to the case inwhich the integrand is not continuous but lower semi-continuous. From a nu-merical point of view, the problem is solved by adapting the level-set method tothis probabilistic framework. In addition to the high computational cost, whichis inherent to optimization problems with uncertainties represented by randomfields, the main difficulty lies in the fact that the integrand in the cost func-tionals measuring risk awareness is discontinuous. Similarly to [21], a smoothapproximation of this functional is introduced. Then, the continuous gradientof the approximated functional is explicitly computed. Although at the theo-retical level both the approximated cost functional and its gradient are smooth,at the numerical level they are not. Hence, the numerical approximation, byusing stochastic collocation methods (as the one considered in [13]), of the cor-responding integrals in the random domain may lead to incorrect results if afew collocations points are located in the unknown discontinuities. The use ofvery fine stochastic grids is unaffordable due to the high computational burden.This difficulty is addressed in this work by the use of an adaptive anisotropicPolynomial Chaos (PC) expansion and anisotropic sparse grid method for un-certainty propagation in combination with a Monte-Carlo (MC) method for thenumerical approximation of the cost functional and its sensitivity. Stochasticexpansion methods have been previously applied in RTO to estimate statisticalmoments [16, 29, 30], and they have shown recently their effectiveness in morecomplex problems, such as in RBTO to estimate failure probabilities and theirsensitivities [31, 32]. This work complements the previous studies [21, 32] byinvestigating the use of adaptive anisotropic PC approaches to efficiently solveRATO problems under random field uncertainties. Finally, the proposed ap-proach is applied to several two-dimensional benchmarks where uncertaintiesare incorporated in loads and material properties along with different probabil-ity distribution types and random fields.

The paper is organized as follows. Section 2 is devoted to the problemstatement including the proper formulation of the RATO problems as well asthe proof of the existence result under volume and geometrical constraints onthe set of admissible designs. The proposed numerical resolution of RATOproblems for continuous structures using the level-set method is presented insection 3. Section 4 is devoted to the uncertainty modeling and the anisotropicPC approach used for the numerical integration over failure regions. Somenumerical experiments are presented in Section 5 to verify the effectiveness ofthe proposed development. Such experiments include uncertainty in loads andmaterial properties represented by random fields. Finally, some conclusion and

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concluding remarks are presented in section 6.

2. Problem statement

2.1. The state law

Let (Ω,F ,P) be a complete probability space, and let O ⊂ Rd (d = 2 or 3 inapplications) be a bounded Lipschitz domain whose boundary ∂O is decomposedinto three disjoint parts

∂O = ΓD ∪ ΓN ∪ Γ0, |ΓD| > 0, (1)

where | · | stands for the Lebesgue measure. Consider the system of linearelasticity with random input data

−div(Ae(u(x, ω))) = f in O × Ω,

u(x, ω) = 0 on ΓD × Ω,

(Ae(u(x, ω))) · n = g on ΓN × Ω,

(Ae(u(x, ω))) · n = 0 on Γ0 × Ω,

(2)

where e (u) =(∇uT +∇u

)/2 is the strain tensor, n is the unit outward normal

vector to ∂O, and A is the material Hooke’s law, defined for any symmetricmatrix ζ by

Aζ = 2µζ + λ(Trζ)Id,

where λ = λ(x, ω) and µ = µ(x, ω) are the Lame moduli of the material, whichmay depend on both x = (x1, · · · , xd) ∈ O and ω ∈ Ω. In the same vein,the volume and surface loads f and g depend on a spatial variable x and on arandom event ω ∈ Ω, i.e. f = f(x, ω) and g = g(x, ω). The divergence (div)and the gradient ∇ operators in (2) involve only derivatives with respect to thespatial variable x.

Since the domain O is changing during the optimization process, λ, µ and fmust be known for all possible configurations of O. Thus, a working boundeddomain D is introduced, which contains all admissible domains O and satisfiesthat ΓD ∪ ΓN ⊂ ∂D. Therefore, it is assumed that λ, µ and f are defined inthe bounded working domain D. ΓN is kept fixed during optimization. Hence,g(ω) ∈ L2 (ΓN )

da.s. ω ∈ Ω.

From now on, if X is a Banach space and 1 ≤ p ≤ ∞, LpP (Ω;X) denotesthe Lebesgue-Bochner space composed of all (equivalence classes of) stronglymeasurable functions h : Ω→ X whose norm

‖h‖LpP (Ω;X) =

(∫Ω‖h (·, ω) ‖pX dP (ω)

)1/p, p <∞

ess supω∈Ω‖h (·, ω) ‖X , p =∞

is finite. In order for problem (2) to be well-posed, the following assumptionson the uncertain input parameters of system (2) are considered:

5

(A1) λ(x, ω), µ(x, ω) ∈ L∞P (Ω;L∞(D)) and there exist positive constants µmin,µmax, λmin, λmax such that

0 < µmin ≤ µ(x, ω) ≤ µmax <∞ a.e. x ∈ D, a.s. ω ∈ Ω,

and

0 < λmin ≤ 2µ(x, ω) + dλ(x, ω) ≤ λmax <∞ a.e. x ∈ D, a.s. ω ∈ Ω,

(A2) f = f(x, ω) ∈ L2P(Ω;L2(D)d

),

(A3) g = g(x, ω) ∈ L2P(Ω;L2(ΓN )d

).

To introduce an appropriate functional framework for problem (2), the Hilbertspace

VO =v ∈ H1(O)d : v|ΓD = 0 in the sense of traces

,

equipped with the usual H1(O)d-norm, is considered.A weak solution of (2) is a random field u ∈ L2

P (Ω;VO) such that∫Ω

∫O Ae(u(x, ω)) : e(v(x, ω)) dxdP(ω) =∫

Ω

[∫O f · v dx+

∫ΓN

g · v ds]dP(ω), ∀v ∈ L2

P (Ω;VO) .(3)

A straightforward application of the Lax-Milgram lemma allows one to statethe well-posedness of (2). Moreover, there exists c > 0 such that

‖u‖L2P(Ω;VO) ≤ c

(‖f‖L2

P(Ω;L2(O)d) + ‖g‖L2P(Ω;L2(ΓN )d)

)and

‖u(ω)‖VO ≤ c(‖f(ω)‖L2(O)d + ‖g(ω)‖L2(ΓN )d

), a.s. ω ∈ Ω. (4)

Notice that, thanks to assumption (A1), the constant c = c(O, µmin, λmin)in (4) does not depend on ω ∈ Ω.

2.2. The set of admissible shapes

The ε-cone property, introduced by Chenais [33], is presented below. Let apoint y ∈ Rd, a vector ξ ∈ Rd and ε > 0 be given. Consider the cone

C(y, ξ, ε) =z ∈ Rd : (z − y, ξ) ≥ cos ε|z − y|, 0 < |z − y| < ε

.

An open set O ⊂ Rd is said to have the ε-cone property if for all x ∈ ∂O thereexists a unit vector ξx such that ∀y ∈ O ∩B(x, ε), the property C(y, ξx, ε) ⊂ Oholds.

Consider the class

Oε = O open, O ⊂ D, O has the ε-cone property .

6

For instance, the bounded, open and convex sets are some of the domains inthe class Oε for ε small enough [34, Prop. 2.4.4]. Moreover, if O satisfies the ε-cone property, then ∂O is uniformly Lipschitz, where the geometrical constantswhich are involved in the uniform Lipschitz character of O depend only on ε.And conversely, a uniform Lipschitz domain O satisfies the ε-cone property fora positive ε which depends on the constants bounding the uniform Lipschitzcharacter of O (see the proof of [34, Th. 2.4.7 and Remark 2.4.8]). Imposing avolume constraint V0 on an admissible domain, the class of admissible domainsis

Uad = O ∈ Oε, ΓD ∪ ΓN ⊂ ∂O, |O| = V0 , with 0 < V0 < |D|.

The class of admissible shapes is composed of open and bounded sets O offixed volume V0, i.e. |O| = V0 > 0, where | · | stands for the Lebesgue measure.In addition, the boundary of O satisfies the condition (1) with ΓD and ΓN fixedso that Γ0 is the part of the boundary to be optimized.

2.3. Formulation and existence of solution of the Risk-Averse Topology Opti-mization problem

For each realization ω ∈ Ω and for a given admissible domain O, considerthe compliance

J(O, ω) =

∫Of(x, ω) · uO(x, ω) dx+

∫ΓN

g(x, ω) · uO(y, ω) ds, (5)

where uO = uO(x, ω) is a weak solution of (2).Throughout the paper, if X ∈ L1(Ω) is a real-valued random variable, its

expected value is denoted by

E(X) =

∫Ω

X(ω) dP(ω).

The variance is widely used as a measure of deviation in the context of RTO[12, 14, 35]. This, however, could not be the best choice in some minimizationproblems because the variance treats the excess over the mean equally as theshortfall. This is the case when one is interested in minimizing the expectedloss related with damages caused by catastrophic failures [20]. Accordingly, fora given threshold value η ∈ R, the following cost functional (which is typicallyreferred to as Excess Probability [21] in the context of RATO) is considered:

JEPη (O) = P ω ∈ Ω : J(O, ω) > η = E (H (J(O, ω)− η)) (6)

where

H(z) =

0, z ≤ 01, z > 0

7

is the Heaviside function. The following optimization problem is investigated inthis paper:

(PEPη ) minJEPη (O) : O ∈ Uad

.

It is well known that if an admissible domain is assumed to be only Lebesguemeasurable the deterministic shape optimization problem may be ill-posed dueto the lack of closure of the set of admissible domains with respect to the weak-?topology in L∞(D; 0, 1). To overcome this difficulty, additional conditions,such as smoothness, geometrical or topology constraints, are imposed on theclass of admissible designs. In this work, the standard ε-cone assumption [33] isconsidered. To prove the existence of a solution for (PEPη ), the set of admissibledomains Uad is equipped with an appropriate topology so that the compactnessof Uad and the continuity of JEPη (O) are ensured. Following [34, Definition2.2.3], let Onn≥1 and O be (Lebesgue) measurable sets of Rd. It is said thatOn converges to O in the sense of characteristic functions as n→∞ if

1On −→ 1O in Lploc(Rd)∀p ∈ [1,∞[,

where 1B stands for the characteristic function of a measurable set B ⊂ Rd.The class of domains Oε enjoys the following compactness property: let

Onn≥1 be a sequence in the class Oε. Then, up to a subsequence, still labelledby n, On converges to some O? ∈ Oε, in the sense of characteristic functions(also in the sense of Hausdorff distance and in the sense of compacts sets). Werefer the readers to [34, Th. 2.4.10] for more details on this passage.

Another very important property that the domains in the class Oε satisfy isthe following uniform extension property [33, Th. II.1] : there exists a positiveconstant K, which depends only on ε, such that for all O ∈ Oε there exists alinear and continuous extension operator

PO : H1 (O)→ H1(Rd), with ‖PO‖ ≤ K. (7)

We are now in a position to prove the following existence result:

Theorem 2.1. Problem (PEPη ) is well posed, i.e., it has, at least, one solution.

Proof. Let On ⊂ Uad be a minimizing sequence. Due to the compactnessproperty of Oε mentioned above there exists O? ∈ Oε and a subsequence, stilllabelled by n, such that On converges to O?. Moreover, thanks to the conver-gence in the sense of characteristic functions, O? satisfies the volume constraintand hence O? ∈ Uad. The domain O? is the candidate to be the minimizer weare looking for.

Similarly to [13, Th. 2.1], the following point-wise (w.r.t. ω ∈ Ω) conver-gence holds:

J (On, ω)→ J (O?, ω) as n→∞, a.s. ω ∈ Ω. (8)

8

Indeed, let un (x, ω) and uO? (x, ω) be the solutions to (2) associated to Onand O?, respectively. Denoting by un (·, ω) := POn (un) (·, ω) the extension ofun (·.ω) to Rd, by (4) and (7) one has

‖un(ω)‖VD ≤ ‖POn‖‖un(ω)‖VOn≤ Kc

(‖f(ω)‖L2(D)d + ‖g(ω)‖L2(ΓN )d

),

a.s. ω ∈ Ω, which proves that un(ω) is bounded in VD. Thus, by extractinga subsequence, it converges weakly in VD, and strongly in L2(D)d to someu?(ω) ∈ VD. Let us prove that

u?|O?(ω) = uO?(ω), ω ∈ Ω.

From the definition of un, it follows that∫On

Ae(un) : e(v) dx =

∫On

f · v dx+

∫ΓN

g · v dx ∀v ∈ VD, a.s. ω ∈ Ω,

which is equivalent to∫D

1OnAe(un) : e(v) dx =

∫D

1Onf · v dx+

∫ΓN

g · v dx ∀v ∈ VD, a.s. ω ∈ Ω.

The weak convergence of un in VD and the strong convergence in L2(D) of thecharacteristic functions permit to take the limit to get∫D

1O?Ae(u?) : e(v) dx =

∫D

1O?f · v dx+

∫ΓN

g · v dx ∀v ∈ VD, a.s. ω ∈ Ω,

and also, thanks to the extension property (7),∫O?Ae(u?) : e(v) dx =

∫O?f · v dx+

∫ΓN

g · v dx , ∀v ∈ VO? .

This proves that u?|O?(ω) = uO?(ω) a.s. ω ∈ Ω. Since this is valid for anysubsequence, the whole sequence un converges to uO? . Let us consider thecompliance in the form

J (On, ω) =

∫On

f · un dx+

∫ΓN

g · un ds =

∫D

1Onf · un dx+

∫ΓN

g · un ds.

Since 1Onf → 1O?f strongly in L2 (D) and un u? weakly in VD, the limit inthis expression can be taken to obtain (8).

Since Fn(ω) := H (J(On, ω)− η) are integrable and non-negative, by Fatou’slemma, ∫

Ω

lim infn→∞

Fn(ω) dP(ω) ≤ lim infn→∞

∫Ω

Fn(ω) dP(ω). (9)

By (8) and the lower semi-continuity of the Heaviside function,

H (J(O?, ω)− η) ≤ lim infn→∞

Fn(ω),

9

Integrating in both sides of this expression and using (9) yields∫Ω

H (J(O?, ω)− η) dP(ω) ≤∫

Ω

lim infn→∞

Fn(ω) dP(ω)

≤ lim infn→∞

∫Ω

Fn(ω) dP(Ω).

Since On is a minimizing sequence, the result follows.

Remark 1. We notice that the ε-cone condition in the class of admissible de-signs has been introduced in order to obtain the preceding existence result. Asis usual, ε-cone condition is not taking into account in the numerical resolutionapproach, which is presented in the next section.

3. Numerical resolution of (PEPη) via the level-set method

The numerical resolution of (PEPη ) is addressed by extending the determin-istic level-set method, as introduced in [36], to the probabilistic setting describedin the preceding sections. This is a gradient-based algorithm and hence onlylocal minimizers may be computed. For the sake of completeness, we start byrecalling the notion of shape derivative.

3.1. A brief review on shape derivative

The concept of shape derivative was introduced by Hadamard [37]. Here wefollow the approach of Murat and Simon [38]. Other classical references on thistopic are [39, 34, 40].

Given a reference domain O, we consider perturbations of this domain of theform Oθ = (I + θ) (O), where I is the identity operator and θ ∈W 1,∞ (Rd;Rd).It is known that, for θ small enough, (I + θ) is a diffeomorphism.

Now let J (O) be a (shape) functional. The shape derivative of J (O) isdefined as the Frechet derivative in W 1,∞ (Rd;Rd) at 0 of the mapping θ →J (Oθ), i.e.,

J (Oθ) = J (O) + J ′ (O) (θ) + o (θ) , with limθ→0

|o (θ) |‖θ‖ = 0,

where J ′ (O) is a continuous linear form on W 1,∞ (Rd;Rd).Hadamard’s structure theorem states that J ′ (O) (θ) depends only on the

normal trace θ · n on the boundary. In particular, the following result is veryuseful to compute shape derivatives. See [39, Propositions 6.22 and 6.24] for aproof.Theorem 3.1. Let O be a smooth bounded open domain. Let F1(x) ∈W 1,1

(Rd)

and F2(x) ∈W 2,1(Rd). The shape functional

I (O) =

∫OF1 (x) dx+

∫∂O

F2 (x) ds (10)

10

is differentiable at O and its shape derivative is given by

I ′ (O) (θ) =

∫∂O

θ · n(F1 +

∂F2

∂n+ div(n) · F2

)ds, (11)

where div(n) is the mean curvature of ∂O.

3.2. Explicit (formal) computation of the continuous shape derivative

The volume constraint, as given in the definition of the admissible shapes, isincorporated in the functional cost through an augmented Lagrangian method.According to [41], the augmented Lagrangian function is constructed as

JLEPη (O) =∫

ΩH (J(O, ω)− η) dP(ω)

+L1

(∫O dx− V0

)+ 1

2L2

(∫O dx− V0

)2,

(12)

being L1 and L2 the Lagrange multiplier and the penalty parameter, respec-tively. The Lagrange multiplier is updated at each iteration n according to

L(n+1)1 = Ln1 +

1

Ln2

(∫Odx− V0

). (13)

The penalty parameter L2 is updated as Ln+12 = αLn2 , where α ∈ (0, 1). The

penalty parameter is updated until a minimum value is reached.Since the cost functional JLEPη (O) has a discontinuous integrand, the follow-

ing smooth approximation (see, e.g., [22]) is considered

JL,εEPη(O) =

∫Ω

(1 + e−

2ε (J(O,ω)−η)

)−1

dP(ω)

+L1

(∫O dx− V0

)+ 1

2L2

(∫O dx− V0

)2,

(14)

with 0 < ε 1.

Remark 2. The choice of the parameter ε is a delicate issue. If the value ofε is too high, the accuracy of the Heaviside approximation is degraded. On thecontrary, if the value of ε is too small the values of the sensitivities may beclose to zero when a few number of sampling points are taken in the transitionregions, i.e. transitions between zero and non-zero probability regions. Followingthe guidelines indicated in [31], the parameter ε is adaptively updated during theoptimization in the numerical experiments. This is done by selecting the valueof ε as a function of the standard deviation of the cost functional.

Let θ ∈W 1,∞ (Rd;Rd) be the deformation domain vector field. Since ∂O =ΓD ∪ ΓN ∪ Γ0, where ΓD and ΓN are fixed and Γ0 is the part of the boundaryto be optimized, the vector field θ vanishes on ΓD ∪ ΓN .

11

Proposition 3.1. The shape derivative of JL,εEPη(O) (θ) is expressed as(

JL,εEPη

)′(O)(θ) =

∫Γ0θ · n

[∫ΩC1 (ω) (2f · u−Ae(u) : e(u)) dP(ω)

]ds

+L1

∫Γ0θ · nds+ 1

L2

(∫O dx− V0

) ∫Γ0θ · nds,

(15)where the random variable C1 (ω) is given by

C1 (ω) =2

ε

(1 + e−

2ε (J(O,ω)−η)

)−2

e−2ε (J(O,ω)−η). (16)

Sketch of the proof. Although the computation of the shape derivative of(14) is standard, a formal proof by using the Lagrangian method due to Cea[42] is presented below for the sake of completeness.

To begin with, the Lagrangian

L (O, u, p) =∫

Ω

(1 + e

− 2ε

(∫O f ·u dx+

∫ΓN

g·u ds−η))−1

dP(ω)

+∫O∫

ΩAe (u) : e (p) dP (ω) dx

−∫O∫

Ωp · f dP (ω) dx

−∫

ΓN

∫Ωp · g dP (ω) ds,

(17)

which is defined for u, p ∈ L2P(Ω;H1

(Rd;Rd

))such that u = p = 0 on ΓD, is

considered. For a given O, let (u, p) be a stationary point of L. Equating to0 the partial derivative of L with respect to u in the direction v ∈ L2

P (Ω;VO)evaluated at (O, u, p) yields the adjoint equation of the optimization problem.Indeed, a straightforward computation yields

0 =< ∂L∂u (O, u, p) , v >

=∫

ΩC1 (ω)

(∫O f · v dx+

∫ΓN

g · v ds)dP(ω)

+∫O∫

ΩAe (v) : e (p) dP (ω) dx,

(18)

where C1 (ω) is given by (16). After integrating by parts and by taking v withcompact support in O × Ω gives

−div(Ae(p)) = −C1 (ω) f in O × Ω. (19)

Similarly, varying the trace of v(·, ω) on ΓN and on Γ0 yields the boundaryconditions

Ae(p) · n = −C1 (ω) g on ΓN × Ω (20)

andAe(p) · n = 0 on Γ0 × Ω. (21)

Therefore, p ∈ L2P (Ω;VO) solves the adjoint equation

−div(Ae(p(x, ω))) = −C1 (ω) f(x, ω) in O × Ω,p(x, ω) = 0 on ΓD × Ω,(Ae(p(x, ω))) · n = −C1 (ω) g(x, ω) on ΓN × Ω,(Ae(p(x, ω))) · n = 0 on Γ0 × Ω,

(22)

12

and consequently p(x, ω) = −C1(ω)u(x, ω).A direct computation shows that < ∂L

∂p (O, u, p) , v >= 0 reduces to the

direct problem (2).Next, since u = u(Ω) solves (2), for p ∈ L2

P(Ω;H1

(Rd;Rd

))such that p = 0

on ΓD, one has

L (Ω, u, p) =

∫Ω

(1 + e

− 2ε

(∫O f ·u dx+

∫ΓN

g·u ds−η))−1

dP(ω).

Differentiating this expression with respect to O in the direction θ and applyingthe chain rule theorem yields(∫

Ω

(1 + e

− 2ε

(∫O f ·u dx+

∫ΓN

g·u ds−η))−1

dP(ω)

)′(θ)

= ∂L∂Ω (Ω, u, p) (θ) + < ∂L

∂u (Ω, u, p) , u′(Ω) > .

Taking p = p, solution to the adjoint system (22), the last term vanishes. Thus,the first term in the right-hand side of (15) is then formally obtained by com-puting the partial derivative of L (O, u, p) with respect to O in the direction θ,by using (11), and by taking into account that θ = 0 on ΓD ∪ ΓN .

Finally, by using again formula (11) and the chain rule theorem one easilygets the third and fourth terms in the right-hand side of (15) associated to L1

and L2.

Remark 3. Although in this paper we focus on the compliance objective func-tion (5), other types of cost functionals may be treated in a similar way. Forinstance, another typical choice is

J2 (O, ω) =

(∫Oκ (x) |u(x, ω)− u0|α dx

)1/α

, (23)

which represents a least square error compared to a target displacement. Here,α ≥ 2, u0 ∈ Lα (D) is a prescribed target, and κ ∈ L∞ (D) is a non-negativeweighting factor. In particular, up to the introduction of a suitable adjointequation, the shape derivative of (a smooth approximation of) the cost func-tional J2 (O) =

∫ΩH (J2 (O, ω)− η) dP(ω) may be computed by using the same

arguments as in the preceding result.

3.3. Description of the numerical algorithm

Following an original idea by Osher and Sethian [43], which was importedto shape optimization in [36, 44], the unknown domain O is described throughthe zero level set of a function ψ = ψ(t, x) defined on the working domain D asfollows

∀x ∈ D, ∀t ∈ (0, T ) ,

ψ(x, t) < 0 if x ∈ O,ψ(x, t) = 0 if x ∈ ∂O,ψ(x, t) > 0 if x /∈ O.

(24)

13

Here, t stands for a fictitious time that accounts for the step parameter inthe descent algorithm. Hence, the domain O = Ot is evolving in time duringthe optimization process. In the numerical context, the level-set function ψ isdiscretized at the vertices of a simplicial mesh of D. During the optimizationprocess, ψ solves the Hamilton-Jacobi equation

∂ψ

∂t+ (θ · n) |∇ψ| = 0, t ∈ (0, T ) , x ∈ D, (25)

Equation (25), which is complemented with Neumann type boundary conditions,is solved in the hole domain D by using the so-called “ersatz material” approach.Precisely, the hole D\O is filled by a weak phase that mimics the void, but at thesame time avoids the singularity of the rigidity matrix. Thus, an elasticity tensorA∗ (x) which equals A in O and 10−3 ·A in D \ O is introduced. Decomposingthe boundary of D as ∂D0 ∪ ΓD ∪ ΓN , the displacement field u(x, ω) solves

−div(A∗e(u(x, ω))) = f in D × Ω,

u(x, ω) = 0 on ΓD × Ω,

(A∗e(u(x, ω))) · n = g on ΓN × Ω,

(A∗e(u(x, ω))) · n = 0 on ∂D0 × Ω,

(26)

The chosen descent direction in (25) is θ = −vn, with

v =∫

ΩC1 (ω) (2f · u−Ae(u) : e(u)) dP(ω)

+L1 + 1L2

(∫O dx− V0

),

(27)

which has been previously computed in Proposition 3.1. Note that since thevector displacement u is defined throughout the domain D, the velocity field vis also defined in D and not only on the boundary ∂O. As it is usual in thiscontext, the equation (25) is solved using an explicit first order upwind scheme.To avoid singularities, the level-set function is periodically reinitialized. Thereaders are referred to [36] for more details on this passage.

The crucial step is the numerical approximation of the advection velocity(27), which requires the numerical computation of the random state u(x, ω), i.e.the solution to (26). It is important to emphasize that although the randomvariable C1 (ω), which appears in (27), is (theoretically) smooth with respectto the random event ω, for ε small, numerically it is not. The same problemappears when evaluating the cost functional JL,εEPη

(O) at each iteration of thedescent algorithm. As a consequence, stochastic collocation methods are notthe best choice to compute those integrals and it is more convenient to use aMonte Carlo (MC) sampling strategy. However, the use of a direct MC methodrequires the numerical resolution of (2) at a large number of sampling pointsωk ∈ Ω and at each step of the descent method. This makes a direct MC methodunaffordable from a computational point of view. For this reason, and similarlyto [31], the MC method is combined with a Polynomial Chaos (PC) methodfor uncertainty propagation. More precisely, u(x, ω) is approximated with a PCexpansion method. Then, the MC method is applied to this approximation tocompute the integrals in the random domain which appear in the cost functionalJL,εEPη

(O), and in its sensitivity. Next section provides details on all these issues.

14

4. Numerical integration over failure regions using anisotropic poly-nomial chaos expansion

This section is devoted to the numerical integration over the failure regions.A key point is the fact that the integrand in the risk averse formulation is notcontinuous, which prevent the use of the stochastic collocation method (well-suited for smooth problems). In order to deal with this fact, MC methods canbe used at the cost of increasing the computational burden significantly. Thiscomputation challenge is addressed by the combination of MC method (used fornumerical integration) and a PC expansion (used as an emulator) for approxi-mating the smooth solution of the elasticity system. In addition, the coefficientsin the PC expansion are computed with an anisotropic, non-intrusive, stochasticcollocation method, which can take advantage of the parallelizable character ofcollocation methods.

4.1. Uncertainty modeling

The uncertainties acting in the PDE input data with spatial variations arecharacterized by random fields. In order to address numerically the problem,the random fields are discretized according to the following assumption:

(A4) Finite dimensional noise: the random input data of (2), i.e. Lamecoefficients and loads, depend on a finite number N of real-valued randomvariables YnNn=1.

A typical choice to fulfill the assumption (A4) is the truncated K-L expansion[45] of a second-order random field. Indeed, let U : D × Ω → R be such thatU ∈ L2

P(Ω;L2 (D)

). Denote by U (x) := E (U(x, ·)) =

∫ΩU (x, ω) dP(ω) the

mean function of U , and by

C (x, x′) := E[(U(x, ·)− U(x, ·)

) (U(x′, ·)− U(x′, ·)

)]its covariance function. Then, the random field U(x, ω) admits the so-calledKarhunen-Loeve (K-L) expansion

U(x, ω) = U(x) +

∞∑n=1

√γnbn(x)Yn(ω) (28)

where Yn are pairwise uncorrelated random variables with zero mean and unitvariance, and γn and bn are the eigenvalues and eigenvectors, respectively, ofthe compact and self-adjoint operator

ψ 7→∫D

C(x, x′)ψ(x′) dx′, ψ ∈ L2(D).

If the random field is Gaussian, then Yn ∼ N(0, 1) are independent and iden-tically distributed standard Gaussian variables. The sum in (28) converges inL2P(Ω;L2 (D)

). For numerical simulation purposes, the series (28) is always

15

truncated at some large enough term N , i.e., the random field U(x, ω) is ap-proximated as

U(x, ω) ≈ UN (x, ω) = U(x) +

N∑n=1

√γnbn(x)Yn(ω). (29)

The convergence rate of UN depends on the decay of the eigenvalues γn, whichin turn depend on the smoothness of it associated covariance function. See [46]and the references therein for more details.

The represention (29) is well-suited to represent uncertainty in the loads fand g. In the case of a surface load g, as in the problem under consideration,the spatial domain for the random field is ΓN . However, since Lame coefficientsare positive in nature, uncertainty in material properties cannot be representedby a truncated K-L expansion of a Gaussian field. Instead, log-normal randomfields are considered in this paper, which take the form

a(x, ω) ≈ aN (x, ω) = exp (η + ξUN (x, ω)) , a = λ and/or µ, (30)

where UN is as in (29), and η, ξ are location and scale parameters. Yet, fromthe theoretical point of view, Gaussian variables cannot be used for modelinguncertainty of Lame coefficients. This is due to the lower and upper uniformboundedness condition imposed in (A1). However, in practice, a Gaussian vari-able may be accurately approximated considering its truncation at some largeenough lower and upper points. In addition, some regularity on the eigenfunc-tions bn(x) should be imposed. In the case of Gaussian random fields with acovariance function continuous in D ×D, the eigenfunctions bn(x) are continu-ous and bounded in D×D. We again refer the readers to [46] for more details.

4.2. Anisotropic non-intrusive PC expansion for uncertainty propagation

Since the random variables YnNn=1 are mapped from the sample space Ωto R, by using a standard localization argument, the variational formulation of(2) can be rewritten as

∫O Ae(u(x, y)) : e(ϕ(x)) dx =∫O f(x, y) · ϕ(x) dx+

∫ΓN

g(x, y) · ϕ(x) ds, ∀ϕ ∈ VO, ρ− a.e. in Λ,(31)

where denoting Λn = Yn (Ω) with 1 ≤ n ≤ N , Λ =∏Nn=1 Λn ⊂ RN and ρ = ρ(y)

the joint probability density function of Y = (Y1, · · · , YN ), which is assumed to

both exist and to factorize as ρ (y) =∏Nn=1 ρn (yn), y = (y1, · · · , yn) ∈ Λ. The

solution u = u(x, y) to (31) belongs to the space L2ρ (Λ;VO), which is defined as

L2ρ (Λ;VO) :=

h : Λ→ VO,

∫Λ

‖h(y)‖2VOρ(y) dy <∞

.

The reader is referred to [46, Chapter 9] for more details on this passage.

16

Following [31, 47, 48], let ψpn (yn)∞pn=1 be an orthonormal basis of the

space L2ρn (Λn) :=

h : Λn → R,

∫Λn|h(yn)|2ρn(yn) dyn <∞

composed of a

suitable class of orthonormal polynomials. Since L2ρ (Λ) =

⊗Nn=1 L

2ρn (Λn), a

multivariate orthonormal polynomial basis of L2ρ (Λ) is constructed as

ψp (y) =

N∏n=1

ψpn (yn)

p=(p1,··· ,pN )∈NN

.

Let now ` ∈ N+ be an index indicating the level of approximation in therandom space L2

ρ (Λ).A primary drawback of PC expansion is the well known curse of dimen-

sionality, according to which the number of unknown coefficients increases asthe space of random variables becomes of higher dimension. In this context,adaptive sparse PC expansion [49] has been proposed to significantly reduce thecomputational cost. Additionally, the number of terms required by the K-L ex-pansion in (29) can lead to unaffordable computational problem when full tensorproduct rules are used to compute the PC coefficients. Note that if the samenumber of points R is used in each direction, then the total number of pointsin a full tensor rule is RN . To address this issue, the isotropic sparse grid ap-proaches [50, 51, 52] permit to reduce the number of collocation points keepingthe level of accuracy for problems whose random variables weight equally in thesolution. However, the convergence rate is deteriorated for highly anisotropicproblems [52]. This is the case of K-L expansions in which random variablescorresponding to large eigenvalues have more influence than those associated tolow eigenvalues. In this case, an anisotropic sparse tensor product can lead tohighly accurate solutions while reducing drastically the computational require-ments. In this work, an adaptive anisotropic PC approach is presented to reducethe computational cost involved in RATO problem with random fields.

This method is composed of the following main steps: (i) based on the decayproperties of the eigenvalues of the K-L expansion, a vector of weights β is chosen(see (33)). (ii) Both β and the level ` determine a multi-index set (32), whichis used to select the degree of polynomials in the different stochastic directions.(iii) From this multi-index, a finite dimensional space (34) approximating L2

ρ (Λ)is introduced. (iv) Since the space L2

ρ (Λ;VO), where the solution to (31) livesin, is isomorphic to VO ⊗ L2

ρ (Λ), the solution to (31) is expressed as in (35),where the coefficients in this expansion are computed by using an anisotropicstochastic collocation method. Next, details on these issues are described.

Given a vector of weights β = (β1, · · · , βN ) ∈ NN+ for the different stochasticdirections, the multi-index set

Iβ (`,N) =

p = (p1, · · · , pN ) ∈ NN+ :

N∑n=1

pnβn ≤ `βmin

(32)

with βmin = min1≤n≤N βn is considered. The optimal choice of β depends onthe analyticity properties of the solution of (31) with respect to the parameter

17

y ∈ Λ. For the case of Gaussian random fields the n-th term of β is taken as

βn =1

2√

2γn‖bn‖L∞(D)

, (33)

where (γn, bn (x)) is the eigenpair in the K-L expansion (29).Finally, the anisotropic approximation multivariate polynomial space is

PIβ(`,N) (Λ) = span ψp (y) , p ∈ Iβ (`,N) . (34)

The space PIβ(`,N) (Λ) is referred in the literature as to Anisotropic Total Degreepolynomial space [47]. Thus, an approximated solution u` (x, y) ∈ VO⊗PIβ(`,N)

of (31) is expressed in the form [53, 54]

u = u(x, y) ≈ u` (x, y) =∑

p∈Iβ(`,N)

up (x)ψp (y) , up ∈ VO, (35)

where due to the orthonormality of ψp (y)p∈Iβ(`,N),

up (x) =

∫Λ

u` (x, y)ψp (y) ρ (y) dy ≈∫

Λ

u (x, y)ψp (y) ρ (y) dy. (36)

This latter integral is numerically approximated using the anisotropic sparsegrid collocation method described in [13] considering similar multi-index setsand the weights to those given in (32) and (33), respectively. The level ofquadrature for the collocation method is fixed to `+ 1, which provides enoughaccuracy to integrate the polynomials with order 2p from (36). The anisotropicsparse grid collocation method leads to a set of uncoupled deterministic sub-problems collocated at some stochastic nodes yk ∈ Λ. The shape of the unknowndomain O for each sub-problem is constrained to a bounding box D tessellatedusing a fixed grid of Lagrange Q1 finite elements. From this point on, V hDdenotes a finite element space approximating the Sobolev space VD introducedin Subsection 2.1, where h stands for the size of the spatial mesh.

It remains to analyze how the level of approximation ` is chosen. Since themain goal in this paper is to minimize the cost functional JL,ε,hEPη

(O) and itsderivative, the level of approximation ` is adaptively chosen so as to complywith a prescribed accuracy level δ for those quantities. This is done as follows:

(i) Compute the vector of weights β according to (33).

(ii) Initialize the level ` = 1 and a positive, large enough, integer `.

(iii) Compute an enriched solution uh`

(x, y) of (31) by following the method

described above. This enriched solution (which plays the role of exact so-

lution) is then used to obtain an approximation JL,ε,hEPη,`

(O) of JL,εEPη(O) by

using MC method, where random samplings yk are applied to uh`

(x, y).The domain O used in this computation is the optimal shape of the deter-ministic problem - deterministic meaning that the random input parame-ters of (2) are replaced by their nominal (or mean) values.

18

(iv) Finally, the level ` is linearly increased (from ` = 1 to ` = `opt < `) up tothe stopping criterion

DKL(r`||r`) :=

∫Λ

r`(y) log(r`(y)

r`(y)

)dy ≤ δ, (37)

where r` and r` and are the Probability Density Functions (PDFs) ofthe approximation and the enriched value of J(O, ω), respectively. Theprobability density at y is computed using Kernel Density Smoothing as:

r`(y) =

nMC∑i=1

wiK$(y − si) (38)

where si are the values in the nMC Monte Carlo sample, Kw(y) is thezero-centered kernel function with bandwith $, and wi = 1/nMC is theweigth of each point. In this work a Gaussian kernel, which is implementedthrough the “ksdensity” function of MATLAB, is used:

K$(y − si) =1√

2π$e−

1

2

(y − si$

). (39)

Following the Silverman’s rule [55] the bandwith is $ = σMC1.06n(−1/5)MC ,

being σMC the standard deviation of the Monte Carlo sample. DKL(r`||r`)is the so-called Kullback-Leibler divergence [56], which is a measure of howone probability distribution diverges from a second expected probabilitydistribution. Since DKL provides a criterion to check how accurate theapproximation is in terms of PDFs, it is appropriate in our specific situ-ation of the functional JL,ε,hEPη

(O). In the case that the stopping criterion

(37) does not hold for any ` < `, then a larger ` is taken to ensure thatthe enriched solution uh

`(x, y) is good enough. The whole process is then

repeated taking the new value of ` as reference.

The cost functional JL,εEPη(O) and its sensitivity, especially the constant

C1 (ω), as given by (16), are numerically approximated using MC integration.The MC sampling in the random domain Λ is applied to the approximated so-lution u` (x, ·) ∈ L2

ρ (Λ) for which the explicit representation (35) is available.At the begining of the optimization process, the number of MC sampling pointsis adaptively increased up to fulfill the convergence criterion (37).

Remark 4. The use of stochastic collocation methods during the optimizationprocess can notably increase the computational cost. However, these methodsare considered embarrassingly parallel because they have no shared state and canbe executed in complete isolation. For that reason, stochastic collocation meth-ods become an attractive tool for uncertainty propagation tasks with the adventof High Performance Computing (HPC). Task-level parallelism is used for im-plementing the resolution of the state equation at each yk ∈ Λ stochastic node

19

10−3

10−2

10−1

100

0.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

ε/σ(J)

PJ>

10

10−3

10−2

10−1

100

0.06

0.065

0.07

0.075

0.08

0.085

0.09

ε/σ(J)

PJ>

1000

10−3

10−2

10−1

100

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

ε/σ(J)

PJ>

200

(a) (b) (c)

Figure 1: Excess Probability as a function of ε using the optimal designs for (a) beam-to-cantilever (case η = 10), (b) carrier plate (case η = 1000) and (c) Michell-type structure (caseη = 200) benchmarks. Dashed lines indicate probabilities calculated using MC simulationwith 105 samples.

in clustering. This is done by adopting a Master-Worker Pattern, which per-mits to perform simultaneous processing across multiple machines or processesvia a Master and multiple Workers. The Master creates a set of tasks at eachyk ∈ Λ, which consists of the resolution of the state equation using finite ele-ment analyses. The Workers calculate these works in clustering, taking workafter work until the set of tasks has been calculated. After receiving all samplesat the stochastic nodes, the Master evaluates the statistical moments involved inthe objective function and the shape derivative.

5. Numerical experiments

The performance of the proposal for RATO of continuous structures is evalu-ated solving three numerical benchmarks previously used in the context of RTO.The benchmarks have been adapted to RATO problems including random fieldloading and material uncertainty. The first benchmark is used to investigatethe accuracy and efficiency of the proposal through a simple example using onerandom variable. This simple benchmark permits us to carry out a convergencestudy of the calculation of the PDF obtained using non-intrusive adaptive PCexpansion approach and MC method. Additionally, it aims to show the effectof the prescribed threshold η and the loading uncertainty on the optimal designof structures. The second benchmark aims to investigate the performance ofthe proposal when random fields are used to characterize the loading uncer-tainty. The last benchmark permits us to evaluate the proposal in the presenceof random loads and spatially varying random material properties.

All the benchmarks make use of a fixed grid of linear quadrangular LagrangeQ1 finite elements with elasticity modulus E = 1 (in the third experimentE will be a random field with mean equal to 1) and Poisson’s ratio ν = 0.3considering plane stress conditions. The fixed grid approximation, also referredto as “ersatz material” approach [36], classifies the elements as inside I, outside

20

O and boundary B elements according to their position with respect to thezero-level of level set function ψ (See [57] for more details). The Hooke’s law isthen modified as (χe +∆(1− χe))A, where ∆ = 10−3 and χe = V eI /V

e is thevolume ratio between the volume of element V eI enclosed by O and the totalvolume of element V e. The level set function ψ(t, x) is discretized at the nodesof such a fixed grid.

For all the experiments, the parameter ε, used in the approximation of thecost functional, is updated during the optimization. According to [31], a rea-sonable choice consists in the selection, at iteration n, of εn value depending onthe mean or standard deviation of the cost functional. In this vein, a value ofε = 0.01σ(JL,ε,hEPη,`

(On)) is used in this work, being σ(JL,ε,hEPη,`(On)) the standard

deviation of the approximated cost at level ` associated to the design On, whichis obtained at iteration n. Figure 1 shows that the choice of ε following thiscriterion yields reasonable accuracy for the three benchmarks. The coefficientsof the PC expansion are computed using the anisotropic sparse grid schemepresented in [13] with a quadrature level ` + 1 and the weights given in (33).Since the considered random variables in the examples are Gaussian, a non-nested quadrature rule, whose collocation nodes are determined by the roots ofHermite polynomials, are used.

The Lagrange multiplier and the penalty parameter of the augmented La-grangian functions [13, 31] are initialized heuristically according to

L1 = 0.1 · JL,ε,hEPη,`(O)/V0 and L2 =

1

abs(0.01 · L1/V0),

where O is the initial design. The penalty parameter is augmented, after everyfive iterations, multiplying its previous value by 0.8 until a minimum value of10−3 is reached.

In all the numerical experiments the gradient descent algorithm stops at thefirst n ∈ N for which the two following conditions are satisfied:

max1≤i≤5

|JL,ε,hEPη,`

(On)− JL,ε,hEPη,`(On−i)|

≤ 0.01 · JL,ε,hEPη,`

(On)

and|V n − V0| ≤ 0.005.

The resolution of the direct and adjoint state equations at each stochasticnode is computed using a cluster with 8 nodes. Such nodes are equipped with2 Intel Xeon E5-2620 at 2 GHz and 32 GB of RAM memory.

5.1. Numerical Experiment 1: Beam-to-cantilever design

The beam-to-cantilever problem consists of the shape optimization of a two-dimensional cantilever under uncertain loading conditions. Such a benchmarkis widely used [10, 14, 58] to show the effects of uncertain loading in the robustoptimal design. The left edge of the cantilever is anchored and a unit force withuncertainty in direction, centered in the horizontal line, is applied at the middle

21

g(φ) = 1

Dx = 1

hx

hy

Dy = 2φ ∼ N(0, π/12)

(a) (b)

Figure 2: Beam-to-cantilever problem: (a) The design domain and boundary conditions, and(b) the initialization of level-set function.

of the right edge. The design domain is a 1x2 rectangle which is tessellatedusing a 64x128 regular mesh of quadrangular elements (hx = hy = 0.0156).

The uncertain loading, the boundary conditions and the tessellation of thedesign domain are depicted in Figure 2(a), whereas the initialization of thelevel-set function is shown in Figure 2(b). The configuration and the numericalresolution of the RATO problem is as follows. The volume constraint is set tothe 30% of the initial design domain. The direction φ of the unit load followsa Gaussian distribution centered at the horizontal line with mean µφ = 0 andstandard deviation σφ = π/12.

A convergence study of the PDF obtained using non-intrusive adaptive PCexpansion (PCE) method and MC simulation is performed. The convergence forthe MC approach is depicted in Figures 3(a) and Figures 4(a), where the levelof the PCE is ¯ = 10 and the number of samples of MC simulation (nsamples)is varied from 102 to 106. Figure 3(b) shows the different PDFs obtained byvarying the level ` of the PC expansion from 2 to 4. In this case, the numberof samples of MC simulation is fixed to 106 and the PCE with level ¯ = 10 isconsidered as a reference. The differences depicted in Figure 3 are quantifiedby means of the Kullback-Leibler divergence (37) considering a level ¯= 10 forthe MC method and a level ¯ = 10 and number of samples 106 as a referencefor the PCE method. The results, depicted in Figure 4, reveal that a value ofDKL = 1.5e − 4 is obtained for a level ` = 4 of PCE method. In the light ofthese results, the integrals involved in the cost and sensitivities are evaluatedusing non-intrusive PCE method with level ` = 4.

The deterministic optimization required 86 iterations to converge whereasthe risk-averse optimization (for case η = 10) required 63 iterations to converge.

22

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Compliance

Probabilityden

sity

MC (nsamples = 106)

MC (nsamples = 105)

MC (nsamples = 104)

MC (nsamples = 103)

MC (nsamples = 102)

Compliance0 5 10 15 20 25 30

Pro

bability

den

sity

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5PCE

!7= 10

"PCE (`=2)PCE (`=3)PCE (`=4)

(a) (b)

Figure 3: Beam-to-cantilever problem (case η = 5): PDF of compliance calculated (a) usingPCE-based MC method with different sets of samples and (b) using PCE method with differentlevels `.

102

103

104

10510

−3

10−2

10−1

100

101

Number of Samples

|DKL|

`2 2.5 3 3.5 4

jDK

Lj

10-4

10-3

10-2

10-1

(a) (b)

Figure 4: The beam-to-cantilever problem: Kullback-Leibler divergence for the calculation ofPDF using (a) MC method and (b) an adaptive PCE with level ¯= 10 considered as reference.

23

Deterministic η = 5 η = 6 η = 7 η = 9 η = 10

(a) (b) (c) (d) (e) (f)

Figure 5: The beam-to-cantilever problem: (a) DTO design and (b-f) RATO designs fordifferent η values.

Each iteration of the risk-averse optimization required to solve four differentoccurrences of the state equation, which were performed concurrently at eachworker of the cluster. The resolution of the linear system of equations of elas-ticity was performed using a conjugate gradient method preconditioning with aV-cycle geometric multigrid [59, 60] using six levels and a damping factor for Ja-cobi smoothing w = 0.4. The time per iteration of the risk-averse optimizationusing four workers was 56.5 ms, which is slightly greater than the time requiredby the deterministic optimization 53.7 ms.

The Deterministic Topology Optimization (DTO) design (classical beam)for the horizontal (φ = 0) unit-load is shown in Figure 5(a), whereas the RATOdesigns are shown from Figure 5(b) to Figure 5(f) for various magnitudes of theparameter η. One can observe that the resulting designs incorporate stiffnessin the vertical direction where the uncertain loading is introduced. The risk isfirstly minimized by means of topology modifications, increasing the redundancyby adding paths to transmit the load to the support, and then by thickening thecantilever members. It is worth mentioning that such a behavior is very similarto the one observed by the authors in [13] by using a RTO formulation.

Table 1 shows the probability of exceeding different magnitudes of compli-ance for deterministic and risk-averse designs. The following key conclusions canbe drawn from the information detailed in such a Table. Firstly, the probabilityof exceeding a critical value ηi achieves its minimal value when the quantityP J > ηi is considered as objective function. This is the expected result andthe Excess Probability for each RATO design is highlighted in bold. Secondly,the deterministic design is not always the worst choice when compared to riskaverse designs with η 6= ηi. This is attributed to the shape of the PDF resultingfrom the minimization of the Excess Probability. Figure 6 shows the PDF forthe DTO design and diverse RATO designs using different values of η. Onecan observe that the mean of PDFs is higher as the prescribed threshold ηiincreases. Although the probability of exceeding ηi is reduced, more realiza-

24

Table 1: Excess Probability of optimal design for different values of η and for DTO in beam-to-cantilever benchmark.

Case Compliance P J > 5 P J > 6 P J > 7 P J > 9 P J > 10 σJ

DTO 4.2393 0.7524 0.6269 0.5419 0.4224 0.3763 9.2215η = 5 4.2422 0.7093 0.5274 0.4155 0.2772 0.2280 5.4176η = 6 4.9628 0.9999 0.4632 0.2884 0.1299 0.0872 2.3927η = 7 5.4498 1 0.4861 0.2178 0.0614 0.0311 1.3829η = 9 6.8351 1 1 0.6557 0.0581 0.0249 0.8515η = 10 7.9444 1 1 1 0.1093 0.0225 0.5725

4 6 8 10 12 140

0.5

1

1.5

2

2.5

3

3.5

4

Compliance

Probabilityden

sity

η = 5η = 6η = 7η = 9η = 10Deterministic

Figure 6: PDFs calculated for the DTO design and RATO designs with different values of ηin the beam-to-cantilever benchmark.

tions of the compliance are close to such a threshold value, and thus the meanperformance is degraded. This can lead to deterministic design showing lowerlevels of P J > ηi compared to risk-averse designs obtained with η > ηi val-ues. This conclusion is only pertinent to the problem discussed in this sectionand, as shown in the following examples, is heavily dependent on the impact ofuncertainty in the performance of the deterministic design. Finally, it should benoted that the increment of the mean of PDFs along with the reduction in theExcess Probability involve a reduction of the dispersion in the PDF. This factcan be observed in the last column of Table 1, where the standard deviation ofthe compliance is shown for the different cases analyzed. One can observe thatthe value of the standard deviation is reduced as the threshold η increases. Inaddition to reduce the Excess Probability, the risk-averse formulation providesdesigns which are less sensitive to fluctuations and thus more robust. This factexplains the topological similitudes between the risk-averse designs and the onesobtained in [13] using RTO.

25

D1 = 200

D2=

200

h1

g2

g1(x, w) ≡ Random field

g1

h2

(a) (b)

Figure 7: The carrier plate design problem: (a) The design domain and boundary conditions,and (b) the initialization of level-set function.

5.2. Numerical Experiment 2: Carrier plate design

The carrier plate design problem [14] aims to show RATO of continuumstructures under random field loading uncertainty. The uncertain loading, theboundary conditions and the tessellation of the design domain are shown inFigure 7(a), whereas the initialization of the level-set function is shown in Fig-ure 7(b). The dimension of the domain is a square of side length 200. Thedomain is tessellated using a regular mesh of 128x128 linear quadrangular ele-ments (hx = hy = 1.5625). Three layers of elements at the top of the domain aredefined as non-optimizable to ensure the structure remains attached to the load-ing conditions. The structure is subjected to a uniformly vertically distributedload g2 = 10 acting at its top. Additionally, a random horizontal distributedload g1(x,w) is applied at the top of the structure. The load magnitude is mod-eled as a Gaussian random field with zero mean and isotropic square exponentialcovariance function

C(x, x′) = σ2exp

[−

2∑i=1

(xi − x′i)2

l2

], (40)

where x = (x1, x2), x′ = (x′1, x′2) ∈ ΓN , σ = 0.05 is the standard deviation

and l = 30 is the correlation length. Figure 8(a) shows the eigenvalues ofthe correlation matrix that determines the number of random variables for thecarrier plate experiment. According to the decay rate of the eigenvalues, theK-L expansion is truncated at its fifth term capturing the 60% of the energyfield.

26

100

101

1020

1

2

3

4

5

6

n

λn

100

101

102

103

1040

10

20

30

40

50

60

70

n

λn

(a) (b)

Figure 8: Eigenvalue spectrum for (a) the carrier plate and (b) the Michell-type experiments.

The level of the PC expansion is computed adaptively as described in Sec-tion 4.2 with ¯ = 10 and δ = 1e − 2. For a level ` = 5, the maximum indexesIβ (5, 5) =

5, 4, 4, 3, 3

and the value DKL = 1.2e − 3 are obtained. Using the

anisotropic sparse grid configuration, each iteration of the RATO requires tosolve 1347 different occurrences of the state equation in contrast to the 2203and the 7776 that are required with the isotropic Smolyak’s sparse grid and thefull tensor product respectively. The accuracy of the stochastic approximationis shown in Figure 9, where the PDF obtained using PC expansion is comparedto the obtained using an enriched level ¯ = 10. One can observe in Figure 9that the agreement is satisfactory.

Similarly to the previous example, the volume constraint is set to the 30% ofthe initial design domain. Figures 10(b-d) show the topologies obtained solvingthe RATO problem with thresholds η = 620, η = 1000 and η = 1500, respec-tively. The topologies are compared to the one obtained using a deterministicformulation, which is depicted in Figure 10(a). One can observe that the risk-averse designs differ meaningfully from the DTO design. The risk-averse designsshow a larger span between supports than the deterministic optimal solution.Besides, the material distribution of stochastic designs is further from the sym-metry axis than the deterministic optimal solution, which provides a higherlateral stiffness to such risk-averse designs. In the same vein, the number ofmembers of the risk-averse designs is higher than the deterministic counter-part, which diversifies the load paths. Regarding the risk-averse designs, as thethreshold η increases the topologies evolve by changing the number of holes andthe thickness of the members.

The deterministic optimization required 175 iterations to converge whereasthe risk-averse optimization (for case η = 620) required 153 iterations to con-verge. Each iteration of the risk-averse optimization required to solve the 1347

27

0 1000 2000 3000 4000 50000

1

2

3

4

5

6

710-3

Figure 9: The carrier plate design problem (case η = 1000): Accuracy of PDF estimationusing PC expansion method.

Table 2: Excess Probability of optimal design for different values of η and for DTO in carrierplate design problem.

Case Compliance P J > 620 P J > 1000 P J > 1500 σJ

DTO 335.855 0.9057 0.8511 0.8012 2.553e4η = 620 410.364 0.3051 0.0728 0.0142 254.9993η = 1000 458.509 0.3402 0.0618 0.0103 214.7807η = 1500 525.013 0.5064 0.0790 0.0085 207.9096

different occurrences of the state equation, which were performed concurrentlyat each worker of the cluster. The linear system of equations of elasticity issolved using a V-cycle geometric multigrid preconditioned conjugate gradientmethod with five levels and a damping factor for Jacobi smoothing w = 0.4.The time per iteration of the risk-averse optimization using 96 workers was 1.75sec. whereas the deterministic optimization was 0.101 sec.

The Excess Probabilities for optimal deterministic and risk-averse designsare detailed in Table 2. Similarly to the previous experiment, the probabilityof exceeding a critical value ηi achieves its minimal value when the quantityP J > ηi is considered as objective function. Such values are highlighted inbold. Contrary to previous experiment, the deterministic design provides theworst results for all the cases analyzed. This is attributed to the significantimpact of horizontal random field uncertainty in the stiffness of the deterministicdesign. One can observe that the spreading of the PDF for the DTO design,shown in Figure 11(a), is significantly higher than its risk-averse counterpart,shown in Figure 11(b). Similarly to the previous experiment, the mean of PDFsis incremented as the prescribed threshold ηi increases. This increment comeswith a reduction of the standard deviation of the compliance as shown in the last

28

(a) DTO (b) η = 620

(c) η = 1000 (d) η = 1500

Figure 10: The carrier plate design problem: (a) DTO design and (b-d) RATO designs fordifferent values of η.

29

0 0.5 1 1.5 2 2.5 3 3.5 4

x 105

0

1

2

3

4

5

6

7

8

9x 10

−5

Compliance

Probabilitydensity

Deterministic

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6

7x 10−3

Compliance

Probabilityden

sity

η = 620η = 1000η = 1500

(a) (b)

Figure 11: The carrier plate design problem: PDF of compliance calculated (a) for the DTOdesign (mean value 1.853e4) and (b) for RATO designs with different values of η.

column of Table 2. This behavior shows the capacity of risk-averse formulationto not only minimize the influence of extreme events but also to increase therobustness of the structure reducing the standard deviation of the response.

5.3. Numerical Experiment 3: Michell-type structure design

The Michell-type structure benchmark is used to analyze the risk-averseapproach under combined loading and material uncertainties. The boundaryconditions and the domain discretization are shown in Figure 12(a), whereasthe initialization of the level-set function is shown in Figure 12(b). The designdomain has a roller support in the bottom right-hand corner and a fixed supportin the bottom left-hand corner. The domain is tessellated using a regular gridof 160x64 linear quadrangular elements (hx = hy = 1.25). The structure issubjected to three uncertain concentrated loads applied at its bottom. Themagnitude of the loads are characterized by three independent random variables.These random variables are assumed to follow normal distributions with meanvalues µg1

= µg2= µg3

= 1 and standard deviations σg1= 0.5, σg2

= 0.1 andσg3 = 0.2. Additionally, the uncertainty in the Young’s modulus is modeled by a2D lognormal random field with mean µF = 1 and standard deviation σF = 0.3.Such a lognormal random field is obtained through the transformation

F (x) = exp (η + ξ U(x)) , (41)

with

30

g1

Dy = 80

hx

hy

Dx = 200

g3g2

(a)

(b)

Figure 12: Michell-type structure design problem: (a) The design domain and boundaryconditions, and (b) the initialization of level-set function.

η = log

(µ2F√

µ2F + σF

2

)

ξ =

(log

(1 +

σF2

µ2F

))1/2, (42)

where η and ξ are the location and scale parameters of the lognormal distribu-tion and U is assumed to have zero mean, unit variance and the isotropic squaredexponential covariance function (40), which is defined in the whole spatial do-main. A correlation length lc = 60 is considered in this problem. Figure 8(b)shows the eigenvalues of the correlation matrix that determines the number ofrandom variables for the Michell-type experiment. According to the decay rateof the eigenvalues, the Gaussian random field U is discretized through the trun-cated K-L expansion using four terms, which captures the 55% of the energyfield. The resulting stochastic domain is composed of 7 random variables, 3 ofthem characterizing the uncertain loads and 4 of them characterizing the mate-rial variability. A PC expansion is used to approximate the compliance in thestochastic domain.

The level of the PC expansion is computed adaptively as described in Sec-tion 4.2 with ¯= 10 and δ = 1.0e− 2. For a level ` = 4, the maximum indexesIβ (4, 7) =

4, 4, 4, 4, 3, 2, 2

and the value DKL = 1.8e− 3 are obtained. Using

the anisotropic sparse grid configuration, each iteration of the RATO requiresto solve 2337 different occurrences of the state equation in contrast to the 2437

31

Compliance0 100 200 300 400 500 600

Pro

bability

den

sity

0

0.002

0.004

0.006

0.008

0.01

0.012

PCE (` = 4)

PCE (7= 10)

Figure 13: Accuracy of PDF calculated using PC expansion for case η = 200 in the Michell-type structure design problem.

and the 78125 that are required with the isotropic Smolyak’s sparse grid and thefull tensor product, respectively. The accuracy of the stochastic approximationis shown in Figure 13, where the PDF obtained using PC expansion is comparedto the obtained using an enriched level ¯ = 10. One can observe in Figure 13that the agreement is satisfactory.

The RATO problem is solved with the threshold values η = 100, η = 200and η = 300, obtaining the optimized designs shown in Figure 14(b-d) for atarget volume of the 30% of the initial design domain. One can observe thatthe RATO designs show significant topological differences with respect to theirdeterministic counterpart. As the threshold value η increases, the topologiesevolve modifying the number of holes and increasing the thickness of the mem-bers. One also can observe that the material distribution is oriented to the leftin order to withstand the force with the highest level of uncertainty.

The deterministic optimization required 116 iterations to converge whereasthe risk-averse optimization (for case η = 200) required 121 iterations to con-verge. Each iteration of the risk-averse optimization required to solve the 2337different occurrences of the state equation, which were performed concurrentlyat each worker of the cluster. The linear system of equations of elasticity issolved using a V-cycle geometric multigrid preconditioned conjugate gradientmethod with five levels and a damping factor for Jacobi smoothing w = 0.4.The time per iteration of the risk-averse optimization using 96 workers was 2.5sec. whereas the deterministic optimization was 0.083 sec.

Figure 15 shows the PDFs of the compliance for the deterministic optimaldesign and the risk-averse designs with different values of η. As occurred in theprevious experiments, the mean compliance for RATO designs is incrementedas the prescribed threshold ηi increases. The Excess Probability of compliancefor optimal deterministic and risk-averse designs is also detailed in Table 3.

32

(a) DTO (b) η = 100

(c) η = 200 (d) η = 300

Figure 14: The Michell-type structure design problem: (a) DTO design and (b-d) RATOdesigns for different values of η.

These data show that the probability of exceeding a critical value ηi achievesits minimal value when the quantity P J > ηi is considered as an objectivefunction. One also can observe a reduction in the standard deviation as thethreshold value η increases.

5.4. Discussion

The numerical experiments have shown the effectiveness of the proposal toaddress RATO problems involving smooth random fields not only in the loadingsource but also in material properties. The combination of Adaptive AnisotropicPC expansion and Sparse Grids permits to notably reduce the computationalcost of problems involving random fields with a moderate number of randomvariables. Furthermore, the embarrassingly parallel nature of these techniquesfacilitates the use of HPC on parallel and distributed architectures.

Some conclusions can be drawn from the numerical results. Firstly, the min-imization of the Excess Probability is accomplished at the cost of degrading thestructural performance under nominal conditions. Secondly, despite the differ-ences in formulation, the RATO and RTO show clear similitudes in their optimalresults. For all the experiments carried out in this work, the minimization ofthe risk of exceeding a prescribed threshold is accompanied by an incrementof the structural robustness. However, whereas in RTO the robust designs areobtained modifying a weighting factor that balances the expected value and thestandard deviation [13], which is subjectively selected a priori by the designer,in RATO the optimal designs are controlled by a parameter η, which is relatedto the structural performance, and hence, it is a more objective measure.

33

0 100 200 300 400 500 6000

0.002

0.004

0.006

0.008

0.01

0.012

Compliance

Probabilityden

sity

Deterministicη = 200η = 300η = 100

200 250 300 350 400 450 500 5500

0.5

1

1.5

2

2.5x 10−3

ComplianceProbabilityden

sity

Deterministicη = 200η = 300η = 100

(a) (b)

Figure 15: The Michell-type structure design problem: (a) PDFs of DTO design and RATOdesigns with different values of η, and (b) detailed view of the PDF for values of compliancehigher than 200.

Table 3: The Michell-type structure design problem: Excess Probability of DTO design andRATO designs with different values of η.

Case Compliance P J > 100 P J > 200 P J > 300 σJ

DTO 104.218 0.6278 0.0516 0.0015 43.8265η = 100 104.412 0.6198 0.0554 0.0019 44.8327η = 200 106.033 0.6563 0.0459 0.0017 41.5052η = 300 117.812 0.7969 0.0660 0.0010 40.6750

34

6. Conclusion

This work has presented an adaptive anisotropic stochastic expansion frame-work to address the risk-averse structural topology optimization problem in thecontinuous weak form of the linear system of elasticity with random fields. Akey difficulty is that the risk-averse topology optimization process makes use ofthe analytic expressions for the continuous shape derivatives, which require theintegration over failure regions. This issue is addressed both from the theoreticaland numerical viewpoints. The proposed RATO approach provides a generalframework that is neither limited to the use of discrete random variables northe discretization method. Theoretically, the proper stochastic problem formu-lation is ensured by the proof of the existence of an optimal shape on the set ofadmissible shapes. Numerically, the Excess Probability is efficiently computedby combining a non-intrusive anisotropic PC approach with a MC samplingmethod for uncertainty propagation. From a computational point of view, theanisotropic nature of the K-L expansion is exploited to reduce the number ofcoefficients and stochastic collocation points required by the stochastic expan-sion. Additionally, the non-intrusive nature of the method is preserved enablingthe use of task-level parallelism to address the computational burden of RATOproblem. The effectiveness of the proposed approach is demonstrated solvingbenchmarks subjected to loading and material uncertainties, including randomvariables and smooth random fields. The results show that risk-averse androbust formulations have strong similitudes. However, in RATO formulationsthe optimized design only depends on the threshold η, which is related with thestructural performance, and hence, its choice is less arbitrary than the weightingfactor in RTO.

7. Acknowledgements

This research was partially supported by the AEI/FEDER and UE under thecontract DPI2016-77538-R and by the “Fundacion Seneca-Agencia de Cienciay Tecnologıa de la Region de Murcia” under the contract 19274/PI/14. Ad-ditionally, the third and fourth authors were supported by the “Ministerio deEconomıa y Competitividad” under the contract MTM2013-47053-P. We alsogratefully acknowledge the support of NVIDIA Corporation with the hardwaredonations used for this research.

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