BLACK-BOX MODEL EPISTEMIC UNCERTAINTY AT … · AN AIRCRAFT POWER-PLANT INTEGRATION CASE STUDY ... els of inherent uncertainty in the modelling and simulation tools ... Black-Box

BLACK-BOX MODEL EPISTEMIC UNCERTAINTY AT EARLYDESIGN STAGE. AN AIRCRAFT POWER-PLANT

INTEGRATION CASE STUDY

Arturo Molina-Cristóbal∗, Marco Nunez∗, Marin D. Guenov∗,Timo Laudan∗∗ , Thierry Druot∗∗

∗Cranfield University, Cranfield MK43 0AL,UK∗∗Airbus Operations S.A.S, 31060 Toulouse Cedex, [email protected]; [email protected]

Keywords: uncertainty modelling, epistemic uncertainty, collaborative robust design optimisation,power-plant integration, aircraft design.

Abstract

Presented is a novel numerical approach formodelling and propagating model epistemic un-certainty in simulation workflows composedof [black-box] deterministic models. This isachieved through the introduction of a Randomi-sation Treatment. Specifically, the design pa-rameters computed by particular deterministicsimulation models are numerically perturbed ac-cording to statistical criteria determined on thegrounds of prior design experience and/or otherconsiderations. Established uncertainty propaga-tion methods can thus be deployed to estimatethe resulting statistical behaviour of the simula-tion outputs. The proposed approach is demon-strated via an example of industrial relevancebased on an aircraft power-plant integration casestudy. The latter provides a wider context wherea reduction of epistemic uncertainty is achieveddue to the closer design collaboration betweenthe airframe and engine manufacturers, and inparticular, due to the faster introduction of up-to-date and more comprehensive design informa-tion. It is demonstrated that the proposed ap-proach, combined with appropriate collaborationprocesses leads to more robust solutions.

1 Introduction

Early design studies of complex products suchas aircraft are generally aimed at rapid definition(synthesis) and evaluation (analysis) of highlyidealized concepts rather than accurate and de-tailed representations of the artefact. These stud-ies, however, are performed with significant lev-els of inherent uncertainty in the modelling andsimulation tools/models involved at this stage.Uncertainty itself is undesired as it diminishes theconfidence on the computed results, which can becompensated either by overdesign or by invest-ment in further design iterations.

There are different types of uncertainty anda common practice in the engineering and otherfields has been to distinguish between two ma-jor classes: aleatory and epistemic uncertainty[4][10]. The former is representative of the phys-ical variability that is inherent in the (design) pa-rameters describing the system of concern or itsenvironment. Epistemic uncertainty on the otherhand is related to the lack of knowledge or in-formation. In modelling and simulation it gen-erally arises from the mathematical simplifica-tions/assumptions underlying the development ofsimulation models, the inclusion of incompleteexperimental data, or the incorporation of sub-jective design judgement and experience. Unlikealeatory uncertainty, it is possible to reduce epis-

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A. MOLINA-CRISTÓBAL, M. NUNEZ, M. D. GUENOV, T. LAUDAN, T. DRUOT

temic uncertainty through the addition of furtherknowledge if/when this is available, for exam-ple, under the form of calibration coefficients orhigher-fidelity simulation models.

A methodological perspective and an exten-sive (epistemic) uncertainty classification can befound in Padulo and Guenov [8]. In this workwe concentrate on a particular subset of epistemicuncertainty, namely the one associated with com-putational models describing a physical systemsuch as aircraft at early design stages. The objec-tive is to develop an approach for modelling andpropagating such uncertainty through the compu-tational systems so that it can be taken into ac-count for (design) decision making.

The rest of the paper is organised as fol-lows. Essential background information, defi-nitions and formulations related to uncertaintymanagement are presented in the next section.The methodology for model epistemic uncer-tainty management under the restrictions outlinedabove is presented in section 3. The applicationof the proposed method in the context of an in-dustrial case study concerning power-plant inte-gration is described in section 4. Finally conclu-sions are drawn and future work is outlined insection 5.

2 Uncertainty Management

Uncertainty management includes modelling,propagating and assessing the impact of un-certainty sources embedded in the simulationmodels [9]. A typical uncertainty managementschema is divided into three steps [5][6]: 1) un-certainty quantification, in which the sources ofuncertainty are formally identified, qualified andquantified; 2) uncertainty propagation, wherean estimation of the statistical behaviour of thesimulation outputs is computed through an ad-equate propagation of the modelled uncertain-ties in simulation workflows; 3) robust assess-ment, which allows to assess the robustness ofthe simulation results with respect to any consid-ered uncertainty. A typical uncertainty manage-ment schema is depicted in Fig. 1, whereas anoverview of each step is presented in the follow-ing sub-sections.

1)Uncertainty

Quantification

2) Uncertainty

propagation

3) Robust Assessment:

Robust Design

Optimisation

Model

𝑦1

𝑦2 𝑥1

𝑥2

Fig. 1 Example of a typical uncertainty manage-ment schema.

2.1 Uncertainty Quantification

Notwithstanding previous research efforts, thequantification of epistemic uncertainty is still anon-going issue in both academia and industry[4][7]. Its characterisation by probabilistic ap-proaches in general is not straightforward be-cause of the difficulty in inferring any statisticalinformation due to the intrinsic lack of knowl-edge. Prior proposed methods generally requirepolynomial approximations of the model output[7], which in practice might be inconceivable forlarge multidisciplinary models, or would allowto consider only a limited number of modellingtechniques (e.g., symmetric or uniform distribu-tions) [9][10]. The approach proposed in section3 resorts to a numerical perturbation of requestedsimulation outputs on the basis of statistical crite-ria defined by the designer according to prior ex-perience and/or assumptions. Conceptually simi-lar to modelling aleatory uncertainty, it enables torepresent such type of epistemic uncertainty viaany desirable probability density function (PDF),which can thus be propagated in computationalworkflows via established algorithms.

2.2 Uncertainty Propagation

Once the uncertainty affecting simulation param-eters and/or models has been adequately quanti-fied, it needs to be propagated through the con-sidered computational workflow in order to es-timate the statistical behaviour of the simulationoutputs. Suitable algorithms need to be selected

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Black-Box Model Epistemic Uncertainty at Early Design stage.An Aircraft Power-Plant Integration Case Study

to achieve this while guaranteeing the satisfactionof a number of simulation requirements (e.g., re-quested numerical accuracy, computational time,simulation solvers, etc.). In this context, in-teroperability becomes an important considera-tion, especially in collaborative enterprise simu-lations, where models implemented in differentenvironments need to be integrated together toconduct multidisciplinary and multi-fidelity stud-ies. To this end, and accounting also for the im-portance of intellectual property protection, weutilise the concept of a black-box model. Sucha model can be, for example, a compiled codewith known inputs, outputs and functional per-formance, but whose internal implementation isnot necessarily accessible. Among other suitablealgorithms (e.g., Monte Carlo Simulation, Mo-ment Method, Polynomial Chaos, Stochastic Ex-pansion, etc.), the Univariate Reduced Quadra-ture (URQ) method [11][12] has been adopted.The URQ method is inspired on the sigma-pointtechniques and provides an estimate of the simu-lation outputs mean and variance to a higher ac-curacy when contrasted with alternative methodsof comparable cost. Among its other qualities, itis able to handle several probability density func-tions (symmetric and non-symmetric distribu-tions) which allows to model diverse types of un-certainty. Moreover, the URQ method relies on adeterministic approximation of the statistical be-haviour of the simulation outputs, which makes itcompatible with well-known algorithms for con-ducting different simulation activities (e.g., de-sign optimisation, design of experiments, sensi-tivity analysis, etc.). A more detailed descriptionof the URQ method is presented in section 3.

2.3 Robust Assessment

The third step of uncertainty management inmodelling and simulation activities is aimed atexploiting uncertainty quantification and uncer-tainty propagation for assessing the robustness ofthe simulation results. To achieve this, the prob-lem to be investigated needs to be formally statedin such a way that it is representative of and con-sistent with the considered design requirements,assumptions, sought scope, and deployed numer-

ical methods. Considered in this paper is theapplication of uncertainty management schemasfor conducting optimisation studies under uncer-tainty, which can be generically formulated asfollows:

minµx∈ℜn

F[ f j(x,p)],

subject to: P(gi(x,p)≤ 0)≥ Pgi,with: P(xlbk ≤ µxk ≤ xubk)≥ Pxk .

(1)

where:• j = 1,2, ...,J, i = 1,2, ..., I, k = 1,2, ...,n;• p is a vector of design parameters;• x is the vector of n design variables with

mean µx and affected by uncertainty;• F are suitable functions that express the de-

pendence of the probability density func-tion of the J objectives f j on the input mul-tivariate distribution;• Pgi and Pxk are the desired probabilities of

satisfying the ith constraint gi and the vari-able bounds defined for the kth design vari-able xk, respectively.

The approach adopted here is to express thefunctions F via a weighted sum function [12],which leads to the reformulation of Equation (1)as follows:

minµx∈ℜn

Fj = µ f j + k f jσ f j ,

subject to: G j = µgi + kgiσgi ≤ 0,with: xlb +kxσx < µx < xub−kxσx.

(2)

where Fj and Gi are utility (loss) functions thataccount simultaneously for the mean and vari-ance of the objective f j and gi constraint func-tions, which need to be estimated by propagat-ing adequately the uncertainty affecting particu-lar design parameters or simulation models. Pa-rameters kx depend on the n desired probabili-ties Pxk and the statistical properties assumed formodelling the uncertainty affecting the design pa-rameters. The coefficients k f j and kgi can be es-tablished on the basis of the desired probabilitiesPf j and Pgi , along with the adopted robust de-sign criterion among the two possible choices of

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Table 1 Applicability of the proposed approach.Source of epistemic Randomisation Randomisation Treatment not

uncertainty Treatment Applicable applicableRequirements Insufficient elicitation Incomplete elicitation

definition (e.g., constraint boundary (e.g., missing requirement)set arbitrary)

Design assumptions Calibration/correction of Reduction in numberparametric equations, of relevant

refinement of operating simulation variablesscenarios/environments

Phenomenon Limited scope of the Limitedparameterisation mathematical model (e.g., Understanding/Misinterpretation

limited number of of the phenomenonoperating conditions)

Computational Known/estimated Unknown or unquantifiableaccuracy accuracy of the selected computational errors

algorithm/solver

quantile and tail conditional expectation (TCE)metrics. Further details and a practical explana-tion of the two criteria can be found in Paduloand Guenov [12].

3 Modelling and Propagation of EpistemicUncertainty

Described in this section is the proposed ap-proach for handling the model epistemic un-certainty affecting specific design parametersof simulations conducted at preliminary designstage. The key concept is based on modellingsuch type of uncertainty via a numerical per-turbation with statistical properties defined ac-cording to design experience and/or assumptions.This is achieved through a numerical method thatis described in the next sub-section and will behereafter referred in this paper as Randomisa-tion Treatment. The integration of the proposedtreatment with the URQ uncertainty propagationmethod is described in sub-section 3.2.

3.1 Randomisation treatment

Low-fidelity models are deployed at early designstage to rapidly generate a high-level product de-scription and to compare different design alterna-tives by conducting various fast simulation stud-ies. In turn, it is necessary to adequately ac-

knowledge and manage the epistemic uncertaintystemming from the lack of knowledge inherent inearly design, which can be attributed to a numberof different sources, such as:

• Requirements definition. An incorrect orincomplete elicitation and documentationof design requirements could preclude theadequate development of simulation mod-els to support the design of a product whichfully satisfies relevant customer needs andexpectations.

• Design assumptions. Different assump-tions are normally taken into considerationthroughout the development of the modelsto simplify the physics behind the phenom-ena to be simulated. Examples of com-mon practice are the reduction of the num-ber of relevant simulation variables, thecalibration/correction of parametric equa-tions with experimental data, generalisa-tion of operating scenarios and environ-ments (e.g., initial conditions, boundaryconditions, etc.).

• Phenomenon parameterisation. A limitedunderstanding or misinterpretation of thesimulated phenomenon can prevent devel-oping a mathematical model that appropri-

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ately captures its behaviour for each oper-ating scenario under investigation.

• Computational accuracy. The selection ofparticular algorithms in the implementa-tion of models impacts on the accuracy ofsimulation results. The latter in generalis also dependant on the choice of solversalong with their corresponding setup pa-rameters for executing the individual mod-els and simulation workflows.

It has to be emphasised at this stage that theproposed approach can handle only subsets of theabove epistemic uncertainties. Presented in Table1 are example scenarios for which the proposedapproach can and cannot be applied, respectively.

The Randomisation Treatment mentionedabove is a numerical perturbation used duringthe uncertainty quantification stage. Thus out-puts deterministically computed through simula-tion models affected by epistemic uncertainty arestochastically parameterised via Random Vari-ables with predefined statistical properties. Moreprecisely, each Random Variable (RV) associatedwith a particular model output will have a unitmean, together with minimum and maximumvariations defined according to prior design expe-rience and/or assumptions. In this way, the defi-nition of each Random Variable can be formallystated via adequate probability density functions(PDFs). This process is conceptualised in Equa-tion (3) and Fig. 2 by considering a generic out-put y and its corresponding RVy,

ystochastic = RVy · ydeterministic(xy) (3)

where xy is the vector of design variables onwhich y is dependent.

It is important to note that the applicationof the Randomisation Treatment allows to modeland encapsulate in its output the epistemic uncer-tainty associated with specific simulation models.In other words, it will transform specific deter-ministic simulation outputs onto stochastic vari-ables with precise statistical properties dependingon the probability density functions of their cor-responding Random Variables. Such uncertainty

Simulation Model

yx

Randomisation

Treatment

yy ,

RVy

ystochastic =

y

Fig. 2 Randomisation Treatment deployment ona simulation model.

can thus be propagated through simulation work-flows via conventional numerical methods. De-picted in Fig. 3 is an example of a Randomisa-tion Treatment application on an illustrative sim-ulation workflow.

Described in the next sub-section is the inte-gration of the proposed approach with the Uni-variate Reduced Quadrature (URQ) method foruncertainty propagation [12].

3.2 Epistemic Uncertainty Propagation viathe URQ method

A number of numerical methods can be consid-ered for propagating model epistemic uncertaintyand handled via the aforementioned Randomisa-tion Treatment. As mentioned in Section 2, theUnivariate Reduced Quadrature (URQ) method[12] has been chosen for uncertainty propaga-tion. It allows to estimate the statistical proper-ties (mean µ and standard deviation σ) of selectedsimulation outputs. This is generically illustratedin Fig. 3, where the stochastic behaviours at-tributed to y1 and y2 through the RandomisationTreatment need to be propagated downstream inthe workflow to approximate the resulting meanand variance of yout . Such an approach canbe generalised to manage the epistemic uncer-tainty affecting r simulation parameters throughthe definition of an equivalent number of RandomVariables. An accurate estimation of the statisti-cal behaviour of the simulation outputs can thusbe computed with the URQ method via a sam-pling stencil of 2r + 1 workflow evaluations. Itis important to note that the computational ef-fort grows linearly with r. This is particularly

5


= simulation models

y1 deterministic

y1 stochastic =

= Randomisation Treatment

...

= further simulation models

yout stochastic =

x4 deterministic

x1 deterministic

...

x3 deterministic x2 deterministic

[x5 ,…, xn] deterministic

outout, yy

11, yy

RVy 1 RVy 2

= y2 stochastic

22, yy

y2 deterministic

Fig. 3 Conceptual example of the Randomisa-tion Treatment deployment in a generic simula-tion workflow.

efficient compared to other strategies that requirethousands of evaluations, such as the traditionalMonte Carlo Simulation method, as conceptuallyillustrated in Fig. 4.

Illustrated in Fig. 4 is also the ability of theURQ method to handle a number of differentsymmetric and non-symmetric PDFs (e.g., Gaus-sian, Triangular, Gamma, Beta, etc.). For exam-ple, consider two models that allow the compu-tation of the simulation parameters y1 and y2, asdepicted in Fig. 3. The epistemic uncertainty af-fecting such models is modelled through the for-mulation of the corresponding Random VariablesRV1 and RV2. The first is formulated as a non-symmetrical triangular distribution with a givenminimum and maximum variation from a nomi-nal value, denoted by a and b respectively. Thesecond is formulated as a Gaussian distributionwith a precise standard deviation. The propaga-tion of uncertainty via the URQ method wouldthus require five evaluations of the workflow tocompute an estimation of the mean and stan-dard deviation of the simulation output yout cor-responding to any evaluated design point x. Theexact location of the evaluation stencil points isdefined according to the scalar distances h±1 and

a 1 b

1

URQ Points MCS sample points

ℎ1−𝜎1 ℎ1

+𝜎1

ℎ2+𝜎2

ℎ2−𝜎2

𝑹𝑽𝟏

𝑹𝑽𝟐

Random Variable Space

Fig. 4 Conceptual comparison of the samplingevaluations required by MCS and URQ on ageneric random variable space RV1 RV2.

h±2 , which are functions of the first four statisticalmoments of the random variables. A completedescription of the URQ method, is presented inPadulo et al. [11].

4 Application Example: an Aircraft Power-Plant Integration Case Study

The proposed model epistemic uncertainty man-agement method is demonstrated in this sec-tion via an application of industrial relevance.It is based on an aircraft power-plant integra-tion scenario in which scheduled design itera-tions are progressively conducted as up-to-datedesign information becomes available from theengine manufacturer. In particular, the introduc-tion of surrogate models in simulation studiesis adopted in robust design optimisation (RDO)studies aimed at managing the convergence of en-gine requirements.

4.1 Case study description

The case study is based on the “Prelimi-nary Multi-Disciplinary Power-Plant Design”test case defined in the EU FP7 CRESCENDOproject [1][2]. It sets the scope for robust multi-disciplinary optimisation studies for the prelimi-nary design of aircraft power-plant. A particularattention was given to the interaction between air-

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Table 2 Relevant case study nomenclature.FNslst Sea-level engine thrust [N]Awing Wing area [m2]

ar Wing aspect ratioBPR By-pass ratios f c Specific fuel consumption [lb/lbd/hr]LoD Lift over drag coefficient

MWE Maximum weight empty [kg]to f l Take-off field length [m]vapp Approach speed [kt]T TC Time to climb [min]OEI One engine inoperative

MTOW Maximum Take-off Weight [kg]Fuel Total fuel for mission [kg]

frame and engine manufacturers. This involvedfinding the best compromise between the bene-fits (e.g., thrust and power) and drawbacks (e.g.,weight, fuel consumption, etc.) associated withdifferent engine concepts. The relevant nomen-clature and the design optimisation formulationare given in Table 2 and Table 3, respectively.

The evaluation of alternative design solutionsconducted throughout the optimisation processwas computed by a set of simple models dedi-cated to aircraft conceptual design named SIM-CAD, which was provided by one of the in-dustrial partners. SIMCAD allows to evalu-ate conventional aircraft configurations accord-ing to arbitrary Top Level Aircraft Requirements(TLARs) by running a nested mass-mission loop,required to achieve the convergence of MTOWand Fuel.

The addressed scenario focuses on the de-sign iterations triggered by the progressive re-lease of more detailed design information by theengine manufacturer. From a business perspec-tive, a secure exchange of design data and mod-els between airframe and engine manufacturersis therefore necessary while at the same timeguaranteeing the protection of their correspond-ing intellectual property (IP). One of the solu-tions investigated to achieve this is through theuse of surrogate and black-box simulation mod-els. Specifically, the models in the abovemen-tioned aircraft sizing code SIMCAD were han-dled as black-boxes, whereas a more accurate

Table 3 Aircraft design optimisation formulation.Objective

minimise MTOWConstraints

to f l-standard ≤ 1700 mto f l-hot&high ≤ 3200 m

vapp ≤ 137 ktClimb Ceiling ≥ 33000 ftCruise Ceiling ≥ 35000 ft

Buffeting Ceiling ≥ 37000 ftOEI Ceiling ≥ 16000 ft

T TC ≤ 25 minDesign Variables

FNslst = [70000, 160000] NAwing = [100,200] m2

ar =[7, 12]BPR = [6, 10]

model of the engine was provided as a krigingsurrogate model [3]. The investigation of the con-vergence of engine requirements was thus carriedout by considering two design iterations. Thefirst iteration was conducted by means of a sim-ulation architecture entirely based on SIMCAD.The second iteration included the incorporationof the engine surrogate model which replacedspecific simulation models within SIMCAD, thusacting as a means for reducing the epistemic un-certainty affecting the computation of particularsimulation parameters. The considered set of un-certain parameters and their associated RandomVariables is presented in Table 4, where each un-certainty is represented as a percentage variationfrom a nominal value (i.e., the value computedfor its corresponding parameter). Such variationis assumed on the grounds of prior experienceand is reduced from the first to the second itera-tion due to the refinement of simulation accuracyintroduced by the engine surrogate model.

4.2 Robust design optimisation

A weighted sum strategy based on Equation (2)along with the proposed approach for the man-agement of epistemic uncertainty were adoptedto tackle the optimisation problem presented inTable 3. All the uncertainties affecting the sim-

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Table 4 Set of simulation parameters computed by models affected by epistemic uncertainty. Note thatin the second design iteration the parameters EngineWeight, s f c and FanDiameter are assumed not tobe affected by uncertainty.

Simulation Uncertainty of Uncertainty of Correspondingparameters parameter parameter Random

computation in computation in VariableDesign Iteration 1 Design Iteration 2

MaxTrustT/O [-10% , +10%] [-5%, +5%] RVKmtoMaxT hrustClimb [-10% , +10%] [-5%, +5%] RVKmclMaxT hrustCruise [-10% , +10%] [-5%, +5%] RVKmcr

EngineWeight [-5% , +5%] - RVMpwps f c [-5% , +5%] - RVs f c

FanDiameter [-1% , +1%] - RVdnacLoD [-1% , +0.5%] [-0.5%, +0.5%] RVLoD

MWE [-2% , +2%] [-1%, +1%] RVMWE

ulation parameters in Table 4 were modelled viaGaussian distributions, except for LoD in the firstdesign iteration, for which a triangular distribu-tion was considered. This is because the latter re-flects better the designer’s assumption for asym-metric LoD variation from the nominal value. Asummary of the uncertainty modelling via Ran-dom Variables adopted for the two design itera-tions presented in this paper is provided in Table5.

The statistical properties of the objective andconstraints were estimated through the quantilemetric [12], without considering any particularassumption on their respective distribution, andrequesting a 75% satisfaction probability.

The execution of the aforementioned optimi-sation problem represents a clear example of sim-ulation studies requiring a hierarchical applica-tion of relevant numerical treatments on a sim-ulation workflow assembled from a given set ofcomputational models. This has been addressedvia specific functionalities provided in the inno-vative model-based design tool AirCADia [13],which enables an interactive formulation and ex-ecution of black-box simulation studies withoutthe need for a laborious and specialised lowerlevel programming. Illustrated in Fig. 5 is aschematic representation of the considered sim-ulation architecture. It is possible to note howthe optimisation treatment (loop) encapsulatesthe uncertainty propagation treatment, which in

turn is applied onto the simulation workflow afterachieving the convergence of MTOW and Fuel.

...

Npax

RVMWE

FNslst

...

Range Mach

Awing BPR

ar

RVdnac RVsfc RVMpwp

RVKmcr RVKmlc

RVKmto RVLoD * * *

[ MTOW, tofl-standard, tofl-hot&high, Vapp, Climb Ceiling, … , TTC ]

Op

tim

isat

ion L

oo

p

Un

cert

ain

ty P

rop

agat

ion

Lo

op

MTO

W &

Fu

el L

oo

p

Fig. 5 Illustration of the considered simulationarchitecture. Identified with an asterisk are thethree Random Variables assumed not to be af-fected by uncertainty in the second design iter-ation.

The results obtained in AirCADia via theURQ method coupled with an embedded evo-lutionary algorithm are presented in tabular andgraphical form in Table 6, Table 7 and Fig. 6,respectively.

The power-plant robust design optimisationproblem described above represents an exampleof simulation studies, based on the integration of

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Table 5 Set of Random Variables modelled as epistemic uncertainty.Random Uncertainty Modelling in Uncertainty Modelling inVariable Design Iteration 1 Design Iteration 2RVKmto Gaussian with: µ = 1;σ = 0.1/3 Gaussian with: µ = 1;σ = 0.05/3RVKmcl Gaussian with: µ = 1;σ = 0.1/3 Gaussian with: µ = 1;σ = 0.05/3RVKmcr Gaussian with: µ = 1;σ = 0.1/3 Gaussian with: µ = 1;σ = 0.05/3RVMpwp Gaussian with: µ = 1;σ = 0.05/3 - -RVs f c Gaussian with: µ = 1;σ = 0.05/3 - -RVdnac Gaussian with: µ = 1;σ = 0.01/3 - -RVLoD Triangular with: a=0.99, µ = 1,b = 1.005 Gaussian with: µ = 1;σ = 0.005/3RVMWE Gaussian with: µ = 1;σ = 0.02/3 Gaussian with: µ = 1;σ = 0.01/3

Table 7 Objective and constraint values of the optimal points obtained for the robust design optimisationformulation in Table 3.

Design Iteration 1 Design Iteration 2Performance Mean Standard Mean Standard

Deviation DeviationMTOW [kg] 90136.5341 675.4364 87289.473 204.964

MaxT hrustT/O [lb f ] 28482.5293 949.4176 26980.308 449.671@M 0.25; ISA+15; 0ft.MaxT hrustClimb [lb f ] 7358.9591 245.2986 6970.834 116.180@ M 0.76; ISA; 35kft.MaxT hrustCruise [lb f ] 6847.9203 228.264 6486.748 108.112@ M 0.76; ISA; 35kft.

MWE [kg] 45214.7322 394.7925 44201.022 170.056

Table 6 Design variable values of the optimalpoints obtained for the robust design optimisationformulation in Table 3.

Design DesignIteration 1 Iteration 2

ar 11.786 11.252Awing[m2] 164.7776 161.8699

BPR 9.5432 6.4771FNslst[N] 154440.7238 146295.2379

mixed-fidelity models. The results show reduc-tion of the uncertainty associated with simulationoutputs due to the incorporation of more accuratehigher-fidelity models such as the engine surro-gate model introduced in the second design iter-ation. It is important to note that the reductionof the uncertainty and the improvement of the so-lution were possible not least because of the re-sults computed in the first design iteration. More

specifically, the latter allowed to identify the re-gion in the design space which contained the de-sired set of optimal solutions. Thus the higher-fidelity (surrogate) engine model was developedfrom a dataset obtained from more accurate en-gine simulations conducted around such optimaldesign points. The insight and knowledge gainedfrom the first design iteration were exploited tolimit the epistemic uncertainty affecting the en-gine simulation model deployed in the second de-sign iteration. The convergence to a more certainset of engine requirements was hence achievedby conducting a second robust design optimisa-tion study based on the refined engine model.

5 Summary and Conclusions

Presented in this paper is a novel numerical ap-proach for management of black-box model epis-temic uncertainty. The proposed approach is seenas most beneficial to the early stages of com-

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Iteration 1 Iteration 22.4

2.6

2.8

3

3.2x 10

4

MaxThrustTO__0ft [lbf]

@ M 0.25; ISA+15 ; 0kft

Thr

ust

[lbf]

Iteration 1 Iteration 26500

7000

7500

8000

8500

MaxThrustClimb [lbf]

@ M 0.76; ISA ; 35kft

Thr

ust

[lbf]

Iteration 1 Iteration 26000

6500

7000

7500

8000

MaxThrustCruise [lbf]

@ M 0.76; ISA ;35kft

Thr

ust

[lbf]

Iteration 1 Iteration 24.35

4.4

4.45

4.5

4.55

4.6

4.65x 10

4 MWE [Kg]

Wei

ght

[Kg]

σ

3σ

σ

3σ

σ

3σ

σ

3σ

Fig. 6 Convergence of model epistemic uncertainty associated with engine requirements by means ofbox plots. The horizontal line in the middle of each box represents the mean value of each engineparameter per iteration, whereas the top and bottom lines of each box identify the range correspondingto a variation of such value within one standard deviation. A variation of three standard deviations isalso depicted via the horizontal lines at the extremes of the vertical lines extending above and below eachbox.

plex systems design, when preliminary simula-tion studies are based on low-fidelity models andconducted on the grounds of various assumptionsand incomplete knowledge. A RandomisationTreatment has been proposed to quantify and for-mally state a class of epistemic uncertainty af-fecting the computation of simulation parame-ters. This is numerically obtained through thedefinition of associated Random Variables withprecise statistical properties. In the proposed ap-proach the uncertainty propagation is enabled viathe integration of the URQ method, which allowsthe efficient estimation of the simulation outputrobustness in terms of mean and standard devia-tion. The application of the proposed approachfor the reduction of epistemic uncertainty in anexample of industrial relevance demonstrated itsvalue in the process of maturing and converging

(engine and airframe) design requirements in acollaborative design environment which respectspartners’ intellectual property. Future work willconcentrate on enabling the modelling of depen-dencies among random variables, as well as onthe extension of the proposed methods to otherclasses of epistemic uncertainty.

AcknowledgmentsThe research leading to the presented results hasbeen carried out within the CRESCENDO In-tegrated Project, which was partially funded bythe European Union Seventh Framework Pro-gramme (FP7/2007-2013) under grant agreementn◦ 234344 http://www.crescendo-fp7.eu/). Theauthors would like to thank all the partners whocontributed to the development of the “Prelim-inary Multi-Disciplinary Power-Plant Design”

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project test case, in particular: Erik Baalbergen,Wim Lammen, Johan Kos, Daniel Kickenweitzand Philipp Kupijai.

References

[1] Collaborative Robust Engineering using Sim-ulation Capability Enabling Next Design Op-timisation (CRESCENDO). (2009-2012), Sev-enth Framework Programme (FP7), Project Ref-erence 234344, www.crescendo-fp7.eu.

[2] Laudan, T. Preliminary Multi-disiplinaryPower-plant Design (PMPD) presentation,CRESCENDO Forum, 19th to 21st June 2012,Toulouse France. Also presented as: Aircraftand Engine manufacturers collaborating to sharerequirements and converge towards an optimaldesign, 3DS Customer Conference. Novemeber6-7,2012. Orlando, Florida, USA, 2012.

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