1 Nomenclature
=  the number of design variables 
=  the number of inequality constraints 
=  the domain of the design variables, i.e., 
=  the objective function, i.e., 
=  the constraints function, i.e., 
=  the prediction of the Gaussian process of a given function build with samples 
=  the uncertainty of the Gaussian process of a given function build with samples 
=  the acquisition function of the objective function 
=  the feasibility criterion of the constraint functions 
=  the approximated feasible domain defined by the feasibility criterion 
=  the low speed maximum lift coefficient 
2 Introduction
Bombardier aviation multidisciplinary optimization (MDO) framework has been continuously evolving over the past decade to ensure that future development programs result in highly efficient and competitive aircraft [Piperni2004]. In order to achieve this vision, Bombardier has developed a multilevel, multifidelity MDO framework to mature an aircraft configuration from early design stage to detailed design [Piperni2013].
The first element of this framework is called Conceptual MDO (CMDO). The main objective of the CMDO framework is to enable the design team to explore the design space assuming a set of realistic constraints established from previous aircraft development programs and marketing requirements. The main deliverable of this process is an aircraft configuration, which includes the initial sizing of the major structural components and systems. This conceptual aircraft configuration then generates a set of design requirements that will be validated using high fidelity frameworks, namely the Preliminary MDO (PMDO) and the Detailed MDO (DMDO). The second element corresponds to the PMDO framework analyzing the aerodynamic and structural characteristics of the wing. Its main objectives are to refine the wing sizing and to validate the wing related conceptual requirements, such as drag polar and wing weight. The two deliverables of this process are (1) a detailed wing geometry, external and internal, and (2) an initial definition of the wing structure.
The context of this research is about the design process of a new aircraft configuration in an industrial setting. The main inputs to initiate the design of a new aircraft platform, as well as inputs to the MDO framework, are a set of requirements from the marketing team (e.g. cruise mach number, design range, field performance and critical airport operations). For each of these sets of requirements, or constraints to the MDO framework, there needs to be a convergence between the CMDO and PMDO framework to ensure a robust initial design point. Since there could be multiple sets of requirements to be analyzed, the design team becomes rapidly limited by the time and number of CPU available.
The CMDO framework is built on a large set of low fidelity modules representing the aircraft disciplines. All these modules include their own set of design variables, which increases the number of dimensions of the optimization problem. Consequently, it makes the thorough exploration of the design space expensive in terms of function calls and CPU time, in the order of days. Similarly, the PMDO framework also includes a large number of design variables required to define the wing geometry and the aerostructural models are time consuming to converge, in the order of weeks. The motivation of this research is then to apply Bayesian Optimization (BO) methods [MockusBayesianmethodsseeking1975] to reduce the number of functions calls and accelerate the convergence time for both MDO frameworks. To do so, the SEGOMOE python toolbox is investigate against historical optimizers used at Bombardier aviation. The main objective of this paper is thus to demonstrate that all the optimizers of the toolbox has good convergence properties and performs well on an industrial test case.
This paper is organized into four sections: (1) detailed description of the models used in the CMDO and PMDO frameworks, (2) introduction to the Bombardier Research Aircraft Configuration (BRAC) case study, (3) definition of the optimizer selected for the analysis of the case study and finally (4) the analysis of the results with the emphasis on the convergence in terms of CPU time and number of evaluations.
3 An industrial multilevel, multifidelity and multidisciplinary optimization process
The MDO framework developed at Bombardier Aviation consists of three levels, corresponding to the three traditional stages of aircraft design: conceptual, preliminary, and detailed. Only a brief description of the first two levels is given here. Refer to [Piperni2013] for more information.
3.1 Conceptual multidisciplinary optimization framework
Conceptual MDO (CMDO) is the first level of the multilevel MDO framework. It accounts for all major aircraftlevel disciplines, such as aerodynamics, structures, systems, weight and balance, performance, stability and control, and economics. Due to the high number of disciplines, the level of fidelity is low and varies from empirical to simple physicsbased methods. For example, in the aerodynamics module, an inhouse method is used to predict the lowspeed maximum lift coefficient while a vortexlattice method coupled with a 2D computational fluid dynamics code is used to compute highspeed trim drag. Similarly, other disciplines can combine mixedfidelity analyses depending on the desired accuracy and turnaround times. Example usages of CMDO are design space exploration, optimization of marketing requirements and objectives, validation of business cases, assessment of technology insertion, down selection of promising configurations, and definition of performance targets [Piperni2013].
Typical design variables of CMDO include the wing area and planform as well as the engine scaling factor in the case of a fixed engine architecture. It is also possible to vary the thicknesstochord ratios of the wing and chordwise positions of the spars at various spanwise locations. Typical constraints include performance parameters such as the balanced field length (BFL), approach speed (), initial cruise altitude (ICA), range for different missions, and the ability to takeoff and land at critical airports. Systems and geometric considerations such as landing gear integration and wingtip chord can also be included. Typical objectives include maximum takeoff weight (MTOW), cost, and climate impact, or any combination thereof. For this study only MTOW is used since it combines the effects of cost and climate impact through fuel burn.
The CMDO framework is implemented as a workflow in Isight [vanisight2010]. Isight is a processintegration, designoptimization software developed and distributed by Dassault Systemes. It comes with its own set of optimizers.
The output of CMDO is an input to Preliminary MDO (PMDO), discussed next.
3.2 Preliminary multidisciplinary optimization framework
The preliminary MDO (PMDO) is the second level of the multilevel MDO framework. Compared to CMDO, the scope of the design space is narrowed, while the fidelity of the underlying analysis tools is increased. PMDO focuses primarily on the coupling and tradeoff between aerodynamics and structures. The objective is to obtain wing outer mold lines (OML) that are aerodynamically efficient in both highspeed and lowspeed flight, as well as to provide adequate performance at critical offdesign conditions. At the same time, the OML must be structurally viable, yield low weight, and satisfy space allocation requirements.
Piperni2013 previously described the PMDO process in detail. Although many of the components remain the same, the framework has continued to evolve, and a complete description falls outside the scope of this paper. In brief, the aircraft surfaces are updated in a parametric CATIA V5 model. Bombardier’s inhouse structured mesh generator, MBGRID [PiperniBoudreau2003], and NavierStokes flow solver, FANSC [Mohamed2009], are used to compute the flow solutions at several highspeed design and static load case conditions. The external loads are transferred to the wing structural mesh generated by AWSOM [Deblois2010], Bombardier’s inhouse wing sizing tool, which minimizes the primary wingbox weight while respecting various margin of safety constraints. The lowspeed characteristics are assessed using the semiempirical Valarezo criteria [Valarezo1994] in conjunction with VSAERO, a 3D panel code [maskew1982program]. Finally, the highspeed drag and structural weight are related through a fractional change equation, as described in [Piperni2007].
The architecture of the PMDO framework has changed significantly with the maturation of BOOST, an adjointbased optimization framework used at Bombardier [Sermeus2010, Reist2019]. The adjoint method is an efficient tool for computing the gradients needed for aerodynamic shape optimization, which involves long running simulation times and large numbers of design variables. In order to integrate the aerodynamic adjoint in a multidisciplinary environment, a hybridadjoint MDO framework was developed [Deblois2013], as depicted in Figure 1. In the first stage, an aerostructural optimization is performed in Isight by varying the wing planform, wing twist, and maximum thickness distribution (by scaling the wing profiles). This is followed by an aeroshape optimization in BOOST, where the drag is minimized by varying the airfoil shapes at fixed thickness to maximum chord ratio and wing spanload. This process is repeated in a sequential manner until convergence is achieved. This paper will examine the toplevel optimizer used to drive the aerostructural optimization, highlighted in bold in Figure 1, which is the current bottleneck in the PMDO process.
3.3 Bombardier research aircraft configuration
The test case considered for this work is representative of a small business aircraft, and is referred to as the Bombardier research aircraft configuration (BRAC). It is based on an early design of the Challenger 300 platform, and was developed to be shared with academic partners and for publication purposes. An image of the baseline BRAC geometry is shown in Figure 2.
4 Bombardier research aircraft configuration optimization case
In this section, the two level optimization process is introduced. First, we present the optimization of the BRAC model into the CMDO framework. Then, the previously obtained result is used in the PMDO of the BRAC model.
4.1 BRAC CMDO problem
The BRAC CMDO problem is based on a modified version of the initial Challenger 300 marketing requirements and objectives. It is thus representative of a typical industrial problem, yet here the design space is purposefully relaxed to test the global optimization capabilities of each optimizer. For confidentiality reasons the exact bounds on the design variables and constraints are not given here; instead, in the following sections their value will be normalized between and .
Formally, the CMDO problem is:
(1) 
where
is the design variable vector of
described in Table 2, the number of design variables, is the inequality constraints described in Table 3 and the associated bounds. Due to confidentiality reason is not given. Finally for the CMDO optimization problem (1), we consider here 12 design variables and 8 constraints.Also note that the longrange cruise (i.e. nominal) mission range is satisfied implicitly by the workflow. The baseline BRAC (see Figure 2) as defined in the CMDO workflow is unfeasible. This is inconsequential, since that initial design is not used as a starting point by any of the optimizers. Section 5 explains which starting point each optimizer uses instead. For industrial reasons, the CMDO of BRAC cannot take more than hours.
Constraint  Constraint Name  Bounds  Constraint  Constraint Name  Bounds 

index  index  index  index  
BFL  Climb performance  
ICA  Highspeed mission range  
Vref  Landing gear spacing  
Excess fuel  Wingtip chord 
4.2 BRAC PMDO problem
As mentioned earlier, the PMDO portion of this study will consider the aerostructural workflow, shown on the left half of Figure 1. Similar to CMDO, the objective is to minimize MTOW, as in problem (1). The wing and winglet design variables considered are listed in Table 4. The wing area and engine size are kept constant at the PMDO level, and the wing area of the CMDO optimum is used as an input to the planform generator. Similarly, the spar locations are fixed using the CMDO result, since the highlift systems integration is not considered. The constraints imposed during PMDO are listed in Table 5. The volume availability factor is a measure of the fuel volume available in the wing compared to the volume required to meet the range requirements. The maximum wingtip twist deformation is a proxy for a dynamics constraint, and is enforced to prevent an overly flexible wing. Again, for confidentiality reasons, the variable and constraint bounds are omitted. Finally for the PMDO optimization problem (1), we consider here 19 design variables and 5 constraints which is representative of a typical PMDO industrial application.
Constraint  Constraint Name  Bounds  Constraints  Constraint Name  Bounds 

index  index  index  index  
Volume availability factor  Landing gear spacing  
retracted  Wingtip twist deformation  
Cruise deck angle 
5 Optimizers
In this section, we introduce six optimizers to solve the following constrained problem:
(2) 
where is the objective function, gives the inequality constraints which are expensive to evaluate and is the vector of design variables. Two of the optimizers are included in the Isight software, which is mandatory to use because of CMDO and PMDO frameworks. We developed an Isight interface for the SEGOMOE python toolbox in which the other optimizers are implemented.
5.1 Bayesian optimizers
This section introduces the constrained Bayesian optimization (CBO) framework [MockusBayesianmethodsseeking1975, JonesEfficientglobaloptimization1998] that aims to solve the optimization problem (2) with a minimal number of calls. To do so, one uses Gaussian Process (GP) (also known as Kriging) [RasmussenGaussianprocessesmachine2006, Krigestatisticalapproachbasic1951] trained with a precomputed design of experiments (DoE) (i.e. set of designs evaluated on the objective and constraints functions) of points. GPs are then used to provide, with a cheap computational cost, a prediction and an associated uncertainty for each point of where can be either and for a given constraint component . Concerning the objective function, these information are combined in an acquisition function [frazier2018tutorial, Bartoliadaptivemodeling2019, WangMaxvalueentropysearch2017] coding the tradeoff between exploration of the highly uncertain domain that can hide a minimum and exploitation of the minimum of the GP prediction. For the constraints, these information are joined to produce a feasibility criterion [frazier2018tutorial, lam2015multifidelity, priem2019use] which is generally explicit. The point , solving the constrained maximization tradeoff subproblem:
(3) 
where is the approximated feasible domain defined by the feasibility criterion , is thus iteratively added to the DoE until a maximum number of iterations max_nb_it is reached. The solution provided by CBO to the problem (2) is eventually the best point in the DoE (i.e with the minimal feasible value of ). The main steps of CBO are finally summarized in Algorithm 1.
In this context, Bartoliadaptivemodeling2019 implemented a CBO python toolbox based on the super efficient global optimization (SEGO) algorithm of SasenaExplorationmetamodelingsampling2002 which has been enhanced by the use of mixture of experts (MOE) [hastieelements2005, bettebghor2011surrogate], the kriging with Partial Least Squares (KPLS) for high dimensional problems [BouhlelImprovingkrigingsurrogates2016], additional acquisition functions (e.g. WB2S [Bartoliadaptivemodeling2019]) for highly multimodal objective functions, supplementary feasibility criteria (e.g. the upper trust bound (UTB) [lam2015multifidelity, priem2019use]) for nonlinear constraint functions and a multiprocessing ability to speedup the optimization process. The toolbox [Bartoliadaptivemodeling2019] is thus named the super efficient global optimization with mixture of experts (SEGOMOE) toolbox. We also developed an interface of the python toolbox to allow its use in the Isight software in which BRAC is implemented.
In this paper, the four following variants of SEGO (available with the SEGOMOE toolbox) are tested to show their common ability on industrial MDO problems:

SEGO [SasenaExplorationmetamodelingsampling2002]: it can be seen as a basic implementation of the super efficient global optimization framework, it was developed for standard constrained optimization problems.

SEGOUTB [priem2019use]: it includes a decreasing upper trust bound in the SEGO framework. In the context of this paper and to encourage the exploration of the feasible domain, we used an exponential decrease for the constraints learning rate, see [priem2019use, Fig. 3].

SEGOMOE [Bartoliadaptivemodeling2019]: it combines the use of multiple models (local GPs) with the SEGO framework in order to mitigate the high nonlinearity of the targeted optimization problem.

SEGOMOEUTB: similarly to SEGOUTB, it combines the use of the UTB strategy and the SEGOMOE framework. Again, we choose to work with an exponential decrease for the constraints learning rate.
We note that for all the four variants we used the WB2S acquisition function combined with the KPLS models as the dimension of the design space is larger than 12.
5.2 Isight optimizers
The Isight software [vanisight2010] is delivered with many readytouse constrained optimizers. In this paper, we focus on two of the popular derivativefree Isight constrained optimizers: the evolutionary optimization algorithm (Evol) [schwefelevolutionsstrategie1975] and a variant of the Pointerdog algorithm (Pointer2) [vanisight2010].
Evol is an evolutionary strategy that mutates iteratively the best known point by adding a normally distributed perturbation to the design variables. The standard deviation of the normal distribution is adapted during the optimization process to solve the regarded problem with a minimal number of evaluations. This simple algorithm is enhanced by different features such as: (1) a repeat calculation check to ensure that all the points computed are different, (2) a standard deviation expansion process when the same point is always evaluated, (3) a consecutive variable search allowing the exploration in a single canonical direction of the design space, and (4) a parallel execution accelerating the optimization process.
The Pointer2 strategy is an optimization framework based on an Isight proprietary algorithm that managed a set of wellknown optimizers: Evol [schwefelevolutionsstrategie1975], the HookeJeeves direct search method [hookedirect1961], the
nonlinear programming by quadratic Lagrangian
(NLPQL) algorithm [schittkowskinlpql1986], the downhill simplex algorithm [neldersimplex1965], the multifunction optimization system tool (MOST) technique [vanisight2010] and themultiobjective particle swarm optimization
(MOPSO) algorithm [coellohandling2004]. The choice of the set of optimizers used in the optimization process is made with respect to a classification system using the information available on the problem (i.e. often given by the user). The best optimizer, and its settings, are then updated thanks to the information collected all along the optimization process. Furthermore, the optimization method can be used in two ways. First, the optimizers and their settings are selected to have the higher improvement leading to the best solution in the shortest time. On the contrary, one searches a robust solution to uncertainties which leads to a slower optimization process due to a higher number of calls. The Pointer2 strategy periodically performs a surrogate based optimization to speed up the convergence. In fact, the Pointer2 is able to solve a wide range of constrained optimization problems and thus allows nonspecialist to solve an optimization problem only providing the objective and constraints functions, the design variables, and a targeted optimization time.6 Results
In this section, we introduce and comment the different tests realized on the BRAC CMDO and PMDO problems. We also recall the aim here is to find the best solution to the optimization problem, the fastest, and the different optimizers are tested in this scope. In the following, the six previously introduced optimizers (see Section 5) are tested on the BRAC CMDO and PMDO. For the sake of comparison, we will produce convergence and parallel plots to assess the efficiency of the SEGOMOE toolbox solvers compared to the Isight optimizers.
6.1 BRAC CMDO results
6.1.1 Tests details
The optimizers, introduced in Section 5, are compared with the following test plan. We perform independent runs for each solver using different initial DoEs build with the Latin hypercube sampling method. For each run, all the solvers are initiated with the same DoE. The size of the initial DoEs is set to (i.e where is the dimension of BRAC CMDO problem). Note that Evol and Pointer2 need a single point to launch the optimization process, we thus provide the point with the best valid value of the DoE. If there is no valid value in the DoE, the point with the minimal constraints violation is provided.
The maximum number of evaluations of the SEGOlike solvers is set to (i.e. meaning a total of evaluations is performed). Concerning Evol and Pointer2, we used the historical options settings of the CMDO framework. The maximum number of evaluations of Evol is set to . For each run, Evol thus performed function evaluations, which takes approximately 8 hours to perform. We respectively fix the maximum allowable job time and the topography type options of Pointer2 to hours and nonlinear. All the remaining options of Evol and Pointer2 are kept as default. Finally, the best solution among the ten optimizations performed for each optimizer is kept to initiate the BRAC PMDO problem as explained in Section 3.2.
6.1.2 Convergence plots
To assess the performance of the introduced optimizers, we build two kinds of convergence plots. The first one, named evaluation convergence plot, displays the average and the standard deviation of the best valid value over the runs for increasing number of evaluations. The best valid value is defined as the best valid value if there is, at least, one valid point in the DoE, otherwise, a penalization replaces the obtained invalid value. The penalization is the highest feasible value of the objective function ever found. Because of confidentiality reasons, we scale the value of the convergence plots between and . The second convergence plot, named time convergence plot, also shows the average and the standard deviation of the best valid value over the runs along the optimization time. The penalization is kept the same as in the evaluation convergence plot.
Figure 5 shows that SEGOlike solvers clearly outperform Evol and Pointer2 in term of the optimum value. In term of number of evaluations, Figure (a)a displays that SEGO is converging the fastest to the optimum value even if SEGOMOE has a fastest convergence rate at the beginning of the optimization. Furthermore, note that SEGOUTB and SEGOMOEUTB are not converging as fast as SEGO and SEGOMOE because of their extensive exploratory behaviour. When looking to the optimization time, Figure (b)b reveals that the SEGOlike solvers converge in approximately hours meanwhile Evol and Pointer2 does not converge after hours.
To conclude, these convergence plots have shown that the SEGOlike solvers provide a better solution in less running CPUtime than Evol and Pointer2. We also note that the use of the UTB option within SEGO and SEGOMOE seems to not lead to any improvement on the obtained performance which suggests that the exploration of the feasible domain was not a difficult task to handle on this test case.
6.1.3 Parallel plots
The parallel plots introduced in the following depict the behaviour of the tested solvers along the optimization process in which targeted values are displayed (e.g., the explored design variables). Here, the plotted values are the number of iterations, the design variables values, the objective function value and the constraints violation. Furthermore, several colors are used to distinguish the reference design in red (i.e. the best feasible design found so far by all the tested optimizers, here SEGO), the optimum design found by the regarded solver in black, the feasible explored designs in green, and the unfeasible designs in blue. Due to the stochastic nature of our tests, we build our parallel plots using a median run in the following way. For each run of the optimizer, we store the best valid objective value; if none of the runs converges to a feasible point, we collect the minimal violation explored by the optimizer. The median run is then selected based on the stored values for all runs.
Figures 9 and 13 display the parallel plots of the median run of the tested optimizers. First, note that all the SEGOlike optimizers are converging to the optimum. On the contrary, Evol and Pointer2 do not converge to the optimum, as depicted by Figures (a)a and (b)b. Furthermore, Evol is not able to explore the entire domain as implied by the grouped green lines of Figure (a)a. Secondly, note the different solutions of the SEGOlike solvers even if they all converge approximately to the same objective function value. Indeed, the values for DV_8 and DV_11 (i.e. wing maximum thicknesstochord ratios) are different from an optimizer to another. This can be due to either a local optimum or an inactive design variable (i.e. the objective function does not change along the design variable). From the expert point of view, the behavior observed on the BRAC CMDO problem is due to inactive design variables.
6.2 PMDO results
6.2.1 Tests details
The optimizers are again compared on the BRAC PMDO problem. Because of the computation time of the BRAC PMDO problem (ca. ), we only performed one optimization for each solver. Each of the optimizers is initiated with the same DoE of sample points using the Latin hypercube sampling method. To mimic the introduced MDO process, we also add to the initial DoE the point corresponding to the best solution of all the BRAC CMDO process. As in Section 6.1.1, Evol and Pointer2 only need a single point to launch the optimization, we thus provide the best solution of all the BRAC CMDO cases.
The maximum number of evaluations of the SEGOlike solvers is set to (i.e. meaning a total of evaluations). The maximum number of evaluations of Evol and Pointer2 is set to to follow historical values used in the PMDO framework, and the parallel batch size is set to for both. In addition, a smooth topography type is selected for Pointer2. All the other settings are kept to default values.
6.2.2 Convergence plots
The convergence plots defined in this Section are slightly different from the ones introduced in Section 6.1.2. Indeed, only one optimization is here performed for each of the optimizers. Thus Figure 16 displays the best valid values of the regarded solvers for increasing time or number of evaluations.
First, Figure (a)a shows that none of the solvers converge to the same optimum value, with the best value obtained with SEGO. Similar values are also achieved with Evol and SEGOMOE. Furthermore, Pointer2 provides the worst value of all the solvers with an optimum at . The other algorithms find better solution values than Pointer2, but worse than SEGO, SEGOMOE and Evol. Note that SEGO and SEGOMOE are able to find a similar solution as Evol with 30 fewer evaluations.
Secondly, in terms of convergence time, Evol and Pointer2 benefit from the batch evaluation of the objective function which is not the case for the SEGOlike solvers. Indeed, the 220 necessary evaluations of Evol to converge are done in less than 40 hours, whereas the SEGOlike solvers need 80 hours for 190 evaluations.
To conclude, Figure 16 shows that the UTB feasibility criterion is not useful for the BRAC CMDO problem, whereas SEGO and SEGOMOE show great performances with a small budget. Moreover, the batch evaluation capability of Evol speeds up the convergence time.
6.2.3 Parallel plots
Here, no run selections are needed for the parallel plots. Indeed, only a single run is performed for the BRAC PMDO problem for each solver. They are produced in the same way as in Section 6.1.3. Figures 20 and 24 show the parallel plots for Evol, Pointer2, SEGO, SEGOMOE, SEGOUTB and SEGOMOEUTB.
As mentioned, SEGO finds the best design, followed closely by Evol and SEGOMOE. These three solutions share similar main planform features, namely the wing span (DV_0) and sweep (DV_1), while there is some variation in the wing twist distribution (DV_47) and maximum thicknesstochord ratios (DV_813), as shown in Figures (a)a and (c)c. Only SEGO converges to the lower thickness bound for the two most outboard profiles (DV_12 and DV_13). The winglet has a secondary impact on the MTOW, and as a result even greater variation is seen in the optimized winglet parameters (DV_1518). The best solutions of the other three optimizers deviate even further from the SEGO optimum. It is possible that all the optimizers converge to different local optima, but the variability in the solutions is more likely due to incomplete convergence of the algorithms, since the number of iterations was limited to meet the time restrictions of an industrial application.
7 Conclusion
The multilevel optimization framework developed at Bombardier Aviation aims to help the design of efficient and competitive aircraft. The two optimization levels, implemented within the Isight software, were discussed in this paper: conceptual and preliminary multidisciplinary optimization (CMDO and PMDO). The use of the readytouse Isight optimizers have shown great results in the past but within a minimum of 8 hours for the CMDO study case and a week for the PMDO one. The SEGOMOE python toolbox was investigated in this paper and led to significant improvements on Bombardier research aircraft configuration test cases. In particular, on the CMDO test case, we showed that all the tested solvers (implemented within the SEGOMOE toolbox) have outperformed two of the best Isight optimizers both in terms of the required computational effort as well as on the CPU running time. On the PMDO test case, the results are more contrasted. SEGO is able to provide a better solution than Evol in less evaluations but with more computational time. This is due to the batch evaluation capability of Evol. The batch evaluation capability of the SEGOMOE toolbox is a future improvement work to perform.
Acknowledgments
This work is part of the activities of ONERA  ISAE  ENAC joint research group in a context of a partnership between ONERA, ISAESUPAERO and Bombardier Aviation. In addition, we thank Jasveer Singh for his work on the BRAC CMDO specification.
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