\section{An industrial multi-level, multi-fidelity and multi-disciplinary 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~\cite{Piperni2013} for more information.

\subsection{Conceptual multi-disciplinary optimization framework}

\textit{Conceptual MDO} (CMDO) is the first level of the multi-level MDO framework.
It accounts for all major aircraft-level 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 physics-based methods.
For example, in the aerodynamics module, an in-house method is used to predict the low-speed maximum lift coefficient $C_{L_{max}}$ while a vortex-lattice method coupled with a 2D computational fluid dynamics code is used to compute high-speed trim drag.
Similarly, other disciplines can combine mixed-fidelity 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~\cite{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 thickness-to-chord ratios of the wing and chordwise positions of the spars at various spanwise locations.
Typical constraints include performance parameters such as the \textit{balanced field length} (BFL), \textit{approach speed} ($V_\textsubscript{ref}$), \textit{initial cruise altitude} (ICA), range for different missions, and the ability to take-off and land at critical airports.
Systems and geometric considerations such as landing gear integration and wingtip chord can also be included.
Typical objectives include \textit{maximum take-off 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 \cite{vanisight2010}. Isight is a process-integration, design-optimization software developed and distributed by Dassault Systemes.
It comes with its own set of optimizers.

The output of CMDO is an input to \textit{Preliminary MDO} (PMDO), discussed next.

\subsection{Preliminary multi-disciplinary optimization framework}
\label{ssec:PMDO}

The \textit{preliminary MDO} (PMDO) is the second level of the multi-level 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 trade-off between aerodynamics and structures.
The objective is to obtain wing \textit{outer mold lines} (OML) that are aerodynamically efficient in both high-speed and low-speed flight, as well as to provide adequate performance at critical off-design conditions.
At the same time, the OML must be structurally viable, yield low weight, and satisfy space allocation requirements.

\citet{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 in-house structured mesh generator, MBGRID~\cite{PiperniBoudreau2003}, and Navier-Stokes flow solver, FANSC~\cite{Mohamed2009}, are used to compute the flow solutions at several high-speed design and static load case conditions.
The external loads are transferred to the wing structural mesh generated by AWSOM~\cite{Deblois2010}, Bombardier’s in-house wing sizing tool, which minimizes the primary wing-box weight while respecting various margin of safety constraints.
The low-speed characteristics are assessed using the semi-empirical Valarezo criteria~\cite{Valarezo1994} in conjunction with VSAERO, a 3D panel code \cite{maskew1982program}.
Finally, the high-speed drag and structural weight are related through a fractional change equation, as described in \cite{Piperni2007}.

The architecture of the PMDO framework has changed significantly with the maturation of BOOST, an adjoint-based optimization framework used at Bombardier~\cite{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 hybrid-adjoint MDO framework was developed~\cite{Deblois2013}, as depicted in Figure~\ref{fig:PMDO_Framework}.
In the first stage, an aero-structural optimization is performed in Isight by varying the wing plan-form, wing twist, and maximum thickness distribution (by scaling the wing profiles).
This is followed by an aero-shape optimization in BOOST, where the drag is minimized by varying the airfoil shapes at fixed thickness to maximum chord ratio and wing span-load.
This process is repeated in a sequential manner until convergence is achieved.
This paper will examine the top-level optimizer used to drive the aero-structural optimization, highlighted in bold in Figure~\ref{fig:PMDO_Framework}, which is the current bottleneck in the PMDO process.

\begin{figure}[htb!]
    \centering
    \includegraphics[width=0.95\textwidth]{Figures/Intro/PMDO_Framework.pdf}
    \caption{Simplified schematic of the hybrid-adjoint PMDO workflow.}
    \label{fig:PMDO_Framework}
\end{figure}

\subsection{Bombardier research aircraft configuration}

The test case considered for this work is representative of a small business aircraft, and is referred to as the \textit{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~\ref{fig:BRAC_3DView}.

\begin{figure}[htb!]
    \centering
    \includegraphics[width=0.5\textwidth]{Figures/Intro/BRAC_3DView.png}
    \caption{The baseline BRAC configuration.}
    \label{fig:BRAC_3DView}
\end{figure}

\section{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.

\subsection{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 $0$ and $1$.

Formally, the CMDO problem is:
\begin{equation}
    \min\limits_{\bm{x}\in [0,1]^d}\left\{ \text{MTOW}(\bm{x}) ~\text{s.t.}~ \bm{g}(\bm{x}) \leq \bm{B} \right\},
    \label{eqn:objective}
\end{equation}
where $\bm{x}$ is the design variable vector of $[0,1]^d$ described in Table \ref{tab:DV_CMDO}, $d$ the number of design variables, $\bm{g}$ is the inequality constraints described in Table \ref{tab:Cst_CMDO} and $\bm{B}$ the associated bounds.
Due to confidentiality reason $\bm{B}$ is not given.
Finally for the CMDO optimization problem (\ref{eqn:objective}), we consider here 12 design variables and 8 constraints.

Also note that the long-range cruise (i.e. nominal) mission range is satisfied implicitly by the workflow.
The baseline BRAC (see Figure \ref{fig:BRAC_3DView}) 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~\ref{sec:opt} explains which starting point each optimizer uses instead.
For industrial reasons, the CMDO of BRAC cannot take more than $8$ hours.

\begin{table}[!ht]
	\centering
	\caption{Design optimization space definition.}
	\centerline{
	\begin{tabular}{l l l | l l l}
		\toprule
		\textbf{Variable} & \multirow{2}{*}{\textbf{Variable Name}} & \textbf{Variable} & \textbf{Variable} & \multirow{2}{*}{\textbf{Variable Name}} & \textbf{Variable} \\
		\textbf{index} & & \textbf{symbol} & \textbf{index} & & \textbf{symbol} \\
		\midrule
		$\bm{x}[0]$ & Engine scaling factor & DV\_0 & $\bm{x}[4-5]$ & Wing rear spar chord-wise locations & DV\_4-5\\
		$\bm{x}[1]$ & Wing aspect ratio & DV\_1 & $\bm{x}[6]$ & Wing sweep & DV\_6\\
		$\bm{x}[2]$ & Wing area & DV\_2  & $\bm{x}[7]$ & Wing taper & DV\_7 \\
		$\bm{x}[3]$ & Wing inboard trailing edge sweep & DV\_3 & $\bm{x}[8-11]$ & Wing maximum thickness-to-chord ratios & DV\_8-11 \\
		\bottomrule
	\end{tabular}}
	\label{tab:DV_CMDO}
\end{table}

\begin{table}[!ht]
	\centering
	\caption{Constraints definition.}
	\begin{tabular}{l l l | l l l}
		\toprule
		\textbf{Constraint} & \multirow{2}{*}{\textbf{Constraint Name}} & \textbf{Bounds} & \textbf{Constraint} & \multirow{2}{*}{\textbf{Constraint Name}} & \textbf{Bounds} \\
		\textbf{index} & & \textbf{index} & \textbf{index} & & \textbf{index} \\
		\midrule
		$\bm{g}[0]$ & BFL & $\bm{B}[0]$ & $\bm{g}[4]$ & Climb performance & $\bm{B}[4]$\\
		$\bm{g}[1]$ & ICA & $\bm{B}[1]$ & $\bm{g}[5]$ & High-speed mission range & $\bm{B}[5]$  \\
		$\bm{g}[2]$ & Vref & $\bm{B}[2]$ & $\bm{g}[6]$ & Landing gear spacing & $\bm{B}[6]$ \\
		$\bm{g}[3]$ & Excess fuel & $\bm{B}[3]$ & $\bm{g}[7]$ & Wingtip chord & $\bm{B}[7]$  \\
		\bottomrule
	\end{tabular}
	\label{tab:Cst_CMDO}
\end{table}

\subsection{BRAC PMDO problem}

As mentioned earlier, the PMDO portion of this study will consider the aero-structural workflow, shown on the left half of Figure~\ref{fig:PMDO_Framework}.
Similar to CMDO, the objective is to minimize MTOW, as in problem \eqref{eqn:objective}.
The wing and winglet design variables considered are listed in Table~\ref{tab:DV_PMDO}.
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 plan-form generator. Similarly, the spar locations are fixed using the CMDO result, since the high-lift systems integration is not considered.
The constraints imposed during PMDO are listed in Table~\ref{tab:Constraints_PMDO}.
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 (\ref{eqn:objective}), we consider here 19 design variables and 5 constraints which is representative of a typical PMDO industrial application.

\begin{table}[!ht]
	\centering
	
	\caption{PMDO design optimization space definition.}
	\centerline{
	\begin{tabular}{l l l | l l l}
		\toprule
		\textbf{Variable} & \multirow{2}{*}{\textbf{Variable Name}} & \textbf{Variable} & \textbf{Variable} & \multirow{2}{*}{\textbf{Variable Name}} & \textbf{Variable} \\
		\textbf{index} & & \textbf{symbol} & \textbf{index} & & \textbf{symbol} \\
		\midrule
		$\bm{x}[0]$ & Wing span & DV\_0 & $\bm{x}[8-13]$ & Wing max thickness-to-chord ratios & DV\_8-13 \\
		$\bm{x}[1]$ & Wing leading edge sweep & DV\_1 &  $\bm{x}[14]$ & Wing inboard trailing edge sweep & DV\_14 \\
		$\bm{x}[2]$ & Wing break loation & DV\_2 & $\bm{x}[15]$ & Winglet cant & DV\_15 \\
		$\bm{x}[3]$ & Wingtip chord & DV\_3 & $\bm{x}[16]$ & Winglet span & DV\_16  \\
		$\bm{x}[4-7]$ & Wing twist distribution & DV\_4-7 & $\bm{x}[17-18]$ & Winglet twist & DV\_17-18  \\	
		\bottomrule
	\end{tabular}}
	\label{tab:DV_PMDO}
\end{table}

\begin{table}[!ht]
	\centering
	\caption{PMDO constraints definition.}
	\begin{tabular}{l l l | l l l}
		\toprule
		\textbf{Constraint} & \multirow{2}{*}{\textbf{Constraint Name}} & \textbf{Bounds} & \textbf{Constraints} & \multirow{2}{*}{\textbf{Constraint Name}} & \textbf{Bounds} \\
		\textbf{index} & & \textbf{index} & \textbf{index} & & \textbf{index} \\
		\midrule
		$\bm{g}[0]$ & Volume availability factor & $\bm{B}[0]$ & $\bm{g}[3]$ & Landing gear spacing & $\bm{B}[3]$\\
		$\bm{g}[1]$ & $C_{L_{max}}$ retracted & $\bm{B}[1]$ & $\bm{g}[4]$ & Wingtip twist deformation & $\bm{B}[4]$  \\
		$\bm{g}[2]$ & Cruise deck angle & $\bm{B}[2]$ &  &  &  \\
		\bottomrule
	\end{tabular}
	\label{tab:Constraints_PMDO}
\end{table}
{"config": "arxiv", "file": "2006.08434/Bombardier_MDO.tex"}