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mdo_intro.tex
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\documentclass[11pt]{article}
\usepackage[breakable]{tcolorbox}
\usepackage{parskip} % Stop auto-indenting (to mimic markdown behaviour)
% Basic figure setup, for now with no caption control since it's done
% automatically by Pandoc (which extracts  syntax from Markdown).
\usepackage{graphicx}
% Maintain compatibility with old templates. Remove in nbconvert 6.0
\let\Oldincludegraphics\includegraphics
% Ensure that by default, figures have no caption (until we provide a
% proper Figure object with a Caption API and a way to capture that
% in the conversion process - todo).
\usepackage{caption}
\DeclareCaptionFormat{nocaption}{}
\captionsetup{format=nocaption,aboveskip=0pt,belowskip=0pt}
\usepackage{float}
\floatplacement{figure}{H} % forces figures to be placed at the correct location
\usepackage{xcolor} % Allow colors to be defined
\usepackage{enumerate} % Needed for markdown enumerations to work
\usepackage{geometry} % Used to adjust the document margins
\usepackage{amsmath} % Equations
\usepackage{amssymb} % Equations
\usepackage{textcomp} % defines textquotesingle
% Hack from http://tex.stackexchange.com/a/47451/13684:
\AtBeginDocument{%
\def\PYZsq{\textquotesingle}% Upright quotes in Pygmentized code
}
\usepackage{upquote} % Upright quotes for verbatim code
\usepackage{eurosym} % defines \euro
\usepackage{iftex}
\ifPDFTeX
\usepackage[T1]{fontenc}
\IfFileExists{alphabeta.sty}{
\usepackage{alphabeta}
}{
\usepackage[mathletters]{ucs}
\usepackage[utf8x]{inputenc}
}
\else
\usepackage{fontspec}
\usepackage{unicode-math}
\fi
\usepackage{fancyvrb} % verbatim replacement that allows latex
\usepackage{grffile} % extends the file name processing of package graphics
% to support a larger range
\makeatletter % fix for old versions of grffile with XeLaTeX
\@ifpackagelater{grffile}{2019/11/01}
{
% Do nothing on new versions
}
{
\def\Gread@@xetex#1{%
\IfFileExists{"\Gin@base".bb}%
{\Gread@eps{\Gin@base.bb}}%
{\Gread@@xetex@aux#1}%
}
}
\makeatother
\usepackage[Export]{adjustbox} % Used to constrain images to a maximum size
\adjustboxset{max size={0.9\linewidth}{0.9\paperheight}}
% The hyperref package gives us a pdf with properly built
% internal navigation ('pdf bookmarks' for the table of contents,
% internal cross-reference links, web links for URLs, etc.)
\usepackage{hyperref}
% The default LaTeX title has an obnoxious amount of whitespace. By default,
% titling removes some of it. It also provides customization options.
\usepackage{titling}
\usepackage{longtable} % longtable support required by pandoc >1.10
\usepackage{booktabs} % table support for pandoc > 1.12.2
\usepackage{array} % table support for pandoc >= 2.11.3
\usepackage{calc} % table minipage width calculation for pandoc >= 2.11.1
\usepackage[inline]{enumitem} % IRkernel/repr support (it uses the enumerate* environment)
\usepackage[normalem]{ulem} % ulem is needed to support strikethroughs (\sout)
% normalem makes italics be italics, not underlines
\usepackage{mathrsfs}
% Colors for the hyperref package
\definecolor{urlcolor}{rgb}{0,.145,.698}
\definecolor{linkcolor}{rgb}{.71,0.21,0.01}
\definecolor{citecolor}{rgb}{.12,.54,.11}
% ANSI colors
\definecolor{ansi-black}{HTML}{3E424D}
\definecolor{ansi-black-intense}{HTML}{282C36}
\definecolor{ansi-red}{HTML}{E75C58}
\definecolor{ansi-red-intense}{HTML}{B22B31}
\definecolor{ansi-green}{HTML}{00A250}
\definecolor{ansi-green-intense}{HTML}{007427}
\definecolor{ansi-yellow}{HTML}{DDB62B}
\definecolor{ansi-yellow-intense}{HTML}{B27D12}
\definecolor{ansi-blue}{HTML}{208FFB}
\definecolor{ansi-blue-intense}{HTML}{0065CA}
\definecolor{ansi-magenta}{HTML}{D160C4}
\definecolor{ansi-magenta-intense}{HTML}{A03196}
\definecolor{ansi-cyan}{HTML}{60C6C8}
\definecolor{ansi-cyan-intense}{HTML}{258F8F}
\definecolor{ansi-white}{HTML}{C5C1B4}
\definecolor{ansi-white-intense}{HTML}{A1A6B2}
\definecolor{ansi-default-inverse-fg}{HTML}{FFFFFF}
\definecolor{ansi-default-inverse-bg}{HTML}{000000}
% common color for the border for error outputs.
\definecolor{outerrorbackground}{HTML}{FFDFDF}
% commands and environments needed by pandoc snippets
% extracted from the output of `pandoc -s`
\providecommand{\tightlist}{%
\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
\DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}}
% Add ',fontsize=\small' for more characters per line
\newenvironment{Shaded}{}{}
\newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
\newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}}
\newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}}
\newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}}
\newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}}
\newcommand{\RegionMarkerTok}[1]{{#1}}
\newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\NormalTok}[1]{{#1}}
% Additional commands for more recent versions of Pandoc
\newcommand{\ConstantTok}[1]{\textcolor[rgb]{0.53,0.00,0.00}{{#1}}}
\newcommand{\SpecialCharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\VerbatimStringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\SpecialStringTok}[1]{\textcolor[rgb]{0.73,0.40,0.53}{{#1}}}
\newcommand{\ImportTok}[1]{{#1}}
\newcommand{\DocumentationTok}[1]{\textcolor[rgb]{0.73,0.13,0.13}{\textit{{#1}}}}
\newcommand{\AnnotationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\CommentVarTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\VariableTok}[1]{\textcolor[rgb]{0.10,0.09,0.49}{{#1}}}
\newcommand{\ControlFlowTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
\newcommand{\OperatorTok}[1]{\textcolor[rgb]{0.40,0.40,0.40}{{#1}}}
\newcommand{\BuiltInTok}[1]{{#1}}
\newcommand{\ExtensionTok}[1]{{#1}}
\newcommand{\PreprocessorTok}[1]{\textcolor[rgb]{0.74,0.48,0.00}{{#1}}}
\newcommand{\AttributeTok}[1]{\textcolor[rgb]{0.49,0.56,0.16}{{#1}}}
\newcommand{\InformationTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
\newcommand{\WarningTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textbf{\textit{{#1}}}}}
% Define a nice break command that doesn't care if a line doesn't already
% exist.
\def\br{\hspace*{\fill} \\* }
% Math Jax compatibility definitions
\def\gt{>}
\def\lt{<}
\let\Oldtex\TeX
\let\Oldlatex\LaTeX
\renewcommand{\TeX}{\textrm{\Oldtex}}
\renewcommand{\LaTeX}{\textrm{\Oldlatex}}
% Document parameters
% Document title
\title{mdo\_intro}
% Pygments definitions
\makeatletter
\def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax%
\let\PY@ul=\relax \let\PY@tc=\relax%
\let\PY@bc=\relax \let\PY@ff=\relax}
\def\PY@tok#1{\csname PY@tok@#1\endcsname}
\def\PY@toks#1+{\ifx\relax#1\empty\else%
\PY@tok{#1}\expandafter\PY@toks\fi}
\def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{%
\PY@it{\PY@bf{\PY@ff{#1}}}}}}}
\def\PY#1#2{\PY@reset\PY@toks#1+\relax+\PY@do{#2}}
\@namedef{PY@tok@w}{\def\PY@tc##1{\textcolor[rgb]{0.73,0.73,0.73}{##1}}}
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\begin{document}
\maketitle
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{1}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k+kn}{import} \PY{n+nn}{numpy} \PY{k}{as} \PY{n+nn}{np}
\end{Verbatim}
\end{tcolorbox}
\hypertarget{intro-to-multidisciplinary-analysis-mda}{%
\section{Intro to multidisciplinary analysis
(MDA)}\label{intro-to-multidisciplinary-analysis-mda}}
The motivation for multidisciplinary design optimization (MDO) comes
from the large number of multidisciplinary analyses (MDAs) that are
common in engineering. The most famous MDA in engineering is the
aero-structural analysis problem. We will illustrate using the example
of a wing subject to aero-structural loads. The aerodynamic forces
deflect the wing which changes its shape, which in turn changes the
aerodynamic behaviour of the wing.
We will look at a simplified example of a wing modeled as a cantilever
beam, made of Euler-Bernoulli beam sections, subject to aerodynamic
forces.
\hypertarget{illustrative-example-structure-fluids-interaction-problem}{%
\subsection{Illustrative example: Structure-fluids interaction
problem}\label{illustrative-example-structure-fluids-interaction-problem}}
\begin{longtable}[]{@{}c@{}}
\toprule
\endhead
\begin{minipage}[t]{0.97\columnwidth}\centering
\begin{figure}
\centering
\includegraphics[height=5cm]{images/wing_structure_fluid.pdf}
\end{figure}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.97\columnwidth}\centering
Fig.0 Aerostructural wing model showing the aerodynamic state variables
(circulations \(\Gamma\)) on the left and structural state variables
(displacements \(\mathbf{d}_z\) and rotations \(\mathbf{d}_\theta\)) on
the right.\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.97\columnwidth}\centering
\emph{From Martins and Ning}\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}
\hypertarget{structural-analysis}{%
\subsubsection{Structural Analysis}\label{structural-analysis}}
Structural analysis can be performed using finite element software to
solve for the structural displacements \(\mathbf{d}\) given some
boundary conditions such as forces \(\mathbf{q}\). These forces may come
from an aerodynamic analysis and are related to the computational fluid
dynamics state variables \(\hat{\Gamma}\). In this example,
\(\hat{\Gamma}\) is the circulation vortex strength on a wing. The hat
denotes that these values are estimates of the state variables when the
analysis has converged and are called \emph{coupling variables}.
\begin{equation}
K\mathbf{d} - \mathbf{q}(\hat{\Gamma}) = \mathbf{0}
\tag{1}
\end{equation}
Solving the above equation is done by specialized finite element
software and returns a solution \({\mathbf{d}}\) that estimates the
displacements. In other words, the FEA program can be thought of as a
function \(\mathcal{U}_1(\hat{\Gamma})\) that returns the displacements
\({\mathbf{d}}\) given some aerodynamic forces
\(\mathbf{q}(\hat{\Gamma})\).
The residual for this discipline becomes
\(r_1(\hat{\mathbf{d}};\hat{\mathbf{q}}) = \hat{\mathbf{d}} - \mathcal{U}_1(\hat{\Gamma}) = \hat{\mathbf{d}} - {\mathbf{d}} = \mathbf{0}\)
and solving it returns an estimate of displacements that will cause the
analysis to converge \(\hat{\mathbf{d}}\). These estimates will be
passed back to the aerodynamic analysis.
\hypertarget{aerodynamic-analysis}{%
\subsubsection{Aerodynamic Analysis}\label{aerodynamic-analysis}}
The aerodynamic analysis involves solving the following system of
equations using computational fluid dynamics (CFD).
\begin{equation}
A(\hat{\mathbf{d}})\Gamma - \mathbf{v}(\hat{\mathbf{d}}) = \mathbf{0}
\tag{2}
\end{equation}
where \(\Gamma\) is the state vector we want to solve for representing
the circulation vortex strength. \(\mathbf{v}\) is a vector of boundary
conditions (similar to the vector of applied loads \(\mathbf{q}\) in the
structural analysis). \(A\) is the matrix of aerodynamic influence
coefficients (similar to the global stiffness matrix \(K\) from the
structural analysis). However, unlike \(K\), \(A\) depends on the shape
of the wing which in turn depends on the displacements estimated by the
structural analysis \(\hat{\mathbf{d}}\).
The residual for this discipline becomes
\(r_2(\hat{\Gamma};\hat{\mathbf{d}}) = \hat{\Gamma} - \mathcal{U}_2(\hat{\mathbf{d}}) = \hat{\Gamma} - {\Gamma} = \mathbf{0}\),
where \(\mathcal{U}_2(\hat{\mathbf{d}})\) involves solving Equation (2)
for the circulation vortex strength \(\Gamma\).
The estimated coupling variable \(\hat{\Gamma}\) is passed back to the
aerodynamic analysis. The previous analysis can be described by the
diagram below.
\begin{longtable}[]{@{}c@{}}
\toprule
\endhead
\begin{minipage}[t]{0.97\columnwidth}\centering
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{images/MDA_diagram_aero.png}
\end{figure}\strut
\end{minipage}\tabularnewline
Fig.1 Aerostructural multidisciplinary analysis (MDA)
diagram.\tabularnewline
\bottomrule
\end{longtable}
The XDSM for the aforementioned aero-structural analysis problem is
shown below:
The XDSM for aerostructural example looks like this:
\begin{longtable}[]{@{}c@{}}
\toprule
\endhead
\begin{minipage}[t]{0.97\columnwidth}\centering
\begin{figure}
\centering
\includegraphics[height=6cm]{xdsm/aerostructural.pdf}
\end{figure}\strut
\end{minipage}\tabularnewline
Fig.2 XDSM for Aerostructural multidisciplinary analysis
(MDA).\tabularnewline
\bottomrule
\end{longtable}
\hypertarget{numerical-mda-example}{%
\subsection{Numerical MDA example}\label{numerical-mda-example}}
Consider the following coupled optimization problem
\begin{equation*}
\begin{aligned}
& \underset{\mathbf{x} = \left[u,v,w\right]^\mathrm{T}}{\text{minimize}}
& & f(u,v;a,b) = u+v+a+b\\
& \text{subject to}
& & g(w;b) = w + b -10 \leq 0\\
& & & 0 \leq u,v,w \leq 10\\
& \text{while solving}
& & r_1(a;b) = a - \left(\log(u) + \log(v) + \log(b)\right) = 0\\
& & & r_2(b;a) = b - \left(u^{-1} + w^{-1} + a^{-1}\right) = 0\\
& \text{for a given $u$, $v$, and $w$}
\end{aligned}
\tag{3}
\end{equation*}
Solving for the state variables \(a\) and \(b\) involves solving the set
of equations below. Equation (4) is an example of an MDA (just like the
aerostructural problem earlier):
\begin{equation*}
\begin{aligned}
& r_1(a;b) = a - \left(\log(u) + \log(v) + \log(b)\right) = 0\\
& r_2(b;a) = b - \left(u^{-1} + w^{-1} + a^{-1}\right) = 0\\
\end{aligned}
\tag{4}
\end{equation*}
which cannot be solved explicitly. To prove this, Let us try to find an
explicit expression for \(a\) in terms of \(b\).
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{2}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k+kn}{import} \PY{n+nn}{sympy} \PY{k}{as} \PY{n+nn}{sym}
\PY{k+kn}{from} \PY{n+nn}{sympy} \PY{k+kn}{import} \PY{n}{pprint}\PY{p}{,} \PY{n}{latex}\PY{p}{,} \PY{n}{root}
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\PY{n}{a} \PY{o}{=} \PY{n}{sym}\PY{o}{.}\PY{n}{Symbol}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{a}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{b} \PY{o}{=} \PY{n}{sym}\PY{o}{.}\PY{n}{Symbol}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{b}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{u} \PY{o}{=} \PY{n}{sym}\PY{o}{.}\PY{n}{Symbol}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{u}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{v} \PY{o}{=} \PY{n}{sym}\PY{o}{.}\PY{n}{Symbol}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{v}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
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\PY{n}{r1} \PY{o}{=} \PY{n}{a} \PY{o}{\PYZhy{}} \PY{n}{sym}\PY{o}{.}\PY{n}{log}\PY{p}{(}\PY{n}{u}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{n}{sym}\PY{o}{.}\PY{n}{log}\PY{p}{(}\PY{n}{v}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{n}{sym}\PY{o}{.}\PY{n}{log}\PY{p}{(}\PY{n}{b}\PY{p}{)}
\PY{n}{r2} \PY{o}{=} \PY{n}{b} \PY{o}{\PYZhy{}} \PY{l+m+mi}{1}\PY{o}{/}\PY{n}{u} \PY{o}{\PYZhy{}} \PY{l+m+mi}{1}\PY{o}{/}\PY{n}{w} \PY{o}{\PYZhy{}} \PY{l+m+mi}{1}\PY{o}{/}\PY{n}{a}
\PY{n}{eq1} \PY{o}{=} \PY{n}{a} \PY{o}{\PYZhy{}} \PY{n}{sym}\PY{o}{.}\PY{n}{log}\PY{p}{(}\PY{n}{u}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{n}{sym}\PY{o}{.}\PY{n}{log}\PY{p}{(}\PY{n}{v}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{n}{sym}\PY{o}{.}\PY{n}{log}\PY{p}{(}\PY{l+m+mi}{1}\PY{o}{/}\PY{n}{u} \PY{o}{+} \PY{l+m+mi}{1}\PY{o}{/}\PY{n}{w} \PY{o}{+} \PY{l+m+mi}{1}\PY{o}{/}\PY{n}{a}\PY{p}{)}
\PY{c+c1}{\PYZsh{} sym.solve(eq1,a) \PYZsh{} trying to solve for a explicitly will result in an error!}
\end{Verbatim}
\end{tcolorbox}
Instead, we focus on solving for \(a\) and \(b\) such that the residuals
\(r_1=0\) and \(r_2=0\) iteratively using a Multidisciplinary Analysis
(MDA) for a fixed value of \(u\), \(v\), and \(w\). First let us
slightly change the notation of the governing equations:
\begin{align*}
& r_1(\hat{a};\hat{b}) = \hat{a} - \mathcal{U}_1(\hat{b}) = 0\\
& r_2(\hat{b};\hat{a}) = \hat{b} - \mathcal{U}_2(\hat{a}) = 0\\
\end{align*}
\(\mathcal{U}_1(\hat{b})\) and \(\mathcal{U}_2(\hat{a})\) are considered
the discipline solvers and could represent the FEA and CFD programs in
the aerostructural problem.
\hypertarget{the-nonlinear-block-gaussseidel-and-block-jacobi-mda-algorithms}{%
\subsection{The Nonlinear block Gauss--Seidel and block Jacobi MDA
algorithms}\label{the-nonlinear-block-gaussseidel-and-block-jacobi-mda-algorithms}}
The MDA approach we will use is called the \emph{block Gauss--Seidel
algorithm} and has an XDSM similar to that of the aerostructural
problem. 1) Starting with an initial guess for \(\hat{a}\) and
\(\hat{b}\), \(\hat{a}^{(0)}\) and \(\hat{b}^{(0)}\), respectively, 2)
we solve discipline 1 for \(\hat{a}\). 3) We feed \(\hat{a}\) into
discipline 2, 4) and then solve for \(\hat{b}\) and then go back to step
2.
We repeat the above process until
\(\hat{a}_\mathrm{new} - \hat{a}_\mathrm{old} \leq \varepsilon\), where
\(\varepsilon\) is some tolerance. We can also solve for \(\hat{a}\) and
\(\hat{b}\) simultaneously rather than serially resulting in the block
Jacobi MDA.
The XDSM for the above MDAs is shown below:
\begin{longtable}[]{@{}c@{}}
\toprule
\endhead
\begin{minipage}[t]{0.97\columnwidth}\centering
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{xdsm/gauss.pdf}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{xdsm/gauss.pdf}
\end{figure}
\end{minipage}\tabularnewline
Fig.3 block Gauss--Seidel and Block Jacobi MDAs.\tabularnewline
\bottomrule
\end{longtable}
This algorithm is implemented below in Python.
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\prompt{In}{incolor}{3}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{c+c1}{\PYZsh{} these are your coupled disciplines}
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\PY{n}{U2} \PY{o}{=} \PY{k}{lambda} \PY{n}{uj}\PY{p}{:} \PY{l+m+mi}{1}\PY{o}{/}\PY{n}{u} \PY{o}{+} \PY{l+m+mi}{1}\PY{o}{/}\PY{n}{w} \PY{o}{+} \PY{l+m+mi}{1}\PY{o}{/}\PY{p}{(}\PY{n}{uj}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{+}\PY{l+m+mf}{1e\PYZhy{}6}\PY{p}{)}
\PY{n}{u} \PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{v} \PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{w} \PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{a0}\PY{o}{=}\PY{l+m+mi}{1}
\PY{n}{b0}\PY{o}{=}\PY{l+m+mi}{1}
\PY{n}{uk} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{array}\PY{p}{(}\PY{p}{[}\PY{n}{a0}\PY{p}{,}\PY{n}{b0}\PY{p}{]}\PY{p}{)}
\PY{n}{k} \PY{o}{=} \PY{l+m+mi}{0}
\PY{n}{e} \PY{o}{=} \PY{l+m+mf}{1e\PYZhy{}6}
\PY{n}{disciplines} \PY{o}{=} \PY{p}{[}\PY{n}{U1}\PY{p}{,}\PY{n}{U2}\PY{p}{]}
\PY{k}{while} \PY{k+kc}{True}\PY{p}{:}
\PY{n}{uk\PYZus{}old} \PY{o}{=} \PY{n}{uk}\PY{o}{.}\PY{n}{copy}\PY{p}{(}\PY{p}{)}
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\PY{n}{mask}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{=} \PY{k+kc}{False} \PY{c+c1}{\PYZsh{} select all state variables except the current one (u\PYZus{}i)}
\PY{n}{uk}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{=} \PY{n}{discipline}\PY{p}{(}\PY{n}{uk}\PY{p}{[}\PY{n}{mask}\PY{p}{]}\PY{p}{)}
\PY{k}{if} \PY{n}{np}\PY{o}{.}\PY{n}{linalg}\PY{o}{.}\PY{n}{norm}\PY{p}{(}\PY{n}{uk} \PY{o}{\PYZhy{}} \PY{n}{uk\PYZus{}old}\PY{p}{)} \PY{o}{\PYZlt{}}\PY{o}{=} \PY{n}{e} \PY{o+ow}{or} \PY{n}{k} \PY{o}{\PYZgt{}} \PY{l+m+mi}{10}\PY{p}{:}
\PY{k}{break}
\PY{n}{k}\PY{o}{+}\PY{o}{=}\PY{l+m+mi}{1}
\PY{n+nb}{print}\PY{p}{(}\PY{n}{uk}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{Verbatim}[commandchars=\\\{\}]
[ 0 1000002]
[13 2]
[ 0 1000002]
[13 2]
[ 0 1000002]
[13 2]
[ 0 1000002]
[13 2]
[ 0 1000002]
[13 2]
[ 0 1000002]
\end{Verbatim}
We can see that the previous algorithm did not converge. It started to
oscillate. We can use an alternative algorithm that is based on the
Newton-Raphson scheme to handle the coupling between the disciplines.
\hypertarget{the-reduced-space-hierarchical-newton-solver}{%
\subsection{The reduced-space hierarchical Newton
solver}\label{the-reduced-space-hierarchical-newton-solver}}
This algorithm is unequivocally also called \emph{Newton's Method}. As
with the Newton-Raphson algorithm, it requires derivative information to
find the coupling variables \(\hat{a}\) and \(\hat{b}\) at which
convergence occurs. The algorithm goes something like this:
\begin{enumerate}
\def\labelenumi{\arabic{enumi})}
\tightlist
\item
Starting with an initial guess for \(\hat{a}\) and \(\hat{b}\),
\(\hat{a}^{(0)}\) and \(\hat{b}^{(0)}\), respectively,
\item
we solve discipline 1 for \(\hat{a}\). We also find the gradient of
the discipline analysis \(\mathcal{U}(\hat{b})\) with respect to
\(\hat{b}\). In other words,
\(\dfrac{\partial\mathcal{U}_1}{\partial\hat{b}}\) (all coupling
variables except \(\hat{a}\))
\item
In parallel, we solve discipline 2 for \(\hat{b}\). We also find the
gradient with respect to \(\hat{a}\),
\(\dfrac{\partial\mathcal{U}_2}{\partial\hat{b}}\) (all coupling
variables except \(\hat{b}\))
\item
We solve the following system of equations for \(\Delta\hat{a}\) and
\(\Delta\hat{b}\): \[
\begin{bmatrix}
1 & \left.\dfrac{\partial\mathcal{U}_1}{\partial\hat{b}}\right|_{\hat{b}} \\
\left.\dfrac{\partial\mathcal{U}_2}{\partial\hat{a}}\right|_{\hat{a}} & 1 \\
\end{bmatrix}
\begin{bmatrix}
\Delta\hat{a}\\
\Delta\hat{b}\\
\end{bmatrix} = -
\begin{bmatrix}
\hat{a} - \mathcal{U}_1(\hat{b})\\
\hat{b} - \mathcal{U}_2(\hat{a})\\
\end{bmatrix}
\]
\item
we update \(\hat{a}\) and \(\hat{b}\) according to
\(\hat{a} \gets \hat{a} + \Delta\hat{a}\) and
\(\hat{a} \gets \hat{a} + \Delta\hat{a}\), respectively and go back to
step 2.
\end{enumerate}
The XDSM for the above MDA is shown below:
\begin{longtable}[]{@{}c@{}}
\toprule
\endhead
\begin{minipage}[t]{0.97\columnwidth}\centering
\begin{figure}
\centering
\includegraphics[width=0.9\textwidth]{xdsm/newton.pdf}
\end{figure}
\end{minipage}\tabularnewline
Fig.4 Reduced-space hierarchical Newton solver.\tabularnewline
\bottomrule
\end{longtable}
This algorithm is implemented below in Python.
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{4}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{c+c1}{\PYZsh{} these are your coupled disciplines}
\PY{n}{U1} \PY{o}{=} \PY{k}{lambda} \PY{n}{uj}\PY{p}{:} \PY{n}{np}\PY{o}{.}\PY{n}{log}\PY{p}{(}\PY{n}{u}\PY{p}{)} \PY{o}{+} \PY{n}{np}\PY{o}{.}\PY{n}{log}\PY{p}{(}\PY{n}{v}\PY{p}{)} \PY{o}{+} \PY{n}{np}\PY{o}{.}\PY{n}{log}\PY{p}{(}\PY{n}{uj}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{)}
\PY{n}{U2} \PY{o}{=} \PY{k}{lambda} \PY{n}{uj}\PY{p}{:} \PY{l+m+mi}{1}\PY{o}{/}\PY{n}{u} \PY{o}{+} \PY{l+m+mi}{1}\PY{o}{/}\PY{n}{w} \PY{o}{+} \PY{l+m+mi}{1}\PY{o}{/}\PY{p}{(}\PY{n}{uj}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{+}\PY{l+m+mf}{1e\PYZhy{}6}\PY{p}{)}
\PY{c+c1}{\PYZsh{} These are the Jacobians of your disciplines}
\PY{n}{dU1} \PY{o}{=} \PY{k}{lambda} \PY{n}{uj}\PY{p}{:} \PY{n}{np}\PY{o}{.}\PY{n}{array}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{1}\PY{o}{/}\PY{p}{(}\PY{n}{uj}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{+}\PY{l+m+mf}{1e\PYZhy{}6}\PY{p}{)}\PY{p}{]}\PY{p}{)}
\PY{n}{dU2} \PY{o}{=} \PY{k}{lambda} \PY{n}{uj}\PY{p}{:} \PY{n}{np}\PY{o}{.}\PY{n}{array}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{o}{/}\PY{p}{(}\PY{n}{uj}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{+}\PY{l+m+mf}{1e\PYZhy{}6}\PY{p}{)}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
\PY{n}{u} \PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{v} \PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{w} \PY{o}{=} \PY{l+m+mi}{1}
\PY{n}{a0}\PY{o}{=}\PY{l+m+mi}{100}
\PY{n}{b0}\PY{o}{=}\PY{l+m+mi}{100}
\PY{n}{uk} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{array}\PY{p}{(}\PY{p}{[}\PY{n}{a0}\PY{p}{,}\PY{n}{b0}\PY{p}{]}\PY{p}{)}
\PY{n}{k} \PY{o}{=} \PY{l+m+mi}{0}
\PY{n}{e} \PY{o}{=} \PY{l+m+mf}{1e\PYZhy{}6}
\PY{n}{disciplines} \PY{o}{=} \PY{p}{[}\PY{n}{U1}\PY{p}{,}\PY{n}{U2}\PY{p}{]}
\PY{n}{Jacobians} \PY{o}{=} \PY{p}{[}\PY{n}{dU1}\PY{p}{,}\PY{n}{dU2}\PY{p}{]}
\PY{k}{while} \PY{k+kc}{True}\PY{p}{:}
\PY{n}{U} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros\PYZus{}like}\PY{p}{(}\PY{n}{uk}\PY{p}{)}
\PY{n}{drdU} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{ones}\PY{p}{(}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{uk}\PY{p}{)}\PY{p}{,}\PY{n+nb}{len}\PY{p}{(}\PY{n}{uk}\PY{p}{)}\PY{p}{)}\PY{p}{)}
\PY{k}{for} \PY{n}{i}\PY{p}{,}\PY{p}{(}\PY{n}{discipline}\PY{p}{,}\PY{n}{Jacobian}\PY{p}{)} \PY{o+ow}{in} \PY{n+nb}{enumerate}\PY{p}{(}\PY{n+nb}{zip}\PY{p}{(}\PY{n}{disciplines}\PY{p}{,}\PY{n}{Jacobians}\PY{p}{)}\PY{p}{)}\PY{p}{:}
\PY{n}{mask} \PY{o}{=} \PY{p}{[}\PY{k+kc}{True}\PY{p}{,}\PY{p}{]}\PY{o}{*}\PY{n+nb}{len}\PY{p}{(}\PY{n}{disciplines}\PY{p}{)}
\PY{n}{mask}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{=} \PY{k+kc}{False} \PY{c+c1}{\PYZsh{} select all state variables but current one}
\PY{n}{U}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{=} \PY{n}{discipline}\PY{p}{(}\PY{n}{uk}\PY{p}{[}\PY{n}{mask}\PY{p}{]}\PY{p}{)} \PY{c+c1}{\PYZsh{} estimate coupling variables}
\PY{n}{drdU}\PY{p}{[}\PY{n}{i}\PY{p}{,}\PY{n}{mask}\PY{p}{]} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{n}{Jacobian}\PY{p}{(}\PY{n}{uk}\PY{p}{[}\PY{n}{mask}\PY{p}{]}\PY{p}{)} \PY{c+c1}{\PYZsh{} estimate gradient of coupling variables}
\PY{n}{r} \PY{o}{=} \PY{n}{uk} \PY{o}{\PYZhy{}} \PY{n}{U} \PY{c+c1}{\PYZsh{} residuals}
\PY{k}{if} \PY{n}{np}\PY{o}{.}\PY{n}{linalg}\PY{o}{.}\PY{n}{norm}\PY{p}{(}\PY{n}{r}\PY{p}{)} \PY{o}{\PYZlt{}}\PY{o}{=} \PY{n}{e} \PY{o+ow}{or} \PY{n}{k} \PY{o}{\PYZgt{}} \PY{l+m+mi}{100}\PY{p}{:}
\PY{k}{break}
\PY{c+c1}{\PYZsh{} Newton\PYZhy{}Raphson step}
\PY{n}{du} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linalg}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{n}{drdU}\PY{p}{)} \PY{o}{@} \PY{o}{\PYZhy{}}\PY{n}{r}
\PY{n}{uk} \PY{o}{=} \PY{n}{uk} \PY{o}{+} \PY{n}{du}
\PY{n}{k}\PY{o}{+}\PY{o}{=}\PY{l+m+mi}{1}
\PY{n+nb}{print}\PY{p}{(}\PY{n}{uk}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{Verbatim}[commandchars=\\\{\}]
[3.02009699 2.00969799]
[0.96977098 2.55590709]
[1.07893819 2.91509142]
[1.07507004 2.93015913]
[1.07506292 2.93017725]
\end{Verbatim}
We found a solution for \(a\) and \(b\). You can plug those values into
your calculator and check that they satisfy the Equations in (4)
yourself!
However, we had to provide the Jacobian of the analysis functions
\(\dfrac{\partial\mathcal{U}}{\partial\hat{\mathbf{u}}}\) with respect
to the coupling variables
\(\hat{\mathbf{u}} = \begin{bmatrix} \hat{a} & \hat{b} \end{bmatrix}^\mathrm{T}\)
for the solver to converge. When \(\mathcal{U}_1\) and \(\mathcal{U}_2\)
are blackboxes (as in the aerostructural problem at the beginning of
this notebook), we seldom have access to residuals, let alone gradient
information! In other words, we sort of cheated.
\hypertarget{multidisciplinary-design-optimization}{%
\section{Multidisciplinary design
optimization}\label{multidisciplinary-design-optimization}}
Since the MDA seems to converge when using the reduced Newton solver, we
will use it to solve the optimization problem (1) at the beginning of
this notebook. Solving the optimization problem using an MDA is known as
Multidisciplinary feasible (MDF) approach.
\hypertarget{multidisciplinary-feasible-mdf-optimization}{%
\subsection{Multidisciplinary feasible (MDF)
optimization}\label{multidisciplinary-feasible-mdf-optimization}}
\begin{equation*}
\begin{aligned}
& \underset{u,v,w}{\text{minimize}}
& & f(u,v,w) = u+v+\hat{a}(u,v,w)+\hat{b}(u,v,w)\\
& \text{subject to}
& & g(u,v,w) = w + \hat{b}(u,v,w) -10 \leq 0\\
& & & 0 \leq u,v,w \leq 10\\
\end{aligned}
\tag{5}
\end{equation*}
The above optimization problem is called a Multi-disciplinary feasible
(MDF) MDO problem since it relies on a monolithic MDA. Note that we have
switched out \(a\) and \(b\) by their MDA counterparts \(\hat{a}\) and
\(\hat{b}\) which need to be solved for during every optimization loop!
We can use a blackbox optimization algorithm to solve the problem, where
the blackbox in this case is the MDA we developed in the previous
section. We define the blackbox below.
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{5}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{c+c1}{\PYZsh{} these are your coupled disciplines}
\PY{n}{U1} \PY{o}{=} \PY{k}{lambda} \PY{n}{x}\PY{p}{,}\PY{n}{uj}\PY{p}{:} \PY{n}{np}\PY{o}{.}\PY{n}{log}\PY{p}{(}\PY{n}{x}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{+}\PY{l+m+mf}{1e\PYZhy{}6}\PY{p}{)} \PY{o}{+} \PY{n}{np}\PY{o}{.}\PY{n}{log}\PY{p}{(}\PY{n}{x}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{o}{+}\PY{l+m+mf}{1e\PYZhy{}6}\PY{p}{)} \PY{o}{+} \PY{n}{np}\PY{o}{.}\PY{n}{log}\PY{p}{(}\PY{n}{uj}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{+}\PY{l+m+mf}{1e\PYZhy{}6}\PY{p}{)}
\PY{n}{U2} \PY{o}{=} \PY{k}{lambda} \PY{n}{x}\PY{p}{,}\PY{n}{uj}\PY{p}{:} \PY{l+m+mi}{1}\PY{o}{/}\PY{p}{(}\PY{n}{x}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{+}\PY{l+m+mf}{1e\PYZhy{}6}\PY{p}{)} \PY{o}{+} \PY{l+m+mi}{1}\PY{o}{/}\PY{p}{(}\PY{n}{x}\PY{p}{[}\PY{l+m+mi}{2}\PY{p}{]}\PY{o}{+}\PY{l+m+mf}{1e\PYZhy{}6}\PY{p}{)} \PY{o}{+} \PY{l+m+mi}{1}\PY{o}{/}\PY{p}{(}\PY{n}{uj}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{+}\PY{l+m+mf}{1e\PYZhy{}6}\PY{p}{)}
\PY{c+c1}{\PYZsh{} These are the Jacobians of your disciplines}
\PY{n}{dU1} \PY{o}{=} \PY{k}{lambda} \PY{n}{x}\PY{p}{,}\PY{n}{uj}\PY{p}{:} \PY{n}{np}\PY{o}{.}\PY{n}{array}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{1}\PY{o}{/}\PY{p}{(}\PY{n}{uj}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{+}\PY{l+m+mf}{1e\PYZhy{}6}\PY{p}{)}\PY{p}{]}\PY{p}{)}
\PY{n}{dU2} \PY{o}{=} \PY{k}{lambda} \PY{n}{x}\PY{p}{,}\PY{n}{uj}\PY{p}{:} \PY{n}{np}\PY{o}{.}\PY{n}{array}\PY{p}{(}\PY{p}{[}\PY{o}{\PYZhy{}}\PY{l+m+mi}{1}\PY{o}{/}\PY{p}{(}\PY{n}{uj}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{o}{+}\PY{l+m+mf}{1e\PYZhy{}6}\PY{p}{)}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
\PY{k}{def} \PY{n+nf}{my\PYZus{}MDA}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{u0}\PY{p}{,}\PY{n}{disciplines}\PY{p}{,}\PY{n}{Jacobians}\PY{p}{,}\PY{n}{epsilon}\PY{o}{=}\PY{l+m+mf}{1e\PYZhy{}6}\PY{p}{,}\PY{n}{k\PYZus{}max}\PY{o}{=}\PY{l+m+mi}{100}\PY{p}{)}\PY{p}{:}
\PY{n}{uk} \PY{o}{=} \PY{n}{u0}
\PY{n}{k} \PY{o}{=} \PY{l+m+mi}{0}
\PY{n}{disciplines} \PY{o}{=} \PY{p}{[}\PY{n}{U1}\PY{p}{,}\PY{n}{U2}\PY{p}{]}
\PY{n}{Jacobians} \PY{o}{=} \PY{p}{[}\PY{n}{dU1}\PY{p}{,}\PY{n}{dU2}\PY{p}{]}
\PY{k}{while} \PY{k+kc}{True}\PY{p}{:}
\PY{n}{U} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{zeros\PYZus{}like}\PY{p}{(}\PY{n}{uk}\PY{p}{)}
\PY{n}{drdU} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{ones}\PY{p}{(}\PY{p}{(}\PY{n+nb}{len}\PY{p}{(}\PY{n}{uk}\PY{p}{)}\PY{p}{,}\PY{n+nb}{len}\PY{p}{(}\PY{n}{uk}\PY{p}{)}\PY{p}{)}\PY{p}{)}
\PY{k}{for} \PY{n}{i}\PY{p}{,}\PY{p}{(}\PY{n}{discipline}\PY{p}{,}\PY{n}{Jacobian}\PY{p}{)} \PY{o+ow}{in} \PY{n+nb}{enumerate}\PY{p}{(}\PY{n+nb}{zip}\PY{p}{(}\PY{n}{disciplines}\PY{p}{,}\PY{n}{Jacobians}\PY{p}{)}\PY{p}{)}\PY{p}{:}
\PY{n}{mask} \PY{o}{=} \PY{p}{[}\PY{k+kc}{True}\PY{p}{,}\PY{p}{]}\PY{o}{*}\PY{n+nb}{len}\PY{p}{(}\PY{n}{disciplines}\PY{p}{)}
\PY{n}{mask}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{=} \PY{k+kc}{False} \PY{c+c1}{\PYZsh{} select all state variables but current one}
\PY{n}{U}\PY{p}{[}\PY{n}{i}\PY{p}{]} \PY{o}{=} \PY{n}{discipline}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{uk}\PY{p}{[}\PY{n}{mask}\PY{p}{]}\PY{p}{)} \PY{c+c1}{\PYZsh{} estimate coupling variables}
\PY{n}{drdU}\PY{p}{[}\PY{n}{i}\PY{p}{,}\PY{n}{mask}\PY{p}{]} \PY{o}{=} \PY{o}{\PYZhy{}}\PY{n}{Jacobian}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{uk}\PY{p}{[}\PY{n}{mask}\PY{p}{]}\PY{p}{)} \PY{c+c1}{\PYZsh{} estimate gradient of coupling variables}
\PY{n}{r} \PY{o}{=} \PY{n}{uk} \PY{o}{\PYZhy{}} \PY{n}{U} \PY{c+c1}{\PYZsh{} residuals}
\PY{k}{if} \PY{n}{np}\PY{o}{.}\PY{n}{linalg}\PY{o}{.}\PY{n}{norm}\PY{p}{(}\PY{n}{r}\PY{p}{)} \PY{o}{\PYZlt{}}\PY{o}{=} \PY{n}{epsilon} \PY{o+ow}{or} \PY{n}{k} \PY{o}{\PYZgt{}} \PY{n}{k\PYZus{}max}\PY{p}{:}
\PY{k}{break}
\PY{c+c1}{\PYZsh{} Newton\PYZhy{}Raphson step}
\PY{n}{du} \PY{o}{=} \PY{n}{np}\PY{o}{.}\PY{n}{linalg}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{n}{drdU}\PY{p}{)} \PY{o}{@} \PY{o}{\PYZhy{}}\PY{n}{r}
\PY{n}{uk} \PY{o}{=} \PY{n}{uk} \PY{o}{+} \PY{n}{du}
\PY{n}{k}\PY{o}{+}\PY{o}{=}\PY{l+m+mi}{1}
\PY{k}{return} \PY{n}{uk}
\PY{n}{u} \PY{o}{=} \PY{l+m+mf}{0.0}
\PY{n}{v} \PY{o}{=} \PY{l+m+mf}{0.0}
\PY{n}{w} \PY{o}{=} \PY{l+m+mf}{0.0}
\PY{n}{x} \PY{o}{=} \PY{p}{[}\PY{n}{u}\PY{p}{,}\PY{n}{v}\PY{p}{,}\PY{n}{w}\PY{p}{]}
\PY{n}{state0}\PY{o}{=}\PY{n}{np}\PY{o}{.}\PY{n}{array}\PY{p}{(}\PY{p}{[}\PY{l+m+mi}{100}\PY{p}{,}\PY{l+m+mi}{100}\PY{p}{]}\PY{p}{)}
\PY{n}{disciplines} \PY{o}{=} \PY{p}{[}\PY{n}{U1}\PY{p}{,}\PY{n}{U2}\PY{p}{]}
\PY{n}{Jacobians} \PY{o}{=} \PY{p}{[}\PY{n}{dU1}\PY{p}{,}\PY{n}{dU2}\PY{p}{]}
\PY{n}{u\PYZus{}hat} \PY{o}{=} \PY{n}{my\PYZus{}MDA}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{state0}\PY{p}{,}\PY{n}{disciplines}\PY{p}{,}\PY{n}{Jacobians}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{n}{u\PYZus{}hat}\PY{p}{)}
\end{Verbatim}
\end{tcolorbox}
\begin{Verbatim}[commandchars=\\\{\}]
[-1.31223634e+01 1.99999992e+06]
\end{Verbatim}
A blackbox that returns the objective and constraint value can now be
defined as follows:
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{6}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k}{def} \PY{n+nf}{my\PYZus{}bb}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{args}\PY{p}{)}\PY{p}{:}
\PY{n}{state0} \PY{o}{=} \PY{n}{args}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}
\PY{n}{disciplines} \PY{o}{=} \PY{n}{args}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}
\PY{n}{Jacobians} \PY{o}{=} \PY{n}{args}\PY{p}{[}\PY{l+m+mi}{2}\PY{p}{]}
\PY{n}{u\PYZus{}hat} \PY{o}{=} \PY{n}{my\PYZus{}MDA}\PY{p}{(}\PY{n}{x}\PY{p}{,}\PY{n}{state0}\PY{p}{,}\PY{n}{disciplines}\PY{p}{,}\PY{n}{Jacobians}\PY{p}{)}
\PY{n}{f} \PY{o}{=} \PY{n}{x}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]} \PY{o}{+} \PY{n}{x}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]} \PY{o}{+} \PY{n}{u\PYZus{}hat}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]} \PY{o}{+} \PY{n}{u\PYZus{}hat}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}
\PY{n}{g} \PY{o}{=} \PY{n}{x}\PY{p}{[}\PY{l+m+mi}{2}\PY{p}{]} \PY{o}{+} \PY{n}{u\PYZus{}hat}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]} \PY{o}{\PYZhy{}} \PY{l+m+mf}{10.0}
\PY{k}{return} \PY{p}{[}\PY{n}{f}\PY{p}{,}\PY{p}{[}\PY{n}{g}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{n}{u\PYZus{}hat}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,}\PY{o}{\PYZhy{}}\PY{n}{u\PYZus{}hat}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{]}\PY{p}{]}
\end{Verbatim}
\end{tcolorbox}
The blackbox above is what we can feed to
\href{https://ahmed-bayoumy.github.io/OMADS/}{\texttt{OMADS}}.
\begin{tcolorbox}[breakable, size=fbox, boxrule=1pt, pad at break*=1mm,colback=cellbackground, colframe=cellborder]
\prompt{In}{incolor}{7}{\boxspacing}
\begin{Verbatim}[commandchars=\\\{\}]
\PY{k+kn}{import} \PY{n+nn}{OMADS}
\PY{c+c1}{\PYZsh{} Optimization setup}
\PY{n}{u0} \PY{o}{=} \PY{l+m+mf}{1.0}\PY{p}{;} \PY{n}{v0} \PY{o}{=} \PY{l+m+mf}{1.0}\PY{p}{;} \PY{n}{w0} \PY{o}{=} \PY{l+m+mf}{1.0}
\PY{n}{x0} \PY{o}{=} \PY{p}{[}\PY{n}{u0}\PY{p}{,}\PY{n}{v0}\PY{p}{,}\PY{n}{w0}\PY{p}{]}
\PY{n}{f}\PY{p}{,}\PY{n}{g} \PY{o}{=} \PY{n}{my\PYZus{}bb}\PY{p}{(}\PY{n}{x0}\PY{p}{,}\PY{n}{args}\PY{o}{=}\PY{p}{[}\PY{n}{state0}\PY{p}{,}\PY{n}{disciplines}\PY{p}{,}\PY{n}{Jacobians}\PY{p}{]}\PY{p}{)} \PY{c+c1}{\PYZsh{} its a good idea to test the blackbox function first at the initial guess}
\PY{n}{lb} \PY{o}{=} \PY{p}{[}\PY{l+m+mf}{0.0}\PY{p}{,} \PY{l+m+mf}{0.0}\PY{p}{,} \PY{l+m+mf}{0.0}\PY{p}{]}
\PY{n}{ub} \PY{o}{=} \PY{p}{[}\PY{l+m+mf}{10.}\PY{p}{,} \PY{l+m+mf}{10.}\PY{p}{,} \PY{l+m+mf}{10.}\PY{p}{]}
\PY{n+nb}{eval} \PY{o}{=} \PY{p}{\PYZob{}}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{blackbox}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{n}{my\PYZus{}bb}\PY{p}{,}
\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{constants}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{p}{[}\PY{n}{state0}\PY{p}{,}\PY{n}{disciplines}\PY{p}{,}\PY{n}{Jacobians}\PY{p}{]}\PY{p}{\PYZcb{}} \PY{c+c1}{\PYZsh{} you can pass blackbox specific options using the ``constants`` key}
\PY{n}{param} \PY{o}{=} \PY{p}{\PYZob{}}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{baseline}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{n}{x0}\PY{p}{,}
\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{lb}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{n}{lb}\PY{p}{,}
\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ub}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{n}{ub}\PY{p}{,}
\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{var\PYZus{}names}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{u}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{v}\PY{l+s+s2}{\PYZdq{}}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{w}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{,}
\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{scaling}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{l+m+mf}{10.0}\PY{p}{,}
\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{post\PYZus{}dir}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{./post}\PY{l+s+s2}{\PYZdq{}}\PY{p}{\PYZcb{}} \PY{c+c1}{\PYZsh{} these are OMADS specific options. You can modify them as necessary}
\PY{n}{options} \PY{o}{=} \PY{p}{\PYZob{}}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{seed}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{l+m+mi}{0}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{budget}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{l+m+mi}{100000}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{tol}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{l+m+mf}{1e\PYZhy{}12}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{display}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{k+kc}{False}\PY{p}{\PYZcb{}}
\PY{n}{data} \PY{o}{=} \PY{p}{\PYZob{}}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{evaluator}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{n+nb}{eval}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{param}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:} \PY{n}{param}\PY{p}{,} \PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{options}\PY{l+s+s2}{\PYZdq{}}\PY{p}{:}\PY{n}{options}\PY{p}{\PYZcb{}}
\PY{n}{out} \PY{o}{=} \PY{p}{\PYZob{}}\PY{p}{\PYZcb{}}
\PY{c+c1}{\PYZsh{} out is a dictionary that will hold output data of the final solution. The out dictionary has three keys: \PYZdq{}xmin\PYZdq{}, \PYZdq{}fmin\PYZdq{} and \PYZdq{}hmin\PYZdq{}}
\PY{n}{out} \PY{o}{=} \PY{n}{OMADS}\PY{o}{.}\PY{n}{main}\PY{p}{(}\PY{n}{data}\PY{p}{)}
\PY{n}{fopt}\PY{p}{,}\PY{n}{gopt} \PY{o}{=} \PY{n}{my\PYZus{}bb}\PY{p}{(}\PY{n}{out}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{xmin}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{,}\PY{n}{args}\PY{o}{=}\PY{p}{[}\PY{n}{state0}\PY{p}{,}\PY{n}{disciplines}\PY{p}{,}\PY{n}{Jacobians}\PY{p}{]}\PY{p}{)} \PY{c+c1}{\PYZsh{} evaluate blackbox at optimizer}
\PY{n}{uopt} \PY{o}{=} \PY{n}{my\PYZus{}MDA}\PY{p}{(}\PY{n}{out}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{xmin}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{,}\PY{n}{state0}\PY{p}{,}\PY{n}{disciplines}\PY{p}{,}\PY{n}{Jacobians}\PY{p}{)} \PY{c+c1}{\PYZsh{} evaluate MDA at optimizer}
\PY{k+kn}{import} \PY{n+nn}{json}
\PY{k}{for} \PY{n}{key}\PY{p}{,}\PY{n}{value} \PY{o+ow}{in} \PY{n}{out}\PY{o}{.}\PY{n}{items}\PY{p}{(}\PY{p}{)}\PY{p}{:}
\PY{n+nb}{print}\PY{p}{(}\PY{n}{key}\PY{p}{,}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{ : }\PY{l+s+s2}{\PYZdq{}}\PY{p}{,}\PY{n}{value}\PY{p}{)}
\PY{n+nb}{print}\PY{p}{(}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{===========================}\PY{l+s+s2}{\PYZdq{}}\PY{p}{)}
\PY{n}{f\PYZus{}tex} \PY{o}{=} \PY{l+s+sa}{r}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}f(}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mathbf}\PY{l+s+si}{\PYZob{}x\PYZcb{}}\PY{l+s+s2}{\PYZca{}*) = }\PY{l+s+si}{\PYZpc{}.4f}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZdq{}} \PY{o}{\PYZpc{}}\PY{p}{(}\PY{n}{out}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{fmin}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{)}
\PY{n}{display}\PY{p}{(}\PY{n}{Latex}\PY{p}{(}\PY{n}{f\PYZus{}tex}\PY{p}{)}\PY{p}{)} \PY{c+c1}{\PYZsh{} minimum}
\PY{n}{x\PYZus{}tex} \PY{o}{=} \PY{l+s+sa}{r}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mathbf}\PY{l+s+si}{\PYZob{}x\PYZcb{}}\PY{l+s+s2}{\PYZca{}*\PYZdl{} = \PYZdl{}[}\PY{l+s+si}{\PYZpc{}.4f}\PY{l+s+s2}{\PYZti{}\PYZti{}}\PY{l+s+si}{\PYZpc{}.4f}\PY{l+s+s2}{\PYZti{}\PYZti{}}\PY{l+s+si}{\PYZpc{}.4f}\PY{l+s+s2}{]\PYZca{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mathrm}\PY{l+s+si}{\PYZob{}T\PYZcb{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZdq{}} \PY{o}{\PYZpc{}}\PY{p}{(}\PY{n}{out}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{xmin}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,}\PY{n}{out}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{xmin}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{,}\PY{n}{out}\PY{p}{[}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{xmin}\PY{l+s+s2}{\PYZdq{}}\PY{p}{]}\PY{p}{[}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
\PY{n}{display}\PY{p}{(}\PY{n}{Latex}\PY{p}{(}\PY{n}{x\PYZus{}tex}\PY{p}{)}\PY{p}{)} \PY{c+c1}{\PYZsh{} optimizer}
\PY{n}{g\PYZus{}tex} \PY{o}{=} \PY{l+s+sa}{r}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}g(}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mathbf}\PY{l+s+si}{\PYZob{}x\PYZcb{}}\PY{l+s+s2}{\PYZca{}*)\PYZdl{} = \PYZdl{}}\PY{l+s+si}{\PYZpc{}.4f}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZdq{}} \PY{o}{\PYZpc{}}\PY{p}{(}\PY{n}{gopt}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{)}
\PY{n}{display}\PY{p}{(}\PY{n}{Latex}\PY{p}{(}\PY{n}{g\PYZus{}tex}\PY{p}{)}\PY{p}{)} \PY{c+c1}{\PYZsh{} constraints}
\PY{n}{u\PYZus{}tex} \PY{o}{=} \PY{l+s+sa}{r}\PY{l+s+s2}{\PYZdq{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{hat}\PY{l+s+s2}{\PYZob{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mathbf}\PY{l+s+si}{\PYZob{}u\PYZcb{}}\PY{l+s+s2}{\PYZcb{}(}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mathbf}\PY{l+s+si}{\PYZob{}x\PYZcb{}}\PY{l+s+s2}{\PYZca{}*)\PYZdl{} = \PYZdl{}[}\PY{l+s+si}{\PYZpc{}.4f}\PY{l+s+s2}{\PYZti{}\PYZti{}}\PY{l+s+si}{\PYZpc{}.4f}\PY{l+s+s2}{]\PYZca{}}\PY{l+s+s2}{\PYZbs{}}\PY{l+s+s2}{mathrm}\PY{l+s+si}{\PYZob{}T\PYZcb{}}\PY{l+s+s2}{\PYZdl{}}\PY{l+s+s2}{\PYZdq{}} \PY{o}{\PYZpc{}}\PY{p}{(}\PY{n}{uopt}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{,}\PY{n}{uopt}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
\PY{n}{display}\PY{p}{(}\PY{n}{Latex}\PY{p}{(}\PY{n}{u\PYZus{}tex}\PY{p}{)}\PY{p}{)} \PY{c+c1}{\PYZsh{} constraints}
\PY{c+c1}{\PYZsh{} print(f\PYZus{}tex)}
\PY{c+c1}{\PYZsh{} print(x\PYZus{}tex)}
\PY{c+c1}{\PYZsh{} print(g\PYZus{}tex)}
\PY{c+c1}{\PYZsh{} print(u\PYZus{}tex)}
\end{Verbatim}
\end{tcolorbox}
\begin{Verbatim}[commandchars=\\\{\}]
xmin : [1.1874999999927240424, 0.74511669576168060303, 7.6562655009329319]
fmin : 5.005735228721652552
hmin : 9.996378214847634363e-19
nbb\_evals : 469
niterations : 78
nb\_success : 27
psize : 9.094947017729282379e-13
psuccess : 7.2759576141834259033e-12
pmax : 2.0
===========================
\end{Verbatim}
$f(\mathbf{x}^*) = 5.0057$
$\mathbf{x}^*$ = $[1.1875~~0.7451~~7.6563]^\mathrm{T}$
$g(\mathbf{x}^*)$ = $0.0000$