\chapter{Conclusions}
\label{chap:conclusions}
We have taken, over the course of this work, a tour through a rather wide expanse of material, so by way of conclusion we present a short summary of the main paradigms we have explored and the principal results we have uncovered.

\subsection*{Part I: Ionization}

\subsubsection[Analytical R-matrix Theory]{Analytical $R$-Matrix theory}
In chapter~\ref{chap:R-matrix} we lay down the groundwork for the ionization parts of this thesis, reviewing in detail the construction of the Analytical $R$-Matrix theory of photoionization. We showed how one can split space into an inner and an outer region, using the inner region as a source term for the Schrödinger equation on the outer region, and how to employ the ionic and Coulomb-corrected continuum wavefunctions to produce a simple trajectory-based description of the photoionization, both on the single-active-electron direct channels as well as for the multi-electron correlation-driven mechanism. In addition, we presented in section~\ref{sec:molecular-shape-factors} simple analytical formulas for a suitable model of a molecular orbital with nontrivial geometry.

\subsubsection{Multi-channel geometrical effects}
In chapter~\ref{chap:multi-channel} we built on the multi-electron Analytical $R$-Matrix expressions for the direct and correlation-driven yields for photoelectrons that leave behind an excited ion, probing the ionization of a suitable model of an aligned carbon dioxide molecule, with a nontrivial combination of orbital geometries. There we showed how to implement a suitable modification of the saddle-point approximation for the correlation-driven geometrical factors, giving clean and physically transparent expressions.

Finally, we showed how the nontrivial angular distributions obtained in this geometry admit a simple interpretation as double-slit diffraction fringes that originate in a pair of `slits' caused by the correlation-driven interaction with the ion inside the tunnelling~barrier.

\subsubsection{Analytical continuation of electrostatic potentials}
In chapter~\ref{chap:complex-space-potentials} we examined in detail one of the crucial ingredients of the multi-electron ARM yield for correlation-driven ionization, the correlation interaction electrostatic potentials
\begin{equation}
\Vnm{\vbr}=\matrixel**{n}{\sum_{j=1}^{N-1} \frac{1-\delta_{nm}}{\| \vbr - \hat{\vbr}_j\|} }{m}
,
\backtag{e4-correlation-interaction-potential-initial}
\end{equation}
and their analytical continuations into complex positions $\vbr$. We examined the naive analytical continuation for an elementary gaussian orbital via direct numerical integration, and we found it wanting, as it does not obey the Cauchy-Riemann equations.

We then focussed on the behaviour of simpler models -- exponential and gaussian type orbitals --, for which we can find exact expressions for the potentials, and therefore examine their analytical continuations directly. Here we find that, for points that are `real enough', in the sense that
\begin{equation}
\Re(\vbr^2)>0,
\backtag{e4-re-r2-less-than-0}
\end{equation}
the different models agree surprisingly well, but that immediately upon leaving that region the potential for the gaussian-type orbital catastrophically diverges.

This means, then, that as long as we keep our evaluations of $\Vnm{\rl(t)}$ to trajectories that obey \eqref{e4-re-r2-less-than-0} throughout, we can be rather confident that the analytical continuation is accurate, even if using gaussian-based quantum chemical calculations. Moreover, as we showed in chapter~\ref{chap:quantum-orbits}, it is in fact possible to choose trajectories that adhere to this constraint.

On the other hand, the other region -- the points that are so imaginary that $\Re(\vbr^2)<0$~-- is much more challenging. As we argued, it is difficult to obtain any information, even of a qualitative type, about the behaviour of the potential in this region, even though there are formal existence theorems that guarantee us the existence of analytical continuations of  our interaction potential.

\subsubsection{Quantum orbits in complex time and complex space}
In chapter~\ref{chap:quantum-orbits} we explored the origin and meaning of the imaginary part of the ARM trajectory, which is of the form
\begin{equation}
\rl(t) = \int_{\ts}^{t} \left[\vbp+\vba(\tau) \right] \: \d\tau
,
\backtag{e5-laser-driven-trajectory}
\end{equation}
and is generally complex-valued for real times $t$. This imaginary part emerges directly from the Schrödinger equation, and more particularly from the boundary matching of our eikonal Volkov states with the WKB asymptotic expressions for the states of the system in the inner ARM region, and it is a crucial ingredient in allowing ARM do describe the Coulomb enhancement of ionization.

However, this imaginary part of the position also combines with the branch cut of the square root in the Coulomb potential,
\begin{equation}
U(\rl(t)) = -\frac{1}{\sqrt{\rl(t)^2}},
\backtag{e5-coulomb-potential-at-the-trajectory}
\end{equation}
to imprint a series of branch cuts on the complex time plane, which we need to integrate over to obtain the ARM yield. Moreover, these branch cuts can and do intersect the standard contour along the real axis, which needs to be accordingly modified, ideally in an algorithmic way which can be performed by a computer to calculate photoelectron spectra.

This modification is possible by hinging on the concept of a time of closest approach, that is, times $\tca$ that obey equations of the type
\begin{equation}
\rl(\tca)\cdot\vbv(\tca)=0,
\backtag{e5-tca-equation}
\end{equation}
and which are always present as saddle points in the middle of any gate formed by two branch cuts. These have a fascinatingly rich geometry, with the complex-valued quantum versions forming multiple sheets of a surface encasing the corresponding classical version, and they show interesting topological transitions -- with strong effects on the requirements for the ARM integration path -- at soft recollisions, where the laser-driven trajectory has a turning point close to the ion.

Moreover, we were able to successfully and algorithmically choose the appropriate $\tca$s to use as waypoints by requiring that they be `out of the tunnelling barrier', having positive real part of the kinetic energy $\vbv(t)^2$, after which it is possible to programmatically choose appropriate integration paths that avoid complex, and fast-changing, configurations of branch cuts. In addition, this navigation algorithm is also automatically able to steer us clear of the problematic regions of chapter~\ref{chap:complex-space-potentials}.

\subsubsection{Low-Energy Structures and Near-Zero Energy Structures}
In chapter~\ref{chap:LES-NZES}, after a review of the available experimental evidence on the Low-Energy and Near-Zero Energy Structures of mid-IR photoionization, we brought our branch-cut navigation algorithm from chapter~\ref{chap:quantum-orbits} to bear on experiment, by using it to analyse the LES regions, which emerge naturally from one of the unavoidable (but resolvable) difficulties of the algorithm: the soft recollisions, where multiple branch cuts come into close proximity, interact, and undergo topological transitions.

Further, these LES peaks, which are well understood to be associated with these soft recollisions, are joined within the ARM formalism by a dynamically equivalent set at much lower energy, which approaches the ion on a forwards turning point, after an integral number of periods, instead of the more usual backwards turning points after a half-integral number of laser cycles.

We study the peaks caused by this new series of trajectories, showing that it produces a peak at energies consistent with those observed for the NZES, and that it produces qualitatively similar transverse photoelectron to those observed in high-resolution experiments. Moreover, the identification of this series as a contributor to the NZES opens clear ways to test this mechanism experimentally: since the new series only needs to advance by the tunnel exit over a laser period, instead of by the laser oscillation quiver radius $\zquiv=F/\omega^2$, its momentum and energy scale as
\begin{equation}
\pzsr
\approx \frac{\zexit}{\Delta t}
= \frac{I_p/F}{(n+1)\pi/\omega}
\propto \frac{I_p\omega}{F}
%
\quad\text{and}\quad
%
\frac12\left(\pzsr\right)^2
\sim \frac{I_p^2}{U_p}
\sim I_p\gamma^2,
\backtagtwo{e6-pzsr-odd-n-summary}{e6-odd-n-energy-scaling}
\end{equation}
respectively. Thus, the new series scales inversely with the ponderomotive potential $U_p$ (as opposed to the known series, which scales as $\frac12\left(\pzsr\right)^2 \propto U_p$), which means that it should be possible to probe the role of the new series by using experiments in harder targets with higher ionization potentials.

\section*{Part II: High-order harmonic generation}

\subsubsection{Conservation of spin angular momentum in bicircular HHG}
After reviewing the standard theory of high-order harmonic generation in chapter~\ref{chap:HHG-intro}, we turned in chapter~\ref{chap:spin-HHG} to the generation of high-order harmonics by bicircular fields -- counter-rotating circularly polarized fields, one at the fundamental at $\SI{800}{nm}$ and one detuned from its second harmonic, at $\SI{410}{nm}$. We reviewed the experimental evidence that establishes clear selection rules coming from the conservation of spin angular momentum, together with the breakdown of these selection rules when the polarization of one of the drivers is degraded from circular through linear.

We then provided a suitable photon-picture model that is able to explain the observed harmonic emission while retaining a parametric picture of harmonic generation which conserves spin angular momentum on a per-channel basis~-- and, indeed, while maintaining consistency with known results when applied to the lowest-order channel, which reduces to perturbative four-wave mixing.

Our model, based essentially on arguments lowest-order perturbation theory by separating an elliptical driver into circular components and treating them separately, is surprisingly effective at predicting the dependence of the harmonic emission on the ellipticity of the driver. Moreover, it correctly matches the results of a numerical experiment, which in principle can also be experimentally realized, where the two circular components of an elliptical field are taken as separate and detuned independently

\subsubsection{Nondipole effects in HHG through noncollinear bicircular beams}
Finally, in chapter~\ref{chap:nondipole-HHG}, we turned to the generation of harmonics in fields that are particularly strong, or at a very long wavelength, or both, looking to extend the harmonic cutoff past the barrier posed by the magnetic field of the driving laser as the velocity of the continuum electron increases.

Here we proposed a simple and flexible scheme for addressing the continuum electron's motion along the direction, by combining two counter-rotating circularly polarized fields of the same frequency in a non-collinear configuration to produce a field with a forwards ellipticity that acts in the same direction as the magnetic Lorentz force.

We then extended the existing beyond-dipole Strong-Field Approximation HHG formalism to deal with arbitrary beam configurations, showing that to do so one needs to use slightly more complex nondipole Volkov states, and related this to a nontrivial average force acting on charged particles across the focus.

Finally, with this extended beyond-dipole SFA in hand, we showed that the field configuration can indeed help recover the harmonic emission from its exponential quenching at the hands of the magnetic Lorentz force, and, moreover, that it can be used at much lower intensities and wavelengths to produce even harmonics that can be used to demonstrate the presence of the effect, for the first time in HHG, using currently available laser sources.


\chapter{Astronomy}

\section{Reading sky map to find celestial objects}

% https://en.wikipedia.org/wiki/Celestial_sphere

Celestial sphere.
The sky looks as if it were projected to a spherical screen.
If a star is far enough, it will look as if it were fixed in the sky.

You find a star in the sky.
You write a letter to your friend.
How do you write where that star is?
How do you explain to him which direction he should look at?

You use the \emph{equatorial coordinate system}.

\emph{Right ascension}, \emph{declination}, and \emph{epoch}.

Example: Alpha Centauri A.
Right ascension 14 h 39 m 35.06311 s.
Declination \(-60\deg\) 50' 15.0992".
Epoch J2000.

% https://en.wikipedia.org/wiki/Epoch_(astronomy)#Julian_years_and_J2000
J2000 is the Gregorian date 2000-01-01 12:00 TT (terrestrial time).

Star chart, star map, sky map

% https://en.wikipedia.org/wiki/Celestial_coordinate_system

Celestial coordinate system

% https://en.wikipedia.org/wiki/Star_chart

% https://en.wikipedia.org/wiki/Celestial_coordinate_system

Equatorial coordinate system

% https://en.wikipedia.org/wiki/Alpha_Centauri

\section{Distance}

% https://en.wikipedia.org/wiki/Parsec

1 au (astronomical unit) is roughly the distance between the Sun and the Earth.
It is about 150 million km.

Parsec is a unit of length.
\( 648000/\pi \).
1 pc is about 3.26 ly.

A \emph{light year} is the distance traveled by light in one year.
\emph{Light year} (ly) is a unit of \emph{distance}, not time.
1 au is about 6 light minutes.

\section{Objects}

A \emph{planet} is?

% https://en.wikipedia.org/wiki/Stellar_evolution
A \emph{star} is a luminous sphere of plasma held together by its own gravity.
("Star", Wikipedia)
Every star begins from collapsing clouds of gas and dust.
A \emph{protostar} is ...
A \emph{main-sequence star} is ...

A \emph{solar system} is?

A \emph{galaxy} is?

A \emph{nebula} is?

A \emph{constellation} is?

A \emph{satellite} is?

A \emph{moon} is?

A \emph{comet} is?

An \emph{asteroid} is?

A \emph{supernova} is?

A \emph{brown dwarf} is?

A \emph{white dwarf} is?

A \emph{black hole} is?

\section{Cosmology}

\section{Cosmogony}


\chapter{Actuator Disc Method}

\section{Coordinate Systems}

\subsection{Rotor Axis System}

Origin of the Rotor Axis System is coincident with the rotor hub center, the x-axis is positive in forward direction, the y-axis is positive right and z-axis is positive downward direction and it is coincident with the rotor shaft axis.

\begin{figure}[h!]
  \centering
  \includegraphics[width=110mm]{eps/coordinate_system_RAS.eps}
  \caption{Rotor Axis System}
\end{figure}

\subsection{Rotor-Wind Axis System}

Rotor-Wind Axis System is very much like Rotor Axis System, the only difference is that it is rotated about z-axis in such a manner that x-axis points directly into relative wind, so there is no lateral airspeed component.

\subsection{Control Axis System}

For most purposes, using the Rotor Axis System causes unnecessary complications. It is convenient to use no cyclic feathering axes system. \cite{GessowMyers1985} Origin of the Control Axis System is coincident with the origin of the Rotor Axis System, but it is rotated by angles of the swashplate roll and pitch so there is no cyclic feathering in this coordinate system.

\begin{figure}[h!]
  \centering
  \includegraphics[width=120mm]{eps/rotor_planes.eps}
  \caption{Rotor reference planes}
\end{figure}

\subsection{Disc Axis System}

Origin of the Disc Axis System is coincident with the origin of the Rotor Axis System, but it is rotated by angles of the rotor cone roll and pitch in such a manner that z-axis is perpendicular to the tip path plane so there is no cyclic flapping in this coordinate system.

\subsection{Control-Wind Axis System}

Control-Wind Axis System is very much like Control Axis System, the only difference is that it is rotated about z-axis in such a manner that x-axis points directly into relative wind, so there is no lateral airspeed component.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Assumptions}

Following assumptions are made for the purpose of modeling helicopter rotor aerodynamics:
\begin{itemize}
  \item[---] forces and moments generated by the rotor are considered to be quasi-steady,
  \item[---] rotor lift force is a linear function of blade incidence angle and drag force is a quadratic function of lift, \cite{Padfield2007}
  \item[---] rotor blades have 3 degrees of freedom movement,
  \item[---] inflow is uniformly distributed over rotor disc, \cite{Padfield2007}
  \item[---] reversed flow effects are ignored,
  \item[---] airflow is considered to be quasi-steady and incompressible,
  \item[---] thrust is considered to be parallel to the z-axis of the Control Axis System and magnitude of the thrust is considered to be magnitude of the resulting rotor force. \cite{GessowMyers1985}
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Momentum Theory}

\subsection{Momentum Theory for Axial Flight}

Mass flow through the rotor disc, momentum change and change in kinetic energy are given by the following formulas. \cite{Padfield2007}
\begin{align}
  \label{eq-aero-mass-flow}
  \dot m
  &=
  \rho A_1 V_C = \rho A_R \left( V_C + V_i \right)
  =
  \rho A_{i \infty} \left( V_C + V_{i \infty}  \right) \\
  \label{eq-aero-thrust-1}
  T
  &=
  \dot m \left( V_C + V_{i \infty} \right)
  -
  \dot m V_C = \dot m V_{ i \infty } \\
  T \left( V_C + V_{ i \infty } \right)
  &=
  \frac{1}{2} \dot m \left( V_C + V_{ i \infty } \right)^2
  -
  \frac{1}{2} \dot m V_C^2
  =
  \frac{1}{2} \dot m \left( 2V_C V_{ i \infty } + V_{ i \infty }^2 \right)
\end{align}

Where:
\begin{description}[align=right,labelwidth=3cm]
  \item [$A_1 = \pi R_1^2$] [m\textsuperscript{2}] control volume section area
  \item [$A_R = \pi R^2$] [m\textsuperscript{2}] rotor disc area
  \item [$A_{\infty} = \pi R_{\infty}^2$] [m\textsuperscript{2}] far wake slipstream section area
  \item [$\dot m$] [kg/s] mass flow
  \item [$V_C$] [m/s] climb velocity
  \item [$V_i$] [m/s] induced velocity
  \item [$V_{i \infty}$] [m/s] far wake induced velocity
  \item [$T$] [N] rotor thrust
\end{description}

\begin{figure}[h!]
  \centering
  \includegraphics[width=51mm]{eps/momentum_theory_axial.eps}
  \caption{Flow trough a rotor in axial flight}
\end{figure}

From these relationships it can be deduced that induced velocity in the far wake is twice the rotor inflow. \cite{Padfield2007}
\begin{equation}
  \label{eq-aero-indeced-vel}
  V_{i \infty} = 2 V_i
\end{equation}

Substituting equations (\ref{eq-aero-mass-flow}) and (\ref{eq-aero-indeced-vel}) into (\ref{eq-aero-thrust-1}) rotor thrust is given as follows:
\begin{equation}
  \label{eq-aero-thrust-climb}
  T = 2 \rho A_R \left( V_C + V_i \right) V_i
\end{equation}

In hover flight, this equation can be expressed as:
\begin{equation}
  \label{eq-aero-thrust-hover}
  T = 2 \rho A_R V_{ih}^2
\end{equation}

Transforming equations (\ref{eq-aero-thrust-climb}) and (\ref{eq-aero-thrust-hover}) gives:
\begin{align}
  V_i    &= \frac{T}{ 2 \rho A_R \left( V_C + V_i \right) } \\
  V_{ih} &= \sqrt{ \frac{T}{ 2 \rho A_R } }
\end{align}

Writing velocities in normalized form:
\begin{align}
  \label{eq-aero-lambda-i}
  \lambda_i    &= \frac{ V_i }    { \Omega_R R } \\
  \label{eq-aero-lambda-ih}
  \lambda_{ih} &= \frac{ V_{ih} } { \Omega_R R } \\
  \mu_C        &= \frac{ V_C }    { \Omega_R R }
\end{align}

Rotor thrust coefficient is:
\begin{equation}
  \label{eq-aero-thrust-coef}
  C_T = \frac{ T }{ \rho A_R \Omega_R^2 R^2 }
\end{equation}

Then equations (\ref{eq-aero-lambda-i}) and (\ref{eq-aero-lambda-ih}) can be expressed as:
\begin{align}
  \lambda_i    &= \frac{ C_T }{ 2 \left( \mu_C + \lambda_i \right) } \\
  \lambda_{ih} &= \sqrt{ \frac{ C_T }{ 2 } }
\end{align}

Combining these equations gives:
\begin{equation}
  \lambda_{ih}^2 = \lambda_i \left( \mu_C + \lambda_i \right)
\end{equation}

This equation can be transformed into following form:
\begin{equation}
  \label{eq-aero-lambda-i-2}
  \lambda_i
  =
  - \frac{\mu_C}{2}
  +
  \sqrt{ \left( \frac{\mu_C}{2} \right)^2 + \lambda_{ih}^2 }
\end{equation}

For descent velocity $V_D = - V_C$ formula (\ref{eq-aero-lambda-i-2}) can be written as:
\begin{equation}
  \lambda_i
  =
  \frac{\mu_D}{2}
  -
  \sqrt{ \left( \frac{\mu_D}{2} \right)^2 - \lambda_{ih}^2 }
\end{equation}

This relationship is valid only in windmill brake state where the wake is fully established and the flow is upwards. \cite{Padfield2007} It can be assumed that such a condition occurs when descent velocity is two times greater than induced velocity in hover. \cite{Stepniewski1984}

Young’s approximation can be used to determine induced velocity outside range of momentum theory application. \cite{Padfield2007}
\begin{align}
  \lambda_i
  =
  \lambda_{ih} \left( 1 + \frac{\mu_D}{\lambda_{ih}} \right)
  &\mathrm{~for~} 0 \leq \mu_D \leq -1.5 \lambda_{ih} \\
  \lambda_i
  =
  \lambda_{ih} \left( 7 - 3 \frac{\mu_D}{\lambda_{ih}} \right)
  &\mathrm{~for~} -1.5 \lambda_{ih} < \mu_D \leq -2 \lambda_{ih}
\end{align}

\subsection{Momentum Theory in Forward Flight}

In forward flight induced velocity in the far wake is twice the flow at the rotor. \cite{Padfield2007} Expression for thrust is given as follows:
\begin{equation}
  T = \dot m 2 V_i = \left( \rho A_R V' \right) 2 V_i
\end{equation}

\begin{figure}
  \centering
  \includegraphics[width=120mm]{eps/momentum_theory_forward.eps}
  \caption{Flow trough a rotor in forward flight}
\end{figure}

Transforming this equation for induced velocity gives:
\begin{equation}
  \label{eq-aero-lambda-i-3}
  V_i = \frac{T}{ 2 \rho A_R V' }
\end{equation}

Where $V'$ is the resultant velocity at the rotor.
\begin{equation}
  \label{eq-aero-vel-rotor}
  V' = \sqrt{
    V^2 \cos^2 \alpha_R
    + \left( V \sin \alpha_R - V_i \right)^2
  }
\end{equation}

Writing velocities in normalized form:
\begin{align}
  \label{eq-aero-lambda-i-4}
  \lambda_i &= \frac{V_i}    { \Omega_R R } \\
  \label{eq-aero-mu-x}
  \mu_X     &= \frac{u_{rw}} { \Omega_R R } \\
  \label{eq-aero-mu-z}
  \mu_Z     &= \frac{w_{rw}} { \Omega_R R }
\end{align}

Where:
\begin{align}
  u_{rw} &= V \cos \alpha_R \\
  w_{rw} &= V \sin \alpha_R
\end{align}

Substituting equations (\ref{eq-aero-lambda-i-4}), (\ref{eq-aero-mu-x}), (\ref{eq-aero-mu-z}) and (\ref{eq-aero-vel-rotor}) into (\ref{eq-aero-lambda-i-3}) formula for the normalized induced velocity can be expressed as follows.
\begin{equation}
  \label{eq-aero-lambda-i-5}
  \lambda_i
  =
  \frac{C_T}{ 2 \sqrt{ \mu_X^2 + \left( \mu_Z - \lambda_i \right)^2 } }
\end{equation}

In high speed flight summing helicopter translational velocity and velocity due to rotor shaft rotation causes strong non-uniformities of rotor induced velocity. Glauert’s model is used to describe this phenomena. \cite{Padfield2007, Bramwell2001}
\begin{equation}
  \lambda_i \left( r, \Psi \right)
  =
  \lambda_{i0} + \frac{r}{R} \lambda_{1c} \cos \Psi
\end{equation}

Where:
\begin{align}
  \lambda_{1c} &= \lambda_{i0} \tan \left( \frac{\chi}{2} \right)
  \mathrm{~for~} \chi < \frac{\pi}{2} \\
  \lambda_{1c} &= \lambda_{i0} \cot \left( \frac{\chi}{2} \right)
  \mathrm{~for~} \chi > \frac{\pi}{2}
\end{align}

The wake angle is given by the following formula:
\begin{equation}
  \chi = \arctan \left( \frac{\mu}{ \lambda_{i0} - \mu_Z } \right)
\end{equation}

Where $\lambda_{i0}$ is given by formula (\ref{eq-aero-lambda-i-5}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Forces Acting on the Blade Segment}

Determining forces and moments acting on segment of the blade is made, assuming that blade is composed of aerodynamically independent, narrow strips of elements. \cite{Stepniewski1984} High aspect ratio of the blade justifies usage of two-dimensional flow, while lift loss at the blade tip and root can be accounted by using tip-loss factor. \cite{Padfield2007, Stepniewski1984, Bramwell2001}

Control-Wind Axis System is used to determine forces and moments generated by the rotor, such computed forces and moments are the transformed to the Body Axis System.

\begin{figure}[h!]
  \centering
  \includegraphics[width=120mm]{eps/blade_element_theory_01.eps}
  \caption{Rotor blade element}
\end{figure}

Lift and drag acting on a blade section is given by the following expressions:
\begin{align}
  \label{eq-aero-blade-section-lift}
  dL &= \frac{1}{2} \rho U^2 \left( r, \Psi \right) C_L c_b dr \\
  \label{eq-aero-blade-section-drag}
  dD &= \frac{1}{2} \rho U^2 \left( r, \Psi \right) C_D c_b dr
\end{align}

Lift and drag coefficients are given as follows: \cite{Padfield2007}
\begin{align}
  \label{eq-aero-blade-section-lift-coef}
  C_L &= a \alpha_{BE} \left( r, \Psi \right) \\
  C_D &= \delta_0 + \delta_2 C_T^2
\end{align}

Blade section angle of attack is given by the following formula.
\begin{equation}
  \alpha_{BE} \left( r , \Psi \right) = \theta + \phi \left( r , \Psi \right)
\end{equation}

Where:
\begin{equation}
  \phi \left( r , \Psi \right)
  =
  \arctan \frac{ U_P \left( r, \Psi \right) }{ U_T \left( r, \Psi \right) }
\end{equation}

Linear velocity of the rotor hub and angular velocity expressed Rotor Axis System are given by the following formulas: \cite{Stepniewski1984, Bramwell2001}
\begin{align}
  {\vec V}_{RH,r}
  &=
  {\boldsymbol T} \left( \varepsilon \right)
  \left(
    {\vec V}_{O,b} + {\vec \omega}_b \times {\vec r}_{RH,b}
  \right) \\
  {\vec \omega}_r
  &=
  {\boldsymbol T} \left( \varepsilon \right)
  {\vec \omega}_b
\end{align}

Rotattion matrix ${\boldsymbol T} \left( \varepsilon \right)$ is given as:
\begin{equation}
  {\boldsymbol T} \left( \varepsilon \right)
  =
  \left[
    \begin{matrix}
       \cos \varepsilon & 0 & \sin \varepsilon \\
                        0 & 1 &              0 \\
      -\sin \varepsilon & 0 & \cos \varepsilon \\
    \end{matrix}
  \right]
\end{equation}

\begin{figure}
  \centering
  \includegraphics[width=100mm]{eps/blade_element_theory_02.eps}
  \caption{Velocity components at the blade element}
\end{figure}

Following formulas can be used to transform this values to Control Axis System:
\begin{align}
  {\vec V}_{RH,c}
  &=
  {\boldsymbol T} \left( \theta_{1c}, \theta_{1s} \right)
  {\vec V}_{RH,r} \\
  {\vec \omega}_c
  &=
  {\boldsymbol T} \left( \theta_{1c}, \theta_{1s} \right)
  {\vec \omega}_r
\end{align}

Where rotation matrices are:

--- for counterclockwise direction of rotor:
\begin{equation}
  {\boldsymbol T} \left( \theta_{1c}, \theta_{1s} \right)
  =
  \left[
    \begin{matrix}
      1 &                0 &                 0 \\
      0 & \cos \theta_{1c} & -\sin \theta_{1c} \\
      0 & \sin \theta_{1c} &  \cos \theta_{1c} \\
    \end{matrix}
  \right]
  \left[
    \begin{matrix}
      \cos \theta_{1s} & 0 & -\sin \theta_{1s} \\
                    0 & 1 &                 0 \\
      \sin \theta_{1s} & 0 &  \cos \theta_{1s} \\
    \end{matrix}
  \right]
\end{equation}

--- for clockwise direction of rotor:
\begin{equation}
  {\boldsymbol T} \left( \theta_{1c}, \theta_{1s} \right)
  =
  \left[
    \begin{matrix}
      1 &                 0 &                0 \\
      0 &  \cos \theta_{1c} & \sin \theta_{1c} \\
      0 & -\sin \theta_{1c} & \cos \theta_{1c} \\
    \end{matrix}
  \right]
  \left[
    \begin{matrix}
      \cos \theta_{1s} & 0 & -\sin \theta_{1s} \\
                    0 & 1 &                 0 \\
      \sin \theta_{1s} & 0 &  \cos \theta_{1s} \\
    \end{matrix}
  \right]
\end{equation}

Following formulas can be used to transform linear and angular velocity vector to Control-Wind Axis System:
\begin{align}
  {\vec V}_{RH,cw}
  &=
  {\boldsymbol T} \left( \beta \right) {\vec V}_{RH,c} \\
  {\vec \omega}_{cw}
  &=
  {\boldsymbol T} \left( \beta \right) {\vec \omega}_c
\end{align}

Rotation matrix ${\boldsymbol T} \left( \beta \right)$ is given by formula (\ref{eq-aero-matrix-beta}).

Assuming that flapping angle is positive upwards and writing velocity components as:
\begin{align}
  {\vec V}_{RH,cw} &= \left[ u_{cw}, 0, w_{cw} \right]^T \\
  {\vec \omega}_{cw} &= \left[ p_{cw}, q_{cw}, 0 \right]^T
\end{align}

\begin{figure}
  \centering
  \includegraphics[width=120mm]{eps/blade_element_velocity.eps}
  \caption{Air velocities at the blade element}
\end{figure}

Air velocity at the blade segment is:

--- for counterclockwise direction of rotor:
\begin{gather}
  U_T = \Omega_R r \cos \beta + u_{cw} \sin \Psi \\
  \begin{multlined}[0.88\textwidth]
    U_P =
    w_{cw} \cos \beta - V_i \cos \beta
    - \dot \beta r - u_{cw} \sin \beta \cos \Psi \\
    + p_{cw} r \sin \Psi + q_{cw} r \cos \Psi
  \end{multlined}
\end{gather}

--- for clockwise direction of rotor:
\begin{gather}
  U_T = \Omega_R r \cos \beta + u_{cw} \sin \Psi \\
  \begin{multlined}[0.88\textwidth]
    U_P =
    w_{cw} \cos \beta - V_i \cos \beta
    - \dot \beta r - u_{cw} \sin \beta \cos \Psi \\
    - p_{cw} r \sin \Psi + q_{cw} r \cos \Psi
  \end{multlined}
\end{gather}

Assuming that for small angles:
\begin{align}
  \label{eq-aero-sin-beta-approx}
  \sin \beta \approx \beta \\
  \label{eq-aero-cos-beta-approx}
  \cos \beta \approx 1
\end{align}

This expressions can be simplified to:

--- for counterclockwise direction of rotor:
\begin{align}
  U_T &= \Omega_R r + u_{cw} \sin \Psi \\
  U_P &=
  w_{cw} - V_i - \dot \beta r - u_{cw} \beta \cos \Psi
  + p_{cw} r \sin \Psi + q_{cw} r \cos \Psi
\end{align}

--- for clockwise direction of rotor:
  \begin{align}
    U_T &= \Omega_R r + u_{cw} \sin \Psi \\
    U_P &=
    w_{cw} - V_i - \dot \beta r - u_{cw} \beta \cos \Psi
    - p_{cw} r \sin \Psi + q_{cw} r \cos \Psi
  \end{align}

Using normalized velocities:

\begin{align}
  \mu     &= \frac{u_{cw}}{\Omega_R R} \\
  \lambda &= \frac{w_{cw} - V_i}{\Omega_R R}
\end{align}

Expressions for air velocity at the blade segment can be written in the following form:

--- for counterclockwise direction of rotor:
\begin{align}
  U_T &= \Omega_R r + \mu \Omega_R R \sin \Psi \\
  U_P &=
  \lambda \Omega_R R - \dot \beta r - \mu \Omega_R R \beta \cos \Psi
  + p_{cw} r \sin \Psi + q_{cw} r \cos \Psi
\end{align}

--- for clockwise direction of rotor:
\begin{align}
  U_T &= \Omega_R r + \mu \Omega_R R \sin \Psi \\
  U_P &=
  \lambda \Omega_R R - \dot \beta r - \mu \Omega_R R \beta \cos \Psi
  - p_{cw} r \sin \Psi + q_{cw} r \cos \Psi
\end{align}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Rotor Thrust}

Assuming that for small angles:
\begin{align}
  \label{eq-aero-approx-phi}
  \phi = \arctan &\frac{U_P}{U_T} \approx \frac{U_P}{U_T} \\
  \label{eq-aero-approx-u}
  U  &\approx U_T \\
  \label{eq-aero-approx-dt}
  dT &\approx dL
\end{align}

Expression for the blade segment angle of attack is given as follows:
\begin{equation}
  \alpha_{BE} = \theta + \frac{U_P}{U_T}
\end{equation}

Then expression (\ref{eq-aero-blade-section-lift-coef}) can be written as. \cite{GessowMyers1985}
\begin{equation}
  \label{eq-aero-rotor-list-coef}
  C_L = a \left( \theta + \frac{U_P}{U_T} \right)
\end{equation}

Substituting expression (\ref{eq-aero-blade-section-lift}) and taking into account simplifications (\ref{eq-aero-approx-phi}), (\ref{eq-aero-approx-u}) and (\ref{eq-aero-approx-dt}) then rotor thrust is given by the following formula.
\begin{equation}
  dT
  \approx
  \frac{1}{2} \rho a c_b U_T^2 \left( \theta + \frac{U_P}{U_T} \right) dr
\end{equation}

Transforming this relationship gives:
\begin{equation}
  \label{eq-aero-blade-section-thrust}
  dT
  \approx
  \frac{1}{2} \rho a c_b
  \left( \theta U_T^2 + U_P U_T \right) dr
\end{equation}

Total thrust generated by the rotor of $n_b$ blades can be determined by integrating differential equation (\ref{eq-aero-blade-section-thrust}) first with respect to the azimuth then along the blade span. \cite{GessowMyers1985} Total thrust is given as follows:
\begin{equation}
  \label{eq-aero-thrust-2}
  T = \frac{ n_b }{ 2 \pi } \int_{0}^{2 \pi} \int_{0}^{BR}
  \frac{dT}{dr} dr d\Psi
\end{equation}

Where $B$ is a tip loss factor.

Substituting (\ref{eq-aero-blade-section-thrust}) into (\ref{eq-aero-thrust-2}) gives:
\begin{equation}
  T = \frac{1}{2} \rho a c_b n_b
  \left(
  \theta \frac{1}{2 \pi}
  \int_{0}^{2\pi} \int_{0}^{BR} \frac{U_T^2}{dr} dr d\Psi
  +
  \frac{1}{2\pi} \int_{0}^{2\pi} \int_{0}^{BR} U_P U_T dr d\Psi
  \right)
\end{equation}

Neglecting helicopter angular velocity expressions for $U_T^2$ and $U_P U_T$ can be written as:
\begin{gather}
  \label{eq-aero-u-t-sq}
  U_T^2 =
  r^2 \Omega_R^2 + 2 \Omega_R^2 R r \mu \sin \Psi
  + \mu^2 R^2 \Omega_R^2 \sin^2 \Psi \\
  \label{eq-aero-u-p-u-t}
  \begin{multlined}[0.88\textwidth]
  U_P U_T =
  \lambda \Omega_R^2 R r - \dot \beta \Omega_R r^2
  - \beta \mu \Omega_R^2 R r \cos \Psi \\
  + \lambda \mu \Omega_R^2 R^2 \sin \Psi
  - \dot \beta \mu \Omega_R R r \sin \Psi
  - \beta \mu^2 \Omega_R^2 R^2 \sin \Psi \cos \Psi
  \end{multlined}
\end{gather}

Expression for the blade flapping angle can be written as follows: \cite{GessowMyers1985, Padfield2007, NASA-TT-F-494}
\begin{equation}
  \label{eq-aero-blade-flapping-angle}
  \beta \left( \Psi \right) =
  \beta_0 + \beta_{1c} \cos \Psi + \beta_{1s} \sin \Psi
\end{equation}

Assuming constant rotor revolution speed, blade flapping angle derivatives with respect to time can be written as derivatives with respect to the azimuth. \cite{GessowMyers1985}
\begin{gather}
  \dot \beta = \frac{d\beta}{dt}
  = \frac{d\beta}{d\Psi} \frac{d\Psi}{dt} =
  \bar \beta \Omega_R =
  \Omega_R
  \left( \beta_{1s} \cos \Psi - \beta_{1c} \sin \Psi \right) \\
  \ddot \beta = \frac{d^2\beta}{dt^2}
  = \frac{d^2\beta}{d\Psi^2} \left( \frac{d\Psi}{dt} \right)^2 =
  \bar{\bar \beta} \Omega_R^2
  =
  -\Omega_R^2
  \left( \beta_{1c} \cos \Psi + \beta_{1s} \sin \Psi \right)
\end{gather}

Then expressions for $U_T^2$ and $U_P U_T$ can be written as:
\begin{gather}
  U_T^2 = r^2 \Omega_R^2 + 2 \Omega_R^2 R r \mu \sin \Psi
  + \mu^2 R^2 \Omega_R^2 \sin^2 \Psi \\
  \begin{multlined}[0.88\textwidth]
    U_P U_T = \lambda \Omega_R^2 R r
    - \beta_{1s} \Omega_R^2 r^2 \cos \Psi
    + \beta_{1c} \Omega_R^2 r^2 \sin \Psi \\
    - \beta \mu \Omega_R^2 R r \cos \Psi
    + \lambda \mu \Omega_R^2 R^2 \sin \Psi
    - \beta_{1s} \mu \Omega_R^2 R r \sin \Psi \cos \Psi \\
    + \beta_{1c} \mu \Omega_R^2 R r \sin^2 \Psi
    - \beta \mu^2 \Omega_R^2 R^2 \sin \Psi \cos \Psi
  \end{multlined}
\end{gather}

Knowing that: \cite{GessowMyers1985}
\begin{gather}
  \frac{1}{2\pi} \int_{0}^{2\pi} \sin \Psi d\Psi =
  \frac{1}{2\pi} \int_{0}^{2\pi} \cos \Psi d\Psi = 0 \\
  \frac{1}{2\pi} \int_{0}^{2\pi} \sin^2 \Psi d\Psi =
  \frac{1}{2\pi} \int_{0}^{2\pi} \cos^2 \Psi d\Psi = \frac{1}{2} \\
  \frac{1}{2\pi} \int_{0}^{2\pi} \sin \Psi \cos \Psi d\Psi = 0
\end{gather}

Rotor thrust can be written as:
\begin{equation}
  T = \frac{1}{2} \rho a c_b n_b \Omega_R^2 R^3 B
  \left[
    \frac{\lambda B}{2}
    +
    \frac{\theta}{3} \left( B^2 + \frac{3}{2} \mu^2 \right)
    +
    \frac{\beta_{1c} B}{4} \mu
  \right]
\end{equation}

Using expression (\ref{eq-aero-thrust-coef}) rotor thrust coefficient is given by the following formula:
\begin{equation}
  C_T = \frac{1}{2} a s B
  \left[
    \frac{\lambda B}{2}
    +
    \frac{\theta}{3} \left( B^2 + \frac{3}{2} \mu^2 \right)
    +
    \frac{\beta_{1c} B}{4} \mu
  \right]
\end{equation}

Where $s$ is rotor solidity.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Rotor H Force}

%TODO

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Rotor Y Force}

%TODO

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Rotor Torque}

The torque on a blade element is given by the following formula. \cite{GessowMyers1985, Bramwell2001}
\begin{equation}
  \label{eq-aero-blade-section-torque}
  dQ = r \left( dD \cos \phi - dL \sin \phi \right) dr
\end{equation}

Assuming that for small angles:
\begin{align}
  \sin \phi &\approx \phi \\
  \cos \phi &\approx 1
\end{align}

And assuming that drag coefficient is constant along blade span, expression (\ref{eq-aero-blade-section-torque}) can be written as: \cite{GessowMyers1985, Bramwell2001}
\begin{equation}
  dQ =
  \frac{1}{2} \rho U_T^2 C_D c_b r dr -
  \frac{1}{2} \rho U_T^2 C_L c_b r \phi dr
\end{equation}

Torque due to the profile drag can be expressed as: \cite{Bramwell2001}
\begin{equation}
  Q_p = \frac{n_b}{2\pi}
  \int_{0}^{R} \int_{0}^{2\pi} \frac{1}{2} \rho U_T^2 C_D c_b r d \Psi dr
\end{equation}

Substituting (\ref{eq-aero-u-t-sq}) and integrating this equation first with respect to the azimuth then along the blade span gives:
\begin{equation}
  Q_p = \frac{1}{2} \rho n_b c_b \Omega_R^2 R^4 C_D
  \left( \frac{1}{4} + \frac{1}{4} \mu^2 \right)
\end{equation}

Induced torque is given by the following formula: \cite{Bramwell2001}
\begin{equation}
  Q_i = \frac{n_b}{2\pi} \int_{0}^{R} \int_{0}^{2\pi}
  \frac{1}{2} \rho U_T^2 C_L c_b r \phi d \Psi dr
\end{equation}

Substituting (\ref{eq-aero-approx-phi}) and (\ref{eq-aero-rotor-list-coef}) gives:
\begin{equation}
  \label{eq-aero-q-i}
  Q_i = \frac{n_b}{2\pi} \frac{1}{2} \rho a c_b
  \int_{0}^{R} \int_{0}^{2\pi}
  \left( \theta U_P U_T r + U_P^2 r \right) d \Psi dr
\end{equation}

Neglecting helicopter angular velocity expressions for $U_P^2$ can be written as:
\begin{multline}
  \label{eq-aero-u-p-sq}
  U_P^2 = \dot \beta^2 r^2
  + 2 \beta \dot \beta \mu \Omega_R R r \cos \Psi
  - 2 \dot \beta \lambda \Omega_R R r \\
  + \beta^2 \mu^2 \Omega_R^2 R^2 \cos^2 \Psi
  - 2 \beta \lambda \mu \Omega_R^2 R^2 \cos \Psi
  + \lambda^2 \Omega_R^2 R^2
\end{multline}

Substituting (\ref{eq-aero-u-p-u-t}) and (\ref{eq-aero-u-p-sq}) into (\ref{eq-aero-q-i}) and integrating equation (\ref{eq-aero-q-i}) first with respect to the azimuth then along the blade span gives:
\begin{multline}
Q_i = \frac{1}{2} \rho a c_b n_b \Omega_R^2 R^4
\left[
    \frac{1}{3} \lambda \theta
  + \frac{1}{2} \lambda^2
  - \frac{1}{8} \left( \beta_{1c}^2 + \beta_{1s}^2 \right)
  \right.
  \\
  \left.
  + \frac{1}{2} \mu^2
  \left(
    \frac{\beta_0^2}{2}
    - \frac{3}{8} \beta_{1c}^2
    - \frac{1}{8} \beta_{1s}^2
  \right)
  - \frac{1}{2} \mu \lambda \beta_{1c}
  + \frac{1}{3} \mu \beta_0 \beta_{1s}
\right]
\end{multline}

The total rotor torque is given as follows: \cite{Bramwell2001, NASA-TT-F-494}
\begin{gather}
  Q = Q_p - Q_i \\
  \begin{multlined}[0.88\textwidth]
    Q = \frac{1}{2} \rho a c_b n_b \Omega_R^2 R^4
    \left[
      \frac{C_D}{4a} \left( 1 + \mu^2 \right)
      - \frac{1}{3} \lambda \theta
      - \frac{1}{2} \lambda^2
      + \frac{1}{8} \left( \beta_{1c}^2 + \beta_{1s}^2 \right)
      \right.
      \\
      \left.
      - \frac{1}{2} \mu^2
      \left(
        \frac{\beta_0^2}{2}
        - \frac{3}{8} \beta_{1c}^2
        - \frac{1}{8} \beta_{1s}^2
      \right)
      + \frac{1}{2} \mu \lambda \beta_{1c}
      - \frac{1}{3} \mu \beta_0 \beta_{1s}
    \right]
  \end{multlined}
\end{gather}

Rotor torque coefficient can be written as:
\begin{multline}
  C_Q = \frac{1}{2} a s
  \left[
    \frac{C_D}{4a} \left( 1 + \mu^2 \right)
    - \frac{1}{3} \lambda \theta
    - \frac{1}{2} \lambda^2
    + \frac{1}{8} \left( \beta_{1c}^2 + \beta_{1s}^2 \right)
    \right.
    \\
    \left.
    - \frac{1}{2} \mu^2
    \left(
      \frac{\beta_0^2}{2}
      - \frac{3}{8} \beta_{1c}^2
      - \frac{1}{8} \beta_{1s}^2
    \right)
    + \frac{1}{2} \mu \lambda \beta_{1c}
    - \frac{1}{3} \mu \beta_0 \beta_{1s}
  \right]
\end{multline}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Flapping Coefficients}

Expressions for blades flapping coefficients can be derived from the equation of moments equilibrium about flapping hinge using method described in \cite{NASA-TT-F-494}.

\begin{figure}
  \centering
  \includegraphics[width=100mm]{eps/blade_element_moments.eps}
  \caption{Forces acting on the blade element}
\end{figure}

\begin{minipage}{\textwidth}
  Moments equilibrium about flapping hinge can be written as follows: \cite{GessowMyers1985}
  \begin{equation}
    \label{eq-aero-blade-moments-equilibrium}
    M_I + M_{CF} + M_C + M_T + M_W = 0
  \end{equation}

  Where:
  \begin{description}[align=right,labelwidth=1.5cm]
    \item [$M_I$]    [N$\cdot$m] moment due to inertia forces of flapping
    \item [$M_{CF}$] [N$\cdot$m] moment due to centrifugal force
    \item [$M_C$]    [N$\cdot$m] moment due to Coriolis force
    \item [$M_T$]    [N$\cdot$m] moment due to thrust
    \item [$M_W$]    [N$\cdot$m] moment due to weight
  \end{description}
\end{minipage}

\subsection{Moments of Inertia Forces}

Assuming that rotor revolution speed is constant, helicopter angular velocities are constant and rotor blades are able to move about flapping hinge axis while neglecting helicopter yaw motion and blades pitching and lagging motion then centrifugal force can be written as:
\begin{equation}
  dF_{CF} = m_b \Omega_R^2 r dr
\end{equation}

Component of the Coriolis force laying on the flapping plane is given:

--- for counterclockwise direction of rotor:
\begin{equation}
  dF_C =
    2 m_b p_{cw} \Omega_R r \cos \Psi dr
  - 2 m_b q_{cw} \Omega_R r \sin \Psi dr
\end{equation}

--- for clockwise direction of rotor:
\begin{equation}
  dF_C =
  - 2 m_b p_{cw} \Omega_R r \cos \Psi dr
  - 2 m_b q_{cw} \Omega_R r \sin \Psi dr
\end{equation}

Moment of inertia forces of flapping is given by the following formula:
\begin{equation}
  M_I = - \ddot \beta \int_{0}^{R} m_b r^2 dr
\end{equation}

Assuming that rotor blades are homogeneous rods blade first moment of mass and moment of inertia can be written as: \cite{NASA-TT-F-494}
\begin{align}
  \label{eq-aero-j-b}
  I_b &\approx \int_{0}^{R} m_b r^2 dr \\
  \label{eq-aero-s-b}
  S_b &\approx \int_{0}^{R} m_b r dr
\end{align}

Hence:
\begin{equation}
  M_I = - \ddot \beta I_b
\end{equation}

Taking into account (\ref{eq-aero-sin-beta-approx}) and (\ref{eq-aero-cos-beta-approx}) moment of centrifugal forces is:
\begin{equation}
  \label{eq-aero-m-cf}
  M_{CF} =
  - \int_{0}^{R} \beta m_b \Omega_R^2 r^2 dr =
  - \Omega_R^2 \beta \int_{0}^{R} m_b r^2 dr
\end{equation}

Substituting (\ref{eq-aero-j-b}) into (\ref{eq-aero-m-cf}) gives:
\begin{equation}
  M_{CF} = - \Omega_R^2 \beta I_b
\end{equation}

Moment of Coriolis forces can be writes as:

--- for counterclockwise direction of rotor:
\begin{multline}
  M_C =
    2 \int_{0}^{R} m_b p_{cw} \Omega_R r^2 \cos \Psi dr
  - 2 \int_{0}^{R} m_b q_{cw} \Omega_R r^2 \sin \Psi dr
  = \\ =
  2 p_{cw} \Omega_R I_b \cos \Psi - 2 q_{cw} \Omega_R I_b \sin \Psi
\end{multline}

--- for clockwise direction of rotor:
\begin{multline}
  M_C =
  - 2 \int_{0}^{R} m_b p_{cw} \Omega_R r^2 \cos \Psi dr
  - 2 \int_{0}^{R} m_b q_{cw} \Omega_R r^2 \sin \Psi dr
  = \\ =
  -2 p_{cw} \Omega_R I_b \cos \Psi - 2 q_{cw} \Omega_R I_b \sin \Psi
\end{multline}

Using approximation (\ref{eq-aero-s-b}) moment due to weight can expressed as:
\begin{equation}
  M_W = -g \int_{0}^{R} m_b r dr = -g S_b
\end{equation}

\subsection{Moment of Thrust}

Using equation (\ref{eq-aero-blade-section-thrust}) expression for moment of thrust about flapping hinge can be written as follows:
\begin{equation}
  M_T =
  \int_{0}^{BR} dT r =
  \frac{1}{2} \rho a c_b
  \int_{0}^{BR} \left( \theta U_T^2 + U_P U_T \right) dr
\end{equation}

\subsection{Equilibrium of Moments about Flapping Hinge}

Substituting into (\ref{eq-aero-blade-moments-equilibrium}) expressions for moments of thrust, weight, inertia, centrifugal and Coriolis forces moments equilibrium equation is given as:

--- for counterclockwise direction of rotor:
\begin{gather}
  \begin{multlined}[0.88\textwidth]
    \int_{0}^{BR} dT r
    - \ddot \beta I_b
    - \beta \Omega_R^2 I_b
    + 2 p_{cw} \Omega_R I_b \cos \Psi \\
    - 2 q_{cw} \Omega_R I_b \sin \Psi
    - g S_b
    = 0
  \end{multlined}
  \\
  \begin{multlined}[0.88\textwidth]
    \label{eq-aero-equilibrium-blade-moments-ccw-1}
    - I_b \ddot \beta
    - I_b \beta \Omega_R^2
    + \int_{0}^{BR} dT r
    = 2 I_b q_{cw} \Omega_R \sin \Psi \\
    - 2 I_b p_{cw} \Omega_R \cos \Psi
    + g S_b
  \end{multlined}
\end{gather}

--- for clockwise direction of rotor:
\begin{gather}
  \begin{multlined}[0.88\textwidth]
    \int_{0}^{BR} dT r
    - \ddot \beta I_b
    - \beta \Omega_R^2 I_b
    - 2 p_{cw} \Omega_R I_b \cos \Psi \\
    - 2 q_{cw} \Omega_R I_b \sin \Psi
    - g S_b
    = 0
  \end{multlined}
  \\
  \begin{multlined}[0.88\textwidth]
    \label{eq-aero-equilibrium-blade-moments-cw-1}
    - I_b \ddot \beta
    - I_b \beta \Omega_R^2
    + \int_{0}^{BR} dT r
    = 2 I_b q_{cw} \Omega_R \sin \Psi \\
    + 2 I_b p_{cw} \Omega_R \cos \Psi
    + g S_b
  \end{multlined}
\end{gather}

Dividing both sides of equations (\ref{eq-aero-equilibrium-blade-moments-ccw-1}) and (\ref{eq-aero-equilibrium-blade-moments-cw-1}) by $I_b \Omega_R^2$ gives:
\begin{gather}
  \label{eq-aero-equilibrium-blade-moments-ccw-2}
  \bar{\bar \beta} + \beta
  = \frac{1}{ I_b \Omega_R^2 } \int_{0}^{BR} dT r
  - 2 \frac{q_{cw}}{\Omega_R} \sin \Psi
  + 2 \frac{p_{cw}}{\Omega_R} \cos \Psi
  - \frac{g S_b}{I_b \Omega_R^2} \\
  \label{eq-aero-equilibrium-blade-moments-cw-2}
    \bar{\bar \beta} + \beta
  = \frac{1}{ I_b \Omega_R^2 } \int_{0}^{BR} dT r
  - 2 \frac{q_{cw}}{\Omega_R} \sin \Psi
  - 2 \frac{p_{cw}}{\Omega_R} \cos \Psi
  - \frac{g S_b}{I_b \Omega_R^2}
\end{gather}

Substituting expressions (\ref{eq-aero-u-t-sq}) and (\ref{eq-aero-u-p-u-t}) into (\ref{eq-aero-equilibrium-blade-moments-ccw-2}) and (\ref{eq-aero-equilibrium-blade-moments-cw-2}) gives:
\begin{multline}
  \label{eq-aero-blade-moment-thrust-ccw}
  M_T =
  \int_{0}^{BR} dT r =
  \frac{1}{2} \rho a c_b
  \int_{0}^{BR} \left( \theta U_T^2 + U_P U_T \right) r dr = \\
  =
  \frac{1}{2} \rho a c_b R^4 \Omega_R^2
  \left(
    \frac{B^4}{4} \theta
  + \frac{2}{3} B^3 \theta \mu \sin \Psi
  + \frac{B^2}{2} \theta \mu^2 \sin^2 \Psi
  \right.
  \\
  + \frac{B^3}{3} \lambda
  - \frac{B^4}{4} \bar \beta
  - \frac{B^3}{3} \beta \mu \cos \Psi
  + \frac{B^4}{4} \frac{p_{cw}}{\Omega_R} \sin \Psi \\
  + \frac{B^2}{2} \lambda \mu \sin \Psi
  - \frac{B^2}{2} \beta \mu^2 \sin \Psi \cos \Psi
  - \frac{B^3}{3} \bar \beta \mu \sin \Psi
  \\
  \left.
  + \frac{B^3}{3} \frac{p_{cw}}{\Omega_R} \mu \sin^2 \Psi
  + \frac{B^4}{4} \frac{q_{cw}}{\Omega_R} \cos \Psi
  + \frac{B^3}{3} \frac{q_{cw}}{\Omega_R} \mu \sin \Psi \cos \Psi
  \right)
\end{multline}
\begin{multline}
  \label{eq-aero-blade-moment-thrust-cw}
  M_T =
  \int_{0}^{BR} dT r =
  \frac{1}{2} \rho a c_b
  \int_{0}^{BR} \left( \theta U_T^2 + U_P U_T \right) r dr = \\
  =
  \frac{1}{2} \rho a c_b R^4 \Omega_R^2
  \left(
    \frac{B^4}{4} \theta
  + \frac{2}{3} B^3 \theta \mu \sin \Psi
  + \frac{B^2}{2} \theta \mu^2 \sin^2 \Psi
  \right.
  \\
  + \frac{B^3}{3} \lambda
  - \frac{B^4}{4} \bar \beta
  - \frac{B^3}{3} \beta \mu \cos \Psi
  - \frac{B^4}{4} \frac{p_{cw}}{\Omega_R} \sin \Psi \\
  + \frac{B^2}{2} \lambda \mu \sin \Psi
  - \frac{B^2}{2} \beta \mu^2 \sin \Psi \cos \Psi
  - \frac{B^3}{3} \bar \beta \mu \sin \Psi
  \\
  \left.
  - \frac{B^3}{3} \frac{p_{cw}}{\Omega_R} \mu \sin^2 \Psi
  + \frac{B^4}{4} \frac{q_{cw}}{\Omega_R} \cos \Psi
  + \frac{B^3}{3} \frac{q_{cw}}{\Omega_R} \mu \sin \Psi \cos \Psi
  \right)
\end{multline}

Knowing that:
\begin{align}
  \label{eq-aero-trigonometric-4}
  \sin \Psi \cos \Psi &= \frac{ \sin \left( 2 \Psi \right) }{2} \\
  \label{eq-aero-trigonometric-5}
  \sin^2 \Psi &= \frac{ 1 - \cos \left( 2 \Psi \right) }{2} \\
  \label{eq-aero-trigonometric-6}
  \cos^2 \Psi &= \frac{ 1 + \cos \left( 2 \Psi \right) }{2} \\
  \label{eq-aero-trigonometric-7}
  \cos \Psi \sin \left( 2 \Psi \right) &=
  \frac{ \sin \Psi + \sin \left( 3 \Psi \right) }{2} \\
  \label{eq-aero-trigonometric-8}
  \sin \Psi \sin \left( 2 \Psi \right) &=
  \frac{ \cos \Psi - \cos \left( 3 \Psi \right) }{2}
\end{align}

Substituting (\ref{eq-aero-blade-moment-thrust-ccw}) into (\ref{eq-aero-equilibrium-blade-moments-ccw-2}) and (\ref{eq-aero-blade-moment-thrust-cw}) into (\ref{eq-aero-equilibrium-blade-moments-cw-2}) gives:
\begin{multline}
  \label{eq-aero-flapping-coefs-ccw-1}
  \bar{\bar \beta} + \bar \beta \frac{\gamma}{2}
  \left( \frac{B^4}{4} + \frac{B^3}{3} \mu \sin \Psi \right)
  + \beta \left(
    1 + \frac{B^3}{6} \gamma \mu \cos \Psi
    + \frac{B^2}{8} \gamma \mu^2 \sin \left( 2 \Psi \right)
  \right)
  = \\ =
  \frac{\gamma}{2}
  \left(
      \frac{B^3}{3} \lambda
    + \frac{B^4}{4} \frac{p_{cw}}{\Omega_R} \sin \Psi
    + \frac{B^2}{2} \lambda \mu \sin \Psi
    + \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
    - \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu \cos \left( 2 \Psi \right)
    \right.
    \\
    + \frac{B^4}{4} \frac{q_{cw}}{\Omega_R} \cos \Psi
    + \frac{B^3}{6} \frac{q_{cw}}{\Omega_R} \mu \sin \left( 2 \Psi \right)
    + \frac{B^4}{4} \theta
    \\
    \left.
    + \frac{2}{3} B^3 \theta \mu \sin \Psi
    + \frac{B^2}{4} \theta \mu^2
    - \frac{B^2}{4} \theta \mu^2 \cos \left( 2 \Psi \right)
  \right)
  \\
  - 2 \frac{q_{cw}}{\Omega_R} \sin \Psi
  + 2 \frac{p_{cw}}{\Omega_R} \cos \Psi
  - \frac{ g S_b }{ I_b \Omega_R^2 }
\end{multline}
\begin{multline}
  \label{eq-aero-flapping-coefs-cw-1}
  \bar{\bar \beta} + \bar \beta \frac{\gamma}{2}
  \left( \frac{B^4}{4} + \frac{B^3}{3} \mu \sin \Psi \right)
  + \beta \left(
    1 + \frac{B^3}{6} \gamma \mu \cos \Psi
    + \frac{B^2}{8} \gamma \mu^2 \sin \left( 2 \Psi \right)
  \right)
  = \\ =
  \frac{\gamma}{2}
  \left(
      \frac{B^3}{3} \lambda
    - \frac{B^4}{4} \frac{p_{cw}}{\Omega_R} \sin \Psi
    + \frac{B^2}{2} \lambda \mu \sin \Psi
    - \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
    + \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu \cos \left( 2 \Psi \right)
    \right.
    \\
    + \frac{B^4}{4} \frac{q_{cw}}{\Omega_R} \cos \Psi
    + \frac{B^3}{6} \frac{q_{cw}}{\Omega_R} \mu \sin \left( 2 \Psi \right)
    + \frac{B^4}{4} \theta
    \\
    \left.
    + \frac{2}{3} B^3 \theta \mu \sin \Psi
    + \frac{B^2}{4} \theta \mu^2
    - \frac{B^2}{4} \theta \mu^2 \cos \left( 2 \Psi \right)
  \right)
  \\
  - 2 \frac{q_{cw}}{\Omega_R} \sin \Psi
  - 2 \frac{p_{cw}}{\Omega_R} \cos \Psi
  - \frac{ g S_b }{ I_b \Omega_R^2 }
\end{multline}

Where $\gamma$ is a blade Lock number.

Transforming equations (\ref{eq-aero-flapping-coefs-ccw-1}) and (\ref{eq-aero-flapping-coefs-cw-1}) gives:
\begin{multline}
  \label{eq-aero-flapping-coefs-ccw-2}
  \bar{\bar \beta} + \bar \beta \frac{\gamma}{2}
  \left( \frac{B^4}{4} + \frac{B^3}{3} \mu \sin \Psi \right)
  + \beta \left(
    1 + \frac{B^3}{6} \gamma \mu \cos \Psi
    + \frac{B^2}{8} \gamma \mu^2 \sin \left( 2 \Psi \right)
  \right)
  = \\ =
  \left[
    \frac{\gamma}{2}
    \left(
        \frac{B^3}{3} \lambda
      + \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
      + \frac{B^4}{4} \theta
      + \frac{B^2}{4} \theta \mu^2
    \right)
    - \frac{ g S_b }{ I_b \Omega_R^2 }
  \right]
  \\
  + \left[
    \frac{\gamma}{2}
    \left( \frac{B^4}{4} \frac{q_{cw}}{\Omega_R} \right)
    + 2 \frac{p_{cw}}{\Omega_R}
  \right] \cos \Psi
  \\
  + \left[
    \frac{\gamma}{2}
    \left(
        \frac{B^4}{4} \frac{p_{cw}}{\Omega_R}
      + \frac{B^2}{2} \lambda \mu
      + \frac{2}{3} B^3 \theta \mu
    \right)
    - 2 \frac{q_{cw}}{\Omega_R}
  \right] \sin \Psi
  \\
  + \left[
    \frac{\gamma}{2}
    \left(
      - \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
      - \frac{B^2}{4} \theta \mu^2
    \right)
  \right] \cos \left( 2 \Psi \right)
  \\
  + \left[
    \frac{\gamma}{2}
    \left( \frac{B^3}{6} \frac{q_{cw}}{\Omega_R} \mu \right)
  \right] \sin \left( 2 \Psi \right)
\end{multline}

\vfill

\begin{multline}
  \label{eq-aero-flapping-coefs-cw-2}
  \bar{\bar \beta} + \bar \beta \frac{\gamma}{2}
  \left( \frac{B^4}{4} + \frac{B^3}{3} \mu \sin \Psi \right)
  + \beta \left(
    1 + \frac{B^3}{6} \gamma \mu \cos \Psi
    + \frac{B^2}{8} \gamma \mu^2 \sin \left( 2 \Psi \right)
  \right)
  = \\ =
  \left[
    \frac{\gamma}{2}
    \left(
        \frac{B^3}{3} \lambda
      - \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
      + \frac{B^4}{4} \theta
      + \frac{B^2}{4} \theta \mu^2
    \right)
    - \frac{ g S_b }{ I_b \Omega_R^2 }
  \right]
  \\
  + \left[
    \frac{\gamma}{2}
    \left( \frac{B^4}{4} \frac{q_{cw}}{\Omega_R} \right)
    - 2 \frac{p_{cw}}{\Omega_R}
  \right] \cos \Psi
  \\
  + \left[
    \frac{\gamma}{2}
    \left(
      - \frac{B^4}{4} \frac{p_{cw}}{\Omega_R}
      + \frac{B^2}{2} \lambda \mu
      + \frac{2}{3} B^3 \theta \mu
    \right)
    - 2 \frac{q_{cw}}{\Omega_R}
  \right] \sin \Psi
  \\
  + \left[
    \frac{\gamma}{2}
    \left(
        \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
      - \frac{B^2}{4} \theta \mu^2
    \right)
  \right] \cos \left( 2 \Psi \right)
  \\
  + \left[
    \frac{\gamma}{2}
    \left( \frac{B^3}{6} \frac{q_{cw}}{\Omega_R} \mu \right)
  \right] \sin \left( 2 \Psi \right)
\end{multline}

\vfill

Differentiating equation (\ref{eq-aero-blade-flapping-angle}) gives:
\begin{align}
  \label{eq-aero-beta-prim}
  \bar \beta &=
  \left( \beta_{1s} \cos \Psi - \beta_{1c} \sin \Psi \right) \\
  \label{eq-aero-beta-bis}
  \bar{\bar \beta} &= -
  \left( \beta_{1c} \cos \Psi + \beta_{1s} \sin \Psi  \right)
\end{align}

Substituting expressions (\ref{eq-aero-beta-prim}) and (\ref{eq-aero-beta-bis}) into equations (\ref{eq-aero-flapping-coefs-ccw-2}) and (\ref{eq-aero-flapping-coefs-cw-2}) as well as using trigonometric identities (\ref{eq-aero-trigonometric-4}), (\ref{eq-aero-trigonometric-5}), (\ref{eq-aero-trigonometric-6}), (\ref{eq-aero-trigonometric-7}) and (\ref{eq-aero-trigonometric-8}) gives:
\begin{gather}
  \label{eq-aero-flapping-coefs-ccw-3}
  \begin{multlined}[0.88\textwidth]
    \beta_0
    +
    \beta_{1s} \gamma \frac{B^2}{8}
    \left( \frac{\mu^2}{2} + B^2 \right) \cos \Psi
    +
    \beta_{1c} \gamma \frac{B^2}{8}
    \left( \frac{\mu^2}{2} - B^2 \right) \sin \Psi
    \\
    +
    \beta_{1c} \gamma \frac{B^3}{6} \mu \cos \left( 2 \Psi \right)
    +
    \left(
        \beta_{1s} \gamma \frac{B^3}{6} \mu
      + \beta_0 \gamma \frac{B^2}{8} \mu^2
    \right) \sin \left( 2 \Psi \right)
    \\
    +
    \beta_{1c} \gamma \frac{B^2}{16} \mu^2 \sin \left( 3 \Psi \right)
    -
    \beta_{1s} \gamma \frac{B^2}{16} \mu^2 \cos \left( 3 \Psi \right)
    = \\ =
    \left[
      \frac{\gamma}{2}
      \left(
          \frac{B^3}{3} \lambda
        + \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
        + \frac{B^4}{4} \theta
        + \frac{B^2}{4} \theta \mu^2
      \right) - \frac{ g S_b }{ I_b \Omega_R^2 }
    \right]
    \\
    + \left[
      \frac{\gamma}{2}
      \left(
          \frac{B^4}{4} \frac{q_{cw}}{\Omega_R}
        - \frac{B^3}{3} \beta_0 \mu
      \right) + 2 \frac{p_{cw}}{\Omega_R}
    \right] \cos \Psi
    \\
    + \left[
      \frac{\gamma}{2}
      \left(
          \frac{B^4}{4} \frac{p_{cw}}{\Omega_R}
        + \frac{B^2}{2} \lambda \mu
        + \frac{2}{3} B^3 \theta \mu
        \right) - 2 \frac{q_{cw}}{\Omega_R}
    \right] \sin \Psi
    \\
    + \left[
      \frac{\gamma}{2}
      \left(
        - \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
        - \frac{B^2}{4} \theta \mu^2
      \right)
    \right] \cos \left( 2 \Psi \right)
    \\
    + \left[
      \frac{\gamma}{2}
      \left( \frac{B^3}{6} \frac{q_{cw}}{\Omega_R} \mu \right)
    \right] \sin \left( 2 \Psi \right)
  \end{multlined}
  \\
  \label{eq-aero-flapping-coefs-cw-3}
  \begin{multlined}[0.88\textwidth]
    \beta_0
    +
    \beta_{1s} \gamma \frac{B^2}{8}
    \left( \frac{\mu^2}{2} + B^2 \right) \cos \Psi
    +
    \beta_{1c} \gamma \frac{B^2}{8}
    \left( \frac{\mu^2}{2} - B^2 \right) \sin \Psi
    \\
    +
    \beta_{1c} \gamma \frac{B^3}{6} \mu \cos \left( 2 \Psi \right)
    +
    \left(
        \beta_{1s} \gamma \frac{B^3}{6} \mu
      + \beta_0 \gamma \frac{B^2}{8} \mu^2
    \right) \sin \left( 2 \Psi \right)
    \\
    -
    \beta_{1s} \gamma \frac{B^2}{16} \mu^2 \cos \left( 3 \Psi \right)
    +
    \beta_{1c} \gamma \frac{B^2}{16} \mu^2 \sin \left( 3 \Psi \right)
    = \\ =
    \left[
      \frac{\gamma}{2}
      \left(
          \frac{B^3}{3} \lambda
        - \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
        + \frac{B^4}{4} \theta
        + \frac{B^2}{4} \theta \mu^2
      \right) - \frac{ g S_b }{ I_b \Omega_R^2 }
    \right]
    \\
    + \left[
      \frac{\gamma}{2}
      \left(
          \frac{B^4}{4} \frac{q_{cw}}{\Omega_R}
        - \frac{B^3}{3} \beta_0 \mu
      \right) - 2 \frac{p_{cw}}{\Omega_R}
    \right] \cos \Psi
    \\
    + \left[
      \frac{\gamma}{2}
      \left(
        - \frac{B^4}{4} \frac{p_{cw}}{\Omega_R}
        + \frac{B^2}{2} \lambda \mu
        + \frac{2}{3} B^3 \theta \mu
        \right) - 2 \frac{q_{cw}}{\Omega_R}
    \right] \sin \Psi
    \\
    + \left[
      \frac{\gamma}{2}
      \left(
          \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
        - \frac{B^2}{4} \theta \mu^2
      \right)
    \right] \cos \left( 2 \Psi \right)
    \\
    + \left[
      \frac{\gamma}{2}
      \left( \frac{B^3}{6} \frac{q_{cw}}{\Omega_R} \mu \right)
    \right] \sin \left( 2 \Psi \right)
  \end{multlined}
\end{gather}

Neglecting all blade flapping harmonics above the first \cite{GessowMyers1985} equations (\ref{eq-aero-flapping-coefs-ccw-3}) and (\ref{eq-aero-flapping-coefs-cw-3}) can be simplified as follows:

\begin{multline}
  \label{eq-aero-flapping-coefs-ccw-4}
  \beta_0
  +
  \beta_{1s} \gamma \frac{B^2}{8}
  \left( \frac{\mu^2}{2} + B^2 \right) \cos \Psi
  +
  \beta_{1c} \gamma \frac{B^2}{8}
  \left( \frac{\mu^2}{2} - B^2 \right) \sin \Psi
  = \\ =
  \left[
    \frac{\gamma}{2}
    \left(
        \frac{B^3}{3} \lambda
      + \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
      + \frac{B^4}{4} \theta
      + \frac{B^2}{4} \theta \mu^2
    \right) - \frac{ g S_b }{ I_b \Omega_R^2 }
  \right]
  \\
  + \left[
    \frac{\gamma}{2}
    \left(
      \frac{B^4}{4} \frac{q_{cw}}{\Omega_R}
      - \frac{B^3}{3} \beta_0 \mu
    \right) + 2 \frac{p_{cw}}{\Omega_R}
  \right] \cos \Psi
  \\
  + \left[
    \frac{\gamma}{2}
    \left(
        \frac{B^4}{4} \frac{p_{cw}}{\Omega_R}
      + \frac{B^2}{2} \lambda \mu
      + \frac{2}{3} B^3 \theta \mu
    \right) - 2 \frac{q_{cw}}{\Omega_R}
  \right] \sin \Psi
\end{multline}

\begin{multline}
  \label{eq-aero-flapping-coefs-cw-4}
  \beta_0
  +
  \beta_{1s} \gamma \frac{B^2}{8}
  \left( \frac{\mu^2}{2} + B^2 \right) \cos \Psi
  +
  \beta_{1c} \gamma \frac{B^2}{8}
  \left( \frac{\mu^2}{2} - B^2 \right) \sin \Psi
  = \\ =
  \left[
    \frac{\gamma}{2}
    \left(
        \frac{B^3}{3} \lambda
      - \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
      + \frac{B^4}{4} \theta
      + \frac{B^2}{4} \theta \mu^2
    \right) - \frac{ g S_b }{ I_b \Omega_R^2 }
  \right]
  \\
  + \left[
    \frac{\gamma}{2}
    \left(
      \frac{B^4}{4} \frac{q_{cw}}{\Omega_R}
      - \frac{B^3}{3} \beta_0 \mu
    \right) - 2 \frac{p_{cw}}{\Omega_R}
  \right] \cos \Psi
  \\
  + \left[
    \frac{\gamma}{2}
    \left(
      - \frac{B^4}{4} \frac{p_{cw}}{\Omega_R}
      + \frac{B^2}{2} \lambda \mu
      + \frac{2}{3} B^3 \theta \mu
    \right) - 2 \frac{q_{cw}}{\Omega_R}
  \right] \sin \Psi
\end{multline}

\clearpage

\subsection{Final Form}

Equations (\ref{eq-aero-flapping-coefs-ccw-4}) and (\ref{eq-aero-flapping-coefs-cw-4}) can be transformed to get blade flapping coefficients:

--- for counterclockwise direction of rotor:
\begin{align}
  \beta_0 &=
  \frac{\gamma}{2}
  \left(
      \frac{B^3}{3} \lambda
    + \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
    + \frac{B^4}{4} \theta
    + \frac{B^2}{4} \theta \mu^2
  \right) - \frac{ g S_b }{ I_b \Omega_R^2 }
  \\
  \beta_{1c} &=
  2 \mu \left( \lambda + \frac{4}{3} B \theta \right)
  \cfrac{1}{\cfrac{\mu^2}{2} - B^2}
  +
  \left(
      B^4 \frac{p_{cw}}{\Omega_R}
    - 16 \frac{q_{cw}}{\gamma \Omega_R}
  \right)
  \cfrac{1}{B^2\left(\cfrac{\mu^2}{2}-B^2\right)}
  \\
  \beta_{1s} &=
  - \frac{4}{3} \beta_0 \mu
  \cfrac{B}{\cfrac{\mu^2}{2} + B^2}
  +
  \left(
      B^4 \frac{q_{cw}}{\Omega_R}
    + 16 \frac{p_{cw}}{\gamma \Omega_R}
  \right)
  \cfrac{1}{B^2 \left(\cfrac{\mu^2}{2}+B^2\right)}
\end{align}

--- for clockwise direction of rotor:
\begin{align}
  \beta_0 &=
  \frac{\gamma}{2}
  \left(
      \frac{B^3}{3} \lambda
    - \frac{B^3}{6} \frac{p_{cw}}{\Omega_R} \mu
    + \frac{B^4}{4} \theta
    + \frac{B^2}{4} \theta \mu^2
  \right) - \frac{ g S_b }{ I_b \Omega_R^2 }
  \\
  \beta_{1c} &=
  2 \mu \left( \lambda + \frac{4}{3} B \theta \right)
  \cfrac{1}{\cfrac{\mu^2}{2} - B^2}
  -
  \left(
      B^4 \frac{p_{cw}}{\Omega_R}
    + 16 \frac{q_{cw}}{\gamma \Omega_R}
  \right)
  \cfrac{1}{B^2\left(\cfrac{\mu^2}{2}-B^2\right)}
  \\
  \beta_{1s} &=
  - \frac{4}{3} \beta_0 \mu
  \cfrac{B}{\cfrac{\mu^2}{2} + B^2}
  +
  \left(
      B^4 \frac{q_{cw}}{\Omega_R}
    - 16 \frac{p_{cw}}{\gamma \Omega_R}
  \right)
  \cfrac{1}{B^2 \left(\cfrac{\mu^2}{2}+B^2\right)}
\end{align}


\graphicspath{ {imgs/} }
\documentclass[main.tex]{subfiles}
\externaldocument{02data.tex}
\externaldocument{04methods.tex}
\begin{document}
\chapter{Model}\label{chap:model}
With the building blocks defined in the previous chapter it is possible to construct a complete model for the task of lung nodule detection. This section describes in detail the used model and the learning process that was used to train it.

\section{Network Architecture}
The model is inspired by the network presented by Huang~\cite{huang2017lung}. It has three 3D convolutional layers and three dense layers. The full architecture can be seen in Figure~\ref{fig:net_struct} and is in the following sections explained from the input to output.

\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{net_struct.png}
\end{center}
\caption{Architecture of the neural network. Each of the convolutional layers is composed of a 3D convolution layer with the respective filter size followed by a batch normalization and a pooling layer. The pool size is $(2,2,2)$ with a stride of $(1,1,1)$. The structure of the neural network resembles the one described by Huang~\cite{huang2017lung}.}
\label{fig:net_struct}
\end{figure}

\subsection{Input}
The input to the network are the patches that have been cut and stored from the complete lung scan as described in Chapter~\ref{chap:data} with a shape of $(50x50x5)$. The patches are randomly augmented by flipping them in x and y plane (examples can be seen in \ref{fig:input}). The augmentation is applied to make the learned classification more robust against distortions in the input and aiding in generalization. This makes sense in the specific scenario since the nodules are growing in different shapes and locations in the lung and flipping them is not producing an impossible input to the network. There is also an additional parameter that allows for scaling the input in the $x,y$ plane.

\begin{figure}
\begin{center}
\includegraphics[scale=0.25]{patches-health.png}
\includegraphics[scale=0.25]{patches-nodule.png}
\end{center}
\caption{Input data for the cases of healthy and nodule patches. The image is taken from Tensorboard and shows in the case of nodules the random permutation of the input data.}
\label{fig:input}
\end{figure}

\subsection{Hidden Layers}
The convolutional part has three convolutional layers with $40$, $20$ and $20$ kernels each. The kernel size is $(3,3,3)$. This is in accordance with Huang et al.'s~\cite{huang2017lung} implementation. Each of them is followed by a batch normalization layer. Batch normalization is in TensorFlow implemented as described by Ioffe and Szegedy~\cite{ioffe2015batch}. The input of it is fed into a max-pooling layer with a pool size of $(2,2,2)$. The output of the last convolutional layer is flattened and fed to the dense layers. Two dense layers with $64$ neurons each and a ReLU activation function are used in this model. Their output is finally combined in two neurons, forming a 2D output of the network.

\subsection{Output}
The output of the network is the activation of the final two neurons. The class of the input is determined by the neuron with the higher activation in an one-hot design. Given the index of the maximum, is $0$ encoding a healthy $1$ encoding a nodule patch.

\section{Training}
During training, the network is operated with batches of the input data. Regularization methods used for this network include batch normalization and dropout. Batch normalization is already described in Section~\ref{ss:convlayer}. Dropout is another regularization method which during training drops random neurons of the network - training effectively several models at once, as discovered by Srivastava et al.~\cite{srivastava2014dropout}, which should increase the overall performance of the network. During training, no improved performance could be observed when applying dropout throughout the whole network. It was rather harmful if applied to the convolutional layers. Thus, the final model uses dropout only in the fully connected layers. The loss of the network is then computed by the softmax cross-entropy between the labels in their one-hot form and the output of the two neurons at the end of the network. The Adam Optimizer is used on this loss for adapting the network parameters. For each epoch, the training is done on the complete training set with randomly permuted batches of size $10$. The training time is set to 3 full days.

\end{document}


\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{amssymb,amsfonts,amsmath}
\usepackage{graphicx}
\usepackage{caption}
\usepackage{subfig}
\usepackage{color}
\usepackage[affil-it]{authblk}
\usepackage{multirow}
\usepackage{fullpage}
\usepackage{booktabs}
\usepackage{pdfsync}

%\usepackage[nofiglist, notablist, nomarkers]{endfloat}
\input{utils.tex}
%\title{Probabilistic learning of confounding factors in genetical
%genomics studies}
\title{Derivations and mathematical details of GPmix}
\author[1]{Christoph Lippert, Oliver Stegle}

\affil[1]{Department Empirical Inference,
Max Planck Institutes T\"ubingen, Germany}
\date{}

\captionsetup[subfloat]{listofformat=parens}
\newcommand{\OLI}[1]{{\color{blue}\fbox{OLI} #1}}
\makeatletter
\newcommand{\rmnum}[1]{\romannumeral #1}
\newcommand{\Rmnum}[1]{\expandafter\@slowromancap\romannumeral #1@}
\makeatother

\newcommand{\fix}{\marginpar{FIX}}
\newcommand{\new}{\marginpar{NEW}}

%\nipsfinalcopy % Uncomment for camera-ready version
%\newcommand{\B}[1]{\bm{#1}} Christoph:removed this to make things consistent
\newcommand{\B}[1]{{\bf{#1}}}
\newcommand{\Exp}{\mathbb{E}}
\newcommand{\norma}[1]{\mathcal{N}\left(#1\right)}
\newcommand{\eref}[1]{(\ref{#1})}
\renewcommand{\R}{\mathbb{R}}
\newcommand\norm[1]{\left\Vert {#1} \right\Vert}
\newcommand\rank{\mathrm{rank}}
\newcommand\tr{\mathrm{Tr}}
\newcommand\Normal[3]{\normal{#1}{{#2}\;;\;{#3}}}
\newcommand\ve[1]{\text{vec}\left(#1\right)}
\newcommand\Real{\mathbb{R}}
\newcommand{\Ykron}{\B{U}_R^\T\B{Y}\B{U}_C}
\newcommand{\XWAkron}[1]{\B{U}_R^\T\B{X}_{#1}\B{W}_{#1}\B{A}_{#1}\B{U}_C}
\newcommand{\XWAkronT}[1]{\B{U}_C^\T\B{A}_{#1}^\T\B{W}_{#1}^{\T}\B{X}_{#1}^{\T}\B{U}_R}
\begin{document}
\maketitle

\section{Some definitions}
\begin{itemize}
\item $\B{A}=\B{U}\B{S}\B{U}^{\T}$ is the eigenvalue decomposition of
  the symmetric $D$-by-$D$ matrix $\B{A}$, where $\B{U}$ is an
  $D$-by-$D$ orthonormal matrix, holding the $D$ eigenvectors of
  $\B{A}$ and $\B{S}$ is an $D$-by-$D$ diagonal matrix, holding the
  corresponding eigenvalues of $\B{A}$ as diagonal entries.
\item $\B{A}\odot\B{B}$ is the pointwise or Hadamard product of $\B{A}$ and $\B{B}$.
\item $\B{A}\otimes\B{B}$ is the Kronecker product of $\B{A}$ and $\B{B}$.
\item $\B{Y}\in\mathbb{R}^{N\times G}$ is the matrix holding all samples, having $N$ rows and $G$ columns.
\end{itemize}

\section{Kronecker testing strategies and models}
$\B{A}_j\in\Real^{C\times M_j}$ is a matrix, that replicates the
$j$-th fixed effects matrix $\B{X}_j\in\Real^{R\times D_j}$. $\B{A}_j^\T$
typically would be a binary matrix, but could in principle be
anything. Using different versions of $\B{A}_j^\T$ corresponds to chosing
a testing strategy. For example, when $\B{A}_j^\T$ is the $C\times C$
Identity matrix, then one would fit an independent weight to every
column of $\B{Y}$, when $\B{A}^\T_j$ is a row-vector of ones, then one
would fit a single joint weight to all columns of $\B{Y}$.
\begin{equation}
\Normal{\ve{\B{Y}}}{\sum_j\B{A}^{\T}_j\otimes\B{X}_j\ve{\B{W}_j}}{\B{C}\otimes\B{R}
+ \sigma^2\B{I}}
\end{equation}
As long as $D_j\leq R$, $M_j\leq C$, the rank of $\B{X}_j$ is $D_j$
and the rank of $\B{A}_j$ is $M_j$, and the rank of
$\left[\B{X}_1...\B{X}_J\right]$ is $\sum_{j=1}^JM_j$ the number of
degrees of freedom of a single $\B{W}_j$ is $D_j \cdot M_j$
(sufficient condition).
\section{Efficient computation of tensor GP models}
\begin{equation}
\log\Normal{\ve{\B{Y}}}{\sum_{j=1}^J\B{A}^{\T}_j\otimes\B{X}_j\ve{\B{W}_j}}{\B{C}\otimes\B{R}
+ \sigma^2\B{I}}
\end{equation}
Apply the vec-trick to the mean term:
\begin{equation}
\log\Normal{\ve{\B{Y}}}{\ve{\sum_{j=1}^J\B{X}_j\B{W}_j\B{A}_j}}{\B{C}\otimes\B{R}
+ \sigma^2\B{I}}
\end{equation}
\begin{equation}
-\frac{C\cdot R}{2}\log(2\pi)-\frac{1}{2}\log|\B{C}\otimes\B{R} +
\sigma^2\B{I}|
-\frac{1}{2}\ve{\B{Y} -\sum_{j=1}^J\B{X}_j\B{W}_j\B{A}_j}^\T\left(\B{C}\otimes\B{R} +
\sigma^2\B{I}\right)^{-1}\ve{\B{Y} -\sum_{j=1}^J\B{X}_j\B{W}_j\B{A}_j}
\end{equation}
\subsection{Derivative of the squared form wrt. $\B{W}$}
\begin{equation}
\frac{\partial}{\partial [\B{W}_k]_{ab}}\left(-\frac{1}{2}\ve{\B{Y} -\sum_{j=1}^J\B{X}_j\B{W}_j\B{A}_j}^\T\left(\B{C}\otimes\B{R} +
\sigma^2\B{I}\right)^{-1}\ve{\B{Y} -\sum_{j=1}^J\B{X}_j\B{W}_j\B{A}_j}\right)
\end{equation}
We define matrix $\B{D}\in\Real^{R\times C}$, such that $\ve{\B{D}} = \diag\left(\B{S}_C\otimes\B{S}_R +
\sigma^2\B{I}\right)^{-1}$ and rotate the data:
\begin{equation}
\frac{\partial}{\partial [\B{W}_k]_{ab}}
-\frac{1}{2}\tr\left(\left(\Ykron
    -\sum_{j=1}^J\XWAkron{j}\right)^\T\left(\left(\Ykron
      -\sum_{j=1}^J\XWAkron{j}\right)\odot \B{D}\right)\right)
\end{equation}
\begin{eqnarray}\nonumber
\frac{\partial}{\partial [\B{W}_k]_{ab}}
-\frac{1}{2}\tr\left(\Ykron^\T (\Ykron \odot \B{D})\right)
    +\tr\left( \sum_{j=1}^J\XWAkronT{j}(\Ykron \odot \B{D})\right) \\
-\tr\left( \sum_{j=1}^J\sum_{i=j+1}^J\XWAkronT{j}(\XWAkron{i} \odot
  \B{D}) \right)
-\frac{1}{2}\tr\left(\sum_{j=1}^J\XWAkronT{j}(\XWAkron{j} \odot \B{D}) \right)
\end{eqnarray}
Leaving out all terms without $\B{W}_k$:
\begin{eqnarray}\nonumber
\frac{\partial}{\partial [\B{W}_k]_{ab}}
    \tr\left( \XWAkronT{k}(\Ykron \odot \B{D})\right) \\
-\tr\left( \sum_{j\neq k}\XWAkronT{k}(\XWAkron{j} \odot
  \B{D}) \right)
-\frac{1}{2}\tr\left(\XWAkronT{k}(\XWAkron{k} \odot \B{D}) \right)
\end{eqnarray}

\begin{eqnarray}\nonumber
    \tr\left( {\B{U}_C^\T[\B{A}_{k}]_{b:}^{\T}[\B{X}_{k}]_{:a}^{\T}\B{U}_R} (\Ykron \odot \B{D})\right) \\
-\tr\left( \sum_{j\neq k}{\B{U}_C^\T[\B{A}_{k}]_{b:}^{\T}[\B{X}_{k}]_{:a}^{\T}\B{U}_R} (\XWAkron{j} \odot
  \B{D}) \right)
-\tr\left({\B{U}_C^\T[\B{A}_{k}]_{b:}^{\T}[\B{X}_{k}]_{:a}^{\T}\B{U}_R} (\XWAkron{k} \odot \B{D}) \right)
\end{eqnarray}
Moving around the terms in the trace and combining the last two traces
into one sum:
\begin{eqnarray}\nonumber
    \tr\left( [\B{X}_{k}]_{:a}^{\T}\B{U}_R (\Ykron \odot \B{D}) \B{U}_C^\T[\B{A}_{k}]_{b:}^{\T}\right) \\
-\tr\left( \sum_{j=1}^J{[\B{X}_{k}]_{:a}^{\T}\B{U}_R} (\XWAkron{j} \odot
  \B{D}) \B{U}_C^\T[\B{A}_{k}]_{b:}^{\T}\right)
\end{eqnarray}
Observing that this is a scalar, we can eliminate the trace and re-arrange:
\begin{eqnarray}
    [\B{X}_{k}]_{:a}^{\T}\B{U}_R ((\Ykron-\sum_{j=1}^J \XWAkron{j}) \odot \B{D}) \B{U}_C^\T[\B{A}_{k}]_{b:}^{\T} \\
\end{eqnarray}
Stacking together these terms for all $a$ and $b$, the gradient becomes
\begin{eqnarray}
    \B{X}_{k}^{\T}\B{U}_R ((\Ykron-\sum_{j=1}^J \XWAkron{j}) \odot \B{D}) \B{U}_C^\T\B{A}_{k}^{\T} \\
\end{eqnarray}

\newpage
\end{document}


\documentclass[american,ignorenonframetext,notheorems,aspectratio=1610]{beamer}
\usetheme[compress]{Madrid}
\usecolortheme{albatross}
\usepackage{tikz,tikzscale}
\usetikzlibrary{snakes}
\input{mixed/fig/tikzsettings}
\def\restrict{r}
\def\prolongate{p}

\usepackage{../mathsim}
\usepackage{times}
\usepackage{xr}
\externaldocument{main}
\usepackage{mfirstuc}
\usepackage{mathtools}
\mathtoolsset{showonlyrefs}

\def\footnote#1{}
\def\putindex#1{#1}
\title{Finite Elements}
\author{Guido Kanschat}
\date{\today}
\begin{document}
\frame{\maketitle}
\frame{\frametitle{Overview}\tableofcontents[hideallsubsections]}
\section{Elliptic PDE and Their Weak Formulation}
\frame{\sectoc}
\subsection{Elliptic boundary value problems}

\frame {\input{blocks/Notation-coordinates.tex}}
\frame {\input{blocks/Notation-partial-derivative.tex}}
\frame {\input{blocks/Notation-elim-coord.tex}
  \input{blocks/Definition-lin-pde-2order.tex}}
\frame {\input{blocks/Definition-poisson-eqn.tex}}
\frame {\input{blocks/Definition-domain.tex}}
\frame {\input{blocks/Definition-boundary-conditions.tex}}
\frame {\input{blocks/Definition-dirichlet-problem-differential.tex}}
\frame {\input{blocks/Theorem-Dirichlet-principle.tex}}
\frame {\input{blocks/Theorem-Dirichlet-variational-principle.tex}
  \input{blocks/Corollary-Dirichlet-uniqueness.tex}}
\frame {\input{blocks/Lemma-reduction-to-zero-bc.tex}}
\frame {\input{blocks/Notation-l2.tex}}
\frame {\input{blocks/Lemma-Friedrichs-continuous.tex}
  \input{blocks/Problem-Friedrichs.tex}}
\frame {\input{blocks/Lemma-h1-norm.tex}}
\frame {\input{blocks/Lemma-Dirichlet-energy-boundedness.tex}
  \input{blocks/Lemma-minimizing-sequence.tex}}
\frame {\input{blocks/Definition-h10.tex}
  \input{blocks/Lemma-Friedrichs-h1.tex}}
\frame {\input{blocks/Definition-weak-formulation.tex}
  \input{blocks/Theorem-weak-unique-solution-1.tex}}

\frame {\input{blocks/Lemma-neumann-weak.tex}}
\frame {\input{blocks/Definition-natural-bc.tex}}
\frame {\input{blocks/Lemma-mixed-bc-weak.tex}}

\subsection{Hilbert Spaces and Bilinear Forms}
\frame{\tableofcontents[currentsection,subsectionstyle=show/shaded/hide]}

\frame {\input{blocks/Definition-inner-product.tex}}
\frame {\input{blocks/Theorem-bcs-inequality.tex}
  \input{blocks/Lemma-inner-product-norm.tex}}
\frame {\input{blocks/Definition-complete.tex}}
\frame {\input{blocks/Definition-Banach-hilbert.tex}}
\frame {\input{blocks/Definition-orthogonal.tex}}
\frame {\input{blocks/Definition-orthogonal-complement.tex}}
\frame {\input{blocks/Lemma-orthogonal-closed.tex}}
\frame {\input{blocks/Theorem-orthogonal-complement.tex}}
% \frame {\input{blocks/Corollary-ortho-density.tex}}
\frame {\input{blocks/Definition-ortho-projection.tex}}
\frame {\input{blocks/Definition-dual-space.tex}}
\frame {\input{blocks/Theorem-Riesz-representation.tex}}
\frame {\input{blocks/Definition-bilinear-form.tex}}
\frame {\input{blocks/Lemma-pde-bilinear.tex}}
\frame {\input{blocks/Lemma-lax-milgram.tex}}
\frame {\input{blocks/Lemma-weak-well-posed.tex}
  \input{blocks/Definition-elliptic.tex}}

\subsection{Fast facts on Sobolev spaces}
\frame{\tableofcontents[currentsection,subsectionstyle=show/shaded/hide]}

\frame {\input{blocks/Notation-multi-index.tex}}
\frame {\input{blocks/Definition-distributional-derivative.tex}}
\frame {\input{blocks/Definition-Wkp.tex}
  \input{blocks/Corollary-wkp-embedding.tex}}
\frame {\input{blocks/Definition-hkp.tex}
  \input{blocks/Theorem-meyers-serrin.tex}}

\frame {\input{blocks/Definition-boundary-smoothness.tex}}
\frame {\input{blocks/Definition-continuous-embedding.tex}}
\frame {\input{blocks/Theorem-sobolev-embedding.tex}}
\frame {\input{blocks/Lemma-trace-continuous.tex}
  \input{blocks/Theorem-trace.tex}}
\frame {\input{blocks/Definition-hoelder-spaces.tex}}
\frame {\input{blocks/Theorem-hoelder-embedding.tex}
  \input{blocks/Corollary-sobolev-continuous.tex}}

\subsection{Properties of solutions}
\frame{\tableofcontents[currentsection,subsectionstyle=show/shaded/hide]}

\frame {\input{blocks/Definition-wkp-loc.tex}
  \input{blocks/Theorem-gt-8-8.tex}
  \input{blocks/Theorem-gt-8-10.tex}
  \input{blocks/Corollary-gt-8-10.tex}}
\frame {\input{blocks/Theorem-gt-8-13.tex}
  \input{blocks/Corollary-h2-solution-bvp.tex}
  \input{blocks/Remark-classical-smooth.tex}
  \input{blocks/Remark-classical-convex.tex}}
\frame {\input{blocks/Theorem-kondratev.tex}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conforming Finite Element Methods}
\frame{\sectoc}
\subsection{Meshes, shape functions, and degrees of freedom}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\frame {\input{blocks/Definition-facets.tex}
  \input{blocks/Definition-mesh.tex}}
\frame {\input{blocks/Definition-finite-element.tex}
\input{blocks/Notation-dofs.tex}}
\frame {\input{blocks/Definition-node-topology.tex}
  \input{blocks/Definition-fe-space.tex}}
\frame {\input{blocks/Notation-global-local.tex}}
\frame {\input{blocks/Definition-local-global.tex}}
\frame {\input{blocks/Lemma-fe-support.tex}}
\frame {\input{blocks/Lemma-mesh-continuity.tex}
  \input{blocks/Lemma-nodal-continuity.tex}}

\frame {\input{blocks/Definition-barycentric-coordinates.tex}}
\frame {\input{blocks/Lemma-barycentric-affine.tex}
  \input{blocks/Corollary-barycentric-interpolation.tex}}

\begin{frame}
  \frametitle{The $P_1$ element in barycentric coordinates}
  \begin{columns}
    \begin{column}{.5\textwidth}
      \begin{center}
        \includegraphics[width=.6\textwidth]{mixed/fig/p1-p.tikz}
      \end{center}
    \end{column}
    \begin{column}{.5\textwidth}
      \begin{gather*}
        \phi_i = \lambda_i,
        \quad i=0,1,2
      \end{gather*}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{The $P_2$ element in barycentric coordinates}
  \begin{columns}
    \begin{column}{.5\textwidth}
      \begin{center}
        \includegraphics[width=.6\textwidth]{mixed/fig/p2-p.tikz}
      \end{center}
    \end{column}
    \begin{column}{.5\textwidth}
      \begin{xalignat*}2
        \phi_{ii} &= 2\lambda_i^2 - \lambda_i,
        &i&=0,1,2\\
        \phi_{ij} &= 4\lambda_i\lambda_j
        &j&\neq i
      \end{xalignat*}
    \end{column}
  \end{columns}
\end{frame}

\begin{frame}
  \frametitle{The $P_3$ element in barycentric coordinates}
  \begin{columns}
    \begin{column}{.5\textwidth}
      \begin{center}
        \includegraphics[width=.6\textwidth]{mixed/fig/p3-p.tikz}
      \end{center}
    \end{column}
    \begin{column}{.5\textwidth}
      \begin{xalignat*}2
        \phi_{iii} &= \tfrac12 \lambda_i(3\lambda_i-1)(3\lambda_i-2)
        &i&=0,1,2\\
        \phi_{ij} &= \tfrac92\lambda_i\lambda_j(3\lambda_j-1)
        &j&\neq i\\
        \phi_0 &= 27\lambda_0\lambda_1\lambda_2
      \end{xalignat*}
    \end{column}
  \end{columns}
\end{frame}

\frame {\input{blocks/Definition-galerkin-approximation.tex}}
\frame {\input{blocks/Corollary-galerkin-equations.tex}}
\frame {\input{blocks/Lemma-discrete-lax-milgram.tex}
  \input{blocks/Lemma-cea.tex}}
\frame {\input{blocks/Lemma-fe-matrix.tex}}
\frame {\input{blocks/Algorithm-matrix-assembling.tex}}

\frame {\input{blocks/Definition-tensor-product-polynomials.tex}}
\frame {\input{blocks/Lemma-tensor-product-node-functionals.tex}}

\begin{frame}
  \frametitle{Example: $Q_2$ shape functions}
  \begin{center}
    \includegraphics[height=.27\textheight]{graph/shape0}
    \includegraphics[height=.27\textheight]{graph/shape1}
    \includegraphics[height=.27\textheight]{graph/shape2}

    \includegraphics[height=.27\textheight]{graph/shape3}
    \includegraphics[height=.27\textheight]{graph/shape4}
    \includegraphics[height=.27\textheight]{graph/shape5}

    \includegraphics[height=.27\textheight]{graph/shape6}
    \includegraphics[height=.27\textheight]{graph/shape7}
    \includegraphics[height=.27\textheight]{graph/shape8}
  \end{center}
\end{frame}

\frame {\input{blocks/Lemma-tensor-product-trace.tex}}

\begin{frame}
  \frametitle{Example: $Q_2$ continuous basis (selection)}
  \begin{center}
    \includegraphics[height=.27\textheight]{graph/cgbasis1-02}
    \includegraphics[height=.27\textheight]{graph/cgbasis1-03}
    \includegraphics[height=.27\textheight]{graph/cgbasis1-15}
    \includegraphics[height=.27\textheight]{graph/cgbasis1-16}

    \includegraphics[height=.27\textheight]{graph/cgbasis1-07}
    \includegraphics[height=.27\textheight]{graph/cgbasis1-13}
    \includegraphics[height=.27\textheight]{graph/cgbasis1-18}
    \includegraphics[height=.27\textheight]{graph/cgbasis1-22}

    \includegraphics[height=.27\textheight]{graph/cgbasis1-17}
    \includegraphics[height=.27\textheight]{graph/cgbasis1-23}
    \includegraphics[height=.27\textheight]{graph/cgbasis1-20}
    \includegraphics[height=.27\textheight]{graph/cgbasis1-24}
  \end{center}
\end{frame}

\begin{frame}
  \frametitle{Example: $Q_2$ discontinuous basis (selection)}
  \begin{center}
    \includegraphics[height=.27\textheight]{graph/dgbasis1-08}
    \includegraphics[height=.27\textheight]{graph/dgbasis1-15}
    \includegraphics[height=.27\textheight]{graph/dgbasis1-20}
    \includegraphics[height=.27\textheight]{graph/dgbasis1-27}

    \includegraphics[height=.27\textheight]{graph/dgbasis1-07}
    \includegraphics[height=.27\textheight]{graph/dgbasis1-19}
    \includegraphics[height=.27\textheight]{graph/dgbasis1-23}
    \includegraphics[height=.27\textheight]{graph/dgbasis1-30}

    \includegraphics[height=.27\textheight]{graph/dgbasis1-26}
    \includegraphics[height=.27\textheight]{graph/dgbasis1-33}
    \includegraphics[height=.27\textheight]{graph/dgbasis1-24}
    \includegraphics[height=.27\textheight]{graph/dgbasis1-22}
  \end{center}
\end{frame}

\subsubsection{Mapped finite elements}

\frame {\input{blocks/Definition-mapped-mesh.tex}}
\frame {\input{blocks/Example-mapping-linear.tex}}
\frame {\input{blocks/Example-mapping-bilinear.tex}}
\frame {\input{blocks/Definition-mapped-fe.tex}}
\frame {\input{blocks/Lemma-mapped-norms-affine.tex}}
\frame {\input{blocks/Lemma-shape-regular-transformation.tex}}
\frame {\input{blocks/Assumption-mapping-decomposition.tex}}
\frame {\input{blocks/Lemma-scaling-1.tex}}

\subsection{A priori error analysis}
\frame{\subtoc}

\frame {\input{blocks/Lemma-poincare.tex}}
\frame {\input{blocks/Lemma-bramble-hilbert.tex}}
%\frame {\input{blocks/Lemma-b-h-projector.tex}}
\frame {\input{blocks/Definition-mesh-family.tex}}
\frame {\input{blocks/Definition-nodal-interpolation.tex}
  \input{blocks/Lemma-nodal-interpolation.tex}}
\frame {\input{blocks/Definition-broken-sobolev-norm.tex}}
\frame {\input{blocks/Theorem-fe-interpolation.tex}
  \input{blocks/Corollary-fe-approximation.tex}}
\frame {\input{blocks/Theorem-fe-convergence.tex}}

\subsubsection{Estimates of stronger norms}

\frame {\input{blocks/Definition-stronger-norm.tex}}
\frame {\input{blocks/Lemma-inverse-estimate.tex}}
\frame {\input{blocks/Theorem-h2-error.tex}}

\subsubsection{Estimates of weaker norms}

\frame {\input{blocks/Definition-dual-problem.tex}}
\frame {\input{blocks/Lemma-poisson-dual.tex}}
\frame {\input{blocks/Assumption-elliptic-regularity.tex}}
\frame {\input{blocks/Theorem-fe-l2.tex}
  \input {blocks/Corollary-fe-functional.tex}}

\subsubsection{Green's function and maximum norm estimates}

\frame {\input {blocks/Definition-greens-function.tex}
  \input {blocks/Theorem-greens-function-rd.tex}}
\frame {\input {blocks/Lemma-greens-function-domain.tex}}
\frame {\input {blocks/Theorem-greens-function-representation.tex}}
\frame {\input {blocks/Theorem-linfty-error.tex}}

\subsection{A posteriori error analysis}
\frame{\subtoc}

\frame {\input {blocks/Definition-a-posteriori.tex}}
\frame {\input {blocks/Definition-locally-quasi-uniform.tex}}
\frame {\input {blocks/Definition-fem-neighborhood.tex}}
\frame {\input {blocks/Theorem-clement.tex}}
\frame {\input {blocks/Theorem-scott-zhang.tex}}
\frame {\input {blocks/Theorem-schoeberl-interpolation.tex}}
\frame {\input {blocks/Definition-residual.tex}}
\frame {\input {blocks/Lemma-residual-hm1.tex}}
\frame {\input {blocks/Definition-mvl-jmp.tex}}
\frame {\input {blocks/Definition-residual-strong.tex}}
\frame {\input {blocks/Lemma-residual-upper-bound.tex}}
\frame {\input {blocks/Definition-osc.tex}}
\frame {\input {blocks/Definition-residual-estimator.tex}}
\frame {\input {blocks/Theorem-residual-estimate.tex}}

\section{Variational Crimes}
\frame{\sectoc}
\subsection{Numerical quadrature}
\frame {\input {blocks/Lemma-strang-1.tex}}
\frame {\input {blocks/Definition-tensor-product-quadrature.tex}
  \input {blocks/Lemma-tensor-product-gauss.tex}}
\frame {\input {blocks/Assumption-bilinear-form-quadrature.tex}}
\frame {\input {blocks/Definition-bilinear-quadrature.tex}}
\frame {\input {blocks/Lemma-quadrature-stability.tex}}
\frame {\input {blocks/Theorem-quadrature-error-bilinear.tex}}
\frame {\input {blocks/Lemma-product-sobolev-norm.tex}}
\frame {\input {blocks/Theorem-quadrature-error-rhs.tex}}

\section{Solving the Discrete Problem}
\frame{\sectoc}
\subsection{Richardson's method}
\frame {\input {blocks/Definition-spd-condition-number.tex}}
\frame {\input {blocks/Definition-matrix-condition-number.tex}}
\frame {\input {blocks/Lemma-richardson-error-step.tex}}
\frame {\input {blocks/Theorem-richardson-convergence1.tex}}
\frame {\input {blocks/Definition-richardson-method.tex}}
\frame {\input {blocks/Theorem-richardson-convergence2.tex}}
\frame {\input {blocks/Definition-riesz-isomorphism.tex}}
\frame {\input {blocks/Definition-p-richardson-method.tex}}
\subsection{The conjugate gradient method}
\frame{\subtoc}
\frame {\input {blocks/Definition-steepest-descent.tex}}
\frame {\input {blocks/Lemma-steepest-descent.tex}}
\frame {\input {blocks/Definition-cg-method.tex}}
\frame {\input {blocks/Definition-pcg-method.tex}}
\frame {\input {blocks/Definition-krylov-space.tex}}
\frame {\input {blocks/Lemma-cg-orthogonality.tex}}
\frame {\input {blocks/Lemma-cg-minimization.tex}}
\frame {\input {blocks/Theorem-cg-convergence.tex}}

\subsection{Condition numbers of finite element matrices}
\frame{\subtoc}
\frame {\input {blocks/Notation-up-to-a-constant.tex}}
\frame {\input {blocks/Definition-mass-stiffness.tex}}
\frame {\input {blocks/Lemma-condition-number-mass-matrix.tex}}
\frame {\input {blocks/Corollary-refined-condition-number.tex}}
\frame {\input {blocks/Theorem-condition-number-stiffness-matrix.tex}}

\subsection{Multigrid methods}
\frame {\input {blocks/Algorithm-mgm.tex}}

\section{Bibliography}
\frame{\bibliographystyle{alpha}
\bibliography{all}}
\end{document}

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