Update theory manual

documents/theory-manual/document.tex

@@ -25,9 +25,9 @@
 % \allowdisplaybreaks		% Break aligned equations over pages
 
 % Parameters
-\newcommand{\version}{0.8}
+\newcommand{\version}{0.9.0}
 \newcommand{\website}{\href{http://www.virtualbow.org/}{\texttt{\textcolor{blue}{http://www.virtualbow.org/}}}}
-\newcommand{\copyrights}{Copyright (C) 2016-2021 Stefan Pfeifer}
+\newcommand{\copyrights}{Copyright (C) 2016-2022 Stefan Pfeifer}
 
 \begin{document}
 

documents/theory-manual/latex/equations.tex

@@ -738,219 +738,6 @@ \subsection{Floating Frame of Reference}
 \end{equation}
 
 where~$\alpha \in\ ]0,\,\nicefrac{1}{50}]$ is a tunable parameter. Tests with the bow simulation suggest that the choice of~$\alpha$ barely has any influence on the results. Therefore the maximum value~$\alpha = \nicefrac{1}{50}$ has been chosen for numerical reasons.
-
-\newpage
-\subsection{Stiffness Matrix of a Beam Segment}
-
-In this section the analytical stiffness matrix for a curved beam segment as shown in figure~\ref{fig:beam-linear-1} is derived following the approach presented in \cite{bib:curved-beam-stiffness-matrix}.
-
-\begin{figure}[h]
-\centering
-\includegraphics[width=0.5\textwidth]{figures/elements/beam-linear-1}
-\caption{Curved beam segment with two nodes}
-\label{fig:beam-linear-1}
-\end{figure}
-
-The segment has an initial length of~$l$. Its undeformed shape is described by the curve~$x(s)$, $y(s)$, $\varphi(s)$ with~$\varphi$ being the orientation angle measured against the $x$-axis and~$s \in [0,\,l]$ the arc length along the curve.
-
-In order for the segment to be in static equilibrium, the nodal forces~$p_0$,\,\ldots\,$p_5$ have to fulfill the equilibrium conditions
-%
-\begin{align}
-&p_0 + p_3 = 0 \\
-&p_1 + p_4 = 0 \\
-&p_2 + p_5 + p_0\,\Delta y - p1\,\Delta x = 0
-\end{align}
-
-where~$\Delta x = x(l) - x(0)$ and~$\Delta y = y(l) - y(0)$. This can be rearranged into the following matrix equation that relates the forces on nodes A and B to each other,
-
-\begin{equation}
-\underbrace{
-\begin{bmatrix}
-p_3 \\ p_4 \\ p_5
-\end{bmatrix}
-}_{\boldsymbol{p}_B}
-=
-\underbrace{
-\begin{bmatrix}
--1 &  0 & 0 \\
- 0 & -1 & 0 \\
--\Delta y & \Delta x & -1
-\end{bmatrix}
-}_{\boldsymbol{S}_{BA}}
-\underbrace{
-\begin{bmatrix}
-p_0 \\ p_1 \\ p_2
-\end{bmatrix}
-}_{\boldsymbol{p}_A}
-\label{eq:linear-beam-static-relation}
-\end{equation}
-
-\begin{figure}[h]
-\centering
-\includegraphics[width=0.5\textwidth]{figures/elements/beam-linear-2}
-\caption{Cross section forces of the beam segment}
-\label{fig:beam-linear-2}
-\end{figure}
-
-Forces on the cross section:
-
-\begin{align*}
--&N(s)\cos(\varphi(s)) + Q(s)\sin(\varphi(s)) + p_3 = 0 \\
--&N(s)\sin(\varphi(s)) - Q(s)\cos(\varphi(s)) + p_4 = 0 \\
--&M(s) - p_3\,(y(l)-y(s)) + p_4\,(x(l) - x(s)) + p_5 = 0
-\end{align*}
-
-\begin{equation}
-\begin{bmatrix}
-N \\ M
-\end{bmatrix}
-=
-\underbrace{
-\begin{bmatrix}
-\cos(\varphi(s)) & \sin(\varphi(s)) & 0 \\
-y(s) - y(l) & x(l) - x(s) & 1
-\end{bmatrix}
-}_{\boldsymbol{H}_B(s)}
-\underbrace{
-\begin{bmatrix}
-p_3 \\ p_4 \\ p_5
-\end{bmatrix}
-}_{\boldsymbol{p}_B}
-\end{equation}
-
-
-\subsubsection*{Case 1: Flexibility matrix of point B when A is fixed}
-
-\begin{align}
-V_B &= \frac{1}{2}\int_{0}^{l}
-\begin{bmatrix}
-N \\ M
-\end{bmatrix}^\mathrm{T}
-\begin{bmatrix}
-\varepsilon \\ \kappa
-\end{bmatrix}
-ds \\
-&= \frac{1}{2}\int_{0}^{l}
-\begin{bmatrix}
-N \\ M
-\end{bmatrix}^\mathrm{T}
-\begin{bmatrix}
-C_{\varepsilon\varepsilon} & C_{\varepsilon\kappa}\\
-C_{\varepsilon\kappa} & C_{\kappa\kappa}
-\end{bmatrix}^{-1}
-\begin{bmatrix}
-N \\ M
-\end{bmatrix}
-ds \\
-&= \frac{1}{2}\,\boldsymbol{p}_B^\mathrm{T}
-\underbrace{
-\left(\int_{0}^{l} \boldsymbol{H}_B^\mathrm{T}\,
-\boldsymbol{C}^{-1}\boldsymbol{H}_B\,ds\right)
-}_{\boldsymbol{K}_{BB}^{-1}}
-\boldsymbol{p}_B
-\end{align}
-
-According to Castigliano's second theorem the displacement of node B under the external loads~$\boldsymbol{p}_B$ is
-
-\begin{equation}
-\boldsymbol{u}_{B} = \frac{\partial V_B}{\partial \boldsymbol{p}_B} = \boldsymbol{K}_{BB}^{-1}\,\boldsymbol{p}_B
-\end{equation}
-
-with the flexibility matrix~$\boldsymbol{K}_{BB}^{-1}$ which is the inverse of the stiffness matrix.
-
-\subsubsection*{Case 2: Flexibility matrix of point A when B is fixed}
-
-\begin{equation}
-\begin{bmatrix}
-N \\ M
-\end{bmatrix}
-= \boldsymbol{H}_B\,\boldsymbol{p}_B = \boldsymbol{H}_B\,\boldsymbol{S}_{BA}\,\boldsymbol{p}_A
-\end{equation}
-
-\begin{align}
-V_A &= \frac{1}{2}\,\boldsymbol{p}_A^\mathrm{T}
-\left(\int_{0}^{l} \boldsymbol{H}_A^\mathrm{T}\,
-\boldsymbol{C}^{-1}\boldsymbol{H}_A\,ds\right)
-\boldsymbol{p}_A \\
-&= \frac{1}{2}\,\boldsymbol{p}_A^\mathrm{T}
-\boldsymbol{S}_{BA}^\mathrm{T}\left(\int_{0}^{l} \boldsymbol{H}_B^\mathrm{T}\,
-\boldsymbol{C}^{-1}\boldsymbol{H}_B\,ds\right)\boldsymbol{S}_{BA}\,\boldsymbol{p}_A \\
-&= \frac{1}{2}\,\boldsymbol{p}_A^\mathrm{T}
-\underbrace{
-\left(\boldsymbol{S}_{BA}^\mathrm{T}\boldsymbol{K}_{BB}^{-1}\boldsymbol{S}_{BA}\right)
-}_{\boldsymbol{K}_{AA}^{-1}}
-\boldsymbol{p}_A 
-\end{align}
-
-Displacement of node A under the external loads~$\boldsymbol{p}_A$,
-
-\begin{equation}
-\boldsymbol{u}_{A} = \frac{\partial V_A}{\partial \boldsymbol{p}_A} = \boldsymbol{K}_{AA}^{-1}\,\boldsymbol{p}_A
-\end{equation}
-
-\subsubsection*{Non block-diagonal stiffness}
-
-\begin{equation}
-\boldsymbol{p}_B = \boldsymbol{S}_{BA}\,\boldsymbol{p}_A =
-\underbrace{
-\boldsymbol{S}_{BA}\,\boldsymbol{K}_{AA}
-}_{\boldsymbol{K}_{BA}}
-\,\boldsymbol{u}_A
-\end{equation}
-
-Therefore the complete stiffness relation for the beam segment is
-
-\begin{equation}
-\begin{bmatrix}
-\boldsymbol{p}_A \\ \boldsymbol{p}_B
-\end{bmatrix}
-=
-\begin{bmatrix}
-\boldsymbol{K}_{AA} & \boldsymbol{K}_{BA}^\mathrm{T} \\
-\boldsymbol{K}_{BA} & \boldsymbol{K}_{BB}
-\end{bmatrix}
-\begin{bmatrix}
-\boldsymbol{u}_A \\ \boldsymbol{u}_B
-\end{bmatrix}
-\end{equation}
-
-Note that this is an exact result. No simplifications have been made within the linear Euler-Bernoulli beam theory.
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 \newpage
 \section{Contact element}
@@ -1121,3 +908,5 @@ \subsection{Contact Forces}
 \end{equation*}
 
 \subsection{Broadphase Algorithm}
+
+\textcolor{red}{TODO}

documents/theory-manual/latex/introduction.tex

@@ -5,5 +5,5 @@ \chapter{Introduction}
 
 \website.
 
-This documentation, as opposed to the user manual, is about the theoretical foundations and technical details of the software.
-It is meant as a reference for developers and interested users.
+This documentation is about the theoretical foundations and technical details of the software.
+It is still a work in progress and is meant as a reference for developers and interested users.

documents/theory-manual/latex/model.tex

@@ -2,7 +2,7 @@ \chapter{The Bow Model}
 
 A scientific model is a simplification and abstraction of reality, often formulated in a mathematical way.
 A good model reduces the complexity of a real system by including only its most significant aspects and disregarding less significant ones.
-The results of analyzing this simplified model can then be used to draw conclusions about the real system.
+Analyzing this simplified model can then lead to conclusions about the real system that wouldn't have been possible otherwise.
 
 What aspects of reality a model has to reflect depends on the kinds of questions it seeks to answer.
 The first step for developing a bow model is therefore to clarify its scope and intended application.