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notebook.tex
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% Default to the notebook output style
% Inherit from the specified cell style.
\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
% Nicer default font (+ math font) than Computer Modern for most use cases
\usepackage{mathpazo}
% Basic figure setup, for now with no caption control since it's done
% automatically by Pandoc (which extracts  syntax from Markdown).
\usepackage{graphicx}
% We will generate all images so they have a width \maxwidth. This means
% that they will get their normal width if they fit onto the page, but
% are scaled down if they would overflow the margins.
\makeatletter
\def\maxwidth{\ifdim\Gin@nat@width>\linewidth\linewidth
\else\Gin@nat@width\fi}
\makeatother
\let\Oldincludegraphics\includegraphics
% Set max figure width to be 80% of text width, for now hardcoded.
\renewcommand{\includegraphics}[1]{\Oldincludegraphics[width=.8\maxwidth]{#1}}
% Ensure that by default, figures have no caption (until we provide a
% proper Figure object with a Caption API and a way to capture that
% in the conversion process - todo).
\usepackage{caption}
\DeclareCaptionLabelFormat{nolabel}{}
\captionsetup{labelformat=nolabel}
\usepackage{adjustbox} % Used to constrain images to a maximum size
\usepackage{xcolor} % Allow colors to be defined
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\usepackage{geometry} % Used to adjust the document margins
\usepackage{amsmath} % Equations
\usepackage{amssymb} % Equations
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\usepackage{hyperref}
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\usepackage{booktabs} % table support for pandoc > 1.12.2
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\title{numerical\_method}
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\begin{document}
\maketitle
\hypertarget{numerical-method}{%
\subsection{Numerical method}\label{numerical-method}}
Here we will solve few problems by numerical method using Lagrangian
grid which deformed and moved together with system. We will use Wilkins
Method.
\hypertarget{numerical-implementation-of-boundary-conditions}{%
\subsubsection{Numerical implementation of boundary
conditions}\label{numerical-implementation-of-boundary-conditions}}
\hypertarget{basics-of-the-system}{%
\paragraph{Basics of the system}\label{basics-of-the-system}}
Lets consider tho slabs collision problem. Suppose slabs have areas
\(AB\) and \(CD\) on \(x\) axes:
we call \(A\), \(B\), \(C\) and \(D\) \emph{side bars}
We mark here and following formulas in varible \(v^n_i\) top index \(n\)
as time and \(i\) as index of \emph{inner bars} in the grid.
\emph{Bar} has always integer index \(k\) and \emph{node} has index
\(k\pm1/2\)
For \emph{inner bars} we will use:
for speeds:
\(\begin{align} v^{n+1/2}_{i} = v^{n-1/2}_{i} + 2 \Delta t \frac{\Sigma^n_{i+1/2} - \Sigma^n_{i-1/2}}{\phi^n_{i+1/2} + \phi^n_{i - 1/2}} \end{align}\)
for positions:
\(\begin{align} x^{n+1}_i = x^{n}_i + \Delta t v^{n+1/2}_i \\ \end{align}\)
where \(\Delta t\) is time step;
\(\phi^n_{i+1/2} = \rho^n_{i+1/2}(x^n_{i+1} - x^n_{i})\) where \(\rho\)
is density.
We suppose that all values of all variables are know for moment \(t^n\)
and before it.
\hypertarget{collision}{%
\paragraph{Collision}\label{collision}}
Slabs collide at moment \(t^n\), we merge \(B\) and \(C\) and set
following: \(\Sigma_B = \Sigma_C\), where \(\Sigma\) is tension.
Boundary conditions for the \emph{side bar}s we find at moment \(t^n\)
as follows. Let's add a \emph{mock node} \(i - 1/2\) with \emph{bars}
\(i - 1\) and \(i\) which we will use only for computations. This
\emph{mock node} we place to the left corner of the right stab.
And we suposse following for the \emph{mock node}:
\(\begin{align} \Sigma^n_{i-1/2} = 0 ; \phi^n_{i-1/2} = 0 \end{align}\)
% Add a bibliography block to the postdoc
\end{document}