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rAKA akantu
manual-introduction.tex
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\chapter
{
Introduction
}
% \section{Data structures\label{chap:data-structure}}
% \subsection{Vectors\label{sec:vectors}}
% The Vector class is a template class that can store scalar types as Real, UInt,
% Int or bool. A Vector instance is defined by its size and its number of
% component. It also has an identifier and some extra internal variables for
% memory handling purpose.
% \begin{itemize}
% \item The size is the number of tupels stored in the Vector.
% \item The number of component is the number of values stored for each tuple.
% \end{itemize}
% \begin{figure}[!htb]
% \centering
% \includegraphics{figures/vectors}
% \caption{View of a Vector of size n and c compenents, (a) representation of
% the vector, (b) representation of the storage of the same Vector.}
% \label{fig:vectors}
% \end{figure}
% All the data are linerarized in memory in an array called values. This class
% member is public. So for a Vector of size $n$ and $c$ component, to access the
% $j^{th}$ component of th $i^{th}$ tuple, you have to get $values[i * c +
% j]$. You can also access to this value with accessor \code{at(i, j)} or the
% constant accessor \code{get(i, j)}.
% If you want to store a matrix on each tuple you have to linearize it. For
% exemple if you want to store a m * k matrix on each tuple you must specify c = m
% * k. To access a particular value in a matrix of a tuple you will have to do
% something like $values[ i * m * k + j_i * m + j_j]$.
% \subsubsection{Vector storage convention within FE object\label{sec:FE-convention}}
% The point of this section is to describe the convention of storage for vectors
% intented to be passed to {\bf FE} object methods. Indeed a convention of
% necessary since gradient operators or integrtaion loops will use vectors as
% input and ouput. Such vectors will be ordered with a specific convention that
% we intend to describe now.
% For vector objects, the size of the vector is always the number of nodes
% associated. The number of components is related to the order of the tensor
% considered. For a scalar field it is 1, for a vectorial field, the size of the
% vector is the number of components. For a $m\times n$ matrix field the number of
% components is $m\times n$.
% One common operation is the manipulation of continuum fields at the quadrature
% point positions. Here the size of the vector is $mn\_element \times
% nb\_quad\_points$ and the number of components is related to the stored
% object. For instance the method \verb$interpolateOnQuadraturePoints()$ take a
% nodal field stored in a vector($n\_nodes$,$dim$) and return a vector($n\_elem
% \times n\_quads$,$dim$).
% Basic gradient operations, like method \verb$gradientOnQuadraturePoints()$, will
% take as input a vector($n\_nodes$,$dim$) and return a vector($n\_elem \times
% n\_quads$,$dim \times spatial\_dimension$) where spatial dimension is the number
% of dimension in which the domain is embedded.
% In the same way the integration routines expect vector($n\_elem \times
% n\_quads$,$dim$) and will return vector($n\_elem$,$dim$). For non-Gaussian
% integrations, the input by be direction a nodal field. (At present time, only
% gaussian integrators are implemented within akantu).
% Last but not the least is the vectorial assembly process for which accept as
% input vector($n\_elem \times n\_quads \times n\_nodes\_per\_element$,$dim$)
% and will return a vector($n\_nodes$,$dim$) nodal object.\\
% {\bf The general convention is that the number of component shall always be the
% size of the object manipulated at the lowest level. The object could be a
% per-element, of per quadrature point or even per node basis this will also
% apply as shown below. The figures \ref{fig:vector-chain} and
% \ref{fig:interpolate-storage} shows the pattern of the vectors is the case a
% interpolation on quadrature points.}
% \begin{figure}
% \begin{equation}
% \left(
% \begin{array}{c}
% N_1 \\
% \vdots \\
% N_I
% \left\{
% \begin{array}{c}
% a_1 \\
% \vdots \\
% a_{dim} \\
% \end{array}
% \right. \\
% \vdots \\
% N_{nb\_nodes}
% \end{array}
% \right)
% \begin{array}{c}
% \Longrightarrow \\
% interpolate\\
% On\\
% Quadrature\\
% Points\\
% \end{array}
% \left(
% \begin{array}{c}
% E_1 \\
% \vdots \\
% E_e
% \left\{
% \begin{array}{c}
% q_1 \\
% \vdots \\
% \left.
% \begin{array}{c}
% a_1 \\
% \vdots \\
% a_{dim}
% \end{array}
% \right\} q_i \\
% \vdots \\
% q_p \\
% \end{array}
% \right. \\
% \vdots \\
% E_{nb\_elements}
% \end{array}
% \right)
% \end{equation}
% \caption{\label{fig:vector-chain}Pattern of vectors manipulated during
% interpolation on quadrature points. Symbols N (resp. E, q) denotes nodes
% (resp. elements, quadrature points).}
% \end{figure}
% \begin{figure}[!htb]
% \centering
% \includegraphics[width=\textwidth]{figures/interpolate}
% \caption{Input and output vector of interpolateOnQuadraturePoints. The output
% Vector has nb\_quadrature\_points tuples, the quadrature points are grouped
% by elements (p is the number of quadrature points per element).}
% \label{fig:interpolate-storage}
% \end{figure}
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