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vector.rst
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.. _conventions_vector:
Vector representation
=====================
In GooseFEM there are three ways to represent vectors. In particular, a vector field (e.g. the displacement) can be collected:
* per node (denoted "nodevec", :ref:`see below <conventions_storage>`),
* per degree-of-freedom (denoted "dofval", :ref:`see below <conventions_storage>`),
* per element (denoted "elemvec", :ref:`see below <conventions_storage>`).
.. warning::
Care has to be taken in the conversion from one representation to the other as 'down-sizing' can be done in more than one way, see :ref:`conventions_vector_conversion`.
Consider a simple two-dimensional mesh of just two elements, and a displacement vector per node:
.. image:: figures/data-representation.svg
:width: 400px
:align: center
|
Collected per node
------------------
.. math::
\texttt{disp} =
\begin{bmatrix}
u_x^{(0)} & u_y^{(0)} \\
u_x^{(1)} & u_y^{(1)} \\
u_x^{(2)} & u_y^{(2)} \\
u_x^{(3)} & u_y^{(3)} \\
u_x^{(4)} & u_y^{(4)} \\
u_x^{(5)} & u_y^{(5)}
\end{bmatrix}
Collected per degree-of-freedom
-------------------------------
The following definition
.. math::
\texttt{dofs} =
\begin{bmatrix}
0 & 1 \\
2 & 3 \\
4 & 5 \\
6 & 7 \\
8 & 9 \\
10 & 11
\end{bmatrix}
gives:
.. math::
\texttt{u} =
\big[
u_x^{(0)} \,
u_y^{(0)} \,
u_x^{(1)} \,
u_y^{(1)} \,
u_x^{(2)} \,
u_y^{(2)} \,
u_x^{(3)} \,
u_y^{(3)} \,
u_x^{(4)} \,
u_y^{(4)} \,
u_x^{(5)} \,
u_y^{(5)}
\big]^T
Whereby "dofs" can be used to:
* **Reorder** "u" such that is can be easily (even directly) partitioned. For example, consider that all :math:`x`-coordinates are *Prescribed* and all :math:`y`-coordinates are *Unknown*. In particular,
.. math::
\texttt{dofs} =
\begin{bmatrix}
6 & 0 \\
7 & 1 \\
8 & 2 \\
9 & 3 \\
10 & 4 \\
11 & 5
\end{bmatrix}
gives
.. math::
\texttt{u} =
\big[
u_y^{(0)} \,
u_y^{(1)} \,
u_y^{(2)} \,
u_y^{(3)} \,
u_y^{(4)} \,
u_y^{(5)} \, \;
u_x^{(0)} \,
u_x^{(1)} \,
u_x^{(2)} \,
u_x^{(3)} \,
u_x^{(4)} \,
u_x^{(5)}
\big]^T
=
\big[
\texttt{u}_u \, \;
\texttt{u}_p
\big]^T
which allows
.. math::
\texttt{u}_u &= \texttt{u[:6]} \\
\texttt{u}_p &= \texttt{u[6:]}
|
* **Eliminate** dependent nodes. For example, suppose that the displacement of all top nodes is equal to that of the bottom nodes. In this one could:
.. math::
\texttt{dofs} =
\begin{bmatrix}
0 & 1 \\
2 & 3 \\
4 & 5 \\
0 & 1 \\
2 & 3 \\
4 & 5
\end{bmatrix}
\qquad
\rightarrow
\qquad
\texttt{u} =
\begin{bmatrix}
u_0 \\
u_1 \\
u_2 \\
u_3 \\
u_4 \\
u_5
\end{bmatrix}
\quad
\leftrightarrow
\quad
\texttt{disp} =
\begin{bmatrix}
u_0 & u_1 \\
u_2 & u_3 \\
u_4 & u_5 \\
u_0 & u_1 \\
u_2 & u_3 \\
u_4 & u_5
\end{bmatrix}
Collected per element
---------------------
For this example:
.. math::
\texttt{conn} =
\begin{bmatrix}
0 & 1 & 4 & 3 \\
1 & 2 & 5 & 4
\end{bmatrix}
The storage per node proceeds in
.. math::
\texttt{shape(ue)}
&= \left[ n_\text{elements} \times n_\text{nodes-per-element} \times n_\text{dim} \right]
\\
&= \left[ 2 \times 4 \times 2 \right]
In particular:
.. math::
\texttt{ue[0,:,:]} =
\begin{bmatrix}
u_x^{(0)} & u_y^{(0)} \\
u_x^{(1)} & u_y^{(1)} \\
u_x^{(4)} & u_y^{(4)} \\
u_x^{(3)} & u_y^{(3)} \\
\end{bmatrix}
and
.. math::
\texttt{ue[1,:,:]} =
\begin{bmatrix}
u_x^{(1)} & u_y^{(1)} \\
u_x^{(2)} & u_y^{(2)} \\
u_x^{(5)} & u_y^{(5)} \\
u_x^{(4)} & u_y^{(4)} \\
\end{bmatrix}
.. _conventions_vector_conversion:
Conversion
----------
Conversion to a larger representation (up-sizing) can always be done uniquely, however, conversion to a more compact representation (down-sizing) can be done in two ways. In particular:
+---------+---------+-------------------+------------------------------+
| From | To | Function | Remarks |
+=========+=========+===================+==============================+
| dofval | nodevec | asNode(...) | unique |
+---------+---------+-------------------+------------------------------+
| dofval | elemvec | asElement(...) | unique |
+---------+---------+-------------------+------------------------------+
| nodevec | elemvec | asElement(...) | unique |
+---------+---------+-------------------+------------------------------+
| nodevec | dofval | asDofs(...) | overwrites reoccurring items |
+---------+---------+-------------------+------------------------------+
| elemvec | dofval | asDofs(...) | overwrites reoccurring items |
+---------+---------+-------------------+------------------------------+
| elemvec | nodevec | asNode(...) | overwrites reoccurring items |
+---------+---------+-------------------+------------------------------+
| nodevec | dofval | assembleDofs(...) | adds reoccurring items |
+---------+---------+-------------------+------------------------------+
| elemvec | dofval | assembleDofs(...) | adds reoccurring items |
+---------+---------+-------------------+------------------------------+
| elemvec | nodevec | assembleNode(...) | adds reoccurring items |
+---------+---------+-------------------+------------------------------+

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