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Getting Started
===============
Compiling Akantu
----------------
Akantu is a `CMake <https://cmake.org/>`_ project, so to configure it, you can either
follow the usual way::
> cd akantu
> mkdir build
> cd build
> ccmake ..
[ Set the options that you need ]
> make
> make install
Or, use the ``Makefile`` we added for your convenience to
handle the CMake configuration::
> cd akantu
> make config
> make
> make install
All the Akantu options are documented in Appendix app:package-dependencies.
Writing a ``main`` function
---------------------------
Akantu first needs to be initialized. The memory management
included in the core library handles the correct allocation and
de-allocation of vectors, structures and/or objects. Moreover, in
parallel computations, the initialization procedure performs the
communication setup. This is achieved by a pair of functions
(``initialize`` and ``finalize``) that are used as follows::
#include "aka_common.hh"
#include "..."
using namespace akantu;
int main(int argc, char *argv[]) {
initialize("input_file.dat", argc, argv);
// your code ...
finalize();
}
The ``initialize`` function takes the text inpute file and the program
parameters which can be parsed by Akantu in due form (see sect:parser).
Obviously it is necessary to include all files needed in main. In this manual
all provided code implies the usage of ``akantu`` as namespace.
Creating and Loading a Mesh
---------------------------
In its current state, Akantu supports three types of meshes: Gmsh,
Abaqus and Diana. Once a ``Mesh`` object is created with a given
spatial dimension, it can be filled by reading a mesh input file. The
method ``read`` of the class ``Mesh`` infers the mesh type from the
file extension. If a non-standard file extension is used, the mesh
type has to be specified. ::
UInt spatial_dimension = 2;
Mesh mesh(spatial_dimension);
// Reading Gmsh files
mesh.read("my_gmsh_mesh.msh");
mesh.read("my_gmsh_mesh", _miot_gmsh);
// Reading Abaqus files
mesh.read("my_abaqus_mesh.inp");
mesh.read("my_abaqus_mesh", _miot_abaqus);
// Reading Diana files
mesh.read("my_diana_mesh.dat");
mesh.read("my_diana_mesh", _miot_diana);
The Gmsh reader adds the geometrical and physical tags as mesh data.
The physical values are stored as a ``UInt`` data called ``tag_0``, if
a string name is provided it is stored as a ``std::string`` data named
``physical_names``. The geometrical tag is stored as a ``UInt`` data
named ``tag_1``.
The Abaqus reader stores the ``ELSET`` in ElementGroups and the ``NSET``
in NodeGroups. The material assignment can be retrieved from the
``std::string`` mesh data named ``abaqus_material``.
Using Arrays
------------
Data in Akantu can be stored in data containers implemented by the
``Array`` class. In its most basic usage, the ``Array`` class
implemented in \akantu is similar to the ``vector`` class of the
Standard Template Library (STL) for C++. A simple ``Array`` containing
a sequence of ``nb_element`` values (of a given type) can be generated
with::
Array<type> example_array(nb_element);
where ``type`` usually is ``Real``, ``Int``, ``UInt`` or ``bool``.
Each value is associated to an index, so that data can be accessed by
typing::
auto & val = example_array(index);
``Arrays`` can also contain tuples of values for each index. In that
case, the number of components per tuple must be specified at the
``Array`` creation. For example, if we want to create an ``Array`` to
store the coordinates (sequences of three values) of ten nodes, the
appropriate code is the following::
UInt nb_nodes = 10;
UInt spatial_dimension = 3;
Array<Real> position(nb_nodes, spatial_dimension);
In this case the :math:`x` position of the eighth node number will be given
by ``position(7, 0)`` (in C++, numbering starts at 0 and not 1). If
the number of components for the sequences is not specified, the
default value of 1 is used. Here is a list of some basic operations
that can be performed on ``Array``:
- ``resize(size)`` change the size of the ``Array``.
- ``clear()`` set all entries of the ``Array`` to zero.
- ``set(t)`` set all entries of the ``Array`` to ``t``.
- ``copy(const Array<T> & other)`` copy another ``Array`` into the
current one. The two ``Array`` should have the same number of
components.
- ``push_back(tuple)`` append a tuple with the correct number of
components at the end of the ``Array``.
- ``erase(i)`` erase the value at the i-th position.
- ``find(value)`` search ``value`` in the current ``Array``. Return
position index of the first occurence or -1 if not found.
- ``storage()`` Return the address of the allocated memory of the
``Array``.
``Array`` iterators
-------------------
It is very common in \akantu to loop over arrays to perform a specific
treatment. This ranges from geometric calculation on nodal quantities
to tensor algebra (in constitutive laws for example).
The ``Array`` object has the possibility to request iterators
in order to make the writing of loops easier and enhance readability.
For instance, a loop over the nodal coordinates can be performed like::
// accessing the nodal coordinates Array
// with spatial_dimension components
const auto & nodes = mesh.getNodes();
//creating the iterators
auto it = nodes.begin(spatial_dimension);
auto end = nodes.end(spatial_dimension);
for (; it != end; ++it){
const auto & coords = (*it);
// do what you need ....
}
In that example, each ``coords`` is a ``Vector<Real>`` containing
geometrical array of size ``spatial_dimension`` and the iteration is
conveniently performed by the ``Array`` iterator.
With the switch to ``c++14`` this can be also written as::
// accessing the nodal coordinates Array
// with spatial_dimension components
const auto & nodes = mesh.getNodes();
for (const auto & coords : make_view(nodes, spatial_dimension) {
// do what you need ....
}
The ``Array`` object is intensively used to store second order
tensor values. In that case, it should be specified that the returned
object type is a matrix when constructing the iterator. This is done
when calling the ``begin`` function. For instance, assuming that we
have a ``Array`` storing stresses, we can loop over the stored
tensors by::
// creating the iterators
auto it = stresses.begin(spatial_dimension, spatial_dimension);
auto end = stresses.end(spatial_dimension, spatial_dimension);
for (; it != end; ++it){
Matrix<Real> & stress = (*it);
// do what you need ....
}
In that last example, the ``Matrix`` objects are
``spatial_dimension`` :math:`\times` ``spatial_dimension`` matrices.
The light objects ``Matrix`` and ``Vector`` can be used and
combined to do most common linear algebra. If the number of component
is 1, it is possible to use a ``scalar_iterator`` rather than the
vector/matrix one.
In general, a mesh consists of several kinds of elements.
Consequently, the amount of data to be stored can differ for each
element type. The straightforward example is the connectivity array,
namely the sequences of nodes belonging to each element (linear
triangular elements have fewer nodes than, say, rectangular quadratic
elements etc.). A particular data structure called
``ElementTypeMapArray`` is provided to easily manage this kind of
data. It consists of a group of ``Arrays``, each associated to an
element type. The following code can retrieve the
``ElementTypeMapArray`` which stores the connectivity arrays for a
mesh::
const ElementTypeMapArray<UInt> & connectivities =
mesh.getConnectivities();
Then, the specific array associated to a given element type can be obtained by::
const Array<UInt> & connectivity_triangle =
connectivities(_triangle_3);
where the first order 3-node triangular element was used in the presented piece
of code.
Vector & Matrix
```````````````
The ``Array`` iterators as presented in the previous section can be
shaped as ``Vector`` or ``Matrix``. This objects represent 1st and 2nd
order tensors. As such they come with some functionalities that we
will present a bit more into detail in this here.
``Vector<T>``
'''''''''''''
- Accessors:
- ``v(i)`` gives the ``i`` -th component of the vector ``v``
- ``v[i]`` gives the ``i`` -th component of the vector ``v``
- ``v.size()`` gives the number of component
- Level 1: (results are scalars)
- ``v.norm()`` returns the geometrical norm (:math:`L_2`)
- ``v.norm<N>()`` returns the :math:`L_N` norm defined as :math:`\left(\sum_i
|v(i)|^N\right)^{1/N}`. N can take any positive integer value.
There are also some particular values for the most commonly used
norms, ``L_1`` for the Manhattan norm, ``L_2`` for the geometrical
norm and ``L_inf`` for the norm infinity.
- ``v.dot(x)`` return the dot product of ``v`` and ``x``
- ``v.distance(x)`` return the geometrical norm of :math:`v - x`
- Level 2: (results are vectors)
- ``v += s``, ``v -= s``, ``v *= s``, ``v /= s`` those are
element-wise operators that sum, substract, multiply or divide all the
component of ``v`` by the scalar ``s``
- ``v += x``, ``v -= x`` sums or substracts the vector ``x``
to/from ``v``
- ``v.mul(A, x, alpha)`` stores the result of :math:`\alpha \boldsymbol{A} \vec{x}` in ``v``,
:math:`\alpha` is equal to 1 by default
- ``v.solve(A, b)`` stores the result of the resolution of the system :math:`\boldsymbol{A} \vec{x} =
\vec{b}` in ``v``
- ``v.crossProduct(v1, v2)`` computes the cross product of ``v1`` and ``v2`` and
stores the result in ``v``
``Matrix<T>``
'''''''''''''
- Accessors:
- ``A(i, j)`` gives the component :math:`A_{ij}` of the matrix ``A``
- ``A(i)`` gives the :math:`i^{th}` column of the matrix as a ``Vector``
- ``A[k]`` gives the :math:`k^{th}` component of the matrix, matrices are
stored in a column major way, which means that to access :math:`A_{ij}`, :math:`k = i +
j M`
- ``A.rows()`` gives the number of rows of ``A`` (:math:`M`)
- ``A.cols()`` gives the number of columns of ``A`` (:math:`N`)
- ``A.size()`` gives the number of component in the matrix (:math:`M \times N`)
- Level 1: (results are scalars)
- ``A.norm()`` is equivalent to ``A.norm<L_2>()``
- ``A.norm<N>()`` returns the :math:`L_N` norm defined as
:math:`\left(\sum_i\sum_j |A(i,j)|^N\right)^{1/N}`. N can take
any positive integer value. There are also some particular values
for the most commonly used norms, ``L_1`` for the Manhattan
norm, ``L_2`` for the geometrical norm and ``L_inf`` for
the norm infinity.
- ``A.trace()`` return the trace of ``A``
- ``A.det()`` return the determinant of ``A``
- ``A.doubleDot(B)`` return the double dot product of ``A`` and
``B``, :math:`\mat{A}:\mat{B}`
- Level 3: (results are matrices)
- ``A.eye(s)``, ``Matrix<T>::eye(s)`` fills/creates a matrix with
the :math:`s\mat{I}` with :math:`\mat{I}` the identity matrix
- ``A.inverse(B)`` stores :math:`\mat{B}^{-1}` in ``A``
- ``A.transpose()`` returns :math:`\mat{A}^{t}`
- ``A.outerProduct(v1, v2)`` stores :math:`\vec{v_1} \vec{v_2}^{t}` in
``A``
- ``C.mul<t_A, t_B>(A, B, alpha)``: stores the result of the product of
``A`` and code{B} time the scalar ``alpha`` in ``C``. ``t_A``
and ``t_B`` are boolean defining if ``A`` and ``B`` should be
transposed or not.
+----------+----------+--------------+
|``t_A`` |``t_B`` |result |
| | | |
+----------+----------+--------------+
|false |false |:math:`\mat{C}|
| | |= \alpha |
| | |\mat{A} |
| | |\mat{B}` |
| | | |
+----------+----------+--------------+
|false |true |:math:`\mat{C}|
| | |= \alpha |
| | |\mat{A} |
| | |\mat{B}^t` |
| | | |
+----------+----------+--------------+
|true |false |:math:`\mat{C}|
| | |= \alpha |
| | |\mat{A}^t |
| | |\mat{B}` |
| | | |
+----------+----------+--------------+
|true |true |:math:`\mat{C}|
| | |= \alpha |
| | |\mat{A}^t |
| | |\mat{B}^t` |
+----------+----------+--------------+
- ``A.eigs(d, V)`` this method computes the eigenvalues and
eigenvectors of ``A`` and store the results in ``d`` and ``V`` such
that :math:`d(i) = \lambda_i` and :math:`V(i) = \vec{v_i}` with
:math:`\mat{A}\vec{v_i} = \lambda_i\vec{v_i}` and :math:`\lambda_1 > ... > \lambda_i >
... > \lambda_N`

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