# Getting Started¶

## Building `Akantu`

¶

### Dependencies¶

In order to compile `Akantu`

any compiler supporting fully C++14 should work.
In addition some libraries are required:

CMake (>= 3.5.1)

Boost (preprocessor and Spirit)

zlib

blas/lapack

For the python interface:

Python (>=3 is recommended)

pybind11 (if not present the build system will try to download it)

To run parallel simulations:

MPI

Scotch

To use the static or implicit dynamic solvers at least one of the following libraries is needed:

MUMPS (since this is usually compiled in static you also need MUMPS dependencies)

PETSc

To compile the tests and examples:

Gmsh

google-test (if not present the build system will try to download it)

## Using the python interface¶

You can install `Akantu`

using pip:

```
> pip install akantu
```

You can then import the package in a python script as:

```
import akantu
```

The python API is similar to the C++ one, see Reference . If you encouter any problem with the python interface, you are welcome to do a merge request or post an issue on GitLab .

### Tutorials with the python interface¶

To help getting started, several tutorials using the python interface
are available as notebooks with pre-installed version of `Akantu`

on Binder.
The following tutorials are currently available:

## 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
the function `initialize`

that is 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 ...
}
```

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.

## Compiling your simulation¶

The easiest way to compile your simulation is to create a `cmake`

project by
putting all your code in some directory of your choosing. Then, make sure that
you have `cmake`

installed and create a `CMakeLists.txt`

file. An example of
a minimal `CMakeLists.txt`

file would look like this:

```
project(my_simu)
cmake_minimum_required(VERSION 3.0.0)
find_package(Akantu REQUIRED)
add_akantu_simulation(my_simu my_simu.cc)
```

Then create a directory called `build`

and inside it execute ```
cmake
-DAkantu_DIR=<path_to_akantu> -DCMAKE_BUILD_TYPE=RelWithDebInfo ..
```

. If you
installed `Akantu`

in a standard directory such as `/usr/local`

(using
`make install`

), you can omit the `-DAkantu_DIR=<path_to_akantu>`

option.

Other why `path_to_akantu`

is either the folder where you built `Akantu`

if
you did not do a `make install`

, or if you installed `Akantu`

in
`CMAKE_INSTALL_PREFIX`

it is `<CMAKE_INSTALL_PREFIX>/share/cmake/Akantu`

.

Once `cmake`

managed to configure and generate a `makefile`

you can just do
`make`

## Creating and Loading a Mesh¶

In its current state, `Akantu`

supports three types of meshes: Gmsh, Abaqus and
Diana. Once a `akantu::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);
```

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`

.

## Using Arrays¶

Data in `Akantu`

can be stored in data containers implemented by the
`akantu::Array`

class. In its most basic usage, the `Array`

class implemented in akantu is similar to the `std::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 \(x\) 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`

reset the size of the`Array`

to zero. (warningthis changed in > v4.0)

`copy(const Array & other)`

copy another`Array`

into the current one. The two`Arrays`

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();
for (const auto & coords : make_view(nodes, spatial_dimension)) {
// 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.

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 `make_view`

. For instance,
assuming that we have a `Array`

storing stresses, we
can loop over the stored tensors by:

```
for (const auto & stress :
make_view(stresses, spatial_dimension, spatial_dimension)) {
// stress is of type `const Matrix<Real>&`
}
```

In that last example, the `Matrix<Real>`

objects are
`spatial_dimension`

\(\times\) `spatial_dimension`

matrices. The light
objects `Matrix<T>`

and
`Vector<T>`

can be used and combined to do most
common linear algebra. If the number of component is 1, it is possible to use
`make_view`

to this effect.

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<T>`

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<UInt>`

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<T>`

iterators as presented in the previous
section can be shaped as `Vector<T>`

or
`Matrix<T>`

. 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 (\(L_2\))`v.norm<N>()`

returns the \(L_N\) norm defined as \(\left(\sum_i |v(i)|^N\right)^{1/N}\). 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 \(v - x\)

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 \(\alpha \boldsymbol{A} \vec{x}\) in`v`

, \(\alpha\) is equal to 1 by default`v.solve(A, b)`

stores the result of the resolution of the system \(\boldsymbol{A} \vec{x} = \vec{b}\) 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 \(A_{ij}\) of the matrix`A`

`A(i)`

gives the \(i^{th}\) column of the matrix as a`Vector`

`A[k]`

gives the \(k^{th}\) component of the matrix, matrices are stored in a column major way, which means that to access \(A_{ij}\), \(k = i + j M\)`A.rows()`

gives the number of rows of`A`

(\(M\))`A.cols()`

gives the number of columns of`A`

(\(N\))`A.size()`

gives the number of component in the matrix (\(M \times N\))

Level 1: (results are scalars)

`A.norm()`

is equivalent to`A.norm<L_2>()`

`A.norm<N>()`

returns the \(L_N\) norm defined as \(\left(\sum_i\sum_j |A(i,j)|^N\right)^{1/N}\). 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`

, \(\mat{A}:\mat{B}\)

Level 3: (results are matrices)

`A.eye(s)`

,`Matrix<T>::eye(s)`

fills/creates a matrix with the \(s\mat{I}\) with \(\mat{I}\) the identity matrix`A.inverse(B)`

stores \(\mat{B}^{-1}\) in`A`

`A.transpose()`

returns \(\mat{A}^{t}\)`A.outerProduct(v1, v2)`

stores \(\vec{v_1} \vec{v_2}^{t}\) 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

\(\mat{C} = \alpha \mat{A} \mat{B}\)

false

true

\(\mat{C} = \alpha \mat{A} \mat{B}^t\)

true

false

\(\mat{C} = \alpha \mat{A}^t \mat{B}\)

true

true

\(\mat{C} = \alpha \mat{A}^t \mat{B}^t\)

`A.eigs(d, V)`

this method computes the eigenvalues and eigenvectors of`A`

and store the results in`d`

and`V`

such that \(d(i) = \lambda_i\) and \(V(i) = \vec{v_i}\) with \(\mat{A}\vec{v_i} = \lambda_i\vec{v_i}\) and \(\lambda_1 > ... > \lambda_i > ... > \lambda_N\)

## Mesh¶

### Manipulating group of nodes and/or elements¶

`Akantu`

provides the possibility to manipulate subgroups of elements and
nodes. Any `ElementGroup`

and/or
`NodeGroup`

must be managed by a
`GroupManager`

. Such a manager has the role to
associate group objects to names. This is a useful feature, in particular for
the application of the boundary conditions, as will be demonstrated in section
Setting Boundary Conditions. To most general group manager is the `Mesh`

class which inherits from `GroupManager`

.

For instance, the following code shows how to request an element group to a mesh:

```
// request creation of a group of nodes
NodeGroup & my_node_group = mesh.createNodeGroup("my_node_group");
// request creation of a group of elements
ElementGroup & my_element_group = mesh.createElementGroup("my_element_group");
/* fill and use the groups */
```

#### The `NodeGroup`

object¶

A group of nodes is stored in `NodeGroup`

objects. They are quite simple objects which store the indexes of the selected
nodes in a `Array<UInt>`

. Nodes are selected by
adding them when calling `add`

. For instance
you can select nodes having a positive \(X\) coordinate with the following
code:

```
const auto & nodes = mesh.getNodes();
auto & group = mesh.createNodeGroup("XpositiveNode");
for (auto && data : enumerate(make_view(nodes, spatial_dimension))){
auto node = std::get<0>(data);
const auto & position = std::get<1>(data);
if (position(0) > 0) group.add(node);
}
```

#### The `ElementGroup`

object¶

A group of elements is stored in `ElementGroup`

objects. Since a group can contain elements of various
types the `ElementGroup`

object stores indexes
in a `ElementTypeMapArray<UInt>`

object. Then elements can be added to the group by calling `add`

.

For instance, selecting the elements for which the barycenter of the nodes has a positive \(X\) coordinate can be made with:

```
auto & group = mesh.createElementGroup("XpositiveElement");
Vector<Real> barycenter(spatial_dimension);
for_each_element(mesh, [&](auto && element) {
mesh.getBarycenter(element, barycenter);
if (barycenter(_x) > 0.) { group.add(element); }
});
```