This model contains at least the following :cpp:class:`Arrays <akantu::Arrays>`:
- **blocked_dofs** contains a Boolean value for each degree of
freedom specifying whether that degree is blocked or not. A
Dirichlet boundary condition can be prescribed by setting the
**blocked_dofs** value of a degree of freedom to
``true``. The **displacement** is computed for all degrees
of freedom for which the **blocked_dofs** value is set to
``false``. For the remaining degrees of freedom, the imposed
values (zero by default after initialization) are kept.
- **displacement_rotation** contains the generalized displacements (*i.e.* displacements and rotations) of all degrees of freedom. It can be either a computed displacement for free degrees of freedom or an imposed displacement in case of blocked ones (:math:`\vec{u}` in the following).
- **external_force** contains the generalized external forces (forces and moments) applied to the nodes (:math:`\vec{f_{\st{ext}}}` in the following).
- **internal_force** contains the generalized internal forces (forces and moments) applied to the nodes (:math:`\vec{f_{\st{int}}}` in the following).
An example to help understand how to use this model will be presented in the
next section.
.. _sec:structMechMod:setup:
Model Setup
-----------
Initialization
``````````````
The easiest way to initialize the structural mechanics model is:
.. code-block:: c++
model.initFull();
The method :cpp:class:`initFull <akantu::StructuralMechanicsModel::initFull>` computes the shape
functions, initializes the internal vectors mentioned above and allocates the
memory for the stiffness matrix, unlike the solid mechanics model, its default
argument is ``_static``.
Material properties are defined using the :cpp:class:`StructuralMaterial
<akantu::StructuralMaterial>` structure described in
:numref:`tab-structmechmod-strucmaterial`. Such a definition could, for
instance, look like
.. code-block:: c++
StructuralMaterial mat1;
mat.E=3e10;
mat.I=0.0025;
mat.A=0.01;
.. _tab-structmechmod-strucmaterial:
.. table:: Material properties for structural elements defined in the class :cpp:class:`StructuralMaterial <akantu::StructuralMaterial>`.
:align: center
====== ======
Field Description
====== ======
``E`` Young's modulus
``A`` Cross section area
``I`` Second cross sectional moment of inertia (for 2D elements)
``Iy`` ``I`` around beam :math:`y`--axis (for 3D elements)
``Iz`` ``I`` around beam :math:`z`--axis (for 3D elements)
``GJ`` Polar moment of inertia of beam cross section (for 3D elements)
====== ======
Materials can be added to the model's ``element_material`` vector using
.. code-block:: c++
model.addMaterial(mat1);
They are successively numbered and then assigned to specific elements.
As explained before, the Dirichlet boundary conditions are applied through the
array **blocked_dofs**. Two options exist to define Neumann conditions.
If a nodal force is applied, it has to be directly set in the array
**force_momentum**. For loads distributed along the beam length, the
method :cpp:class:`computeForcesFromFunction <akantu::computeForcesFromFunction>` integrates them into nodal forces. The
method takes as input a function describing the distribution of loads along the
beam and a functor :cpp:class:`BoundaryFunctionType <akantu::BoundaryFunctionType>` specifing if the function is expressed in the local coordinates (``_bft_traction_local``) or in the
The :cpp:class:`StructuralMechanicsModel <akantu::StructuralMechanicsModel>` class can perform static analyses of structures. In this case, the equation to solve is the same as for the :cpp:class:`SolidMechanicsModel <akantu::SolidMechanicsModel>` used for static analyses
.. math:: \mat{K} \vec{u} = \vec{f_{\st{ext}}}~,
:label: eqn-structmechmod-static
where :math:`\mat{K}` is the global stiffness matrix, :math:`\vec{u}` the
generalized displacement vector and :math:`\vec{f_{\st{ext}}}` the vector of
generalized external forces applied to the system.
is used. First a model has to be created and initialized.
.. code-block:: c++
StructuralMechanicsModel model(mesh);
model.initFull();
- :cpp:func:`model.initFull <akantu::StructuralMechanicsModel::initFull>` initializes all
internal vectors to zero.
Once the model is created and initialized, the boundary conditions can be set as explained in Section :ref:`sect:structMechMod:boundary`. Boundary conditions will prescribe the external forces or moments for the free degrees of freedom :math:`\vec{f_{\st{ext}}}` and displacements or rotations for the others. To completely define the system represented by equation (:eq:`eqn-structmechmod-static`), the global stiffness matrix :math:`\mat{K}` must be assembled.
.. code-block:: c++
model.assembleStiffnessMatrix();
The computation of the static equilibrium is performed using the same