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Structural Mechanics Model
==========================
Static structural mechanics problems can be handled using the class
:cpp:class:`StructuralMechanicsModel <akantu::StructuralMechanicsModel>`. So
far, ``Akantu`` provides 2D and 3D Bernoulli beam elements :cite:`frey2009`.
This model is instantiated for a given :cpp:class:`Mesh <akantu::Mesh>`, as for
the :cpp:class:`SolidMechanicsModel <akantu::SolidMechanicsModel>`. The model
will create its own :cpp:class:`FEEngine <akantu::FEEngine>` object to compute
the interpolation, gradient, integration and assembly operations. The
:cpp:class:`StructuralMechanicsModel <akantu::StructuralMechanicsModel>`
constructor is called in the following way:
.. code-block:: c++
StructuralMechanicsModel model(mesh, spatial_dimension);
where ``mesh`` is a :cpp:class:`Mesh <akantu::Mesh>` object defining the structure for
which the equations of statics are to be solved, and
``spatial_dimension`` is the dimensionality of the problem. If
``spatial_dimension`` is omitted, the problem is assumed to have
the same dimensionality as the one specified by the mesh.
\note[\ 1]{Dynamic computations are not supported to date.}
\note[\ 2]{Structural meshes are created and loaded as described in
Section~\ref{sect:common:mesh} with ``_miot_gmsh_struct`` instead of ``_miot_gmsh``:}
.. code-block:: c++
Mesh mesh;
mesh.read("structural_mesh.msh", _miot_gmsh_struct);
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::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
Table~\ref{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;
\begin{table}[htb] \centering
\begin{tabular}{cl}
\toprule
Field & Description \\
\midrule
``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)\\
\bottomrule
\end{tabular}
\caption{Material properties for structural elements defined in
the class \code{StructuralMaterial}.}
\label{tab:structMechMod:strucMaterial}
\end{table}
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.
.. code-block:: c++
for (UInt i = 0; i < nb_element_mat_1; ++i) {
model.getElementMaterial(_bernoulli_beam_2)(i,0) = 1;
}
.. _sect:structMechMod:boundary:
Setting Boundary Conditions
```````````````````````````
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
global system of coordinates (``_bft_traction``).
.. code-block:: c++
static void lin_load(double * position, double * load,
Real * normal, UInt surface_id){
memset(load,0,sizeof(Real)*3);
load[1] = position[0]*position[0]-250;
}
int main(){
...
model.computeForcesFromFunction<_bernoulli_beam_2>(lin_load,
_bft_traction_local);
...
}
.. _sect:structMechMod:static:
Static Analysis
---------------
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.
To solve such a problem, the static solver of the
:cpp:class:`StructuralMechanicsModel <akantu::StructuralMechanicsModel>` object
is used. First a model has to be created and initialized.
.. code-block:: c++
StructuralMechanicsModel model(mesh);
model.initFull();
- \item :cpp:class:`model.initFull <akantu::model.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 (:ref:`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
Newton-Raphson algorithm as described in
Section~\ref{sect:smm:static}.
\note{To date, :cpp:class:`StructuralMechanicsModel
<akantu::StructuralMechanicsModel>` handles only constitutively and
geometrically linear problems, the algorithm is therefore guaranteed to converge
in two iterations.}
.. code-block:: c++
model.updateResidual();
model.solve();
\item :cpp:func:`model.updateResidual
<akantu::StructuralMechanicsModel::updateResidual>` assembles the internal
forces and removes them from the external forces.
\item :cpp:class:`model.solve <akantu::model.solve>` solves the :eq:`eqn:structMechMod:static`.
The **increment** vector of the model will contain the new
increment of displacements, and the **displacement_rotation**
vector is also updated to the new displacements.
At the end of the analysis, the final solution is stored in the
**displacement_rotation** vector. A full example of how to solve a structural
mechanics problem is presented in the code
``example/structural_mechanics/bernoulli_beam_2_example.cc``. This example is
composed of a 2D beam, clamped at the left end and supported by two rollers as
shown in :numref:`fig-structMechMod-exem1_1`. The problem is defined by the
applied load :math:`q=6 \text{\kN/m}`, moment :math:`\bar{M} = 3.6 \text{kN m}`,
moments of inertia :math:`I_1 = 250\,000 \text{cm}^4` and :math:`I_2 = 128\,000
\text{cm}^4` and lengths :math:`L_1 = 10\text{m}` and :math:`L_2 = 8\text{m}`.
The resulting rotations at node two and three are :math:`\varphi_2 = 0.001\,167`
and :math:`\varphi_3 = -0.000\,771`.
.. _fig-structMechMod-exem1_1:
.. figure:: figures/beam_example.svg
:align: center
2D beam example
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