\chapter{Solid Mechanics Model\index{SolidMechanicsModel}\label{sect:smm}}

The solid mechanics model is a specific implementation of the
\code{Model} interface dedicated to handle the equations of motion or
equations of equilibrium. The model is created for a given mesh.  It
will create its own \code{FEEngine} object to compute the interpolation,
gradient, integration and assembly operations.  A
\code{SolidMechanicsModel} object can simply be created like this:
\begin{cpp}
  SolidMechanicsModel model(mesh);
\end{cpp}
where \code{mesh} is the mesh for which the equations are to be
solved. A second parameter called \code{spatial\_dimension} can be
added after \code{mesh} if the spatial dimension of the problem is
different than that of the mesh.

This model contains at least the the following six \code{Arrays}:
\begin{description}
\item[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
  \textbf{blocked\_dofs} value of a degree of freedom to \code{true}.
  A Neumann boundary condition can be applied by setting the
  \textbf{blocked\_dofs} value of a degree of freedom to \code{false}.
  The \textbf{displacement}, \textbf{velocity} and
  \textbf{acceleration} are computed for all degrees of freedom for
  which the \textbf{blocked\_dofs} value is set to \code{false}. For
  the remaining degrees of freedom, the imposed values (zero by
  default after initialization) are kept.
\item[displacement] contains the displacements 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
  ($\vec{u}$ in the following).
\item[velocity] contains the velocities of all degrees of freedom.  As
  \textbf{displacement}, it contains computed or imposed velocities
  depending on the nature of the degrees of freedom ($\dot{\vec{u}}$
  in the following).
\item[acceleration] contains the accelerations of all degrees of
  freedom. As \textbf{displacement}, it contains computed or imposed
  accelerations depending on the nature of the degrees of freedom
  ($\ddot{\vec{u}}$ in the following).
\item[force] contains the external forces applied on the nodes
  ($\vec{f}_{\st{ext}}$ in the following).
\item[residual] contains the difference between external and internal
  forces. On blocked degrees of freedom, \textbf{residual} contains
  the support reactions.  ($\vec{r}$ in the following).  It should be
  mentioned that at equilibrium \textbf{residual} should be zero on
  free degrees of freedom.
\end{description}

Some examples to help to understand how to use this model will be
presented in the next sections.

\section{Model Setup}


\subsection{Setting Initial Conditions \label{sect:smm:initial_condition}}

For a unique solution of the equations of motion, initial
displacements and velocities for all degrees of freedom must be
specified:
\begin{eqnarray}
  \vec{u}(t=0) = \vec{u}_0\\
  \dot{\vec u}(t=0) =\vec{v}_0
\end{eqnarray} The solid mechanics model can be initialized as
follows:
\begin{cpp}
  model.initFull()
\end{cpp}
This function initializes the internal arrays and sets them to
zero. Initial displacements and velocities that are not equal to zero
can be prescribed by running a loop over the total number of
nodes. Here, the initial displacement in $x$-direction and the
initial velocity in $y$-direction for all nodes is set to $0.1$ and $1$,
respectively.
\begin{cpp}
Array<Real> & disp = model.getDisplacement();
Array<Real> & velo = model.getVelocity();
for (UInt i = 0; i < mesh.getNbNodes(); ++i) {
  disp(i, 0) = 0.1;
  velo(i, 1) = 1.;
}
\end{cpp}

\subsection{Setting Boundary Conditions\label{sect:smm:boundary}}
This section explains how to impose Dirichlet or Neumann boundary
conditions. A Dirichlet boundary condition specifies the values that
the displacement needs to take for every point $x$ at the boundary
($\Gamma_u$) of the problem domain (Fig.~\ref{fig:smm:boundaries}):
\begin{equation}
  \vec{u} = \bar{\vec u} \quad \forall \vec{x}\in
  \Gamma_{u}
\end{equation}
A Neumann boundary condition imposes the value of the gradient of the
solution at the boundary $\Gamma_t$ of the problem domain
(Fig.~\ref{fig:smm:boundaries}):
\begin{equation}
  \vec{t} = \mat{\sigma} \vec{n} = \bar{\vec t} \quad
  \forall \vec{x}\in \Gamma_{t}
\end{equation}

\begin{figure} \centering
\def\svgwidth{0.5\columnwidth}
  \input{figures/problemDomain.pdf_tex}
  \caption{Problem domain $\Omega$ with boundary in three
    dimensions. The Dirchelet and the Neumann regions of the boundary
    are denoted with $\Gamma_u$ and $\Gamma_t$,
    respecitvely.\label{fig:smm:boundaries}}
  \label{fig:problemDomain}
\end{figure}

Different ways of imposing these boundary conditions exist. A basic
way is to loop over nodes or elements at the boundary and apply local
values. A more advanced method consists of using the notion of the
boundary of the mesh. In the following both ways are presented.

Starting with the basic approach, as mentioned, the Dirichlet boundary
conditions can be applied by looping over the nodes and assigning the
required values. Figure~\ref{fig:smm:dirichlet_bc} shows a beam with a
fixed support on the left side. On the right end of the beam, a load
is applied. At the fixed support, the displacement has a given
value. For this example, the displacements in both the $x$ and the
$y$-direction are set to zero. Implementing this displacement boundary
condition is similar to the implementation of initial displacement
conditions described above. However, in order to impose a displacement
boundary condition for all time steps, the corresponding nodes need to
be marked as boundary nodes as shown in the following code:

\begin{cpp}
Array<bool> & blocked = model.getBlockedDOFs();
const Array<Real> & pos = mesh.getNodes();

UInt nb_nodes = mesh.getNbNodes();

for (UInt i = 0; i < nb_nodes; ++i) {
  if(Math::are_float_equal(pos(i, 0), 0)) {
    blocked(i, 0) = true; //block displacement in x-direction
    blocked(i, 1) = true; //block displacement in y-direction
    disp(i, 0) = 0.; //fixed displacement in x-direction
    disp(i, 1)= 0.; //fixed displacement in y-direction
  }
}
\end{cpp}

\begin{figure}[!htb]
  \centering
  \includegraphics[scale=0.4]{figures/dirichlet}
  \caption{Beam with fixed support.\label{fig:smm:dirichlet_bc}}
\end{figure}

For the more advanced approach, one needs the notion of a boundary in
the mesh. Therefore, the boundary should be created before boundary
condition functors can be applied. Generally the boundary can be
specified from the mesh file or the geometry.  For the first case, the
function \code{createGroupsFromMeshData} is called.  This function
can read any types of mesh data which are provided in the mesh
file. If the mesh file is created with Gmsh, the function takes one
input strings which is either \code{tag\_0}, \code{tag\_1} or
\code{physical\_names}. The first two tags are assigned by Gmsh to
each element which shows the physical group that they belong to. In
Gmsh, it is also possible to consider strings for different groups of
elements. These elements can be separated by giving a string
\code{physical\_names} to the function
\code{createGroupsFromMeshData}.  Boundary conditions can also be
created from the geometry by calling
\code{createBoundaryGroupFromGeometry}. This function gathers all the
elements on the boundary of the geometry.

To apply the required boundary conditions, the function \code{applyBC}
needs to be called on a \code{SolidMechanicsModel}. This function
gets a Dirichlet or Neumann functor and a string which specifies the
desired boundary on which the boundary conditions is to be
applied. The functors specify the type of conditions to apply. Three
built-in functors for Dirichlet exist: \code{FlagOnly, FixedValue,}
and \code{IncrementValue}. The functor \code{FlagOnly} is used if a
point is fixed in a given direction. Therefore, the input parameter to
this functor is only the fixed direction. The \code{FixedValue}
functor is used when a displacement value is applied in a fixed
direction. The \code{IncrementValue} applies an increment to the
displacement in a given direction. The following code shows the
utilization of three functors for the top, bottom and side surface of
the mesh which were already defined in the Gmsh file:

\begin{cpp}
mesh.createGroupsFromMeshData<std::string>("physical_names");

model.applyBC(BC::Dirichlet::FixedValue(13.0, BC::_y), "Top");

model.applyBC(BC::Dirichlet::FlagOnly(BC::_x), "Bottom");

model.applyBC(BC::Dirichlet::IncrementValue(13.0, BC::_x), "Side");
\end{cpp}

To apply a Neumann boundary condition, the applied traction or stress
should be specified before. In case of specifying the traction on the
surface, the functor \code{FromTraction} of Neumann boundary
conditions is called. Otherwise, the functor \code{FromStress} should
be called which gets the stress tensor as an input parameter.

\begin{cpp}
Array<Real> surface_traction(3);
surface_traction(0)=0.0;
surface_traction(1)=0.0;
surface_traction(2)=-1.0;

Array<Real> surface_stress(3, 3, 0.0);
surface_stress(0,0)=0.0;
surface_stress(1,1)=0.0;
surface_stress(2,2)=-1.0;

model.applyBC(BC::Neumann::FromTraction(surface_traction), "Bottom");

model.applyBC(BC::Neumann::FromStress(surface_stress), "Top");
\end{cpp}

If the boundary conditions need to be removed during the simulation, a
functor is called from the Neumann boundary condition to free those
boundary conditions from the desired boundary.

\begin{cpp}
 model.applyBC(BC::Neumann::FreeBoundary(), "Side");
\end{cpp}

User specified functors can also be implemented.  A full example for
setting both initial and boundary conditions can be found in
\shellcode{\examplesdir/boundary\_conditions.cc}.  The problem solved
in this example is shown in Fig.~\ref{fig:smm:bc_and_ic}. It consists
of a plate that is fixed with movable supports on the left and bottom
side. On the right side, a traction, which increases linearly with the
number of time steps, is applied. The initial displacement and
velocity in $x$-direction at all free nodes is zero and two
respectively.
\begin{figure}[!htb]
  \centering
  \includegraphics[scale=0.8]{figures/bc_and_ic_example}
  \caption{Plate on movable supports.\label{fig:smm:bc_and_ic}}
\end{figure}

\subsection{Material Selector\label{sect:smm:materialselector}}

If the user wants to assign different materials to the different
finite elements of choice in \akantu, a material selector has to be
used. By default, \akantu assigns the first valid material in the
material file to all elements present in the model (regular continuum
materials are assigned to the regular elements and cohesive materials
are assigned to cohesive elements or element facets).

To assign different materials to specific elements, mesh data
information such as tag information or specified pyhsical names can be
used. \code{MeshDataMaterialSelector} class uses this information to
assign different materials. With the proper physical name or tag name
and index, different materials can be assigned as demonstrated in the
examples below.

\begin{cpp}
  MeshDataMaterialSelector<std::string> * mat_selector;
  mat_selector = new MeshDataMaterialSelector<std::string>("physical_names", model);
  model.setMaterialSelector(*mat_selector);
\end{cpp}

In this example the physical names specified in a GMSH geometry file will by
used to match the material names in the input file.

Another example would be:

\begin{cpp}
  MeshDataMaterialSelector<UInt> * mat_selector;
  mat_selector = new MeshDataMaterialSelector<UInt>("tag_1", model);
  model.setMaterialSelector(*mat_selector);
\end{cpp}

where \code{tag\_1} of the mesh file is used as the classifier index
and the elements that have index value equal to one will be assigned
to the second material in the material file.

There are four different material selectors pre-defined in
\akantu. \code{MaterialSelector} and \code{DefaultMaterialSelector} is
used to assign a material to regular elements by default. For the
regular elements, as in the example above,
\code{MeshDataMaterialSelector} can be used to assign different
materials to different elements. However, for cohesive elements, it is
not possible to assign multiple cohesive materials automatically and
user has to write a custom class. \akantu has a pre-defined material
selector to assign the first cohesive material by default to the
cohesive elements which is called
\code{DefaultMaterialCohesiveSelector} and it inherits its properties
from \code{DefaultMaterialSelector}. This material selector first
checks if an element is a cohesive element or a facet. If there is,
then the first cohesive material in the material file is assigned to
that element. Otherwise, this material is assigned to the element
facet, to which, a cohesive element might be inserted later. Hence,
this material selector works for the extrinsic cohesive elements which
are dynamically inserted throughout the analysis. If the element of
interest is not a cohesive element or an element facet, then
\code{DefaultMaterialSelector} is used by default.

Apart from the \akantu's default material selectors, users can always
develop their own classes in the main code to tackle various
multi-material assignment situations.

% An application of \code{DefaultMaterialCohesiveSelector} and usage in
% a customly generated material selector class can be seen in
% \shellcode{\examplesdir/cohesive\_element/cohesive\_extrinsic\_IG\_TG/cohesive\_extrinsic\_IG\_TG.cc}.

\IfFileExists{manual-cohesive_elements_insertion.tex}{\input{manual-cohesive_elements_insertion}}{}

\section{Static Analysis\label{sect:smm:static}}

The \code{SolidMechanicsModel} class can handle different analysis
methods, the first one being presented is the static case.  In this
case, the equation to solve is
\begin{equation}
  \label{eqn:smm:static} \mat{K} \vec{u} =
  \vec{f}_{\st{ext}}
\end{equation}
where $\mat{K}$ is the global stiffness matrix, $\vec{u}$ the
displacement vector and $\vec{f}_{\st{ext}}$ the vector of external
forces applied to the system.

To solve such a problem, the static solver of the
\code{SolidMechanicsModel}\index{SolidMechanicsModel} object is used.
First, a model has to be created and initialized.  To create the
model, a mesh (which can be read from a file) is needed, as explained
in Section~\ref{sect:common:mesh}.  Once an instance of a
\code{SolidMechanicsModel} is obtained, the easiest way to initialize
it is to use the \code{initFull}\index{SolidMechanicsModel!initFull}
method by giving the \code{SolidMechanicsModelOptions}. These options
specify the type of analysis to be performed and whether the materials
should be initialized or not.
\begin{cpp}
SolidMechanicsModel model(mesh);
model.initFull(SolidMechanicsModelOptions(_static));
\end{cpp}
Here, a static analysis is chosen by passing the argument
\code{\_static} to the method. By default, the Boolean for no
initialization of the materials is set to false, so that they are
initialized during the \code{initFull}. The method \code{initFull}
also initializes all appropriate vectors to zero.  Once the model is
created and initialized, the boundary conditions can be set as
explained in Section~\ref{sect:smm:boundary}.  Boundary conditions
will prescribe the external forces for some free degrees of freedom
$\vec{f}_{\st{ext}}$ and displacements for some others.  At this point
of the analysis, the function
\code{solveStep}\index{SolidMechanicsModel!solveStep} can be called:
\begin{cpp}
model.solveStep<_scm_newton_raphson_tangent_modified, _scc_residual>(1e-4, 1);
\end{cpp}
This function is templated by the solving method and the convergence
criterion and takes two arguments: the tolerance and the maximum
number of iterations, which are $\num{1e-4}$ and $1$ for this example. The
modified Newton-Raphson method is chosen to solve the system. In this
method, the equilibrium equation (\ref{eqn:smm:static}) is modified in
order to apply a Newton-Raphson convergence algorithm:
\begin{align}\label{eqn:smm:static-newton-raphson}
  \mat{K}^{i+1}\delta\vec{u}^{i+1} &= \vec{r} \\
  &= \vec{f}_{\st{ext}} -\vec{f}_{\st{int}}\\
  &= \vec{f}_{\st{ext}} - \mat{K}^{i} \vec{u}^{i}\\
  \vec{u}^{i+1} &= \vec{u}^{i} + \delta\vec{u}^{i+1}~,\nonumber
\end{align}
where $\delta\vec{u}$ is the increment of displacement to be added
from one iteration to the other, and $i$ is the Newton-Raphson
iteration counter.  By invoking the \code{solveStep} method in the
first step, the global stiffness matrix $\mat{K}$ from
Equation~(\ref{eqn:smm:static}) is automatically assembled. A
Newton-Raphson iteration is subsequently started, $\mat{K}$ is updated
according to the displacement computed at the previous iteration and
one loops until the forces are balanced (\code{\_scc\_residual}), \ie
$||\vec{r}|| < \mbox{\code{\_scc\_residual}}$.  One can also iterate
until the increment of displacement is zero (\code{\_scc\_increment})
which also means that the equilibrium is found.  For a linear elastic
problem, the solution is obtained in one iteration and therefore the
maximum number of iterations can be set to one. But for a non-linear
case, one needs to iterate as long as the norm of the residual exceeds
the tolerance threshold and therefore the maximum number of iterations
has to be higher, e.g.  $100$:
\begin{cpp}
model.solveStep<_scm_newton_raphson_tangent_modified,_scc_residual>(1e-4, 100)
\end{cpp}
At the end of the analysis, the final solution is stored in the
\textbf{displacement} vector.  A full example of how to solve a static
problem is presented in the code \code{\examplesdir/static/static.cc}.
This example is composed of a 2D plate of steel, blocked with rollers
on the left and bottom sides as shown in Figure \ref{fig:smm:static}.
The nodes from the right side of the sample are displaced by $0.01\%$
of the length of the plate.

\begin{figure}[!htb]
  \centering
  \includegraphics[scale=1.05]{figures/static}
  \caption{Numerical setup\label{fig:smm:static}}
\end{figure}

The results of this analysis is depicted in
Figure~\ref{fig:smm:implicit:static_solution}.

\begin{figure}[!htb]
  \centering
  \includegraphics[width=.7\linewidth]{figures/static_analysis}
  \caption{Solution of the static analysis. Left: the initial
condition, right: the solution (deformation magnified 50 times)}
  \label{fig:smm:implicit:static_solution}
\end{figure}

\section{Dynamic Methods} \label{sect:smm:Dynamic_methods}

Different ways to solve the equations of motion are implemented in the
solid mechanics model.  The complete equations that should be solved
are:
\begin{equation}
\label{eqn:equation-motion}
\mat{M}\ddot{\vec{u}} +
\mat{C}\dot{\vec{u}} + \mat{K}\vec{u} = \vec{f}_{\st{ext}}~,
\end{equation}
where $\mat{M}$, $\mat{C}$ and $\mat{K}$ are the mass,
damping and stiffness matrices, respectively.

In the previous section, it has already been discussed how to solve this
equation in the static case, where $\ddot{\vec{u}} = \dot{\vec{u}} = 0$.  Here
the method to solve this equation in the general case will be presented.  For
this purpose, a time discretization has to be specified.  The most common
discretization method in solid mechanics is the Newmark-$\beta$ method, which is
also the default in \akantu.

For the Newmark-$\beta$ method, (\ref{eqn:equation-motion}) becomes a
system of three equations (see \cite{curnier92a} \cite{hughes-83a} for
more details):
\begin{align}
\mat{M} \ddot{\vec{u}}_{n+1} + \mat{C}\dot{\vec{u}}_{n+1} + \mat{K} \vec{u}_{n+1} &={\vec{f}_{\st{ext}}}_{\, n+1}
\label{eqn:equation-motion-discret} \\
\vec{u}_{n+1} &=\vec{u}_{n} + \left(1 - \alpha\right) \Delta t \dot{\vec{u}}_{n} +
\alpha \Delta t \dot{\vec{u}}_{n+1} + \left(\frac{1}{2} -
\alpha\right) \Delta t^2
\ddot{\vec{u}}_{n} \label{eqn:finite-difference-1}\\
\dot{\vec{u}}_{n+1} &= \dot{\vec{u}}_{n} + \left(1 - \beta\right)
\Delta t \ddot{\vec{u}}_{n} + \beta \Delta t
\ddot{\vec{u}}_{n+1} \label{eqn:finite-difference-2}
\end{align}

In these new equations, $\ddot{\vec{u}}_{n}$, $\dot{\vec{u}}_{n}$ and
$\vec{u}_{n}$ are the approximations of $\ddot{\vec{u}}(t_n)$,
$\dot{\vec{u}}(t_n)$ and $\vec{u}(t_n)$.
Equation~(\ref{eqn:equation-motion-discret}) is the equation of motion
discretized in space (finite-element discretization), and equations
(\ref{eqn:finite-difference-1}) and (\ref{eqn:finite-difference-2})
are discretized in both space and time (Newmark discretization).  The
$\alpha$ and $\beta$ parameters determine the stability and the
accuracy of the algorithm. Classical values for $\alpha$ and $\beta$
are usually $\beta = 1/2$ for no numerical damping and $0 < \alpha <
1/2$.

\begin{center}
  \begin{tabular}{cll}
    \toprule
    $\alpha$ & Method ($\beta = 1/2$) & Type\\
    \midrule
    $0$ & central difference & explicit\\
    $1/6$ & Fox-Goodwin (royal road) &implicit\\
    $1/3$ & Linear acceleration &implicit\\
    $1/2$ & Average acceleration (trapezoidal rule)& implicit\\
    \bottomrule
  \end{tabular}
\end{center}

The solution of this system of equations,
(\ref{eqn:equation-motion-discret})-(\ref{eqn:finite-difference-2}) is
split into a predictor and a corrector system of equations.  Moreover,
in the case of a non-linear equations, an iterative algorithm such as
the Newton-Raphson method is applied. The system of equations can be
written as:

\begin{enumerate}
\item \textit{Predictor:}
\begin{align}
  \vec{u}_{n+1}^{0} &= \vec{u}_{n} + \Delta t
  \dot{\vec{u}}_{n} + \frac{\Delta t^2}{2} \ddot{\vec{u}}_{n} \\
  \dot{\vec{u}}_{n+1}^{0} &= \dot{\vec{u}}_{n} + \Delta t
  \ddot{\vec{u}}_{n} \\
  \ddot{\vec{u}}_{n+1}^{0} &= \ddot{\vec{u}}_{n}
\end{align}

\item \textit{Solve:}
\begin{align}
  \left(c \mat{M} + d \mat{C} + e \mat{K}_{n+1}^i\right)
  \vec{w} = {\vec{f}_{\st{ext}}}_{\,n+1} - {\vec{f}_{\st{int}}}_{\,n+1}^i -
  \mat{C} \dot{\vec{u}}_{n+1}^i - \mat{M} \ddot{\vec{u}}_{n+1}^i = \vec{r}_{n+1}^i
\end{align}

\item \textit{Corrector:}
\begin{align}
  \ddot{\vec{u}}_{n+1}^{i+1} &= \ddot{\vec{u}}_{n+1}^{i} +c \vec{w} \\
  \dot{\vec{u}}_{n+1}^{i+1} &= \dot{\vec{u}}_{n+1}^{i} + d\vec{w} \\
  \vec{u}_{n+1}^{i+1} &= \vec{u}_{n+1}^{i} + e \vec{w}
\end{align}
\end{enumerate}

where $i$ is the Newton-Raphson iteration counter and $c$, $d$ and $e$
are parameters depending on the method used to solve the equations

\begin{center}
  \begin{tabular}{lcccc}
    \toprule
    & $\vec{w}$ & $e$ & $d$ & $c$\\
    \midrule
    in acceleration &$ \delta\ddot{\vec{u}}$ & $\alpha \beta\Delta t^2$ &$\beta \Delta t$ &$1$\\
    in velocity & $ \delta\dot{\vec{u}}$& $\frac{1}{\beta} \Delta t$ & $1$ & $\alpha\Delta t$\\
    in displacement &$\delta\vec{u}$ & $ 1$ & $\frac{1}{\alpha} \Delta t$ & $\frac{1}{\alpha \beta} \Delta t^2$\\
    \bottomrule
  \end{tabular}
\end{center}

%\note{If you want to use the implicit solver \akantu should be compiled at least with one sparse matrix solver such as Mumps\cite{mumps}.}


\subsection{Implicit Time Integration}
To solve a problem with an implicit time integration scheme, first a
\code{SolidMechanicsModel} object has to be created and initialized.
Then the initial and boundary conditions have to be set.  Everything
is similar to the example in the static case
(Section~\ref{sect:smm:static}), however, in this case the implicit
dynamic scheme is selected at the initialization of the model.

\begin{cpp}
SolidMechanicsModel model(mesh);
model.initFull(SolidMechanicsModelOptions(_implicit_dynamic));
/*Boundary conditions see Section~%\ref{sect:smm:boundary}% */
\end{cpp}
Because a dynamic simulation is conducted, an integration time step
$\Delta t$ has to be specified. In the case of implicit simulations,
\akantu implements a trapezoidal rule by default.  That is to say
$\alpha = 1/2$ and $\beta = 1/2$ which is unconditionally
stable. Therefore the value of the time step can be chosen arbitrarily
within reason.  \index{SolidMechanicsModel!setTimeStep}
\begin{cpp}
model.setTimeStep(time_step);
\end{cpp}
Since the system has to be solved for a given amount of time, the
method \code{solveStep()}, (which has already been used in the static
example in Section~\ref{sect:smm:static}), is called inside a time
loop:
\begin{cpp}
/// time loop
Real time = 0.;
for (UInt s = 1; time <max_time; ++s, time += time_step) {
  model.solveStep<_scm_newton_raphson_tangent_modified,_scc_increment>(1e-12, 100);
}
\end{cpp}
An example of solid mechanics with an implicit time integration scheme
is presented in
\shellcode{\examplesdir/implicit/implicit\_dynamic.cc}.  This example
consists of a 3D beam of
$\SI{10}{\metre}\,\times\,\SI{1}{\metre}\,\times\,\SI{1}{\metre}$
blocked on one side and is on a roller on the other side.  A constant
force of \SI{5}{\kilo\newton} is applied in its middle.
Figure~\ref{fig:smm:implicit:dynamic} presents the geometry of this
case. The material used is a fictitious linear elastic material with a
density of \SI{1000}{\kilo\gram\per\cubic\metre}, a Young's Modulus of
\SI{120}{\mega\pascal} and Poisson's ratio of $0.3$. These values
were chosen to simplify the analytical solution.

An approximation of the dynamic response of the middle point of the
beam is given by:
\begin{equation}
  \label{eqn:smm:implicit}
  u\left(\frac{L}{2}, t\right)
  = \frac{1}{\pi^4} \left(1 - cos\left(\pi^2 t\right) +
    \frac{1}{81}\left(1 - cos\left(3^2 \pi^2 t\right)\right) +
    \frac{1}{625}\left(1 - cos\left(5^2 \pi^2 t\right)\right)\right)
\end{equation}

\begin{figure}[!htb]
  \centering
  \includegraphics[scale=.6]{figures/implicit_dynamic}
  \caption{Numerical setup}
  \label{fig:smm:implicit:dynamic}
\end{figure}

Figure \ref{fig:smm:implicit:dynamic_solution} presents the deformed
beam at 3 different times during the simulation: time steps 0, 1000 and
2000.

\begin{figure}[!htb]
  \centering
  \setlength{\unitlength}{0.1\textwidth}
  \begin{tikzpicture}
    \node[above right] (img) at (0,0)
    {\includegraphics[width=.6\linewidth]{figures/dynamic_analysis}};
    \node[left] at (0pt,20pt) {$0$}; \node[left] at (0pt,60pt) {$1000$};
    \node[left] at (0pt,100pt) {$2000$};
  \end{tikzpicture}

  \caption{Deformed beam at 3 different times (displacement are
    magnified by a factor 10).}
  \label{fig:smm:implicit:dynamic_solution}
\end{figure}

\subsection{Explicit Time Integration}
\label{ssect:smm:expl-time-integr}

The explicit dynamic time integration scheme is based on the
Newmark-$\beta$ scheme with $\alpha=0$ (see equations
\ref{eqn:equation-motion-discret}-\ref{eqn:finite-difference-2}).  In
\akantu, $\beta$ is defaults to $\beta=1/2$, see section
\ref{sect:smm:Dynamic_methods}.

The initialization of the simulation is similar to the static and
implicit dynamic version.  The model is created from the
\code{SolidMechanicsModel} class.  In the initialization, the explicit
scheme is selected using the \code{\_explicit\_lumped\_mass} constant.

\begin{cpp}
SolidMechanicsModel model(mesh);
model.initFull(SolidMechanicsModelOptions(_explicit_lumped_mass));
\end{cpp}
\index{SolidMechanicsModel!initFull}
\note{Writing \code{model.initFull()} or \code{model.initFull(SolidMechanicsModelOptions());} is
equivalent to use the \code{\_explicit\_lumped\_mass} keyword, as this
is the default case.}

The explicit time integration scheme implemented in \akantu uses a
lumped mass matrix $\mat{M}$ (reducing the computational cost). This
matrix is assembled by distributing the mass of each element onto its
nodes. The resulting $\mat{M}$ is therefore a diagonal matrix stored
in the \textbf{mass} vector of the model.

The explicit integration scheme is conditionally stable. The time step
has to be smaller than the stable time step which is obtained in
\akantu as follows:

\begin{cpp}
time_step = model.getStableTimeStep();
\end{cpp} \index{SolidMechanicsModel!StableTimeStep}

The stable time step is defined as:
\begin{equation}
\label{eqn:smm:explicit:stabletime}
\Delta t_{\st{crit}} = \Delta x \sqrt{\frac{\rho}{2 \mu + \lambda}}
\end{equation}
where $\Delta x$ is a characteristic length (\eg the inradius in the
case of linear triangle element), $\mu$ and $\lambda$ are the first
and second Lame's coefficients and $\rho$ is the density.  The stable
time step corresponds to the time the fastest wave (the compressive
wave) needs to travel the characteristic length of the mesh.
However, it is recommended to impose a time step that is smaller than the stable time step, for
instance, by multiplying the stable time step by a safety factor
smaller than one.

\begin{cpp}
const Real safety_time_factor = 0.8;
Real applied_time_step = time_step * safety_time_factor;
model.setTimeStep(applied_time_step);
\end{cpp}
\index{SolidMechanicsModel!setTimeStep} The initial displacement and
velocity fields are, by default, equal to zero if not given
specifically by the user (see \ref{sect:smm:initial_condition}).

Like in implicit dynamics, a time loop is used in which the
displacement, velocity and acceleration fields are updated at each
time step. The values of these fields are obtained from the
Newmark$-\beta$ equations with $\beta=1/2$ and $\alpha=0$. In \akantu
these computations at each time step are invoked by calling the
function \code{solveStep}:
\begin{cpp}
for (UInt s = 1; (s-1)*applied_time_step < total_time; ++s) {
  model.solveStep();
}
\end{cpp} \index{SolidMechanicsModel!solveStep}
The method
\code{solveStep} wraps the four following functions:
\begin{itemize}
\item \code{model.explicitPred()} allows to compute the displacement
  field at $t+1$ and a part of the velocity field at $t+1$, denoted by
  $\vec{\dot{u}^{\st{p}}}_{n+1}$, which will be used later in the method
  \code{model.explicitCorr()}. The equations are:
  \begin{align}
    \vec{u}_{n+1} &= \vec{u}_{n} + \Delta t
    \vec{\dot{u}}_{n} + \frac{\Delta t^2}{2} \vec{\ddot{u}}_{n}\\
    \vec{\dot{u}^{\st{p}}}_{n+1} &= \vec{\dot{u}}_{n} + \Delta t
    \vec{\ddot{u}}_{n}
    \label{eqn:smm:explicit:onehalfvelocity}
  \end{align}

\item \code{model.updateResidual()} and
  \code{model.updateAcceleration()} compute the acceleration increment
  $\delta \vec{\ddot{u}}$:
  \begin{equation}
    \left(\mat{M} + \frac{1}{2} \Delta t \mat{C}\right)
    \delta \vec{\ddot{u}} = \vec{f_{\st{ext}}} - \vec{f}_{\st{int}\, n+1}
    - \mat{C} \vec{\dot{u}}_{n} - \mat{M} \vec{\ddot{u}}_{n}
  \end{equation}

  \note{The internal force $\vec{f}_{\st{int}\, n+1}$ is computed from
    the displacement $\vec{u}_{n+1}$ based on the constitutive law.}

\item \code{model.explicitCorr()} computes the velocity and
  acceleration fields at $t+1$:
  \begin{align}
    \vec{\dot{u}}_{n+1} &= \vec{\dot{u}^{\st{p}}}_{n+1} + \frac{\Delta t}{2}
    \delta \vec{\ddot{u}} \\ \vec{\ddot{u}}_{n+1} &=
    \vec{\ddot{u}}_{n} + \delta \vec{\ddot{u}}
  \end{align}
\end{itemize}

The use of an explicit time integration scheme is illustrated by the
example:\par
\noindent \shellcode{\examplesdir/explicit/explicit\_dynamic.cc}\par
\noindent This example models the propagation of a wave in a steel beam. The
beam and the applied displacement in the $x$ direction are shown in
Figure~\ref{fig:smm:explicit}.

\begin{figure}[!htb] \centering
  \begin{tikzpicture}
    \coordinate (c) at (0,2);
    \draw[shift={(c)},thick, color=blue] plot [id=x, domain=-5:5, samples=50] ({\x, {(40 * sin(0.1*pi*3*\x) * exp(- (0.1*pi*3*\x)*(0.1*pi*3*\x) / 4))}});
    \draw[shift={(c)},-latex] (-6,0) -- (6,0) node[right, below] {$x$};
    \draw[shift={(c)},-latex] (0,-0.7) -- (0,1) node[right] {$u$};
    \draw[shift={(c)}] (-0.1,0.6) node[left] {$A$}-- (1.5,0.6);

    \coordinate (l) at (0,0.6);
    \draw[shift={(0,-0.7)}] (-5, 0) -- (5,0) -- (5, 1) -- (-5, 1) -- cycle;
    \draw[shift={(l)}, latex-latex] (-5,0)-- (5,0) node [midway, above] {$L$};
    \draw[shift={(l)}] (5,0.2)-- (5,-0.2);
    \draw[shift={(l)}] (-5,0.2)-- (-5,-0.2);

    \coordinate (h) at (5.3,-0.7);
    \draw[shift={(h)}, latex-latex] (0,0)-- (0,1) node [midway, right] {$h$};
    \draw[shift={(h)}] (-0.2,1)-- (0.2,1);
    \draw[shift={(h)}] (-0.2,0)-- (0.2,0);
  \end{tikzpicture}

  \caption{Numerical setup \label{fig:smm:explicit}}
\end{figure}

The length and height of the beam are $L=\SI{10}{\metre}$ and
$h = \SI{1}{\metre}$, respectively.  The material is linear elastic,
homogeneous and isotropic (density:
\SI{7800}{\kilo\gram\per\cubic\metre}, Young's modulus:
\SI{210}{\giga\pascal} and Poisson's ratio: $0.3$).  The imposed
displacement follow a Gaussian function with a maximum amplitude of $A = \SI{0.01}{\meter}$. The
potential, kinetic and total energies are computed.  The safety factor
is equal to $0.8$.

\input{manual-constitutive-laws}

\section{Adding a New Constitutive Law}\index{Material!create a new
material}

There are several constitutive laws in \akantu as described in the
previous Section~\ref{sect:smm:CL}. It is also possible to use a
user-defined material for the simulation. These materials are referred
to as local materials since they are local to the example of the user
and not part of the \akantu library.  To define a new local material,
two files (\code {material\_XXX.hh} and \code{material\_XXX.cc}) have
to be provided where \code{XXX} is the name of the new material. The
header file \code {material\_XXX.hh} defines the interface of your
custom material. Its implementation is provided in the
\code{material\_XXX.cc}. The new law must inherit from the
\code{Material} class or any other existing material class. It is
therefore necessary to include the interface of the parent material
in the header file of your local material and indicate the inheritance
in the declaration of the class:
\begin{cpp}
/* ---------------------------------------------------------------------- */
#include "material.hh"
/* ---------------------------------------------------------------------- */

#ifndef __AKANTU_MATERIAL_XXX_HH__
#define __AKANTU_MATERIAL_XXX_HH__

__BEGIN_AKANTU__

class MaterialXXX : public Material {

/// declare here the interface of your material

};
\end{cpp}
In the header file the user also needs to declare all the members of the new
material. These include the parameters that a read from the
material input file, as well as any other material parameters that will be computed during the simulation and internal variables.\\
In the following the example of adding a new damage material will be presented. In this
case the parameters in the material will consist of the Young's modulus, the
Poisson coefficient, the resistance to damage and the damage threshold. The
material will then from these values compute its Lam\'{e} coefficients and its
bulk modulus. Furthermore, the user has to add a new internal variable
\code{damage} in order to store the amount of damage at each quadrature point in
each step of the simulation. For this specific material the member declaration
inside the class will look as follows:
\begin{cpp}
class LocalMaterialDamage : public Material {

/// declare constructors/destructors here

/// declare methods and accessors here

  /* -------------------------------------------------------------------- */
  /* Class Members                                                        */
  /* -------------------------------------------------------------------- */

  AKANTU_GET_MACRO_BY_ELEMENT_TYPE_CONST(Damage, damage, Real);
private:

  /// the young modulus
  Real E;

  /// Poisson coefficient
  Real nu;

  /// First Lame coefficient
  Real lambda;

  /// Second Lame coefficient (shear modulus)
  Real mu;

  /// resistance to damage
  Real Yd;

  /// damage threshold
  Real Sd;

  /// Bulk modulus
  Real kpa;

  /// damage internal variable
  InternalField<Real> damage;

};
\end{cpp}
In order to enable to print the material parameters at any point in
the user's example file using the standard output stream by typing:
\begin{cpp}
for (UInt m = 0; m  < model.getNbMaterials(); ++m)
  std::cout << model.getMaterial(m) << std::endl;
\end{cpp}
the standard output stream operator has to be redefined. This should be done at the end of the header file:
\begin{cpp}
class LocalMaterialDamage : public Material {

  /// declare here the interace of your material

}:
/* ---------------------------------------------------------------------- */
/* inline functions                                                       */
/* ---------------------------------------------------------------------- */
/// standard output stream operator
inline std::ostream & operator <<(std::ostream & stream, const LocalMaterialDamage & _this)
{
  _this.printself(stream);
  return stream;
}
\end{cpp}
However, the user still needs to register the material parameters that
should be printed out. The registration is done during the call of the
constructor. Like all definitions the implementation of the
constructor has to be written in the \code{material\_XXX.cc}
file. However, the declaration has to be provided in the
\code{material\_XXX.hh} file:
\begin{cpp}
class LocalMaterialDamage : public Material {
  /* -------------------------------------------------------------------- */
  /* Constructors/Destructors                                             */
  /* -------------------------------------------------------------------- */
public:

  LocalMaterialDamage(SolidMechanicsModel & model, const ID & id = "");
};
\end{cpp}
The user can now define the implementation of the constructor in the
\code{material\_XXX.cc} file:
\begin{cpp}
/* ---------------------------------------------------------------------- */
#include "local_material_damage.hh"
#include "solid_mechanics_model.hh"

__BEGIN_AKANTU__

/* ---------------------------------------------------------------------- */
LocalMaterialDamage::LocalMaterialDamage(SolidMechanicsModel & model,
					 const ID & id)  :
  Material(model, id),
  damage("damage", *this) {
  AKANTU_DEBUG_IN();

  this->registerParam("E", E, 0., _pat_parsable, "Young's modulus");
  this->registerParam("nu", nu, 0.5, _pat_parsable, "Poisson's ratio");
  this->registerParam("lambda", lambda, _pat_readable, "First Lame coefficient");
  this->registerParam("mu", mu, _pat_readable, "Second Lame coefficient");
  this->registerParam("kapa", kpa, _pat_readable, "Bulk coefficient");
  this->registerParam("Yd", Yd,   50., _pat_parsmod);
  this->registerParam("Sd", Sd, 5000., _pat_parsmod);

  damage.initialize(1);

  AKANTU_DEBUG_OUT();
}
\end{cpp}
During the intializer list the reference to the model and the material id are
assigned and the constructor of the internal field is called. Inside the scope
of the constructor the internal values have to be initialized and the
parameters, that should be printed out, are registered with the function:
\code{registerParam}\index{Material!registerParam}:
\begin{cpp}
void registerParam(name of the parameter (key in the material file),
		   member variable,
		   default value (optional parameter),
		   access permissions,
		   description);
\end{cpp}
The available access permissions are as follows:
\begin{itemize}
  \item \code{\_pat\_internal}: Parameter can only be output when the material is printed.
  \item \code{\_pat\_writable}: User can write into the parameter. The parameter is output when the material is printed.
  \item \code{\_pat\_readable}: User can read the parameter. The parameter is output when the material is printed.
  \item \code{\_pat\_modifiable}: Parameter is writable and readable.
  \item \code{\_pat\_parsable}: Parameter can be parsed, \textit{i.e.} read from the input file.
  \item \code{\_pat\_parsmod}: Parameter is modifiable and parsable.
\end{itemize}

In order to implement the new constitutive law the user needs to
specify how the additional material parameters, that are not
defined in the input material file, should be calculated. Furthermore,
it has to be defined how stresses and the stable time step should be
computed for the new local material. In the case of implicit
simulations, in addition, the computation of the tangent stiffness needs
to be defined. Therefore, the user needs to redefine the following
functions of the parent material:
\begin{cpp}
void initMaterial();

// for explicit and implicit simulations void
computeStress(ElementType el_type, GhostType ghost_type = _not_ghost);

// for implicit simulations
void computeTangentStiffness(const ElementType & el_type,
			     Array<Real> & tangent_matrix,
			     GhostType ghost_type = _not_ghost);

// for explicit and implicit simulations
Real getStableTimeStep(Real h, const Element & element);
\end{cpp}
In the following a detailed description of these functions is provided:
\begin{itemize}

\item \code{initMaterial}:~ This method is called after the material
  file is fully read and the elements corresponding to each material
  are assigned. Some of the frequently used constant parameters are
  calculated in this method. For example, the Lam\'{e} constants of
  elastic materials can be considered as such parameters.

\item \code{computeStress}:~ In this method, the stresses are
  computed based on the constitutive law as a function of the  strains of the
  quadrature points.  For example, the stresses for the elastic
  material are calculated based on the following formula:
  \begin{equation}
    \label{eqn:smm:constitutive_elastic}
    \mat{\sigma }  =\lambda\mathrm{tr}(\mat{\varepsilon})\mat{I}+2 \mu \mat{\varepsilon}
  \end{equation}

  Therefore, this method contains a loop on all quadrature points
  assigned to the material using the two macros:\par
  \code{MATERIAL\_STRESS\_QUADRATURE\_POINT\_LOOP\_BEGIN}\par
  \code{MATERIAL\_STRESS\_QUADRATURE\_POINT\_LOOP\_END}

  \begin{cpp}
    MATERIAL_STRESS_QUADRATURE_POINT_LOOP_BEGIN(element_type);

    // sigma <- f(grad_u)

    MATERIAL_STRESS_QUADRATURE_POINT_LOOP_END;
  \end{cpp}

  \note{The strain vector in \akantu contains the values of $\nabla
\vec{u}$, i.e. it is really the \emph{displacement gradient},}

\item \code{computeTangentStiffness}:~ This method is called when
  the tangent to the stress-strain curve is desired (see Fig \ref
  {fig:smm:AL:K}).  For example, it is called in the implicit solver
  when the stiffness matrix for the regular elements is assembled
  based on the following formula:
  \begin{equation}
    \label{eqn:smm:constitutive_elasc} \mat{K }
    =\int{\mat{B^T}\mat{D(\varepsilon)}\mat{B}}
  \end{equation}

  Therefore, in this method, the \code{tangent} matrix (\mat{D}) is
  computed for a given strain.

  \note{ The \code{tangent} matrix is a $4^{th}$ order tensor which is
    stored as a matrix in Voigt notation.}

  \begin{figure}[!htb]
    \begin{center}
      \includegraphics[width=0.4\textwidth,keepaspectratio=true]{figures/tangent.pdf}
      \caption{Tangent to the stress-strain curve.}
      \label{fig:smm:AL:K}
    \end{center}
  \end{figure}


\item \code{getStableTimeStep}:~ The stable time step should be
  calculated based on \eqref{eqn:smm:explicit:stabletime} for the
  conditionally stable  methods.  This function depends on the longitudinal wave
  speed which changes for different materials and strains. Therefore,
  the value of this velocity should be defined in this method for
  each new material.

  \note {In case of need, the calculation of the longitudinal and
    shear wave speeds can be done in separate functions
    (\code{getPushWaveSpeed} and \code{getShearWaveSpeed}).
    Therefore, the first function can be called for calculation of the
    stable time step.}
\end{itemize}
Once the declaration and implementation of the new material has been
completed, this material can be used in the user's example by including the header file:
\begin{cpp}
#include "material_XXX.hh"
\end{cpp}
For existing materials, as mentioned in Section~\ref{sect:smm:CL}, by
default, the materials are initialized inside the method
\code{initFull}. If a local material should be used instead, the
initialization of the material has to be postponed until the local
material is registered in the model. Therefore, the model is
initialized with the boolean for skipping the material initialization
equal to true:
\begin{cpp}
/// model initialization
model.initFull(SolidMechanicsModelOptions(_explicit_lumped_mass,true));
\end{cpp}
Once the model has been initialized, the local material needs
to be registered in the model:
\begin{cpp}
model.registerNewCustomMaterials<XXX>("name_of_local_material");
\end{cpp}
Only at this point the material can be initialized:
\begin{cpp}
model.initMaterials();
\end{cpp}
A full example for adding a new damage law can be found in
\shellcode{\examplesdir/new\_material}.

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