Peridynamics with LAMMPS
========================

This Howto is based on the Sandia report 2010-5549 by Michael L. Parks,
Pablo Seleson, Steven J. Plimpton, Richard B. Lehoucq, and
Stewart A. Silling.

Overview
""""""""

Peridynamics is a nonlocal extension of classical continuum mechanics.
The discrete peridynamic model has the same computational structure as a
molecular dynamics model.  This Howto provides a brief overview of the
peridynamic model of a continuum, then discusses how the peridynamic
model is discretized within LAMMPS as described in the original article
:ref:`(Parks) <Parks2>`.  An example problem with comments is also
included.

Quick Start
"""""""""""

The peridynamics styles are included in the optional :ref:`PERI package
<PKG-PERI>`.  If your LAMMPS executable does not already include the
PERI package, you can see the :doc:`build instructions for packages
<Build_package>` for how to enable the package when compiling a custom
version of LAMMPS from source.

Here is a minimal example for setting up a peridynamics simulation.

.. code-block:: LAMMPS

   units         si
   boundary      s s s
   lattice       sc 0.0005
   atom_style    peri
   atom_modify   map array
   neighbor      0.0010 bin
   region        target cylinder y 0.0 0.0 0.0050 -0.0050 0.0 units box
   create_box    1 target
   create_atoms  1 region target

   pair_style    peri/pmb
   pair_coeff    * * 1.6863e22 0.0015001 0.0005 0.25
   set           group all density 2200
   set           group all volume 1.25e-10
   velocity      all set 0.0 0.0 0.0 sum no units box
   fix           1 all nve
   compute       1 all damage/atom
   timestep      1.0e-7

Some notes on this input example:

- peridynamics simulations typically use SI :doc:`units <units>`
- particles must be created on a :doc:`simple cubic lattice <lattice>`
- using the :doc:`atom style peri <atom_style>` is required
- an :doc:`atom map <atom_modify>` is required for indexing particles
- The :doc:`skin distance <neighbor>` used when computing neighbor lists
  should be defined appropriately for your choice of simulation
  parameters. The *skin* should be set to a value such that the
  peridynamic horizon plus the skin distance is larger than the maximum
  possible distance between two bonded particles (before their bond
  breaks). Here it is set to 0.001 meters.
- a :doc:`peridynamics pair style <pair_peri>` is required.  Available
  choices are currently: *peri/eps*, *peri/lps*, *peri/pmb*, and
  *peri/ves*.  The model parameters are set with a :doc:`pair_coeff
  <pair_coeff>` command.
- the mass density and volume fraction for each particle must be
  defined.  This is done with the two :doc:`set <set>` commands for
  *density* and *volume*.  For a simple cubic lattice, the volume of a
  particle should be equal to the cube of the lattice constant, here
  :math:`V_i = \Delta x ^3`.
- with the :doc:`velocity <velocity>` command all particles are initially at rest
- a plain :doc:`velocity-Verlet time integrator <fix_nve>` is used,
  which is algebraically equivalent to a centered difference in time,
  but numerically more stable
- you can compute the damage at the location of each particle with
  :doc:`compute damage/atom <compute_damage_atom>`
- finally, the timestep is set to 0.1 microseconds with the
  :doc:`timestep <timestep>` command.


Peridynamic Model of a Continuum
""""""""""""""""""""""""""""""""

The following is not a complete overview of peridynamics, but a
discussion of only those details specific to the model we have
implemented within LAMMPS. For more on the peridynamic theory, the
reader is referred to :ref:`(Silling 2007) <Silling2007_2>`. To begin,
we define the notation we will use.

Basic Notation
^^^^^^^^^^^^^^

Within the peridynamic literature, the following notational conventions
are generally used. The position of a given point in the reference
configuration is :math:`\textbf{x}`. Let
:math:`\mathbf{u}(\mathbf{x},t)` and :math:`\mathbf{y}(\mathbf{x},t)`
denote the displacement and position, respectively, of the point
:math:`\mathbf{x}` at time :math:`t`. Define the relative position and
displacement vectors of two bonded points :math:`\textbf{x}` and
:math:`\textbf{x}^\prime` as :math:`\mathbf{\xi} = \textbf{x}^\prime -
\textbf{x}` and :math:`\mathbf{\eta} = \textbf{u}(\textbf{x}^\prime,t) -
\textbf{u}(\textbf{x},t)`, respectively. We note here that
:math:`\mathbf{\eta}` is time-dependent, and that :math:`\mathbf{\xi}`
is not. It follows that the relative position of the two bonded points
in the current configuration can be written as :math:`\boldsymbol{\xi} +
\boldsymbol{\eta} =
\mathbf{y}(\mathbf{x}^{\prime},t)-\mathbf{y}(\mathbf{x},t)`.

Peridynamic models are frequently written using *states*, which we
briefly describe here. For the purposes of our discussion, all states
are operators that act on vectors in :math:`\mathbb{R}^3`. For a more
complete discussion of states, see :ref:`(Silling 2007)
<Silling2007_2>`. A *vector state* is an operator whose image is a
vector, and may be viewed as a generalization of a second-rank
tensor. Similarly, a *scalar state* is an operator whose image is a
scalar. Of particular interest is the vector force state
:math:`\underline{\mathbf{T}}\left[ \mathbf{x},t \right]\left<
\mathbf{x}^{\prime}-\mathbf{x} \right>`, which is a mapping, having
units of force per volume squared, of the vector
:math:`\mathbf{x}^{\prime}-\mathbf{x}` to the force vector state
field. The vector state operator :math:`\underline{\mathbf{T}}` may
itself be a function of :math:`\mathbf{x}` and :math:`t`. The
constitutive model is completely contained within
:math:`\underline{\mathbf{T}}`.

In the peridynamic theory, the deformation at a point depends
collectively on all points interacting with that point. Using the
notation of :ref:`(Silling 2007) <Silling2007_2>`, we write the
peridynamic equation of motion as

.. _periNewtonII:

.. math::

   \rho(\mathbf{x}) \ddot{\mathbf{u}}(\mathbf{x},t) =
   \int_{\mathcal{H}_{\mathbf{x}}} \left\{
   \underline{\mathbf{T}}\left[\mathbf{x},t
   \right]\left<\mathbf{x}^{\prime}-\mathbf{x} \right> -
   \underline{\mathbf{T}}\left[\mathbf{x}^{\prime},t
   \right]\left<\mathbf{x}-\mathbf{x}^{\prime} \right> \right\}
   {d}V_{\mathbf{x}^\prime} + \mathbf{b}(\mathbf{x},t), \qquad\qquad\textrm{(1)}

where :math:`\rho` represents the mass density,
:math:`\underline{\mathbf{T}}` the force vector state, and
:math:`\mathbf{b}` an external body force density. A point
:math:`\mathbf{x}` interacts with all the points
:math:`\mathbf{x}^{\prime}` within the neighborhood
:math:`\mathcal{H}_{\mathbf{x}}`, assumed to be a spherical region of
radius :math:`\delta>0` centered at :math:`\mathbf{x}`. :math:`\delta`
is called the *horizon*, and is analogous to the cutoff radius used in
molecular dynamics. Conditions on :math:`\underline{\mathbf{T}}` for
which :ref:`(1) <periNewtonII>` satisfies the balance of linear and angular
momentum are given in :ref:`(Silling 2007) <Silling2007_2>`.

We consider only force vector states that can be written as

.. math::

   \underline{\mathbf{T}} = \underline{t}\,\underline{\mathbf{M}},

with :math:`\underline{t}` a *scalar force state* and
:math:`\underline{\mathbf{M}}` the *deformed direction vector
state*, defined by

.. math::

   \underline{\mathbf{M}}\left< \boldsymbol{\xi} \right> =
   \left\{ \begin{array}{cl}
   \frac{\boldsymbol{\xi} + \boldsymbol{\eta}}{
   \left\Vert \boldsymbol{\xi} + \boldsymbol{\eta} \right\Vert
   } & \left\Vert \boldsymbol{\xi} + \boldsymbol{\eta} \right\Vert \neq 0 \\
   \boldsymbol{0}  & \textrm{otherwise}
   \end{array}
   \right. . \qquad\qquad\textrm{(2)}

Such force states correspond to so-called *ordinary* materials
:ref:`(Silling 2007) <Silling2007_2>`. These are the materials for which
the force between any two interacting points :math:`\textbf{x}` and
:math:`\textbf{x}^\prime` acts along the line between the points.


Linear Peridynamic Solid (LPS) Model
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

We summarize the linear peridynamic solid (LPS) material model. For more
on this model, the reader is referred to :ref:`(Silling 2007)
<Silling2007_2>`.  This model is a nonlocal analogue to a classical
linear elastic isotropic material. The elastic properties of a a
classical linear elastic isotropic material are determined by (for
example) the bulk and shear moduli. For the LPS model, the elastic
properties are analogously determined by the bulk and shear moduli,
along with the horizon :math:`\delta`.

The LPS model has a force scalar state

.. math::

   \underline{t} = \frac{3K\theta}{m}\underline{\omega}\,\underline{x} +
   \alpha \underline{\omega}\,\underline{e}^\mathrm{d}, \qquad\qquad\textrm{(3)}

with :math:`K` the bulk modulus and :math:`\alpha` related to the shear
modulus :math:`G` as

.. math::

   \alpha = \frac{15 G}{m}.

The remaining components of the model are described as follows. Define
the reference position scalar state :math:`\underline{x}` so that
:math:`\underline{x}\left<\boldsymbol{\xi} \right> = \left\Vert
\boldsymbol{\xi} \right\Vert`. Then, the weighted volume :math:`m` is
defined as

.. math::

   m\left[ \mathbf{x} \right] = \int_{\mathcal{H}_\mathbf{x}}
   \underline{\omega} \left<\boldsymbol{\xi}\right>
   \underline{x}\left<\boldsymbol{\xi} \right>
   \underline{x}\left<\boldsymbol{\xi} \right>{d}V_{\boldsymbol{\xi} }.  \qquad\qquad\textrm{(4)}

Let

.. math::

   \underline{e}\left[ \mathbf{x},t \right] \left<\boldsymbol{\xi}
   \right> = \left\Vert \boldsymbol{\xi} + \boldsymbol{\eta}
   \right\Vert - \left\Vert \boldsymbol{\xi} \right\Vert

be the extension scalar state, and

.. math::

   \theta\left[ \mathbf{x}, t \right] = \frac{3}{m\left[ \mathbf{x}
   \right]}\int_{\mathcal{H}_\mathbf{x}} \underline{\omega}
   \left<\boldsymbol{\xi}\right> \underline{x}\left<\boldsymbol{\xi}
   \right> \underline{e}\left[ \mathbf{x},t
   \right]\left<\boldsymbol{\xi} \right>{d}V_{\boldsymbol{\xi}}

be the dilatation. The isotropic and deviatoric parts of the extension
scalar state are defined, respectively, as

.. math::

   \underline{e}^\mathrm{i}=\frac{\theta \underline{x}}{3}, \qquad
   \underline{e}^\mathrm{d} = \underline{e}- \underline{e}^\mathrm{i},


where the arguments of the state functions and the vectors on which they
operate are omitted for simplicity. We note that the LPS model is linear
in the dilatation :math:`\theta`, and in the deviatoric part of the
extension :math:`\underline{e}^\mathrm{d}`.

.. note::

   The weighted volume :math:`m` is time-independent, and does
   not change as bonds break. It is computed with respect to the bond
   family defined at the reference (initial) configuration.

The non-negative scalar state :math:`\underline{\omega}` is an
*influence function* :ref:`(Silling 2007) <Silling2007_2>`. For more on
influence functions, see :ref:`(Seleson 2010) <Seleson2010>`. If an
influence function :math:`\underline{\omega}` depends only upon the
scalar :math:`\left\Vert \boldsymbol{\xi} \right\Vert`, (i.e.,
:math:`\underline{\omega}\left<\boldsymbol{\xi}\right> =
\underline{\omega}\left<\left\Vert \boldsymbol{\xi} \right\Vert\right>`\
), then :math:`\underline{\omega}` is a spherical influence function.
For a spherical influence function, the LPS model is isotropic
:ref:`(Silling 2007) <Silling2007_2>`.

.. note::

   In the LAMMPS implementation of the LPS model, the influence function
   :math:`\underline{\omega}\left<\left\Vert \boldsymbol{\xi}
   \right\Vert\right> = 1 / \left\Vert \boldsymbol{\xi} \right\Vert` is
   used. However, the user can define their own influence function by
   altering the method "influence_function" in the file
   ``pair_peri_lps.cpp``. The LAMMPS peridynamics code permits both
   spherical and non-spherical influence functions (e.g., isotropic and
   non-isotropic materials).


Prototype Microelastic Brittle (PMB) Model
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

We summarize the prototype microelastic brittle (PMB) material
model. For more on this model, the reader is referred to
:ref:`(Silling 2000) <Silling2000_2>` and :ref:`(Silling 2005)
<Silling2005>`.  This model is a special case of the LPS model; see
:ref:`(Seleson 2010) <Seleson2010>` for the derivation. The elastic
properties of the PMB model are determined by the bulk modulus :math:`K`
and the horizon :math:`\delta`.

The PMB model is expressed using the scalar force state field

.. _periPMBState:

.. math::

   \underline{t}\left[ \mathbf{x},t \right]\left< \boldsymbol{\xi} \right> = \frac{1}{2} f\left( \boldsymbol{\eta} ,\boldsymbol{\xi} \right), \qquad\qquad\textrm{(5)}

with :math:`f` a scalar-valued function. We assume that :math:`f` takes
the form

.. math::

   f = c s,

where

.. _peric:

.. math::

   c = \frac{18K}{\pi \delta^4}, \qquad\qquad\textrm{(6)}

with :math:`K` the bulk modulus and :math:`\delta` the horizon, and
:math:`s` the bond stretch, defined as

.. math::

   s(t,\mathbf{\eta},\mathbf{\xi}) = \frac{ \left\Vert {\mathbf{\eta}+\mathbf{\xi}} \right\Vert - \left\Vert {\mathbf{\xi}} \right\Vert }{\left\Vert {\mathbf{\xi}} \right\Vert}.

Bond stretch is a unitless quantity, and identical to a one-dimensional
definition of strain. As such, we see that a bond at its equilibrium
length has stretch :math:`s=0`, and a bond at twice its equilibrium
length has stretch :math:`s=1`.  The constant :math:`c` given above is
appropriate for 3D models only. For more on the origins of the constant
:math:`c`, see :ref:`(Silling 2005) <Silling2005>`. For the derivation
of :math:`c` for 1D and 2D models, see :ref:`(Emmrich) <Emmrich2007>`.

Given :ref:`(5) <periPMBState>`, :ref:`(1) <periNewtonII>` reduces to

.. math::

   \rho(\mathbf{x}) \ddot{\mathbf{u}}(\mathbf{x},t) = \int_{\mathcal{H}_\mathbf{x}} \mathbf{f} \left( \boldsymbol{\eta},\boldsymbol{\xi} \right){d}V_{\boldsymbol{\xi}} + \mathbf{b}(\mathbf{x},t), \qquad\qquad\textrm{(7)}

with

.. math::

   \mathbf{f} \left( \boldsymbol{\eta},  \boldsymbol{\xi}\right) =f \left( \boldsymbol{\eta},  \boldsymbol{\xi}\right)  \frac{\boldsymbol{\xi}+ \boldsymbol{\eta}}{ \left\Vert {\boldsymbol{\xi} + \boldsymbol{\eta}} \right\Vert}.

Unlike the LPS model, the PMB model has a Poisson ratio of
:math:`\nu=1/4` in 3D, and :math:`\nu=1/3` in 2D. This is reflected in
the input for the PMB model, which requires only the bulk modulus of the
material, whereas the LPS model requires both the bulk and shear moduli.

.. _peridamage:

Damage
^^^^^^

Bonds are made to break when they are stretched beyond a given
limit. Once a bond fails, it is failed forever :ref:`(Silling)
<Silling2005>`. Further, new bonds are never created during the course
of a simulation. We discuss only one criterion for bond breaking, called
the *critical stretch* criterion.

Define :math:`\mu` to be the history-dependent scalar
boolean function

.. _perimu:

.. math::

   \mu(t,\mathbf{\eta},\mathbf{\xi}) = \left\{
   \begin{array}{cl}
   1 & \mbox{if $s(t^\prime,\mathbf{\eta},\mathbf{\xi})< \min \left(s_0(t^\prime,\mathbf{\eta},\mathbf{\xi}) , s_0(t^\prime,\mathbf{\eta}^\prime,\mathbf{\xi}^\prime) \right)$ for all $0 \leq t^\prime \leq t$} \\
   0 & \mbox{otherwise}
   \end{array}\right\}.  \qquad\qquad\textrm{(8)}

where :math:`\mathbf{\eta}^\prime = \textbf{u}(\textbf{x}^{\prime
\prime},t) - \textbf{u}(\textbf{x}^\prime,t)` and
:math:`\mathbf{\xi}^\prime = \textbf{x}^{\prime \prime} -
\textbf{x}^\prime`. Here, :math:`s_0(t,\mathbf{\eta},\mathbf{\xi})` is a
critical stretch defined as

.. _peris0:

.. math::

   s_0(t,\mathbf{\eta},\mathbf{\xi}) = s_{00} - \alpha s_{\min}(t,\mathbf{\eta},\mathbf{\xi}), \qquad s_{\min}(t) = \min_{\mathbf{\xi}} s(t,\mathbf{\eta},\mathbf{\xi}), \qquad\qquad\textrm{(9)}

where :math:`s_{00}` and :math:`\alpha` are material-dependent
constants. The history function :math:`\mu` breaks bonds when the
stretch :math:`s` exceeds the critical stretch :math:`s_0`.

Although :math:`s_0(t,\mathbf{\eta},\mathbf{\xi})` is expressed as a
property of a particle, bond breaking must be a symmetric operation for
all particle pairs sharing a bond. That is, particles :math:`\textbf{x}`
and :math:`\textbf{x}^\prime` must utilize the same test when deciding
to break their common bond. This can be done by any method that treats
the particles symmetrically. In the definition of :math:`\mu` above, we
have chosen to take the minimum of the two :math:`s_0` values for
particles :math:`\textbf{x}` and :math:`\textbf{x}^\prime` when
determining if the :math:`\textbf{x}`--:math:`\textbf{x}^\prime` bond
should be broken.

Following :ref:`(Silling) <Silling2005>`, we can define the damage at a
point :math:`\textbf{x}` as

.. _peridamageeq:

.. math::

   \varphi(\textbf{x}, t) = 1 - \frac{\int_{\mathcal{H}_{\textbf{x}}} \mu(t,\mathbf{\eta},\mathbf{\xi}) dV_{\textbf{x}^\prime}
   }{ \int_{\mathcal{H}_{\textbf{x}}} dV_{\textbf{x}^\prime} }.  \qquad\qquad\textrm{(10)}

Discrete Peridynamic Model and LAMMPS Implementation
""""""""""""""""""""""""""""""""""""""""""""""""""""

In LAMMPS, instead of :ref:`(1) <periNewtonII>`, we model this equation of
motion:

.. math::

   \rho(\mathbf{x}) \ddot{\textbf{y}}(\mathbf{x},t) = \int_{\mathcal{H}_{\mathbf{x}}}
   \left\{ \underline{\mathbf{T}}\left[ \mathbf{x},t \right]\left<\mathbf{x}^{\prime}-\mathbf{x} \right>
   - \underline{\mathbf{T}}\left[\mathbf{x}^{\prime},t \right]\left<\mathbf{x}-\mathbf{x}^{\prime} \right> \right\}
     {d}V_{\mathbf{x}^\prime} + \mathbf{b}(\mathbf{x},t),

where we explicitly track and store at each timestep the positions and
not the displacements of the particles. We observe that
:math:`\ddot{\textbf{y}}(\textbf{x}, t) = \ddot{\textbf{x}} +
\ddot{\textbf{u}}(\textbf{x}, t) = \ddot{\textbf{u}}(\textbf{x}, t)`, so
that this is equivalent to :ref:`(1) <periNewtonII>`.

Spatial Discretization
^^^^^^^^^^^^^^^^^^^^^^

The region defining a peridynamic material is discretized into particles
forming a simple cubic lattice with lattice constant :math:`\Delta x`,
where each particle :math:`i` is associated with some volume fraction
:math:`V_i`. For any particle :math:`i`, let :math:`\mathcal{F}_i`
denote the family of particles for which particle :math:`i` shares a
bond in the reference configuration.  That is,

.. _periBondFamily:

.. math::

   \mathcal{F}_i = \{ p ~ | ~ \left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert \leq \delta \}. \qquad\qquad\textrm{(11)}

The discretized equation of motion replaces :ref:`(1) <periNewtonII>` with

.. _peridiscreteNewtonII:

.. math::

   \rho \ddot{\textbf{y}}_i^n =
   \sum_{p \in \mathcal{F}_i}
   \left\{ \underline{\mathbf{T}}\left[ \textbf{x}_i,t \right]\left<\textbf{x}_p^{\prime}-\textbf{x}_i \right>
   - \underline{\mathbf{T}}\left[\textbf{x}_p,t \right]\left<\textbf{x}_i-\textbf{x}_p \right> \right\}
     V_{p} + \textbf{b}_i^n, \qquad\qquad\textrm{(12)}

where :math:`n` is the timestep number and subscripts denote the particle number.

Short-Range Forces
^^^^^^^^^^^^^^^^^^

In the model discussed so far, particles interact only through their
bond forces. A particle with no bonds becomes a free non-interacting
particle. To account for contact forces, short-range forces are
introduced :ref:`(Silling 2007) <Silling2007_2>`. We add to the force in
:ref:`(12) <peridiscreteNewtonII>` the following force

.. math::

   \textbf{f}_S(\textbf{y}_p,\textbf{y}_i) = \min \left\{ 0, \frac{c_S}{\delta}(\left\Vert {\textbf{y}_p-\textbf{y}_i} \right\Vert - d_{pi}) \right\}
   \frac{\textbf{y}_p-\textbf{y}_i}{\left\Vert {\textbf{y}_p-\textbf{y}_i} \right\Vert}, \qquad\qquad\textrm{(13)}

where :math:`d_{pi}` is the short-range interaction distance between
particles :math:`p` and :math:`i`, and :math:`c_S` is a multiple of the
constant :math:`c` from :ref:`(6) <peric>`. Note that the short-range force
is always repulsive, never attractive. In practice, we choose

.. _pericS:

.. math::

   c_S = 15 \frac{18K}{\pi \delta^4}. \qquad\qquad\textrm{(14)}

For the short-range interaction distance, we choose :ref:`(Silling 2007)
<Silling2007_2>`

.. math::

   d_{pi} = \min \left\{ 0.9 \left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert, 1.35 (r_p + r_i) \right\}, \qquad\qquad\textrm{(15)}

where :math:`r_i` is called the *node radius* of particle
:math:`i`. Given a discrete lattice, we choose :math:`r_i` to be half
the lattice constant.

.. note::

   For a simple cubic lattice, :math:`\Delta x = \Delta y = \Delta z`.

Given this definition of :math:`d_{pi}`, contact forces appear only when
particles are under compression.

When accounting for short-range forces, it is convenient to define the
short-range family of particles

.. math::

   \mathcal{F}^S_i = \{ p ~ | ~ \left\Vert {\textbf{y}_p - \textbf{y}_i} \right\Vert \leq d_{pi} \}.


Modification to the Particle Volume
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The right-hand side of :ref:`(12) <peridiscreteNewtonII>` may be thought of as
a midpoint quadrature of :ref:`(1) <periNewtonII>`. To slightly improve the
accuracy of this quadrature, we discuss a modification to the particle
volume used in :ref:`(12) <peridiscreteNewtonII>`. In a situation where two
particles share a bond with :math:`\left\Vert { \textbf{x}_p -
\textbf{x}_i }\right\Vert = \delta`, for example, we suppose that only
approximately half the volume of each particle is "seen" by the other
:ref:`(Silling 2007) <Silling2007>`. When computing the force of each
particle on the other we use :math:`V_p / 2` rather than :math:`V_p` in
:ref:`(12) <peridiscreteNewtonII>`. As such, we introduce a nodal volume
scaling function for all bonded particles where :math:`\delta - r_i \leq
\left\Vert { \textbf{x}_p - \textbf{x}_i } \right\Vert \leq \delta` (see
the Figure below).

We choose to use a linear unitless nodal volume scaling function

.. math::

   \nu(\textbf{x}_p - \textbf{x}_i) = \left\{
   \begin{array}{cl}
   -\frac{1}{2 r_i} \left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert + \left( \frac{\delta}{2 r_i} + \frac{1}{2} \right) & \mbox{if }
   \delta - r_i \leq \left\Vert {\textbf{x}_p - \textbf{x}_i } \right\Vert \leq \delta \\
   1 & \mbox{if } \left\Vert {\textbf{x}_p - \textbf{x}_i } \right\Vert \leq \delta - r_i \\
   0 & \mbox{otherwise}
   \end{array}
   \right\}

If :math:`\left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert = \delta`, :math:`\nu = 0.5`, and if
:math:`\left\Vert {\textbf{x}_p - \textbf{x}_i} \right\Vert = \delta - r_i`, :math:`\nu = 1.0`, for
example.


.. figure:: JPG/pdlammps_fig1.png
   :figwidth: 80%
   :figclass: align-center

   Diagram showing horizon of a particular particle, demonstrating that
   the volume associated with particles near the boundary of the horizon is
   not completely contained within the horizon.

Temporal Discretization
^^^^^^^^^^^^^^^^^^^^^^^

When discretizing time in LAMMPS, we use a velocity-Verlet scheme, where
both the position and velocity of the particle are stored
explicitly. The velocity-Verlet scheme is generally expressed in three
steps. In :ref:`Algorithm 1 <algvelverlet>`, :math:`\rho_i` denotes the
mass density of a particle and :math:`\widetilde{\textbf{f}}_i^n`
denotes the the net force density on particle :math:`i` at timestep
:math:`n`. The LAMMPS command :doc:`fix nve <fix_nve>` performs a
velocity-Verlet integration.

   .. _algvelverlet:

   .. admonition:: Algorithm 1: Velocity Verlet
      :class: tip

      | 1: :math:`\textbf{v}_i^{n + 1/2} = \textbf{v}_i^n + \frac{\Delta t}{2 \rho_i} \widetilde{\textbf{f}}_i^n`
      | 2: :math:`\textbf{y}_i^{n+1}     = \textbf{y}_i^n + \Delta t \textbf{v}_i^{n + 1/2}`
      | 3: :math:`\textbf{v}_i^{n+1} = \textbf{v}_i^{n+1/2} + \frac{\Delta t}{2 \rho_i} \widetilde{\textbf{f}}_i^{n+1}`

Breaking Bonds
^^^^^^^^^^^^^^

During the course of simulation, it may be necessary to break bonds, as
described in the :ref:`Damage section <peridamage>`. Bonds are recorded
as broken in a simulation by removing them from the bond family
:math:`\mathcal{F}_i` (see :ref:`(11) <periBondFamily>`).

A naive implementation would have us first loop over all bonds and
compute :math:`s_{min}` in :ref:`(9) <peris0>`, then loop over all bonds
again and break bonds with a stretch :math:`s > s0` as in
:ref:`(8) <perimu>`, and finally loop over all particles and compute forces
for the next step of :ref:`Algorithm 1 <algvelverlet>`. For reasons of
computational efficiency, we will utilize the values of :math:`s_0` from
the *previous* timestep when deciding to break a bond.

.. note::

   For the first timestep, :math:`s_0` is initialized to
   :math:`\mathbf{\infty}` for all nodes. This means that no bonds may
   be broken until the second timestep. As such, it is recommended that
   the first few timesteps of the peridynamic simulation not involve any
   actions that might result in the breaking of bonds. As a practical
   example, the projectile in the :ref:`commented example below
   <periexample>` is placed such that it does not impact the target
   brittle plate until several timesteps into the simulation.

LPS Pseudocode
^^^^^^^^^^^^^^

A sketch of the LPS model implementation in the PERI package appears in
:ref:`Algorithm 2 <algperilps>`. This algorithm makes use of the
routines in :ref:`Algorithm 3 <algperilpsm>` and :ref:`Algorithm 4
<algperilpstheta>`.

   .. _algperilps:

   .. admonition:: Algorithm 2: LPS Peridynamic Model Pseudocode
      :class: tip

      | Fix :math:`s_{00}`, :math:`\alpha`, horizon :math:`\delta`, bulk modulus :math:`K`, shear modulus :math:`G`, timestep :math:`\Delta t`, and generate initial lattice of particles with lattice constant :math:`\Delta x`. Let there be :math:`N` particles. Define constant :math:`c_S` for repulsive short-range forces.
      | Initialize bonds between all particles :math:`i \neq j` where :math:`\left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert \leq \delta`
      | Initialize weighted volume :math:`m` for all particles using :ref:`Algorithm 3 <algperilpsm>`
      | Initialize :math:`s_0 = \mathbf{\infty}` {*Initialize each entry to MAX_DOUBLE*}
      | **while** not done **do**
      |     Perform step 1 of :ref:`Algorithm 1 <algvelverlet>`, updating velocities of all particles
      |     Perform step 2 of :ref:`Algorithm 1 <algvelverlet>`, updating positions of all particles
      |     :math:`\tilde{s}_0 = \mathbf{\infty}` {*Initialize each entry to MAX_DOUBLE*}
      |     **for** :math:`i=1` to :math:`N` **do**
      |         {Compute short-range forces}
      |         **for all** particles :math:`j \in \mathcal{F}^S_i` (the short-range family of nodes for particle :math:`i`) **do**
      |             :math:`r = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert`
      |             :math:`dr = \min \{ 0, r - d \}` {*Short-range forces are only repulsive, never attractive*}
      |             :math:`k = \frac{c_S}{\delta} V_k dr` {:math:`c_S` *defined in :ref:`(14) <pericS>`*}
      |             :math:`\textbf{f} = \textbf{f} + k \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}`
      |         **end for**
      |     **end for**
      |     Compute the dilatation for each particle using :ref:`Algorithm 4 <algperilpstheta>`
      |     **for** :math:`i=1` to :math:`N` **do**
      |         {Compute bond forces}
      |         **for all** particles :math:`j` sharing an unbroken bond with particle :math:`i` **do**
      |             :math:`e = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert - \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert`
      |             :math:`\omega_+ = \underline{\omega}\left<\textbf{x}_j - \textbf{x}_i\right>` {*Influence function evaluation*}
      |             :math:`\omega_- = \underline{\omega}\left<\textbf{x}_i - \textbf{x}_j\right>` {*Influence function evaluation*}
      |             :math:`\hat{f} = \left[ (3K-5G)\left( \frac{\theta(i)}{m(i)}\omega_+ + \frac{\theta(j)}{m(j)}\omega_- \right) \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert + 15G \left( \frac{\omega_+}{m(i)} + \frac{\omega_-}{m(j)} \right) e \right] \nu(\textbf{x}_j - \textbf{x}_i) V_j`
      |             :math:`\textbf{f} = \textbf{f} + \hat{f} \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}`
      |             **if** :math:`(dr / \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert) > \min(s_0(i), s_0(j))` **then**
      |                Break :math:`i`'s bond with :math:`j` {:math:`j` *'s bond with* :math:`i` *will be broken when this loop iterates on* :math:`j`}
      |             **end if**
      |             :math:`\tilde{s}_0(i) = \min (\tilde{s}_0(i),s_{00}-\alpha(dr / \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert))`
      |         **end for**
      |     **end for**
      |     :math:`s_0 = \tilde{s}_0` {*Store for use in next timestep*}
      |     Perform step 3 of :ref:`Algorithm 1 <algvelverlet>`, updating velocities of all particles
      | **end while**


   .. _algperilpsm:

   .. admonition:: Algorithm 3: Computation of Weighted Volume *m*
      :class: tip

      | **for** :math:`i=1` to :math:`N` **do**
      |     :math:`m(i) = 0.0`
      |     **for all** particles :math:`j` sharing a bond with particle :math:`i` **do**
      |         :math:`m(i) = m(i) + \underline{\omega}\left<\textbf{x}_j - \textbf{x}_i\right> \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert^2 \nu(\textbf{x}_j - \textbf{x}_i) V_j`
      |     **end for**
      | **end for**

   .. _algperilpstheta:

   .. admonition:: Algorithm 4: Computation of Dilatation :math:`\theta`
      :class: tip

      | **for** :math:`i=1` to :math:`N` **do**
      |     :math:`\theta(i) = 0.0`
      |     **for all** particles :math:`j` sharing an unbroken bond with particle :math:`i` **do**
      |         :math:`e = \left\Vert {\textbf{y}_i - \textbf{y}_j} \right\Vert - \left\Vert {\textbf{x}_i - \textbf{x}_j} \right\Vert`
      |         :math:`\theta(i) = \theta(i) + \underline{\omega}\left<\textbf{x}_j - \textbf{x}_i\right> \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert e  \nu(\textbf{x}_j - \textbf{x}_i) V_j`
      |     **end for**
      |     :math:`\theta(i) = \frac{3}{m(i)}\theta(i)`
      | **end for**

PMB Pseudocode
^^^^^^^^^^^^^^

A sketch of the PMB model implementation in the PERI package appears in
:ref:`Algorithm 5 <algperipmb>`.

   .. _algperipmb:

   .. admonition:: Algorithm 5: PMB Peridynamic Model Pseudocode
      :class: tip

      | Fix :math:`s_{00}`, :math:`\alpha`, horizon :math:`\delta`, spring constant :math:`c`, timestep :math:`\Delta t`, and generate initial lattice of particles with lattice constant :math:`\Delta x`. Let there be :math:`N` particles.
      | Initialize bonds between all particles :math:`i \neq j` where :math:`\left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert \leq \delta`
      | Initialize :math:`s_0 = \mathbf{\infty}` {*Initialize each entry to MAX_DOUBLE*}
      | **while** not done **do**
      |     Perform step 1 of :ref:`Algorithm 1 <algvelverlet>`, updating velocities of all particles
      |     Perform step 2 of :ref:`Algorithm 1 <algvelverlet>`, updating positions of all particles
      |     :math:`\tilde{s}_0 = \mathbf{\infty}` {*Initialize each entry to MAX_DOUBLE*}
      |     **for** :math:`i=1` to :math:`N` **do**
      |         {Compute short-range forces}
      |         **for all** particles :math:`j \in \mathcal{F}^S_i` (the short-range family of nodes for particle :math:`i`) **do**
      |             :math:`r = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert`
      |             :math:`dr = \min \{ 0, r - d \}` {*Short-range forces are only repulsive, never attractive*}
      |             :math:`k = \frac{c_S}{\delta} V_k dr` {:math:`c_S` *defined in :ref:`(14) <pericS>`*}
      |             :math:`\textbf{f} = \textbf{f} + k \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}`
      |         **end for**
      |     **end for**
      |     **for** :math:`i=1` to :math:`N` **do**
      |         {Compute bond forces}
      |         **for all** particles :math:`j` sharing an unbroken bond with particle :math:`i` **do**
      |             :math:`r = \left\Vert {\textbf{y}_j - \textbf{y}_i} \right\Vert`
      |             :math:`dr = r - \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert`
      |             :math:`k = \frac{c}{\left\Vert {\textbf{x}_i - \textbf{x}_j} \right\Vert} \nu(\textbf{x}_i - \textbf{x}_j) V_j dr` {:math:`c` *defined in :ref:`(6) <peric>`*}
      |             :math:`\textbf{f} = \textbf{f} + k \frac{\textbf{y}_j-\textbf{y}_i}{\left\Vert {\textbf{y}_j-\textbf{y}_i} \right\Vert}`
      |             **if** :math:`(dr / \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert) > \min(s_0(i), s_0(j))` **then**
      |                 Break :math:`i`'s bond with :math:`j` {:math:`j`\ *'s bond with* :math:`i` *will be broken when this loop iterates on* :math:`j`}
      |             **end if**
      |             :math:`\tilde{s}_0(i) = \min (\tilde{s}_0(i),s_{00}-\alpha(dr / \left\Vert {\textbf{x}_j - \textbf{x}_i} \right\Vert))`
      |         **end for**
      |     **end for**
      |     :math:`s_0 = \tilde{s}_0` {*Store for use in next timestep*}
      |     Perform step 3 of :ref:`Algorithm 1 <algvelverlet>`, updating velocities of all particles
      | **end while**

Damage
^^^^^^

The damage associated with every particle (see :ref:`(10) <peridamageeq>`) can
optionally be computed and output with a LAMMPS data dump. To do this,
your input script must contain the command :doc:`compute damage/atom
<compute_damage_atom>` This enables a LAMMPS per-atom compute to
calculate the damage associated with each particle every time a LAMMPS
:doc:`data dump <dump>` frame is written.

Visualizing Simulation Results
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

There are multiple ways to visualize the simulation results. Typically,
you want to display the particles and color code them by the value
computed with the :doc:`compute damage/atom <compute_damage_atom>`
command.

This can be done, for example, by using the built-in visualizer of the
:doc:`dump image or dump movie <dump_image>` command to create snapshot
images or a movie. Below are example command for using dump image with
the :ref:`example listed below <periexample>` and a set of images
created for steps 300, 600, and 2000 this way.

.. code-block:: LAMMPS

   dump            D2 all image 100 dump.peri.*.png c_C1 type box no 0.0 view 30 60 zoom 1.5 up 0 0 -1  ssao yes 4539 0.6
   dump_modify     D2 pad 5 adiam * 0.001 amap 0.0 1.0 ca 0.1 3 min blue 0.5 yellow max red

.. |periimage1| image:: JPG/dump.peri.300.png
   :width: 32%

.. |periimage2| image:: JPG/dump.peri.600.png
   :width: 32%

.. |periimage3| image:: JPG/dump.peri.2000.png
   :width: 32%

|periimage1|  |periimage2|  |periimage3|

For interactive visualization, the `Ovito <https://ovito.org>`_ is very
convenient to use. Below are steps to create a visualization of the
:ref:`same example from below <periexample>` now using the generated
trajectory in the ``dump.peri`` file.

- Launch Ovito
- File -> Load File -> ``dump.peri``
- Select "-> Particle types" and under "Appearance" set "Display radius:" to 0.0005
- From the "Add modification:" drop down list select "Color coding"
- Under "Color coding" select from the "Color gradient" drop down list "Jet"
- Also under "Color coding" set "Start value:" to 0 and "End value:" to 1
- You can improve the image quality by adding the "Ambient occlusion" modification

.. figure:: JPG/ovito-peri-snap.png
   :figwidth: 80%
   :figclass: align-center

   Screenshot of visualizing a trajectory with Ovito

Pitfalls
^^^^^^^^

**Parallel Scalability**

LAMMPS operates in parallel in a :doc:`spatial-decomposition mode
<Developer_par_part>`, where each processor owns a spatial subdomain of
the overall simulation domain and communicates with its neighboring
processors via distributed-memory message passing (MPI) to acquire ghost
atom information to allow forces on the atoms it owns to be
computed. LAMMPS also uses Verlet neighbor lists which are recomputed
every few timesteps as particles move. On these timesteps, particles
also migrate to new processors as needed. LAMMPS decomposes the overall
simulation domain so that spatial subdomains of nearly equal volume are
assigned to each processor. When each subdomain contains nearly the
same number of particles, this results in a reasonable load balance
among all processors. As is more typical with some peridynamic
simulations, some subdomains may contain many particles while other
subdomains contain few particles, resulting in a load imbalance that
impacts parallel scalability.

**Setting the "skin" distance**

The :doc:`neighbor <neighbor>` command with LAMMPS is used to set the
so-called "skin" distance used when building neighbor lists. All atom
pairs within a cutoff distance equal to the horizon :math:`\delta` plus
the skin distance are stored in the list. Unexpected crashes in LAMMPS
may be due to too small a skin distance. The skin should be set to a
value such that :math:`\delta` plus the skin distance is larger than the
maximum possible distance between two bonded particles. For example, if
:math:`s_{00}` is increased, the skin distance may also need to be
increased.

**"Lost" particles**

All particles are contained within the "simulation box" of LAMMPS. The
boundaries of this box may change with time, or not, depending on how
the LAMMPS :doc:`boundary <boundary>` command has been set. If a
particle drifts outside the simulation box during the course of a
simulation, it is called *lost*.

As an option of the :doc:`themo_modify <thermo_modify>` command of
LAMMPS, the lost keyword determines whether LAMMPS checks for lost atoms
each time it computes thermodynamics and what it does if atoms are
lost. If the value is *ignore*, LAMMPS does not check for lost atoms. If
the value is *error* or *warn*, LAMMPS checks and either issues an error
or warning. The code will exit with an error and continue with a
warning. This can be a useful debugging option. The default behavior of
LAMMPS is to exit with an error if a particle is lost.

The peridynamic module within LAMMPS does not check for lost atoms. If a
particle with unbroken bonds is lost, those bonds are marked as broken
by the remaining particles.

**Defining the peridynamic horizon** :math:`\mathbf{\delta}`

In the :doc:`pair_coeff <pair_coeff>` command, the user must specify the
horizon :math:`\delta`. This argument determines which particles are
bonded when the simulation is initialized. It is recommended that
:math:`\delta` be set to a small fraction of a lattice constant larger than
desired.

For example, if the lattice constant is 0.0005 and you wish to set the
horizon to three times the lattice constant, then set :math:`\delta` to
be 0.0015001, a value slightly larger than three times the lattice
constant. This guarantees that particles three lattice constants away
from each other are still bonded. If :math:`\delta` is set to 0.0015,
for example, floating point error may result in some pairs of particles
three lattice constants apart not being bonded.

**Breaking bonds too early**

For technical reasons, the bonds in the simulation are not created until
the end of the first timestep of the simulation. Therefore, one should
not attempt to break bonds until at least the second step of the
simulation.

Bugs
^^^^

The user is cautioned that this code is a beta release. If you are
confident that you have found a bug in the peridynamic module, please
report it in a `GitHub Issue <https://github.com/lammps/lammps/issues>`
or send an email to the LAMMPS developers. First, check the `New
features and bug fixes <https://www.lammps.org/bug.html>`_ section of
the LAMMPS website site to see if the bug has already been reported or
fixed. If not, the most useful thing you can do for us is to isolate the
problem. Run it on the smallest number of atoms and fewest number of
processors and with the simplest input script that reproduces the
bug. In your message, describe the problem and any ideas you have as to
what is causing it or where in the code the problem might be.  We'll
request your input script and data files if necessary.

Modifying and Extending the Peridynamic Module
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

To add new features or peridynamic potentials to the peridynamic module,
the user is referred to the :doc:`Modifying & extending LAMMPS <Modify>`
section. To develop a new bond-based material, start with the *peri/pmb*
pair style as a template. To develop a new state-based material, start
with the *peri/lps* pair style as a template.

A Numerical Example
"""""""""""""""""""

To introduce the peridynamic implementation within LAMMPS, we replicate
a numerical experiment taken from section 6 of :ref:`(Silling 2005)
<Silling2005>`.

Problem Description and Setup
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

We consider the impact of a rigid sphere on a homogeneous disk of
brittle material. The sphere has diameter :math:`0.01` m and velocity
100 m/s directed normal to the surface of the target. The target
material has density :math:`\rho = 2200` kg/m:math:`^3`. A PMB material
model is used with :math:`K = 14.9` GPa and critical bond stretch
parameters given by :math:`s_{00} = 0.0005` and :math:`\alpha = 0.25`. A
three-dimensional simple cubic lattice is constructed with lattice
constant :math:`0.0005` m and horizon :math:`0.0015` m. (The horizon is
three times the lattice constant.) The target is a cylinder of diameter
:math:`0.074` m and thickness :math:`0.0025` m, and the associated
lattice contains 103,110 particles. Each particle :math:`i` has volume
fraction :math:`V_i = 1.25 \times 10^{-10} \textrm{m}^3`.

The spring constant in the PMB material model is (see :ref:`(6) <peric>`)

.. math::

   c = \frac{18k}{\pi \delta^4} = \frac{18 (14.9 \times 10^9)}{ \pi (1.5 \times 10^{-3})^4} \approx 1.6863 \times 10^{22}.

The CFL analysis from :ref:`(Silling2005) <Silling2005>` shows that a
timestep of :math:`1.0 \times 10^{-7}` is safe.

We observe here that in IEEE double-precision floating point arithmetic
when computing the bond stretch :math:`s(t,\mathbf{\eta},\mathbf{\xi})`
at each iteration where :math:`\left\Vert {\mathbf{\eta}+\mathbf{\xi}}
\right\Vert` is computed during the iteration and :math:`\left\Vert
{\mathbf{\xi}} \right\Vert` was computed and stored for the initial
lattice, it may be that :math:`fl(s) = \varepsilon` with :math:`\left|
\varepsilon \right| \leq \varepsilon_{machine}` for an unstretched
bond. Taking :math:`\varepsilon = 2.220446049250313 \times 10^{-16}`, we
see that the value :math:`c s V_i \approx 4.68 \times 10^{-4}`, computed
when determining :math:`f`, is perhaps larger than we would like,
especially when the true force should be zero. One simple way to avoid
this issue is to insert the following instructions in Algorithm
:ref:`Algorithm 5 <algperipmb>` after instruction 21 (and similarly for
Algorithm :ref:`Algorithm 2 <algperilps>`):

   | **if** :math:`\left| dr \right| < \varepsilon_{machine}` **then**
   |    :math:`dr = 0`
   | **end if**

Qualitatively, this says that displacements from equilibrium on the
order of :math:`10^{-16}`\ m are taken to be exactly zero, a seemingly
reasonable assumption.

The Projectile
^^^^^^^^^^^^^^

The projectile used in the following experiments is not the one used in
:ref:`(Silling 2005) <Silling2005>`. The projectile used here exerts a
force

.. math::

   F(r) = - k_s (r - R)^2

on each atom where :math:`k_s` is a specified force constant, :math:`r` is
the distance from the atom to the center of the indenter, and :math:`R`
is the radius of the projectile. The force is repulsive and :math:`F(r) =
0` for :math:`r > R`. For our problem, the projectile radius :math:`R =
0.05` m, and we have chosen :math:`k_s = 1.0 \times 10^{17}` (compare
with :ref:`(6) <peric>` above).

Writing the LAMMPS Input File
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

We discuss the example input script :ref:`listed below <periexample>`.
In line 2 we specify that SI units are to be used. We specify the
dimension (3) and boundary conditions ("shrink-wrapped") for the
computational domain in lines 3 and 4. In line 5 we specify that
peridynamic particles are to be used for this simulation. In line 7, we
set the "skin" distance used in building the LAMMPS neighbor list. In
line 8 we set the lattice constant (in meters) and in line 10 we define
the spatial region where the target will be placed. In line 12 we
specify a rectangular box enclosing the target region that defines the
simulation domain. Line 14 fills the target region with atoms. Lines 15
and 17 define the peridynamic material model, and lines 19 and 21 set
the particle density and particle volume, respectively. The particle
volume should be set to the cube of the lattice constant for a simple
cubic lattice. Line 23 sets the initial velocity of all particles to
zero. Line 25 instructs LAMMPS to integrate time with velocity-Verlet,
and lines 27-30 create the spherical projectile, sending it with a
velocity of 100 m/s towards the target. Line 32 declares a compute style
for the damage (percentage of broken bonds) associated with each
particle.  Line 33 sets the timestep, line 34 instructs LAMMPS to
provide a screen dump of thermodynamic quantities every 200 timesteps,
and line 35 instructs LAMMPS to create a data file (``dump.peri``) with
a complete snapshot of the system every 100 timesteps. This file can be
used to create still images or movies. Finally, line 36 instructs LAMMPS
to run for 2000 timesteps.

.. _periexample:

.. code-block:: LAMMPS
   :linenos:
   :caption: Peridynamics Example LAMMPS Input Script

   # 3D Peridynamic simulation with projectile"
   units           si
   dimension       3
   boundary        s s s
   atom_style      peri
   atom_modify     map array
   neighbor        0.0010 bin
   lattice         sc 0.0005
   # Create desired target
   region          target cylinder y 0.0 0.0 0.037 -0.0025 0.0 units box
   # Make 1 atom type
   create_box      1 target
   # Create the atoms in the simulation region
   create_atoms    1 region target
   pair_style      peri/pmb
   #               <type1> <type2>    <c>    <horizon>  <s00> <alpha>
   pair_coeff         *       *    1.6863e22 0.0015001 0.0005  0.25
   # Set mass density
   set             group all density 2200
   # volume = lattice constant^3
   set             group all volume 1.25e-10
   # Zero out velocities of particles
   velocity        all set 0.0 0.0 0.0 sum no units box
   # Use velocity-Verlet time integrator
   fix             F1 all nve
   # Construct spherical indenter to shatter target
   variable        y0 equal 0.00510
   variable        vy equal -100
   variable        y equal "v_y0 + step*dt*v_vy"
   fix             F2 all indent 1e17 sphere 0.0000 v_y 0.0000 0.0050 units box
   # Compute damage for each particle
   compute         C1 all damage/atom
   timestep        1.0e-7
   thermo          200
   dump            D1 all custom 100 dump.peri id type x y z c_C1
   run             2000

.. note::

   To use the LPS model, replace line 15 with :doc:`pair_style peri/lps
   <pair_peri>` and modify line 16 accordingly.

Numerical Results and Discussion
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

We ran the :ref:`input script from above <periexample>`. Images of the
disk (projectile not shown) appear in Figure below.  The plot of damage
on the top monolayer was created by coloring each particle according to
its damage.

The symmetry in the computed solution arises because a "perfect" lattice
was used, and a because a perfectly spherical projectile impacted the
lattice at its geometric center. To break the symmetry in the solution,
the nodes in the peridynamic body may be perturbed slightly from the
lattice sites. To do this, the lattice of points can be slightly
perturbed using the :doc:`displace_atoms <displace_atoms>` command.

.. _figperitarget:

.. figure:: JPG/pdlammps_fig2.png
   :figwidth: 80%
   :figclass: align-center

   Target during (a) and after (b,c) impact

------------

.. _Emmrich2007:

**(Emmrich)** Emmrich, Weckner, Commun. Math. Sci., 5, 851-864 (2007),

.. _Parks2:

**(Parks)** Parks, Lehoucq, Plimpton, Silling, Comp Phys Comm, 179(11), 777-783 (2008).

.. _Silling2000_2:

**(Silling 2000)** Silling, J Mech Phys Solids, 48, 175-209 (2000).

.. _Silling2005:

**(Silling 2005)** Silling Askari, Computer and Structures, 83, 1526-1535 (2005).

.. _Silling2007_2:

**(Silling 2007)** Silling, Epton, Weckner, Xu, Askari, J Elasticity, 88, 151-184 (2007).

.. _Seleson2010:

**(Seleson 2010)** Seleson, Parks, Int J Mult Comp Eng 9(6), pp. 689-706, 2011.
