2006-09-22 00:22:34 +08:00
2015-07-30 22:53:28 +08:00
<!DOCTYPE html>
<!-- [if IE 8]><html class="no - js lt - ie9" lang="en" > <![endif] -->
<!-- [if gt IE 8]><! --> < html class = "no-js" lang = "en" > <!-- <![endif] -->
< head >
< meta charset = "utf-8" >
< meta name = "viewport" content = "width=device-width, initial-scale=1.0" >
2015-12-21 23:20:41 +08:00
< title > pair_style gran/hooke command — LAMMPS documentation< / title >
2015-07-30 22:53:28 +08:00
2006-09-22 00:22:34 +08:00
2015-07-30 22:53:28 +08:00
2006-09-22 00:22:34 +08:00
2015-07-30 22:53:28 +08:00
2006-09-22 00:22:34 +08:00
2015-07-30 22:53:28 +08:00
2006-09-22 00:22:34 +08:00
2015-07-30 22:53:28 +08:00
2006-09-22 00:22:34 +08:00
2015-07-30 22:53:28 +08:00
< link rel = "stylesheet" href = "_static/css/theme.css" type = "text/css" / >
2006-09-22 00:22:34 +08:00
2015-07-30 22:53:28 +08:00
< link rel = "stylesheet" href = "_static/sphinxcontrib-images/LightBox2/lightbox2/css/lightbox.css" type = "text/css" / >
2006-09-22 00:22:34 +08:00
2015-07-30 22:53:28 +08:00
2015-12-21 23:20:41 +08:00
< link rel = "top" title = "LAMMPS documentation" href = "index.html" / >
2009-01-09 07:18:29 +08:00
2015-07-30 22:53:28 +08:00
< script src = "_static/js/modernizr.min.js" > < / script >
2009-01-09 07:18:29 +08:00
2015-07-30 22:53:28 +08:00
< / head >
2006-09-22 00:22:34 +08:00
2015-07-30 22:53:28 +08:00
< body class = "wy-body-for-nav" role = "document" >
2006-09-22 00:22:34 +08:00
2015-07-30 22:53:28 +08:00
< div class = "wy-grid-for-nav" >
< nav data-toggle = "wy-nav-shift" class = "wy-nav-side" >
< div class = "wy-side-nav-search" >
< a href = "Manual.html" class = "icon icon-home" > LAMMPS
< / a >
< div role = "search" >
< form id = "rtd-search-form" class = "wy-form" action = "search.html" method = "get" >
< input type = "text" name = "q" placeholder = "Search docs" / >
< input type = "hidden" name = "check_keywords" value = "yes" / >
< input type = "hidden" name = "area" value = "default" / >
< / form >
< / div >
< / div >
< div class = "wy-menu wy-menu-vertical" data-spy = "affix" role = "navigation" aria-label = "main navigation" >
< ul >
< li class = "toctree-l1" > < a class = "reference internal" href = "Section_intro.html" > 1. Introduction< / a > < / li >
< li class = "toctree-l1" > < a class = "reference internal" href = "Section_start.html" > 2. Getting Started< / a > < / li >
< li class = "toctree-l1" > < a class = "reference internal" href = "Section_commands.html" > 3. Commands< / a > < / li >
< li class = "toctree-l1" > < a class = "reference internal" href = "Section_packages.html" > 4. Packages< / a > < / li >
< li class = "toctree-l1" > < a class = "reference internal" href = "Section_accelerate.html" > 5. Accelerating LAMMPS performance< / a > < / li >
< li class = "toctree-l1" > < a class = "reference internal" href = "Section_howto.html" > 6. How-to discussions< / a > < / li >
< li class = "toctree-l1" > < a class = "reference internal" href = "Section_example.html" > 7. Example problems< / a > < / li >
< li class = "toctree-l1" > < a class = "reference internal" href = "Section_perf.html" > 8. Performance & scalability< / a > < / li >
< li class = "toctree-l1" > < a class = "reference internal" href = "Section_tools.html" > 9. Additional tools< / a > < / li >
< li class = "toctree-l1" > < a class = "reference internal" href = "Section_modify.html" > 10. Modifying & extending LAMMPS< / a > < / li >
< li class = "toctree-l1" > < a class = "reference internal" href = "Section_python.html" > 11. Python interface to LAMMPS< / a > < / li >
< li class = "toctree-l1" > < a class = "reference internal" href = "Section_errors.html" > 12. Errors< / a > < / li >
< li class = "toctree-l1" > < a class = "reference internal" href = "Section_history.html" > 13. Future and history< / a > < / li >
< / ul >
< / div >
< / nav >
< section data-toggle = "wy-nav-shift" class = "wy-nav-content-wrap" >
< nav class = "wy-nav-top" role = "navigation" aria-label = "top navigation" >
< i data-toggle = "wy-nav-top" class = "fa fa-bars" > < / i >
< a href = "Manual.html" > LAMMPS< / a >
< / nav >
< div class = "wy-nav-content" >
< div class = "rst-content" >
< div role = "navigation" aria-label = "breadcrumbs navigation" >
< ul class = "wy-breadcrumbs" >
< li > < a href = "Manual.html" > Docs< / a > » < / li >
< li > pair_style gran/hooke command< / li >
< li class = "wy-breadcrumbs-aside" >
< a href = "http://lammps.sandia.gov" > Website< / a >
< a href = "Section_commands.html#comm" > Commands< / a >
< / li >
< / ul >
< hr / >
< / div >
< div role = "main" class = "document" itemscope = "itemscope" itemtype = "http://schema.org/Article" >
< div itemprop = "articleBody" >
< div class = "section" id = "pair-style-gran-hooke-command" >
< span id = "index-0" > < / span > < h1 > pair_style gran/hooke command< a class = "headerlink" href = "#pair-style-gran-hooke-command" title = "Permalink to this headline" > ¶< / a > < / h1 >
< / div >
< div class = "section" id = "pair-style-gran-cuda-command" >
< h1 > pair_style gran/cuda command< a class = "headerlink" href = "#pair-style-gran-cuda-command" title = "Permalink to this headline" > ¶< / a > < / h1 >
< / div >
< div class = "section" id = "pair-style-gran-omp-command" >
< h1 > pair_style gran/omp command< a class = "headerlink" href = "#pair-style-gran-omp-command" title = "Permalink to this headline" > ¶< / a > < / h1 >
< / div >
< div class = "section" id = "pair-style-gran-hooke-history-command" >
< h1 > pair_style gran/hooke/history command< a class = "headerlink" href = "#pair-style-gran-hooke-history-command" title = "Permalink to this headline" > ¶< / a > < / h1 >
< / div >
< div class = "section" id = "pair-style-gran-hooke-history-omp-command" >
< h1 > pair_style gran/hooke/history/omp command< a class = "headerlink" href = "#pair-style-gran-hooke-history-omp-command" title = "Permalink to this headline" > ¶< / a > < / h1 >
< / div >
< div class = "section" id = "pair-style-gran-hertz-history-command" >
< h1 > pair_style gran/hertz/history command< a class = "headerlink" href = "#pair-style-gran-hertz-history-command" title = "Permalink to this headline" > ¶< / a > < / h1 >
< / div >
< div class = "section" id = "pair-style-gran-hertz-history-omp-command" >
< h1 > pair_style gran/hertz/history/omp command< a class = "headerlink" href = "#pair-style-gran-hertz-history-omp-command" title = "Permalink to this headline" > ¶< / a > < / h1 >
< div class = "section" id = "syntax" >
< h2 > Syntax< a class = "headerlink" href = "#syntax" title = "Permalink to this headline" > ¶< / a > < / h2 >
< div class = "highlight-python" > < div class = "highlight" > < pre > pair_style style Kn Kt gamma_n gamma_t xmu dampflag
< / pre > < / div >
< / div >
< ul class = "simple" >
< li > style = < em > gran/hooke< / em > or < em > gran/hooke/history< / em > or < em > gran/hertz/history< / em > < / li >
< li > Kn = elastic constant for normal particle repulsion (force/distance units or pressure units - see discussion below)< / li >
< li > Kt = elastic constant for tangential contact (force/distance units or pressure units - see discussion below)< / li >
< li > gamma_n = damping coefficient for collisions in normal direction (1/time units or 1/time-distance units - see discussion below)< / li >
< li > gamma_t = damping coefficient for collisions in tangential direction (1/time units or 1/time-distance units - see discussion below)< / li >
< li > xmu = static yield criterion (unitless value between 0.0 and 1.0e4)< / li >
< li > dampflag = 0 or 1 if tangential damping force is excluded or included< / li >
< / ul >
2015-12-11 01:23:56 +08:00
< div class = "admonition note" >
< p class = "first admonition-title" > Note< / p >
< p class = "last" > Versions of LAMMPS before 9Jan09 had different style names for
granular force fields. This is to emphasize the fact that the
Hertzian equation has changed to model polydispersity more accurately.
A side effect of the change is that the Kn, Kt, gamma_n, and gamma_t
coefficients in the pair_style command must be specified with
different values in order to reproduce calculations made with earlier
versions of LAMMPS, even for monodisperse systems. See the NOTE below
for details.< / p >
2015-07-30 22:53:28 +08:00
< / div >
< / div >
< div class = "section" id = "examples" >
< h2 > Examples< a class = "headerlink" href = "#examples" title = "Permalink to this headline" > ¶< / a > < / h2 >
< div class = "highlight-python" > < div class = "highlight" > < pre > pair_style gran/hooke/history 200000.0 NULL 50.0 NULL 0.5 1
pair_style gran/hooke 200000.0 70000.0 50.0 30.0 0.5 0
< / pre > < / div >
< / div >
< / div >
< div class = "section" id = "description" >
< h2 > Description< a class = "headerlink" href = "#description" title = "Permalink to this headline" > ¶< / a > < / h2 >
< p > The < em > gran< / em > styles use the following formulas for the frictional force
2009-05-12 21:34:00 +08:00
between two granular particles, as described in
2015-07-30 22:53:28 +08:00
< a class = "reference internal" href = "#brilliantov" > < span > (Brilliantov)< / span > < / a > , < a class = "reference internal" href = "#silbert" > < span > (Silbert)< / span > < / a > , and
< a class = "reference internal" href = "#zhang" > < span > (Zhang)< / span > < / a > , when the distance r between two particles of radii
2009-05-12 21:34:00 +08:00
Ri and Rj is less than their contact distance d = Ri + Rj. There is
2015-07-30 22:53:28 +08:00
no force between the particles when r > d.< / p >
< p > The two Hookean styles use this formula:< / p >
< img alt = "_images/pair_gran_hooke.jpg" class = "align-center" src = "_images/pair_gran_hooke.jpg" / >
< p > The Hertzian style uses this formula:< / p >
< img alt = "_images/pair_gran_hertz.jpg" class = "align-center" src = "_images/pair_gran_hertz.jpg" / >
< p > In both equations the first parenthesized term is the normal force
2009-01-08 07:54:46 +08:00
between the two particles and the second parenthesized term is the
tangential force. The normal force has 2 terms, a contact force and a
damping force. The tangential force also has 2 terms: a shear force
2015-07-30 22:53:28 +08:00
and a damping force. The shear force is a “ history” effect that
2009-01-08 07:54:46 +08:00
accounts for the tangential displacement between the particles for the
duration of the time they are in contact. This term is included in
2015-07-30 22:53:28 +08:00
pair styles < em > hooke/history< / em > and < em > hertz/history< / em > , but is not included
in pair style < em > hooke< / em > . The tangential damping force term is included
in all three pair styles if < em > dampflag< / em > is set to 1; it is not included
if < em > dampflag< / em > is set to 0.< / p >
< p > The other quantities in the equations are as follows:< / p >
< ul class = "simple" >
< li > delta = d - r = overlap distance of 2 particles< / li >
< li > Kn = elastic constant for normal contact< / li >
< li > Kt = elastic constant for tangential contact< / li >
< li > gamma_n = viscoelastic damping constant for normal contact< / li >
< li > gamma_t = viscoelastic damping constant for tangential contact< / li >
< li > m_eff = Mi Mj / (Mi + Mj) = effective mass of 2 particles of mass Mi and Mj< / li >
< li > Delta St = tangential displacement vector between 2 spherical particles which is truncated to satisfy a frictional yield criterion< / li >
< li > n_ij = unit vector along the line connecting the centers of the 2 particles< / li >
< li > Vn = normal component of the relative velocity of the 2 particles< / li >
< li > Vt = tangential component of the relative velocity of the 2 particles< / li >
< / ul >
< p > The Kn, Kt, gamma_n, and gamma_t coefficients are specified as
2009-01-09 07:18:29 +08:00
parameters to the pair_style command. If a NULL is used for Kt, then
a default value is used where Kt = 2/7 Kn. If a NULL is used for
2015-07-30 22:53:28 +08:00
gamma_t, then a default value is used where gamma_t = 1/2 gamma_n.< / p >
< p > The interpretation and units for these 4 coefficients are different in
the Hookean versus Hertzian equations.< / p >
< p > The Hookean model is one where the normal push-back force for two
2009-01-08 07:54:46 +08:00
overlapping particles is a linear function of the overlap distance.
Thus the specified Kn is in units of (force/distance). Note that this
push-back force is independent of absolute particle size (in the
2009-01-09 07:18:29 +08:00
monodisperse case) and of the relative sizes of the two particles (in
the polydisperse case). This model also applies to the other terms in
the force equation so that the specified gamma_n is in units of
(1/time), Kt is in units of (force/distance), and gamma_t is in units
2015-07-30 22:53:28 +08:00
of (1/time).< / p >
< p > The Hertzian model is one where the normal push-back force for two
2009-01-08 07:54:46 +08:00
overlapping particles is proportional to the area of overlap of the
two particles, and is thus a non-linear function of overlap distance.
2009-01-09 07:18:29 +08:00
Thus Kn has units of force per area and is thus specified in units of
(pressure). The effects of absolute particle size (monodispersity)
and relative size (polydispersity) are captured in the radii-dependent
pre-factors. When these pre-factors are carried through to the other
terms in the force equation it means that the specified gamma_n is in
units of (1/(time*distance)), Kt is in units of (pressure), and
2015-07-30 22:53:28 +08:00
gamma_t is in units of (1/(time*distance)).< / p >
< p > Note that in the Hookean case, Kn can be thought of as a linear spring
2009-01-08 07:54:46 +08:00
constant with units of force/distance. In the Hertzian case, Kn is
2009-01-09 07:18:29 +08:00
like a non-linear spring constant with units of force/area or
2015-07-30 22:53:28 +08:00
pressure, and as shown in the < a class = "reference internal" href = "#zhang" > < span > (Zhang)< / span > < / a > paper, Kn = 4G /
2009-01-09 07:18:29 +08:00
(3(1-nu)) where nu = the Poisson ratio, G = shear modulus = E /
2015-07-30 22:53:28 +08:00
(2(1+nu)), and E = Young’ s modulus. Similarly, Kt = 4G / (2-nu).
2012-05-25 23:04:57 +08:00
(NOTE: in an earlier version of the manual, we incorrectly stated that
2015-07-30 22:53:28 +08:00
Kt = 8G / (2-nu).)< / p >
< p > Thus in the Hertzian case Kn and Kt can be set to values that
2009-01-08 07:54:46 +08:00
corresponds to properties of the material being modeled. This is also
true in the Hookean case, except that a spring constant must be chosen
2009-01-09 07:18:29 +08:00
that is appropriate for the absolute size of particles in the model.
Since relative particle sizes are not accounted for, the Hookean
2015-07-30 22:53:28 +08:00
styles may not be a suitable model for polydisperse systems.< / p >
2015-12-11 01:23:56 +08:00
< div class = "admonition note" >
< p class = "first admonition-title" > Note< / p >
< p class = "last" > In versions of LAMMPS before 9Jan09, the equation for Hertzian
interactions did not include the sqrt(RiRj/Ri+Rj) term and thus was
not as accurate for polydisperse systems. For monodisperse systems,
sqrt(RiRj/Ri+Rj) is a constant factor that effectively scales all 4
coefficients: Kn, Kt, gamma_n, gamma_t. Thus you can set the values
of these 4 coefficients appropriately in the current code to reproduce
the results of a previous Hertzian monodisperse calculation. For
example, for the common case of a monodisperse system with particles
of diameter 1, all 4 of these coefficients should now be set 2x larger
than they were previously.< / p >
2015-07-30 22:53:28 +08:00
< / div >
< p > Xmu is also specified in the pair_style command and is the upper limit
2009-01-08 07:54:46 +08:00
of the tangential force through the Coulomb criterion Ft = xmu*Fn,
where Ft and Fn are the total tangential and normal force components
in the formulas above. Thus in the Hookean case, the tangential force
between 2 particles grows according to a tangential spring and
dash-pot model until Ft/Fn = xmu and is then held at Ft = Fn*xmu until
the particles lose contact. In the Hertzian case, a similar analogy
2015-07-30 22:53:28 +08:00
holds, though the spring is no longer linear.< / p >
2015-12-11 01:23:56 +08:00
< div class = "admonition note" >
< p class = "first admonition-title" > Note< / p >
< p class = "last" > Normally, xmu should be specified as a fractional value between
0.0 and 1.0, however LAMMPS allows large values (up to 1.0e4) to allow
for modeling of systems which can sustain very large tangential
forces.< / p >
2015-07-30 22:53:28 +08:00
< / div >
< p > For granular styles there are no additional coefficients to set for
each pair of atom types via the < a class = "reference internal" href = "pair_coeff.html" > < em > pair_coeff< / em > < / a > command.
2009-01-06 06:26:31 +08:00
All settings are global and are made via the pair_style command.
2015-07-30 22:53:28 +08:00
However you must still use the < a class = "reference internal" href = "pair_coeff.html" > < em > pair_coeff< / em > < / a > for all
pairs of granular atom types. For example the command< / p >
< div class = "highlight-python" > < div class = "highlight" > < pre > pair_coeff * *
< / pre > < / div >
< / div >
< p > should be used if all atoms in the simulation interact via a granular
2009-01-08 07:54:46 +08:00
potential (i.e. one of the pair styles above is used). If a granular
2015-07-30 22:53:28 +08:00
potential is used as a sub-style of < a class = "reference internal" href = "pair_hybrid.html" > < em > pair_style hybrid< / em > < / a > , then specific atom types can be used in the
2009-01-06 06:26:31 +08:00
pair_coeff command to determine which atoms interact via a granular
2015-07-30 22:53:28 +08:00
potential.< / p >
< hr class = "docutils" / >
< p > Styles with a < em > cuda< / em > , < em > gpu< / em > , < em > intel< / em > , < em > kk< / em > , < em > omp< / em > , or < em > opt< / em > suffix are
2014-08-15 00:30:25 +08:00
functionally the same as the corresponding style without the suffix.
They have been optimized to run faster, depending on your available
2015-07-30 22:53:28 +08:00
hardware, as discussed in < a class = "reference internal" href = "Section_accelerate.html" > < em > Section_accelerate< / em > < / a >
2014-08-15 00:30:25 +08:00
of the manual. The accelerated styles take the same arguments and
should produce the same results, except for round-off and precision
2015-07-30 22:53:28 +08:00
issues.< / p >
< p > These accelerated styles are part of the USER-CUDA, GPU, USER-INTEL,
2014-08-15 00:30:25 +08:00
KOKKOS, USER-OMP and OPT packages, respectively. They are only
2015-07-30 22:53:28 +08:00
enabled if LAMMPS was built with those packages. See the < a class = "reference internal" href = "Section_start.html#start-3" > < span > Making LAMMPS< / span > < / a > section for more info.< / p >
< p > You can specify the accelerated styles explicitly in your input script
by including their suffix, or you can use the < a class = "reference internal" href = "Section_start.html#start-7" > < span > -suffix command-line switch< / span > < / a > when you invoke LAMMPS, or you can
use the < a class = "reference internal" href = "suffix.html" > < em > suffix< / em > < / a > command in your input script.< / p >
< p > See < a class = "reference internal" href = "Section_accelerate.html" > < em > Section_accelerate< / em > < / a > of the manual for
more instructions on how to use the accelerated styles effectively.< / p >
< hr class = "docutils" / >
< p > < strong > Mixing, shift, table, tail correction, restart, rRESPA info< / strong > :< / p >
< p > The < a class = "reference internal" href = "pair_modify.html" > < em > pair_modify< / em > < / a > mix, shift, table, and tail options
are not relevant for granular pair styles.< / p >
< p > These pair styles write their information to < a class = "reference internal" href = "restart.html" > < em > binary restart files< / em > < / a > , so a pair_style command does not need to be
specified in an input script that reads a restart file.< / p >
< p > These pair styles can only be used via the < em > pair< / em > keyword of the
< a class = "reference internal" href = "run_style.html" > < em > run_style respa< / em > < / a > command. They do not support the
< em > inner< / em > , < em > middle< / em > , < em > outer< / em > keywords.< / p >
< p > The single() function of these pair styles returns 0.0 for the energy
2011-12-01 08:19:11 +08:00
of a pairwise interaction, since energy is not conserved in these
dissipative potentials. It also returns only the normal component of
the pairwise interaction force. However, the single() function also
calculates 4 extra pairwise quantities. The first 3 are the
components of the tangential force between particles I and J, acting
2015-07-30 22:53:28 +08:00
on particle I. < em > P4< / em > is the magnitude of this tangential force. These
extra quantites can be accessed by the < a class = "reference internal" href = "compute_pair_local.html" > < em > compute pair/local< / em > < / a > command, as < em > p1< / em > , < em > p2< / em > , < em > p3< / em > ,
< em > p4< / em > .< / p >
< / div >
< hr class = "docutils" / >
< div class = "section" id = "restrictions" >
< h2 > Restrictions< a class = "headerlink" href = "#restrictions" title = "Permalink to this headline" > ¶< / a > < / h2 >
< blockquote >
< div > none< / div > < / blockquote >
< p > All the granular pair styles are part of the GRANULAR package. It is
only enabled if LAMMPS was built with that package. See the < a class = "reference internal" href = "Section_start.html#start-3" > < span > Making LAMMPS< / span > < / a > section for more info.< / p >
< p > These pair styles require that atoms store torque and angular velocity
(omega) as defined by the < a class = "reference internal" href = "atom_style.html" > < em > atom_style< / em > < / a > . They also
require a per-particle radius is stored. The < em > sphere< / em > atom style does
all of this.< / p >
< p > This pair style requires you to use the < a class = "reference internal" href = "comm_modify.html" > < em > comm_modify vel yes< / em > < / a > command so that velocites are stored by ghost
atoms.< / p >
< p > These pair styles will not restart exactly when using the
< a class = "reference internal" href = "read_restart.html" > < em > read_restart< / em > < / a > command, though they should provide
2012-06-19 07:15:48 +08:00
statistically similar results. This is because the forces they
compute depend on atom velocities. See the
2015-07-30 22:53:28 +08:00
< a class = "reference internal" href = "read_restart.html" > < em > read_restart< / em > < / a > command for more details.< / p >
< / div >
< div class = "section" id = "related-commands" >
< h2 > Related commands< a class = "headerlink" href = "#related-commands" title = "Permalink to this headline" > ¶< / a > < / h2 >
< p > < a class = "reference internal" href = "pair_coeff.html" > < em > pair_coeff< / em > < / a > < / p >
< p > < strong > Default:< / strong > none< / p >
< hr class = "docutils" / >
< p id = "brilliantov" > < strong > (Brilliantov)< / strong > Brilliantov, Spahn, Hertzsch, Poschel, Phys Rev E, 53,
p 5382-5392 (1996).< / p >
< p id = "silbert" > < strong > (Silbert)< / strong > Silbert, Ertas, Grest, Halsey, Levine, Plimpton, Phys Rev
E, 64, p 051302 (2001).< / p >
< p id = "zhang" > < strong > (Zhang)< / strong > Zhang and Makse, Phys Rev E, 72, p 011301 (2005).< / p >
< / div >
< / div >
< / div >
< / div >
< footer >
< hr / >
< div role = "contentinfo" >
< p >
2015-12-11 01:23:56 +08:00
© Copyright 2013 Sandia Corporation.
2015-07-30 22:53:28 +08:00
< / p >
< / div >
Built with < a href = "http://sphinx-doc.org/" > Sphinx< / a > using a < a href = "https://github.com/snide/sphinx_rtd_theme" > theme< / a > provided by < a href = "https://readthedocs.org" > Read the Docs< / a > .
< / footer >
< / div >
< / div >
< / section >
< / div >
< script type = "text/javascript" >
var DOCUMENTATION_OPTIONS = {
URL_ROOT:'./',
2015-12-21 23:20:41 +08:00
VERSION:'',
2015-07-30 22:53:28 +08:00
COLLAPSE_INDEX:false,
FILE_SUFFIX:'.html',
HAS_SOURCE: true
};
< / script >
< script type = "text/javascript" src = "_static/jquery.js" > < / script >
< script type = "text/javascript" src = "_static/underscore.js" > < / script >
< script type = "text/javascript" src = "_static/doctools.js" > < / script >
< script type = "text/javascript" src = "https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" > < / script >
< script type = "text/javascript" src = "_static/sphinxcontrib-images/LightBox2/lightbox2/js/jquery-1.11.0.min.js" > < / script >
< script type = "text/javascript" src = "_static/sphinxcontrib-images/LightBox2/lightbox2/js/lightbox.min.js" > < / script >
< script type = "text/javascript" src = "_static/sphinxcontrib-images/LightBox2/lightbox2-customize/jquery-noconflict.js" > < / script >
< script type = "text/javascript" src = "_static/js/theme.js" > < / script >
< script type = "text/javascript" >
jQuery(function () {
SphinxRtdTheme.StickyNav.enable();
});
< / script >
< / body >
< / html >
