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encode some more mathematical expressions and symbols

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This commit is contained in:
Axel Kohlmeyer
2020-03-01 09:25:20 -05:00
parent 2455e35fd8
commit ff614957c6
3 changed files with 43 additions and 41 deletions
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2
doc/src/Howto_triclinic.rst
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@ -55,7 +55,7 @@ rotation of **A**\ , **B**\ , and **C** and can be computed as follows:
c_z = & |\mathbf{C} \cdot \widehat{(\mathbf{A} \times \mathbf{B})}|\quad = \quad \sqrt{C^2 - {c_x}^2 - {c_y}^2}
where A = \| **A** \| indicates the scalar length of **A**\ . The hat symbol (\^)
indicates the corresponding unit vector. *beta* and *gamma* are angles
indicates the corresponding unit vector. :math:`\beta` and :math:`\gamma` are angles
between the vectors described below. Note that by construction,
**a**\ , **b**\ , and **c** have strictly positive x, y, and z components, respectively.
If it should happen that

70
doc/src/fix_lb_fluid.rst
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@ -137,7 +137,7 @@ calculated as:
where :math:`\mathbf{v}_n` is the velocity of the MD particle,
:math:`\mathbf{u}_f` is the fluid
velocity interpolated to the particle location, and gamma is the force
velocity interpolated to the particle location, and :math:`\gamma` is the force
coupling constant. :math:`\zeta` is a weight assigned to the grid point,
obtained by distributing the particle to the nearest lattice sites.
For this, the user has the choice between a trilinear stencil, which
@ -163,10 +163,10 @@ t_{collision}` is a collision time, chosen such that
:math:`\frac{\tau}{\Delta t_{collision}} = 1` (see :ref:`Mackay and
Denniston <Mackay2>` for full details). In order to calculate :math:`m_u`,
the fluid density is interpolated to the MD particle location, and
multiplied by a volume, node\_area\*dx\_lb, where node\_area
multiplied by a volume, node\_area * :math:`dx_{LB}`, where node\_area
represents the portion of the surface area of the composite object
associated with a given MD particle. By default, node\_area is set
equal to dx\_lb\*dx\_lb; however specific values for given atom types
equal to :math:`dx_{LB}^2`; however specific values for given atom types
can be set using the *setArea* keyword.
The user also has the option of specifying their own value for the
@ -174,7 +174,7 @@ force coupling constant, for all the MD particles associated with the
fix, through the use of the *setGamma* keyword. This may be useful
when modelling porous particles. See :ref:`Mackay et al. <fluid-Mackay>` for a
detailed description of the method by which the user can choose an
appropriate gamma value.
appropriate :math:`\gamma` value.
.. note::
@ -188,7 +188,9 @@ appropriate gamma value.
used to integrate the particle motion. However, if the user specifies
their own value for the force coupling constant, as mentioned in
:ref:`Mackay et al. <fluid-Mackay>`, the built-in LAMMPS integrators may prove to
be unstable. Therefore, we have included our own integrators :doc:`fix lb/rigid/pc/sphere <fix_lb_rigid_pc_sphere>`, and :doc:`fix lb/pc <fix_lb_pc>`, to solve for the particle motion in these
be unstable. Therefore, we have included our own integrators
:doc:`fix lb/rigid/pc/sphere <fix_lb_rigid_pc_sphere>`, and
:doc:`fix lb/pc <fix_lb_pc>`, to solve for the particle motion in these
cases. These integrators should not be used with the
:doc:`lb/viscous <fix_lb_viscous>` fix, as they add hydrodynamic forces
to the particles directly. In addition, they can not be used if the
@ -207,28 +209,29 @@ appropriate gamma value.
location, in order to approximate an infinitely massive particle (see
the dragforce test run for an example).
----------
Inside the fix, parameters are scaled by the lattice-Boltzmann
timestep, dt, grid spacing, dx, and mass unit, dm. dt is set equal to
(nevery\*dt\_MD), where dt\_MD is the MD timestep. By default, dm is set
equal to 1.0, and dx is chosen so that tau/(dt) =
(3\*eta\*dt)/(rho\*dx\^2) is approximately equal to 1. However, the user
has the option of specifying their own values for dm, and dx, by using
timestep, :math:`dt_{LB}`, grid spacing, :math:`dx_{LB}`, and mass unit,
:math:`dm_{LB}`. :math:`dt_{LB}` is set equal to
:math:`\mathrm{nevery}\cdot dt_{MD}`, where :math:`dt_{MD}` is the MD timestep.
By default,
:math:`dm_{LB}` is set equal to 1.0, and :math:`dx_{LB}` is chosen so that
:math:`\frac{\tau}{dt} = \frac{3\eta dt}{\rho dx^2}` is approximately equal to 1.
However, the user has the option of specifying their own values for
:math:`dm_{LB}`, and :math:`dx_{LB}`, by using
the optional keywords *dm*\ , and *dx* respectively.
.. note::
Care must be taken when choosing both a value for dx, and a
simulation domain size. This fix uses the same subdivision of the
simulation domain among processors as the main LAMMPS program. In
Care must be taken when choosing both a value for :math:`dx_{LB}`,
and a simulation domain size. This fix uses the same subdivision of
the simulation domain among processors as the main LAMMPS program. In
order to uniformly cover the simulation domain with lattice sites, the
lengths of the individual LAMMPS sub-domains must all be evenly
divisible by dx. If the simulation domain size is cubic, with equal
lengths in all dimensions, and the default value for dx is used, this
will automatically be satisfied.
divisible by :math:`dx_{LB}`. If the simulation domain size is cubic,
with equal lengths in all dimensions, and the default value for
:math:`dx_{LB}` is used, this will automatically be satisfied.
Physical parameters describing the fluid are specified through
*viscosity*\ , *density*\ , and *a0*\ . If the force coupling constant is
@ -240,27 +243,25 @@ the mass, distance, and time units chosen for the main LAMMPS run, as
they are scaled by the LB timestep, lattice spacing, and mass unit,
inside the fix.
----------
The *setArea* keyword allows the user to associate a surface area with
a given atom type. For example if a spherical composite object of
radius R is represented as a spherical shell of N evenly distributed
MD particles, all of the same type, the surface area per particle
associated with that atom type should be set equal to 4\*pi\*R\^2/N.
associated with that atom type should be set equal to :math:`\frac{4\pi R^2}{N}`.
This keyword should only be used if the force coupling constant,
gamma, is set the default way.
:math:`\gamma`, is set the default way.
The *setGamma* keyword allows the user to specify their own value for
the force coupling constant, gamma, instead of using the default
the force coupling constant, :math:`\gamma`, instead of using the default
value.
The *scaleGamma* keyword should be used in conjunction with the
*setGamma* keyword, when the user wishes to specify different gamma
*setGamma* keyword, when the user wishes to specify different :math:`\gamma`
values for different atom types. This keyword allows the user to
scale the *setGamma* gamma value by a factor, gammaFactor, for a given
atom type.
scale the *setGamma* :math:`\gamma` value by a factor, gammaFactor,
for a given atom type.
The *dx* keyword allows the user to specify a value for the LB grid
spacing.
@ -269,9 +270,10 @@ The *dm* keyword allows the user to specify the LB mass unit.
If the *a0* keyword is used, the value specified is used for the
square of the speed of sound in the fluid. If this keyword is not
present, the speed of sound squared is set equal to (1/3)\*(dx/dt)\^2.
Setting a0 > (dx/dt)\^2 is not allowed, as this may lead to
instabilities.
present, the speed of sound squared is set equal to
:math:`\frac{1}{3}\left(\frac{dx_{LB}}{dt_{LB}}\right)^2`.
Setting :math:`a0 > (\frac{dx_{LB}}{dt_{LB}})^2` is not allowed,
as this may lead to instabilities.
If the *noise* keyword is used, followed by a positive temperature
value, and a positive integer random number seed, a thermal
@ -388,13 +390,13 @@ By default, the force coupling constant is set according to
\gamma = \frac{2m_um_v}{m_u+m_v}\left(\frac{1}{\Delta t_{collision}}\right)
and an area of dx\_lb\^2 per node, used to calculate the fluid mass at
and an area of :math:`dx_{LB}^2` per node, used to calculate the fluid mass at
the particle node location, is assumed.
dx is chosen such that tau/(delta t\_LB) =
(3 eta dt\_LB)/(rho dx\_lb\^2) is approximately equal to 1.
dm is set equal to 1.0.
a0 is set equal to (1/3)\*(dx\_lb/dt\_lb)\^2.
*dx* is chosen such that :math:`\frac{\tau}{dt_{LB}} =
\frac{3\eta dt_{LB}}{\rho dx_{LB}^2}` is approximately equal to 1.
*dm* is set equal to 1.0.
*a0* is set equal to :math:`\frac{1}{3}\left(\frac{dx_{LB}}{dt_{LB}}\right)^2`.
The Peskin stencil is used as the default interpolation method.
The D3Q15 lattice is used for the lattice-Boltzmann algorithm.
If walls are present, they are assumed to be stationary.

12
doc/src/fix_qbmsst.rst
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@ -118,18 +118,18 @@ term, see :ref:`(Qi) <Qi>`, and :math:`E^{tot}_0 = e0` is the
initial total energy.
The *eta* (:math:`\eta`) parameter is a unitless coupling constant
between the shock system and the quantum thermal bath. A small *eta*
between the shock system and the quantum thermal bath. A small :math:`\eta`
value cannot adjust the quantum temperature fast enough during the
temperature ramping period of shock compression while large *eta*
leads to big temperature oscillation. A value of *eta* between 0.3 and
temperature ramping period of shock compression while large :math:`\eta`
leads to big temperature oscillation. A value of :math:`\eta` between 0.3 and
1 is usually appropriate for simulating most systems under shock
compression. We observe that different values of *eta* lead to almost
compression. We observe that different values of :math:`\eta` lead to almost
the same final thermodynamic state behind the shock, as expected.
The quantum temperature is updated every *beta* (:math:`\beta`) steps
with an integration time interval *beta* times longer than the
with an integration time interval :math:`\beta` times longer than the
simulation time step. In that case, :math:`E^{tot}` is taken as its
average over the past *beta* steps. The temperature of the quantum
average over the past :math:`\beta` steps. The temperature of the quantum
thermal bath :math:`T^{qm}` changes dynamically according to
the following equation where :math:`\Delta_t` is the MD time step and
:math:`\gamma` is the friction constant which is equal to the inverse