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Revision 3395 by gezelter, Fri May 16 14:25:26 2008 UTC vs.
Revision 3402 by xsun, Fri May 30 15:58:01 2008 UTC

+for use with liquid simulations, or in cases where there is
+orientational anisotropy in the system (i.e. in lipid bilayer
+simulations).
-+
-+
+\section{Langevin Dynamics (LANGEVINDYNAMICS)\label{LDRB}}
-+
-+
+{\sc oopse} implements a Langevin dynamics integrator in order to
-+
+perform molecular dynamics simulations in implicit solvent
-+
+environments.  This results in substantial performance gains when the
-+
+detailed dynamics of the solvent is not important. Since {\sc oopse}
-+
+also handles rigid bodies of arbitrary composition and shape, the
-+
+Langevin integrator is somewhat more complex than in other simulation
-+
+packages.
-+
-+
+Consider the Langevin equations of motion in generalized coordinates
-+
+\begin{equation}
-+
+{\bf M} \dot{{\bf V}}(t) = {\bf F}_{s}(t) +
-+
+{\bf F}_{f}(t)  + {\bf F}_{r}(t)
-+
+\label{LDGeneralizedForm}
-+
+\end{equation}
-+
+where ${\bf M}$ is a $6 \times 6$ diagonal mass matrix (which
-+
+includes the mass of the rigid body as well as the moments of inertia
-+
+in the body-fixed frame) and ${\bf V}$ is a generalized velocity,
-+
+${\bf V} =
-+
+\left\{{\bf v},{\bf \omega}\right\}$. The right side of
-+
+Eq.~\ref{LDGeneralizedForm} consists of three generalized forces: a
-+
+system force (${\bf F}_{s}$), a frictional or dissipative force (${\bf
-+
+F}_{f}$) and a stochastic force (${\bf F}_{r}$). While the evolution
-+
+of the system in Newtonian mechanics is typically done in the lab
-+
+frame, it is convenient to handle the dynamics of rigid bodies in
-+
+body-fixed frames. Thus the friction and random forces on each
-+
+substructure are calculated in a body-fixed frame and may converted
-+
+back to the lab frame using that substructure's rotation matrix (${\bf
-+
+Q}$):
-+
+\begin{equation}
-+
+{\bf F}_{f,r} =
-+
+\left( \begin{array}{c}
-+
+{\bf f}_{f,r} \\
-+
+{\bf \tau}_{f,r}
-+
+\end{array} \right)
-+
+=
-+
+\left( \begin{array}{c}
-+
+{\bf Q}^{T} {\bf f}^{~b}_{f,r} \\
-+
+{\bf Q}^{T} {\bf \tau}^{~b}_{f,r}
-+
+\end{array} \right)
-+
+\end{equation}
-+
+The body-fixed friction force, ${\bf F}_{f}^{~b}$, is proportional to
-+
+the (body-fixed) velocity at the center of resistance
-+
+${\bf v}_{R}^{~b}$ and the angular velocity ${\bf \omega}$
-+
+\begin{equation}
-+
+{\bf F}_{f}^{~b}(t) = \left( \begin{array}{l}
-+
+ {\bf f}_{f}^{~b}(t) \\
-+
+ {\bf \tau}_{f}^{~b}(t) \\
-+
+ \end{array} \right) =  - \left( \begin{array}{*{20}c}
-+
+   \Xi_{R}^{tt} & \Xi_{R}^{rt} \\
-+
+   \Xi_{R}^{tr} & \Xi_{R}^{rr}    \\
-+
+\end{array} \right)\left( \begin{array}{l}
-+
+ {\bf v}_{R}^{~b}(t) \\
-+
+ {\bf \omega}(t) \\
-+
+ \end{array} \right),
-+
+\end{equation}
-+
+while the random force, ${\bf F}_{r}$, is a Gaussian stochastic
-+
+variable with zero mean and variance,
-+
+\begin{equation}
-+
+\left\langle {{\bf F}_{r}(t) ({\bf F}_{r}(t'))^T } \right\rangle  =
-+
+\left\langle {{\bf F}_{r}^{~b} (t) ({\bf F}_{r}^{~b} (t'))^T } \right\rangle  =
-+
+k_B T \Xi_R \delta(t - t'). \label{eq:randomForce}
-+
+\end{equation}
-+
+$\Xi_R$ is the $6\times6$ resistance tensor at the center of
-+
+resistance.  Once this tensor is known for a given rigid body (as
-+
+described in the previous section) obtaining a stochastic vector that
-+
+has the properties in Eq. (\ref{eq:randomForce}) can be done
-+
+efficiently by carrying out a one-time Cholesky decomposition to
-+
+obtain the square root matrix of the resistance tensor,
-+
+\begin{equation}
-+
+\Xi_R = {\bf S} {\bf S}^{T},
-+
+\label{eq:Cholesky}
-+
+\end{equation}
-+
+where ${\bf S}$ is a lower triangular matrix.\cite{Schlick2002} A
-+
+vector with the statistics required for the random force can then be
-+
+obtained by multiplying ${\bf S}$ onto a random 6-vector ${\bf Z}$ which
-+
+has elements chosen from a Gaussian distribution, such that:
-+
+\begin{equation}
-+
+\langle {\bf Z}_i \rangle = 0, \hspace{1in} \langle {\bf Z}_i \cdot
-+
+{\bf Z}_j \rangle = \frac{2 k_B T}{\delta t} \delta_{ij},
-+
+\end{equation}
-+
+where $\delta t$ is the timestep in use during the simulation. The
-+
+random force, ${\bf F}_{r}^{~b} = {\bf S} {\bf Z}$, can be shown to have the
-+
+correct properties required by Eq. (\ref{eq:randomForce}).
-+
-+
+The equation of motion for the translational velocity of the center of
-+
+mass (${\bf v}$) can be written as
-+
+\begin{equation}
-+
+m \dot{{\bf v}} (t) =  {\bf f}_{s}(t) + {\bf f}_{f}(t) +
-+
+{\bf f}_{r}(t)
-+
+\end{equation}
-+
+Since the frictional and random forces are applied at the center of
-+
+resistance, which generally does not coincide with the center of mass,
-+
+extra torques are exerted at the center of mass. Thus, the net
-+
+body-fixed torque at the center of mass, $\tau^{~b}(t)$,
-+
+is given by
-+
+\begin{equation}
-+
+\tau^{~b} \leftarrow \tau_{s}^{~b} + \tau_{f}^{~b} + \tau_{r}^{~b} + {\bf r}_{MR} \times \left( {\bf f}_{f}^{~b} + {\bf f}_{r}^{~b} \right)
-+
+\end{equation}
-+
+where ${\bf r}_{MR}$ is the vector from the center of mass to the center of
-+
+resistance. Instead of integrating the angular velocity in lab-fixed
-+
+frame, we consider the equation of motion for the angular momentum
-+
+(${\bf j}$) in the body-fixed frame
-+
+\begin{equation}
-+
+\frac{\partial}{\partial t}{\bf j}(t) = \tau^{~b}(t)
-+
+\end{equation}
-+
+Embedding the friction and random forces into the the total force and
-+
+torque, one can integrate the Langevin equations of motion for a rigid
-+
+body of arbitrary shape in a velocity-Verlet style 2-part algorithm,
-+
+where $h = \delta t$:
-+
-+
+{\tt move A:}
-+
+\begin{align*}
-+
+{\bf v}\left(t + h / 2\right)  &\leftarrow  {\bf v}(t)
-+
+    + \frac{h}{2} \left( {\bf f}(t) / m \right), \\
-+
+%
-+
+{\bf r}(t + h) &\leftarrow {\bf r}(t)
-+
+    + h  {\bf v}\left(t + h / 2 \right), \\
-+
+%
-+
+{\bf j}\left(t + h / 2 \right)  &\leftarrow {\bf j}(t)
-+
+    + \frac{h}{2} {\bf \tau}^{~b}(t), \\
-+
+%
-+
+{\bf Q}(t + h) &\leftarrow \mathrm{rotate}\left( h {\bf j}
-+
+    (t + h / 2) \cdot \overleftrightarrow{\mathsf{I}}^{-1} \right).
-+
+\end{align*}
-+
+In this context, $\overleftrightarrow{\mathsf{I}}$ is the diagonal
-+
+moment of inertia tensor, and the $\mathrm{rotate}$ function is the
-+
+reversible product of the three body-fixed rotations,
-+
+\begin{equation}
-+
+\mathrm{rotate}({\bf a}) = \mathsf{G}_x(a_x / 2) \cdot
-+
+\mathsf{G}_y(a_y / 2) \cdot \mathsf{G}_z(a_z) \cdot \mathsf{G}_y(a_y
-+
+/ 2) \cdot \mathsf{G}_x(a_x /2),
-+
+\end{equation}
-+
+where each rotational propagator, $\mathsf{G}_\alpha(\theta)$,
-+
+rotates both the rotation matrix ($\mathbf{Q}$) and the body-fixed
-+
+angular momentum (${\bf j}$) by an angle $\theta$ around body-fixed
-+
+axis $\alpha$,
-+
+\begin{equation}
-+
+\mathsf{G}_\alpha( \theta ) = \left\{
-+
+\begin{array}{lcl}
-+
+\mathbf{Q}(t) & \leftarrow & \mathbf{Q}(0) \cdot \mathsf{R}_\alpha(\theta)^T, \\
-+
+{\bf j}(t) & \leftarrow & \mathsf{R}_\alpha(\theta) \cdot {\bf
-+
+j}(0).
-+
+\end{array}
-+
+\right.
-+
+\end{equation}
-+
+$\mathsf{R}_\alpha$ is a quadratic approximation to the single-axis
-+
+rotation matrix.  For example, in the small-angle limit, the
-+
+rotation matrix around the body-fixed x-axis can be approximated as
-+
+\begin{equation}
-+
+\mathsf{R}_x(\theta) \approx \left(
-+
+\begin{array}{ccc}
-+
+& 0 & 0 \\
-+
+& \frac{1-\theta^2 / 4}{1 + \theta^2 / 4}  & -\frac{\theta}{1+
-+
+\theta^2 / 4} \\
-+
+& \frac{\theta}{1+ \theta^2 / 4} & \frac{1-\theta^2 / 4}{1 +
-+
+\theta^2 / 4}
-+
+\end{array}
-+
+\right).
-+
+\end{equation}
-+
+All other rotations follow in a straightforward manner. After the
-+
+first part of the propagation, the forces and body-fixed torques are
-+
+calculated at the new positions and orientations.  The system forces
-+
+and torques are derivatives of the total potential energy function
-+
+($U$) with respect to the rigid body positions (${\bf r}$) and the
-+
+columns of the transposed rotation matrix ${\bf Q}^T = \left({\bf
-+
+u}_x, {\bf u}_y, {\bf u}_z \right)$:
-+
+{\tt Forces:}
-+
+\begin{align*}
-+
+{\bf f}_{s}(t + h) & \leftarrow
-+
+    - \left(\frac{\partial U}{\partial {\bf r}}\right)_{{\bf r}(t + h)} \\
-+
+%
-+
+{\bf \tau}_{s}(t + h) &\leftarrow {\bf u}(t + h)
-+
+    \times \frac{\partial U}{\partial {\bf u}} \\
-+
+%
-+
+{\bf v}^{b}_{R}(t+h) & \leftarrow \mathbf{Q}(t+h) \cdot \left({\bf v}(t+h) + {\bf \omega}(t+h) \times {\bf r}_{MR} \right) \\
-+
+%
-+
+{\bf f}_{R,f}^{b}(t+h) & \leftarrow - \Xi_{R}^{tt} \cdot
-+
+{\bf v}^{b}_{R}(t+h) - \Xi_{R}^{rt} \cdot {\bf \omega}(t+h) \\
-+
+%
-+
+{\bf \tau}_{R,f}^{b}(t+h) & \leftarrow - \Xi_{R}^{tr} \cdot
-+
+{\bf v}^{b}_{R}(t+h) - \Xi_{R}^{rr} \cdot {\bf \omega}(t+h) \\
-+
+%
-+
+Z & \leftarrow  {\tt GaussianNormal}(2 k_B T / h, 6) \\
-+
+{\bf F}_{R,r}^{b}(t+h) & \leftarrow {\bf S} \cdot Z \\
-+
+%
-+
+{\bf f}(t+h) & \leftarrow {\bf f}_{s}(t+h) + \mathbf{Q}^{T}(t+h)
-+
+\cdot \left({\bf f}_{R,f}^{~b} + {\bf f}_{R,r}^{~b} \right) \\
-+
+%
-+
+\tau(t+h) & \leftarrow \tau_{s}(t+h) + \mathbf{Q}^{T}(t+h) \cdot \left(\tau_{R,f}^{~b} + \tau_{R,r}^{~b} \right) + {\bf r}_{MR} \times \left({\bf f}_{f}(t+h) + {\bf f}_{r}(t+h) \right) \\
-+
+\tau^{~b}(t+h) & \leftarrow \mathbf{Q}(t+h) \cdot \tau(t+h) \\
-+
+\end{align*}
-+
+Frictional (and random) forces and torques must be computed at the
-+
+center of resistance, so there are additional steps required to find
-+
+the body-fixed velocity (${\bf v}_{R}^{~b}$) at this location.  Mapping
-+
+the frictional and random forces at the center of resistance back to
-+
+the center of mass also introduces an additional term in the torque
-+
+one obtains at the center of mass.
-+
+Once the forces and torques have been obtained at the new time step,
-+
+the velocities can be advanced to the same time value.
-+
-+
+{\tt move B:}
-+
+\begin{align*}
-+
+{\bf v}\left(t + h \right)  &\leftarrow  {\bf v}\left(t + h / 2
-+
+\right)
-+
+    + \frac{h}{2} \left( {\bf f}(t + h) / m \right), \\
-+
+%
-+
+{\bf j}\left(t + h \right)  &\leftarrow {\bf j}\left(t + h / 2
-+
+\right)
-+
+    + \frac{h}{2} {\bf \tau}^{~b}(t + h) .
-+
+\end{align*}
-+
-+
+The viscosity of the implicit of solvents must be specified using {\tt
-+
+viscosity} keywords in the meta-data file to use langevin dynamics
-+
+integrator. For simple shaped particles (spheres and ellipsoids), no
-+
+further parameters are necessary. However, there are no analytical
-+
+solutions for composite shaped particles, the approximate methods have
-+
+to be applied to get the resistance tensor. The file which contains
-+
+the information about hydro properties is indicated by {\tt
-+
+HydroPropFile} keyword in meta-data file. The {\tt HydroPropFile} is
-+
+precalculated by {\tt Hydro}.
-+
-+
+\begin{longtable}[c]{ABCD}
-+
+\caption{Meta-data Keywords: Langevin Dynamics Parameters}
-+
+\\
-+
+{\bf keyword} & {\bf units} & {\bf use} & {\bf remarks}  \\ \hline
-+
+\endhead
-+
+\hline
-+
+\endfoot
-+
+{\tt viscosity} & centipoise & Sets the value of viscosity of implicit
-+
+solvents & \\ {\tt HydroPropFile} & string & specifies the resistance
-+
+tensor file & usually a {\tt .diff} file which is precalculated by
-+
+{\sc Hydro}. Not necessory for simple shaped particles (spheres and
-+
+ellipsoids) \\
-+
+{\tt beadSize} & $\mbox{\AA}$ & Sets the diameters of
-+
+beads when {\sc Rough Shell Model} is used to generate the resistance
-+
+tensor file. \\
-+
+\label{table:ldParameters}
-+
+\end{longtable}
-+
-+
+\section{\label{sec:constraints}Constraint Methods}
+\subsection{\label{oopseSec:rattle}The {\sc rattle} Method for Bond
+    &  {\tt -{}-ny=INT}           &  number of unit cells in y\\
+    &  {\tt -{}-nz=INT}            &  number of unit cells in z
+\end{longtable}
-+
-+
+\section{\label{oopseSec:Hydro}Hydro}
-+
+{\tt Hydro} generates {\tt .diff} file which is required when a Langevin
-+
+Dynamics simulation using approximate models (supports Bead Model and
-+
+Rough Shell Model) is performed. To generate the {\tt }.diff file, the
-+
+meta-data file is needed as the input file. The viscosity of the fluid
-+
+flow (solvent) and the temperature of the system have to be defined in
-+
+meta-data file. If the approximate model is {\tt Rough Shell Model},
-+
+the {\tt beadSize} which is the diameter of every beads must be
-+
+specified in meta-data file.
-+
+The options available for Hydro are as follows:
-+
+\begin{longtable}[c]{|EFG|}
-+
+\caption{Hydro Command-line Options}
-+
+\\ \hline
-+
+{\bf option} & {\bf verbose option} & {\bf behavior} \\ \hline
-+
+\endhead
-+
+\hline
-+
+\endfoot
-+
+  -h& {\tt -{}-help}               & Print help and exit\\
-+
+  -V& {\tt -{}-version}            & Print version and exit\\
-+
+  -i& {\tt -{}-input=filename}     & input MetaData (md) file\\
-+
+  -o& {\tt -{}-output=STRING}      & Output file name\\
-+
+   &  {\tt -{}-model=STRING}     & hydrodynamics model (supports
-+
+RoughShell and BeadModel)\\
-+
+  -b&  {\tt -{}-beads}            & generate the beads only,
-+
+hydrodynamics will be performed (default=off)\\
-+
+\end{longtable}
-+
-+
+\chapter{\label{oopseSec:parallelization} Parallel Simulation Implementation}
+Although processor power is continually improving, it is still

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Comparing trunk/oopseDocs/oopseDoc.tex (file contents): Revision 3395 by gezelter, Fri May 16 14:25:26 2008 UTC vs. Revision 3402 by xsun, Fri May 30 15:58:01 2008 UTC

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Comparing trunk/oopseDocs/oopseDoc.tex (file contents):
Revision 3395 by gezelter, Fri May 16 14:25:26 2008 UTC vs.
Revision 3402 by xsun, Fri May 30 15:58:01 2008 UTC