new

[book_gpu.git] / BookGPU / Chapters / chapter14 / ch14.tex

\chapterauthor{Alan Gray and Kevin Stratford}{EPCC, The University of Edinburgh, United Kingdom}

\chapter[Ludwig: multiple GPUs for a fluid lattice Boltzmann
application]{Ludwig: multiple GPUs for a complex fluid lattice Boltzmann
application}

%\putbib[biblio]


\section{Introduction}
The lattice Boltzmann (LB) method \index{Lattice Boltzmann method} (for an overview see, e.g.,
\cite{succi-book}) has become a popular approach to a variety of fluid
dynamics problems.  It provides a way to solve the incompressible,
isothermal Navier-Stokes equations and has the attractive features of
being both explicit in time and local in space. This makes the LB
method well-suited to parallel computation. Many efficient parallel
implementations of the LB method have been undertaken, typically using
a combination of distributed domain decomposition and the Message
Passing Interface (MPI). However, the potential
performance benefits offered by GPUs has motivated a new `mixed-mode'
approach to address very large problems. Here, fine-grained
parallelism is implemented on the GPU, while MPI is reserved for
larger-scale parallelism.  This mixed mode is of increasing interest
to application programmers at a time when many supercomputing services
are moving
toward clusters of GPU accelerated nodes. The design questions which
arise when developing a lattice Boltzmann code for this type of
heterogeneous system are therefore worth studying. Furthermore, similar
questions also recur in many other types of stencil-based algorithms.

The first applications of LB on GPUs were to achieve fluid-like
effects in computer animation, rather than scientific applications per
se.  These early works include simple fluids~\cite{wei2004}, miscible
two-component flow~\cite{zhu2006}, and various image processing tasks
based on the use of partial differential equations~\cite{zhao2007}.
While these early works used relatively low level graphics APIs, the
first CUDA runtime interface implementation was a two-dimensional
simple fluid problem~\cite{toelke2010}.  Following pioneering work on
clusters of GPUs coupled via MPI to study air
pollution~\cite{fan2004}, more recent work has included mixed OpenMP
and CUDA~\cite{myre2011}, Posix threads and CUDA~\cite{obrecht2011},
and MPI and CUDA for increasingly large GPU clusters
\cite{bernaschi2010,xian2011,feichtinger2011}. The heterogeneous
nature of these systems has also spurred interest in approaches
including automatic code generation \cite{walshsaar2012} and auto-tuning
\cite{williams2011} to aid application performance.

Many of these authors make use of
another attractive feature of LB: the ability to include fixed
solid-fluid boundary conditions as a straightforward addition to
the algorithm to study, for example, flow in
porous media. This points to an important
application area for LB methods: that of complex fluids.
Complex fluids include mixtures, surfactants, liquid crystals,
and particle suspensions, and typically require additional physics
beyond the bare Navier-Stokes equations to provide a full
description~\cite{aidun2010}. The representation of this extra
physics raises additional design questions for the application
programmer. Here, we consider the \textit{Ludwig} code \cite{desplat}\index{Ludwig code},
an LB application developed specifically for complex fluids
(\textit{Ludwig} was named for Boltzmann, 1844--1906).
We will present the steps
required to allow \textit{Ludwig} to exploit efficiently both a single
GPU, and also many GPUs in parallel.  We show that
\textit{Ludwig} scales excellently to at least the one thousand GPU level
(the largest resource available at the time of writing) with
indications that the code will scale to much larger systems as
they become available. In addition, we consider the steps required
to represent fully-resolved moving solid particles (usually referred
to as colloids in the context of complex fluids). Such particles
need to have their surface resolved on the lattice scale in order
to include relevant surface physics, and must be able to move: e.g.,
to execute Brownian motion in response to random forces from the
fluid. Standard methods are available to represent such particles
(e.g., \cite{ladd1994,nguyen2002}) which are amenable to effective
domain decomposition and message passing \cite{stratford2008}.
We describe below the additional considerations which arise
for such moving particles when developing an
implementation on a GPU.

In the following section we provide a brief overview of the lattice
Boltzmann method, and mention some of the general issues which can
influence performance in the CPU context.
In Section \ref{ch14:sec:singlegpu}, we then describe the alterations
which are required to exploit the GPU architecture effectively, and
highlight how the resulting code differs from the CPU version. In
Section \ref{ch14:sec:parallelgpu}, we extend this description to
include the steps required to allow exploitation of many GPUs in
parallel while retaining effective scaling. We also present results for
a typical benchmark for a fluid-only problem in Section
\ref{ch14:sec:parallelgpu} to demonstrate the success of the approach.
In Section \ref{ch14:sec:particles}, we describe the design choices
which are relevant to a GPU implementation of moving particles. Finally,
we include a number of general observations on software
engineering and maintenance which arise from our experience.
A summary is provided in Section~\ref{ch14:sec:summary}.


\section{Background}\label{ch14:sec:background}

%\begin{figure}[!t]
%\centering
%\includegraphics[width=12cm]{Chapters/chapter14/figures/basiclattice}
%\caption{The lattice Boltzmann approach discritises space into a 3D lattice (left). The fluid is represented at each lattice site (right), by the {\it distribution} vector: each component corresponding to a specific velocity direction. Here we are interested in the {\it D3Q19} model, in which there are 3 spatial directions and 19 velocity components (including a zero component).}
%\label{ch14:fig:basiclattice}
%\end{figure}


For a general complex fluid problem the starting point is the
fluid velocity field $\mathbf{u}(\mathbf{r})$, whose evolution
obeys the Navier-Stokes equations describing the conservation of mass
(or density $\rho$), and momentum:
\begin{equation}
\rho  [ \partial_t \mathbf{u} + (\mathbf{u}.\mathbf{\nabla})\mathbf{u} ]
= -\nabla p + \eta \nabla^2 \mathbf{u} + \mathbf{f}(\mathbf{r}),
\end{equation}
where $p$ is the isotropic pressure and $\eta$ is the viscosity.
A local force $\mathbf{f}(\mathbf{r})$ provides a means for coupling
to other complex fluid constituents, e.g., it might represent the force
exerted on the fluid by a curved interface between different phases or
components.

The LB approach makes use of a regular three-dimensional
lattice (see Figure \ref{ch14:fig:decomphalo}) with discrete spacing
$\Delta r$. It also makes use of a
discrete velocity space $\mathbf{c}_i$, where the $\mathbf{c}_i$
are chosen to capture the correct symmetries of the Navier-Stokes
equations. A typical choice, used here, is the so-called D3Q19
basis in three dimensions where there is one velocity such that
$\mathbf{c} \Delta t$ is zero, along with six extending to the nearest
neighbour
lattice sites, and twelve extending to the next-nearest neighbour sites
($\Delta t$ being the discrete time step). The fundamental object
in LB is then the distribution function $f_i (\mathbf{r};t)$ whose
moments are related to the local hydrodynamic quantities: the fluid
density, momentum, and stress. The time evolution of the distribution
function is described by a discrete Boltzmann equation
\begin{equation}
f_i(\mathbf{r} + \mathbf{c}_i \Delta t; t) - f_i(\mathbf{r}; t) 
= - {\cal L}_{ij} f_j(\mathbf{r};t).
\end{equation}
It is convenient to think of this in two stages. First, the right hand
side represents the action of a collision operator ${\cal L}_{ij}$,
which is local to each lattice site and relaxes the distribution toward
a local equilibrium at a rate ultimately related to the fluid viscosity.
Second, the left hand side represents a propagation step (sometimes referred
to as streaming step), in which each element $i$ of the distribution is
displaced $\mathbf{c}_i \Delta t$, i.e., one lattice spacing in the
appropriate direction per discrete time step. 

More specifically, we store a vector of 19 double precision floating
point values at each lattice site for the distribution function
$f_i(\mathbf{r};t)$.
The collision operation ${\cal L}_{ij}$, which is local at each lattice
site, may be thought of as follows. A matrix-vector multiplication
${\cal M}_{ij}f_j$ is used to transform the distributions into the
hydrodynamic quantities, where ${\cal M}_{ij}$ is a constant 19x19
matrix related to the choice of
$\mathbf{c}_i$. The non-conserved hydrodynamic quantities are then
relaxed toward their (known) equilibrium values, and are transformed
back to new post-collision distributions via the inverse transformation
${\cal M}^{-1}_{ij}$. This gives rise to the need for a minimum of 2x19$^2$
floating point multiplications per lattice site. (Additional operations are
required to implement, for example, the force $\mathbf{f}(\mathbf{r})$.)
In contrast, the
propagation stage consists solely of moving the distribution values
between lattice sites, and involves no floating point operations.

In the CPU version, the \textit{Ludwig} implementation stores one time
level of distribution values $f_i(\mathbf{r}; t)$. This distribution
is stored as a flat array of C data type \texttt{double}, and laid out
so that all elements of the velocity are contiguous at a given site
(often referred to as ``array-of-structures'' order). This is
appropriate for sums over the distribution required locally at the
collision stage as illustrated schematically in
Listing~\ref{ch14:listing1}: the fact that consecutive loads are from
consecutive memory addresses allows the prefetcher to engage fully.
(The temporary scalar \texttt{a\_tmp} allows caching of the
intermediate accumulated value in the innermost loop.)  A variety of
standard sequential optimisations are relevant for the collision stage
(loop unrolling, inclusion of SIMD operations, and so on
\cite{wellein2006}).  For example, the collision stage in
\textit{Ludwig} includes explicit SIMD code, which is useful if the
compiler on a given platform cannot identify it.  The propagation
stage is separate, and is organised as a ``pull'' (again, various
optimisation have been considered, e.g.,
\cite{pohl2003,mattila2007,wittmann2012}).  No further optimisation is
done here beyond ensuring that the ordering of the discrete velocities
allows memory access to be as efficient as possible. While these
optimisations are important, it should be remembered that for some
complex fluid problems, the hydrodynamics embodied in the LB
calculation is a relatively small part of the total computational cost
(at the 10\%-level in some cases).  This means optimisation effort may
be better concentrated elsewhere.

\begin{lstlisting}[float, label=ch14:listing1,
caption = Collision schematic for CPU.]
/* loop over lattice sites */
for (is = 0; is < nsites; is++) {
  ...
  /* load distribution from ``array-of-structures'' */
  for (i = 0; i < 19; i++)    
    f[i] = f_aos[19*is + i]
  ...
  /* perform matrix-vector multiplication */  
  for (i = 0; i < 19; i++) {    
    a_tmp = 0.0;    
    for (j = 0; j < 19; j++) {      
      a_tmp += f[j]*M[i][j];   
    }
    a[i] = a_tmp;
  }
  ...
}
\end{lstlisting}


\begin{figure}[!t]
\centering
\includegraphics[width=12cm]{Chapters/chapter14/figures/decomphalo}
\caption[The lattice is decomposed between MPI tasks.]{Left: the lattice is decomposed between MPI tasks. For
  clarity we show a 2D decomposition of a 3D lattice, but in practice
  we decompose in all 3 dimensions. Halo cells are added to each
  sub-domain (as shown on the upper right for a single slice) which store
  data retrieved from remote neighbours in the halo exchange. Lower
  right: the D3Q19 velocity set resident on a lattice site;
  highlighted are the 5 ``outgoing'' elements to be transferred in a
  specific direction.}
\label{ch14:fig:decomphalo}
\end{figure}


The regular 3D decomposition is illustrated in Fig.~\ref{ch14:fig:decomphalo}.
Each local sub-domain is surrounded by a halo, or ghost, region of width one
lattice site. While the collision is local, elements of the distribution
must be exchanged at the edges of the domains to facilitate the propagation.
To achieve the full 3D halo exchange, the standard approach of shifting the
relevant data in each co-ordinate direction in turn is adopted. This
requires appropriate synchronisation, i.e., a receive in the the first
co-ordinate direction must be complete before a send in the second direction
involving relevant data can take place, and so on. We note that only
``outgoing'' elements of the distribution need to be sent at each edge.
For the D3Q19 model, this reduces the volume of data traffic from 19 to
5 of the $f_i(\mathbf{r};t)$ per lattice site at each edge. In the CPU
version, the necessary transfers are implemented in place using
a vector of appropriately strided MPI datatypes for each direction.


\section{Single GPU Implementation}\label{ch14:sec:singlegpu}


In this section we describe the steps taken to enable \textit{Ludwig}
for the GPU. There are a number of crucial issues: first, the
minimisation of data traffic between host and device; second, the
optimal mapping of available parallelism onto the architecture and
third, the issue of memory coalescing. We discuss each of these in
turn.

While the most important section of the LB in terms of
floating-point performance is the collision stage, this cannot be the
only consideration for a GPU implementation. It is essential to
offload all computational activity which involves the main data
structures (such as the distribution) to the GPU. This includes
kernels with relatively low computational demand, such as the
propagation stage. All relevant data then remain resident on the GPU,
to avoid expensive host-device data transfer at each iteration of
the algorithm. Such transfers would negate any benefit of GPU
acceleration.  We note that for a complex fluid code, this requirement
can extend to a considerable number of kernels, although we limit the
discussion to collision and propagation for brevity.

To achieve optimal performance, it is vital to exploit fully the
parallelism inherent in the GPU architecture, particularly for
those matrix-vector operations within the collision kernel.
The GPU architecture features a hierarchy of parallelism. At the
lowest level, groups of 32 threads (warps) operate in
lock-step on different data elements: this is SIMD style vector-level
parallelism. Multiple warps are combined into a thread block (in which
communication and synchronisation are possible), and multiple blocks can
run concurrently across the streaming multiprocessors in the GPU
(with no communication or
synchronisation possible across blocks).  To decompose on the GPU, we
must choose which part of the collision to assign to which level of
parallelism.

While there exists parallelism within the matrix-vector operation, the
length of each vector (here, 19) is smaller than the warp size and typical
thread block sizes. So we simply decompose the loop over lattice sites
to all levels of parallelism, i.e., we use a separate CUDA thread for
every lattice site, and each thread performs the full matrix-vector
operation. We find that a block size of 256 performs well on current
devices: we therefore decompose
the lattice into groups of 256 sites, and assign each group to a block
of CUDA threads. As the matrix ${\cal M}_{ij}$ is constant, it is
assigned to the fast {\it constant} on-chip device memory.

For the propagation stage, the GPU implementation adds a second time
level of distribution values. The data-dependencies inherent in the
propagation mean that the in-place propagation of the CPU version
cannot be parallelised effectively without the additional time level.
As both time levels may remain resident on the GPU, this is not a
significant overhead.

An architectural
constraint of GPUs means that optimal global memory bandwidth
is only achieved when data are structured such that threads within a
{\it half-warp} (a group of 16 threads) load data from the same memory
segment in a single transaction: this is memory coalescing. It can be
seen that the array-of-structures ordering used for the distribution
in the CPU code would not be suitable for coalescing; in fact, it would
result in serialised memory accesses and relative poor performance.
To meet the coalescing criteria and allow consecutive threads to read
consecutive memory addresses on the GPU, we transpose the layout of the
distribution so that, for each velocity component, consecutive sites
are contiguous in memory (``structure-of-arrays'' order). A schematic of
the GPU collision code is given in Listing~\ref{ch14:listing2}.

\begin{lstlisting}[float, label=ch14:listing2,
caption = Collision schematic for GPU.]
/* compute current site index 'is' from CUDA thread and block */
/* indices and load distribution from "structure-of-arrays" */

for (i = 0; i < 19; i++)    
  f[i] = f_soa[nsites*i + is]

/* perform matrix-vector multiplication as before */
...
\end{lstlisting}


\textit{Ludwig} was modified to allow a choice of distribution data layout
at compilation time depending on the target architecture: CPU or GPU. We
defer some further comments on software engineering aspects of the code to
the summary.


\section{Multiple GPU Implementation}\label{ch14:sec:parallelgpu}

To execute on multiple GPUs, we use the same domain decomposition
and message passing framework as the CPU version. Within each
sub-domain (allocated to one MPI task) the GPU implementation
proceeds as described in the previous section. The only additional
complication is that halo transfers between GPUs must be staged
through the host (in future, direct GPU to GPU data transfers via
MPI may be possible, obviating the need for these steps). This
means host MPI sends must be preceded by appropriate device to host
transfers and host MPI receives must be followed by corresponding host
to device transfers.

%To execute on multiple GPUs in parallel we decompose the problem using
%the existing high-level parallelisaton framework: each MPI task acts
%as a host to a separate GPU, with each GPU kernel operating on the
%subset of the lattice volume corresponding to the partition assigned
%to that task.  As described in Section \ref{ch14:sec:background}, halo
%data exchanges are required at each timestep.  Significant
%developments are made for the GPU adaptation of this communication
%phase to achieve good inter-GPU communication performance and allow
%scaling to many GPUs.

%The CPU code uses strided MPI datatypes to define the six
%planes of lattice data to be sent or received, and performs the
%communications {\it in-place}, i.e.  there is no explicit buffering of
%data in the application.  For the GPU adaptation, the halo planes must
%be exchanged between GPUs, on which MPI functionality is not
%available, so transfers must be staged through the host CPUs. Buffer
%packing/unpacking of the halo planes is explicitly performed using
%CUDA kernels, and CUDA memory copy calls are used to transfer data
%from a device to its host (using pinned host memory) through the PCI-e
%bus (and vice versa). MPI calls are used to transfer the buffers
%between hosts. The CPU version of Ludwig has a mechanism, using MPI built-in
%data types, to reduce the amount of data communicated by around a
%factor of four by only sending those elements of the distribution
%propagating outward from a local domain. For the GPU adaptation, this
%filtering of data is explicitly performed (again since MPI
%functionality is not available on the GPU) in the buffer
%packing/unpacking kernels.  Special care had to be taken for those
%lattice sites on the corners of the halo planes, which had to be sent
%in more than one direction (since the propagation includes accesses to
%diagonal neighbours).

In practice, this data movement requires additional GPU kernels to
pack and unpack the relevant data before and after corresponding MPI
calls. However, the standard shift algorithm, in which each co-ordinate
direction is treated in turn, does provide some scope for the
overlapping of different operations.
For example, after the data for the first co-ordinate direction have been
retrieved by the host, these can be exchanged using MPI between
hosts at the same time as kernels for packing and retrieving of
data for the second co-ordinate direction are
executed. This
overlapping must respect the synchronisation required to ensure
that data values at the corners of the sub-domain are transferred
correctly.
We use a separate CUDA stream for each co-ordinate direction:
this allows some of the host-device communication time to be
effectively ``hidden'' behind the host-host MPI communication,
resulting in an overall speedup.
The improvement is more pronounced for the smaller local lattice size,
perhaps because of less CPU memory bandwidth contention. The
overlapping is then a particularly valuable aid to strong scaling,
where the local system size decreases as the number of GPUs increases.


%\section{Performance}\label{ch14:sec:performance}

To demonstrate the effectiveness of our approach, we compare the
performance of both CPU and GPU versions of \textit{Ludwig}. To
test the complex fluid nature of the code, the problem is
actually an immiscible fluid mixture which uses a second distribution
function to introduce a composition variable. The interested reader
is referred to \cite{ch14:stratford-jsp2005} for further details. The
largest total problem size used is $2548\times 1764\times 1568$.
The CPU system features a Cray XE6 architecture with 2 16-core AMD Opteron
CPUs per node, and with nodes interconnected using Cray Gemini technology.
For GPU results, a Cray XK6 system is used: this is very similar to the
XE6, but has one CPU per node replaced with an NVIDIA X2090 GPU. Each
node in the GPU system therefore features a single Opteron CPU acting
as a host to a single GPU. The inter-node interconnect architecture is
the same as for the Cray XE6. The GPU performance tests use a prototype
Cray XK6 system with 936 nodes (the largest available at the time of writing).
To provide a fair comparison, we compare scaling on a \textit{per node}
basis. That is, we compare 1 fully occupied 32-core CPU node
(running 32 MPI tasks) with 1 GPU node (host running 1 MPI task,
and 1 device). We believe this is representative of the true ``cost'' of
a simulation in terms of accounting budgets, or electricity.


%\begin{figure}[!t]
%\centering
%\includegraphics[width=10cm]{Chapters/chapter14/figures/length_scale}
%\caption{The dependence of the characteristic length scale of a binary
%  fluid mixture on the simulation timestep number. Superimposed are
%  three snapshots showing increased mixture separation with time:
%  these are obtained through the order parameter and each show one
%  eighth of the total simulation size, which is $1548\times 1764\times 1568$. }\label{ch14:fig:length_scale}
%\end{figure}  

\begin{figure}[!t]
\centering
\includegraphics[width=10cm]{Chapters/chapter14/figures/two_graphs_ud}
\caption[The weak (top) and strong (bottom) scaling of \textit{Ludwig}.]{The weak (top) and strong (bottom) scaling of \textit{Ludwig}.  Closed
  shapes denote results using the CPU version run on the Cray
  XE6 (using two 16-core AMD Interlagos CPUs per node), while open
  shapes denote results using the GPU version on the Cray XK6 (using a
  single NVIDIA X2090 GPU per node). Top: the benchmark time is shown
  where the problem size per node is constant. Bottom: performance is
  shown where, for each of the four problem sizes presented, the
  results are scaled by the lattice volume, and all results are
  normalized by the same arbitrary constant.  }
\label{ch14:fig:scaling}
\end{figure}


%The top of Figure \ref{ch14:fig:scaling} shows the benchmark time,
%where the problem size is increased with the number of nodes. This
%shows weak scaling: the time taken by a perfectly scaling code would
%remain constant. The figure plots the time against the number of {\it
%  nodes}: for the GPU version we utilize the one GPU per Cray XK6 node
%whilst for the CPU version we fully utilize two 16-core CPUs
%per Cray XE6 node, i.e. one GPU is compared with 32 CPU cores.  We use
%a single MPI task per Cray XK6 node to host the GPU, and 32 MPI tasks
%to fully utilize the CPUs in the Cray XE6 node.  Note that our CPU
%version is highly optimised: in particular it has been tuned for
%optimal utilisation of the SIMD vector units in each CPU core.

Figure \ref{ch14:fig:scaling} shows the results of both weak and
strong scaling tests (top and bottom panels, respectively). For
weak scaling, where the local sub-domain size is kept fixed
(here a relatively large 196$^3$ lattice), the time taken by
an ideal code would remain constant. It can be seen that while
scaling of the CPU version shows a pronounced upward slope, the
GPU version scales almost perfectly. The advantage of the GPU
version over the CPU version is the fact that the GPU version
has a factor of 32 fewer MPI tasks per node, so communications
require a significantly smaller number of larger data transfers.
The performance
advantage of the GPU version ranges from a factor of around 1.5 to
around 1.8. Careful study by other authors \cite{williams2011} have
found the absolute performance (in terms of floating point
operations per second, or floating point operations per Watt) to
be remarkably similar between architectures.

Perhaps of more interest is the strong scaling picture (lower
panel in Figure \ref{ch14:fig:scaling}) where the performance
as a function of the number of nodes is measured for fixed problem
size. We consider four different fixed problem sizes on both CPU
(up to 512 nodes shown) and GPU (up to 768 nodes). To allow comparison,
the results are scaled by total system size in each case. For strong
scaling, the disparity in the number of MPI tasks is clearly revealed
in the failure of the CPU version to provide any significant benefit
beyond a modest number of nodes as communication overheads dominate.
In contrast, the GPU version shows
reasonably robust scaling even for the smaller system sizes and good
scaling for the larger systems.

\section{Moving Solid Particles}\label{ch14:sec:particles}

\begin{figure}[t]
\centering
%\includegraphics[width=10cm]{Chapters/chapter14/figures/bbl}
\includegraphics[width=10cm]{Chapters/chapter14/figures/colloid_new}
\caption[A two-dimensional schematic picture of spherical particles on the lattice.]{
A two-dimensional schematic picture of spherical particles on the lattice.
Left: a particle is allowed
to move continuously across the lattice, and the position of the
surface defines fluid lattice sites (light blue) and solid lattice
sites (dark red). The discrete surface is defined by links
where propagation would intersect the surface (arrows). Note the
discrete shape of the two particles is different. Right: post-collision
distributions are reversed at the surface by the process of bounce-back
on links, which replaces the propagation.
%A 2D illustration of colloidal particles (grey circles)
%  interacting with the lattice, where each lattice point is
%  represented by a cross. Left: each particle spans multiple sites.
%  Right: the particle-facing distribution components are moved inside
%  the particle with the direction reversed, such that the ``bounce
%  back'' will be completed when the fluid propagates.
}
\label{ch14:fig:bbl}
\end{figure}


%\begin{figure}[t]
%\centering
%\includegraphics[width=6cm]{Chapters/chapter14/figures/particlemove}
%\caption{A 2D illustration of a colloidal particle moving across the
%    lattice. The old and new positions are represented by the open and
%    filled circles respectively.}
%\label{ch14:fig:particlemove}
%\end{figure}

% BEING NEW SOLID TEXT

The introduction of moving solid particles poses an additional hurdle
to efficient GPU implementation of an LB code such as \textit{Ludwig}.
In this section, we give a brief overview of the essential features
of the CPU implementation, and how the considerations raised in the
previous sections --- maximising parallelism and minimising both
host to device and GPU to GPU data movement --- shape the design decisions
for a GPU implementation. We restrict our discussion to a simple fluid
in this section; additional solid-fluid boundary conditions (e.g.,
wetting at a fluid-fluid-solid contact line) usually arise elsewhere
in the calculation and broadly independent of the hydrodynamic boundary
conditions which are described in what follows.

Moving solid particles (here, spheres) are defined by a centre position
which is allowed to move continuously across the space of the lattice,
and a fixed radius which is typically a few lattice spacings $\Delta r$.
The surface of the particle is defined by a series of \textit{links}
where a discrete velocity propagation $\mathbf{c}_i \Delta t$ would
intercept or cut the spherical shell (see Fig.~\ref{ch14:fig:bbl}).
Hydrodynamic boundary conditions are then implemented via the standard
approach of bounce-back on links \cite{ladd1994, nguyen2002}, where the
relevant post-collision distribution values are reversed at the
propagation stage with an appropriate correction to allow for the
solid body motion. The exchange of momentum at each link must then be
accumulated around the entire particle surface to provide the net
hydrodynamic force
and torque on the sphere. The particle motion can then be updated in a
molecular dynamics-like step.

For the CPU implementation, a number of additional MPI communications
are required: (1) to exchange the centre position, radius, etc of each
particle as it moves and (2), to allow the accumulation of the force
and torque for each particle (the links of which may be distributed
between up to 8 MPI tasks in three dimensions). Appropriate marshaling
of these data can provide an effective parallelisation for relatively
dense particle suspensions of up to  40\% solid volume fraction
\cite{stratford2008}. As a final consideration, fluid distributions
must be removed and replaced consistently as a given particle moves
across the lattice and changes its discrete shape.

With these features in mind, we can identify a number of competing
concerns which are relevant to a GPU implementation:
\begin{enumerate}
\item
Minimisation of host-device data transfer would argue for moving the
entire particle code to the GPU. However, the code in question involves
largely conditional logic (e.g., identifying cut surface links) and
irregular memory accesses (e.g., access to distribution elements around
a spherical particle). These operations seem poorly suited to effective
parallelisation on the GPU. As an additional complication, the sums
required over the particle surface would involve potentially tricky
and inefficient reductions in GPU memory.
\item
The alternative is to retain the relevant code on the CPU. While the
transfer of the entire distribution $f_i(\mathbf{r};t)$ between host
and device at each time step is unconscionable owing to PCIe bus bandwidth
considerations, the transfer of only relevant distribution information
to allow bounce-back on links is possible. At modest solid volumes
only a very small fraction of the distribution data is
involved in bounce-back (e.g., a 2\% solid volume fraction of particles
radius 2.3$\Delta r$ would involve approximately 1\% of the distribution
data). This option also has the advantage that no further host-device
data transfers are necessary to allow the MPI exchanges required for
particle information.
\end{enumerate}


We have implemented the second option as follows. For each sub-domain,
a list of boundary-cutting links is assembled on the CPU which includes
the identity of the relevant element of the distribution. This list,
together with the particle information required to compute the correct
bounce-back term, are transferred to the GPU. The updates to the relevant
elements of the distribution can then take place on the GPU. The
corresponding information to compute the update of the particle dynamics
is returned
to the CPU, where the reduction over the surface links is computed.
The change of particle shape may be dealt with in a similar manner:
the relatively small number of updates required at any one time step
(or however frequently the particle position is updated) can be
marshaled to the GPU as necessary. This preliminary implementation
is found to be effective on problems involving up to 4\%
solid volume fraction.

We note that the CPU version actually avoids the collision calculation
at solid lattice points by consulting a look-up table of solid/fluid
status. On the GPU, it is perhaps preferable to perform the collision
stage everywhere instead of moving the look-up table to the GPU and
introducing the associated logic.
Ultimately, the GPU might favour other boundary methods which treat solid and
fluid on a somewhat more equal basis, for example, the immersed boundary
method \cite{ch14:immersed1,ch14:immersed2,ch14:immersed-lb} or smoothed profile method
\cite{ch14:spm}.
However, the approach adopted here  allows us to exploit
the GPU for the intensive fluid simulation whilst maintaining the complex
code required for particles on the CPU. Overheads of CPU-GPU transfer are
minimised by transferring only those data relevant to the hydrodynamic
interaction implemented via bounce-back on links.

% END NEW SOLID TEXT

%It is of interest, in many situations, to include colloidal particles
%in the simulation **more from Kevin***. These are by represented by
%moving spheres that interact with the fluid through transfer of
%momenta: fluid coming into contact with a particle will bounce off it,
%giving the particle a kick in the process. As illustrated on the left
%of Figure \ref{ch14:fig:bbl} (which shows 2 dimensions only for
%clarity - in practice, the particles will be represented by spheres on
%a 3D lattice, and will be much larger relative to the lattice
%spacing), typically each particle spans multiple lattice sites, and
%therefore ``intercepts'' multiple inter-site links. Ludwig stores
%particle information using nested linked lists: a list of particles,
%each with a list of all the links intercepted by that particle (plus
%other quantities). The fluid-particle interaction proceeds using the
%so-called {\it bounce back on links} procedure during each timestep. As
%illustrated in the right of Figure \ref{ch14:fig:bbl}, for each of the
%intercepting links, those velocity components of the distribution that
%face inwards to each particle are moved from the site
%outside the particle to the site inside the particle, and
%reversed in direction. This is done before the propagation stage,
%which then propagates these same velocity components back out of the
%particles: the velocity bounces back from the particles. The
%momenta transfer from fluid to particles is also determined
%from the distribution values.

%A simplified schematic of the implementation on the CPU is as follows,
%noting that multiplicative coefficients are required:
%{\footnotesize
%\begin{verbatim}
%For each particle:
%  For each link:
%    Calculate coefficient
%    Bounce back fluid using coefficient
%    Update particle momentum
%\end{verbatim}
%}
%This process is relatively inexpensive on the CPU, since the total
%number of particles (and intercepted links) is always relatively
%small, but it does require access to distribution data which we want
%to keep resident on the GPU throughout the simulation. The overhead of
%transferring the full distribution to the CPU and back every timestep
%would be too high. But on the other hand, the full codebase required
%to manage particles is quite complex and comprehensive: completely
%porting it to the GPU, such that the particles effectively also remain
%resident on the GPU, would be a large effort and would cause a major
%diversion of CPU and GPU codebases. Further complications would arise
%when running on multiple GPUs due to the fact that particles can move
%between parallel subdomains. Our solution, which minimises overhead,
%is to keep the particles resident on CPU, but offload only their
%interaction with the fluid to the GPU, as we describe below.

%On the CPU, the code in the above listing can be restructured into
%three distinct stages as (introducing arrays to temporrily store
%indices and data):
%{\footnotesize
%\begin{verbatim}
%1) For each particle, for each link:
%     Store site and velocity indices associated with link
%     Calculate and store coefficient
%2) For each particle, for each link:
%     Update velocity data using stored values
%     Calculate and store particle momenta updates
%3) For each particle, for each link:
%     Update particle momentum using stored values
%\end{verbatim}
%}
%Stage 2 is only one that that accesses fluid data: this can therefore
%be moved to the GPU, while stages 1 and 3 remain on the CPU. Stage 2
%therefore becomes, for the GPU implementation
%{\footnotesize
%\begin{verbatim}
%Transfer coefficient and link index arrays to GPU
%Run CUDA kernel:
%  Update fluid data using coefficient and link index arrays
%  Calculate and store particle momenta updates from fluid data
%Transfer particle momenta updates array back to CPU
%\end{verbatim}
%}
%Since the total number of links involves is relatively small, these
%partial data transfers have minimal overhead.  A complication arises
%due to the fact that the particles move through the lattice, albeit
%relatively slowly, and therefore the list of links that they intercept
%occasionally changes.  When a
%particle moves off a lattice site, the fluid on that site must be
%reconstructed. Similarly, the fluid on the newly obstructed site must
%be removed. For the same reasons given above, we want to avoid moving
%all the code dealing with this complication to the GPU, whilst
%minimising overheads. Since, on any one timestep, the number of
%lattice sites affected is relatively low, we implement functionality
%to perform partial data transfers of the distribution. Only those
%sites affected are packed into a buffer on the GPU, transferred to the
%CPU and unpacked into the CPU version of the distribution. Then, the
%existing code can be used to update those affected sites, before the
%reverse partial data transfer is performed to update the GPU version
%of data.

It is perhaps interesting at this point to make some more general
observations on the software engineering challenge presented when
extending an existing CPU code to the GPU. The already complex
task of maintaining the code in a portable fashion while also
maintaining performance is currently formidable. To help this process,
we have followed a number of basic principles. First,
in order to port to the GPU in an incremental fashion,
we have tried to maintain the modular structure of the CPU where
possible. For each data structure, such as the distribution, a separate
analogue is maintained in both the CPU and GPU memory spaces. However,
the GPU copy does not include the complete CPU structure: in
particular, non-intrinsic datatypes such as MPI datatypes are not
required on the GPU. Functions to marshal data between CPU and GPU
are provided for each data structure, abstracting the underlying
CUDA implementation. (This reasonably lightweight abstraction layer
could also allow an OpenCL implementation to be developed.) 
This makes it easy to switch between the CPU and GPU for different
components in the code, which is useful in development
and testing. GPU functionality can be added incrementally while
retaining a code that runs correctly (albeit slowly due to data
transfer overheads).  Once all relevant components are moved to
the GPU, it becomes possible to remove such data transfers and
keep the entire problem resident on the device.

%To ensure that \textit{Ludwig} remains suitable for CPU-based systems,
%we augment rather than modify the original code. All new GPU-related
%functionality are implemented in an additive manner, mirroring the
%modular structure of the original code. The new GPU modules have
%interfaces similar to the original CPU functions, and the ability to
%switch between versions is achieved at compile-time via C
%pre-processing directives.  All GPU compute kernels are implemented using
%CUDA.  Wrapper routines are developed to specify the
%decomposition, invoke the CUDA kernels and perform the necessary data
%management operations: the latter are facilitated through newly
%developed modules for creation and destruction of device memory
%structures, the copying of data between host and device, and where
%necessary the packing and buffering of data.




\section{Summary}
\label{ch14:sec:summary}

We have described the steps take to implement, on both single and
multiple GPUs, the \textit{Ludwig} code which was originally
designed for complex fluid problems on a conventional CPU
architecture. We have added the necessary functionality using
NVIDIA CUDA and discussed the important changes to the main
data structures. By retaining domain decomposition and message
passing via MPI, we have demonstrated it is possible to scale
complex fluid problems to large numbers of GPUs in parallel.
By following the key design criteria of maximising parallelism
and minimising host-device data transfers, we have confirmed
the mixed MPI-GPU approach is viable for scientific applications.
For the more intricate problem of moving solid particles, we find
it is possible to retain the more serial elements related to
particle link operations on the CPU, while offloading only the
parallel lattice-based operations involving the LB distribution
to the GPU. Again, this minimises host-device movement of data.

From the software engineering viewpoint, some duplication of code
to allow efficient implementation on both host and device is
currently required. This issue might be addressed by approaches
such as automatic kernel generation, but may also be addressed
naturally in time as GPU and CPU hardware converge. Portable
abstractions and APIs, perhaps based on approaches such as OpenCL,
will also facilitate the development and maintenance of portable
codes which also exhibit portable performance (perhaps in conjunction
with automatic tuning approaches).

So, while the challenges in designing portable and efficient scientific
applications remain very real, this work provides some hope the large
clusters of GPU machines can be used effectively for a wide range of
complex fluid problems.


%The \textit{Ludwig} LB application, which specializes
%in complex fluid problems, has undergone the developments required to
%use large-scale parallel GPU accelerated supercomputers effectively.
%The new functionality augments the original code, so that
%\textit{Ludwig} can run on either CPU-based or GPU-accelerated systems. 
%Using the NVIDIA CUDA programming language, all the computational
%kernels were offloaded to the GPU. This included not only those
%responsible for the majority of the compute time, but any that
%accessed the relevant data structures. This allows data to be kept
%resident on the GPU and avoids expensive data transfers. We described
%steps taken to optimise the performance on the GPU architecture by
%reducing off-chip memory accesses and restructuring the data layout
%in memory to ensure the available memory bandwidth was exploited fully.

%A new halo-exchange communication phase for the code was developed to
%allow efficient parallel scaling to many GPUs. The CPU version
%relies on MPI datatype functionality for CPU to CPU communication; we
%have explicitly written buffer packing/unpacking and transfer
%operations to for GPU to GPU data exchange, staged through the host
%CPUs. This includes the filtering of velocity components to only
%transfer those required in a specific direction, hence reducing the
%overall volume of data transferred. We used CUDA stream functionality
%to overlap separate stages within the communication phase, and reduce
%the overall communication time. The effect was measured to be more
%profound for smaller local system sizes, i.e. it is especially useful
%in aiding strong scaling.

%We described work to enable colloidal particles in the simulation with
%minimal overhead. We implemented these in such a way that we avoid a
%major diversion of the CPU and GPU codebases, whilst minimising data
%transfers between the CPU and GPU at each timestep. We keep the
%majority of the (relatively inexpensive) particle related code on the
%CPU, while offloading only those parts responsible for the interaction
%with the fluid to the GPU (where the fluid data resides).

%Our resulting package is able to exploit the largest of GPU
%accelerated architectures to address a wide range of complex problems.



% An example of a floating figure using the graphicx package.
% Note that \label must occur AFTER (or within) \caption.
% For figures, \caption should occur after the \includegraphics.
% Note that IEEEtran v1.7 and later has special internal code that
% is designed to preserve the operation of \label within \caption
% even when the captionsoff option is in effect. However, because
% of issues like this, it may be the safest practice to put all your
% \label just after \caption rather than within \caption{}.
%
% Reminder: the "draftcls" or "draftclsnofoot", not "draft", class
% option should be used if it is desired that the figures are to be
% displayed while in draft mode.
%
%\begin{figure}[!t]
%\centering
%\includegraphics[width=2.5in]{myfigure}
% where an .eps filename suffix will be assumed under latex, 
% and a .pdf suffix will be assumed for pdflatex; or what has been declared
% via \DeclareGraphicsExtensions.
%\caption{Simulation Results}
%\label{fig_sim}
%\end{figure}

% Note that IEEE typically puts floats only at the top, even when this
% results in a large percentage of a column being occupied by floats.


% An example of a double column floating figure using two subfigures.
% (The subfig.sty package must be loaded for this to work.)
% The subfigure \label commands are set within each subfloat command, the
% \label for the overall figure must come after \caption.
% \hfil must be used as a separator to get equal spacing.
% The subfigure.sty package works much the same way, except \subfigure is
% used instead of \subfloat.
%
%\begin{figure*}[!t]
%\centerline{\subfloat[Case I]\includegraphics[width=2.5in]{subfigcase1}%
%\label{fig_first_case}}
%\hfil
%\subfloat[Case II]{\includegraphics[width=2.5in]{subfigcase2}%
%\label{fig_second_case}}}
%\caption{Simulation results}
%\label{fig_sim}
%\end{figure*}
%
% Note that often IEEE papers with subfigures do not employ subfigure
% captions (using the optional argument to \subfloat), but instead will
% reference/describe all of them (a), (b), etc., within the main caption.


% An example of a floating table. Note that, for IEEE style tables, the 
% \caption command should come BEFORE the table. Table text will default to
% \footnotesize as IEEE normally uses this smaller font for tables.
% The \label must come after \caption as always.
%
%\begin{table}[!t]
%% increase table row spacing, adjust to taste
%\renewcommand{\arraystretch}{1.3}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
%\caption{An Example of a Table}
%\label{table_example}
%\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
%\begin{tabular}{|c||c|}
%\hline
%One & Two\\
%\hline
%Three & Four\\
%\hline
%\end{tabular}
%\end{table}


% Note that IEEE does not put floats in the very first column - or typically
% anywhere on the first page for that matter. Also, in-text middle ("here")
% positioning is not used. Most IEEE journals/conferences use top floats
% exclusively. Note that, LaTeX2e, unlike IEEE journals/conferences, places
% footnotes above bottom floats. This can be corrected via the \fnbelowfloat
% command of the stfloats package.


% conference papers do not normally have an appendix


% use section* for acknowledgement
\section*{Acknowledgments}
AG was supported by the CRESTA project, which has received
funding from the European Community's Seventh Framework Programme
(ICT-2011.9.13) under Grant Agreement no. 287703. KS was supported
by UK EPSRC under grant EV/J007404/1.


% trigger a \newpage just before the given reference
% number - used to balance the columns on the last page
% adjust value as needed - may need to be readjusted if
% the document is modified later
%\IEEEtriggeratref{8}
% The "triggered" command can be changed if desired:
%\IEEEtriggercmd{\enlargethispage{-5in}}

% references section

% can use a bibliography generated by BibTeX as a .bbl file
% BibTeX documentation can be easily obtained at:
% http://www.ctan.org/tex-archive/biblio/bibtex/contrib/doc/
% The IEEEtran BibTeX style support page is at:
% http://www.michaelshell.org/tex/ieeetran/bibtex/
%\bibliographystyle{IEEEtran}
% argument is your BibTeX string definitions and bibliography database(s)
%\bibliography{IEEEabrv,../bib/paper}
%
% <OR> manually copy in the resultant .bbl file
% set second argument of \begin to the number of references
% (used to reserve space for the reference number labels box)

\putbib[Chapters/chapter14/biblio14]
%\begin{thebibliography}{1}

%\bibitem{IEEEhowto:kopka}
%H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
%  0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.



%% \bibitem{succi-book}
%% S. Succi, \textit{The lattice Boltzmann equation and beyond},
%% Oxford University Press, Oxford, 2001.

%\bibitem{mpi-standard}
%Message Passing Interface Forum, http://www.mpi-forum.org 

%% \bibitem{desplat}
%% Desplat, J.-C., I. Pagonabarraga, and P. Bladon,
%% \textit{LUDWIG: A parallel lattice-Boltzmann code for complex fluids}.
%% Comput. Phys. Comms., \textbf{134}, 273, 2001.

%% \bibitem{aidun2010}
%% C.K. Aidun and J.R. Clausen,
%% \textit{Lattice Boltzmann method for complex flows},
%% Ann. Rev. Fluid Mech., \textbf{42} 439--472 (2010).

%\bibitem{bray1994}
%A.J. Bray,
%\textit{Theory of phase-ordering kinetics},
%Adv. Phys., \textbf{43} 357--459 (1994).

%\bibitem{kendon2001}
%V.M. Kendon, M.E. Cates, I. Pangonabarraga, J.-C. Desplat, and P. Bladon,
%\textit{Inertial effects in the three-dimensional spinodal decomposition of a
%symmetric binary fluid mixture; a lattice Boltzmann study},
%J. Fluid Mech., \textbf{440}, 147--203 (2001).

%\bibitem{swift1996}
%M.R. Swift, E. Orlandini, W.R. Osborn, and J.M. Yeomans,
%\textit{Lattice Boltzmann simulations of liquid-gas and binary
%fluid systems},
%Phys. Rev. E, \textbf{54}, 5041--5052 (1996).

%% \bibitem{stratford2008}
%% K. Stratford and I. Pagonabarraga,
%% Parallel domain decomposition for lattice Boltzmann with moving particles,
%% \textit{Comput. Math. with Applications} \textbf{55}, 1585 (2008).

%\bibitem{xe6}
%Cray XE6 Product Brochure, available from
%http://www.cray.com/Products/XE/Resources.aspx (2010)

%\bibitem{hector} HECToR home page, http://www.hector.ac.uk

%\bibitem{xk6}
%Cray XK6 Product Brochure, available from
%http://www.cray.com/Products/XK6/XK6.aspx (2011)


%% \bibitem{wei2004}
%% X. Wei, W. Li, K. M\"uller, and A.E. Kaufman,
%% \textit{The lattice Boltzmann method for simulating gaseous phenomena},
%% IEEE Transactions on Visualization and Computer Graphics,
%% \textbf{10}, 164--176 (2004).
%% Apparently first LB via GPU; a serious contribution using single fluid
%%d3q19 single relaxation time.

%% \bibitem{zhu2006}
%% H. Zhu, X. Liu, Y. Liu, and E. Wu,
%% \textit{Simulation of miscible binary mixtures based on lattice Boltzmann method},
%% Comp. Anim. Virtual Worlds, \textbf{17}, 403--410 (2006).
%% Single relaxation (MRT mentioned) time d3q19 apparently sound although
%%not a lot of detail. Pre-cuda so graphics code.

%% \bibitem{zhao2007}
%% Y. Zhao,
%% \textit{Lattice Boltzmann based PDE solver on the GPU},
%% Visual Comput., doi 10.1007/s00371-0070191-y (2007).

%% \bibitem{toelke2010}
%% J. T\"olke,
%% Implementation of a lattice Boltzmann kernel using the compute unified
%% device architecture developed by nVIDIA,
%% Comput. Visual Sci. 13 29--39 (2010).

%% \bibitem{fan2004}
%% Z. Fan, F. Qiu, A. Kaufman, and S. Yoakum-Stover,
%% \textit{GPU cluster for high performance computing},
%% Proceedings of ACM/IEEE Supercomputing Conference, pp. 47--59,
%% IEEE Computer Society Press, Pittsburgh, PA (2004).

%% \bibitem{myre2011}
%% J. Myre, S.D.C. Walsh, D. Lilja, and M.O. Saar,
%% \textit{Performance analysis of single-phase, multiphase, and multicomponent
%% lattice Boltzmann fluid flow simulations on GPU clusters},
%% Concurrency Computat.: Pract. Exper., \textbf{23}, 332--350 (2011).

%% \bibitem{obrecht2011}
%% C. Obrecht, F. Kuznik, B. Tourancheau, and J.-J. Roux,
%% \textit{Multi-GPU implementation of the lattice Boltzmann method},
%% Comput. Math. with Applications,
%% doi:10.1016/j.camwa.2011.02.020 (2011).

%% \bibitem{bernaschi2010}
%% M. Bernaschi, M. Fatica, S. Melchionna, S. Succi, and E. Kaxiras,
%% \textit{A flexible high-performance lattice Boltzmann GPU code for the
%% simulations of fluid flow in complex geometries},
%% Concurrency Computat.: Pract. Exper., \textbf{22}, 1--14 (2010).

%% \bibitem{xian2011}
%% W. Xian and A. Takayuki,
%% \textit{Multi-GPU performance of incompressible flow computation by
%% lattice Boltzmann method on GPU cluster},
%% Parallel Comput., doi:10.1016/j.parco.2011.02.007 (2011). 

%% \bibitem{feichtinger2011}
%% C. Feichtinger, J. Habich, H. K\"ostler, G. Hager, U. R\"ude, and
%% G. Wellein,
%% A flexible patch-based lattice Boltzmann parallelization approach
%% for heterogeneous GPU-CPU clusters,
%% \textit{Parallel Computing} \textbf{37} 536--549 (2011).

%% \bibitem{wellein2006}
%% G. Wellein, T. Zeiser, G Hager, and S. Donath,
%% On the single processor performance of simple lattice Boltzmann kernels,
%% \textit{Computers and Fluids}, \textbf{35}, 910--919 (2006).

%% \bibitem{pohl2003}
%% T. Pohl, M. Kowarschik, J. Wilke, K. Igelberger, and U. R\"ude,
%% Optimization and profiling of the cache performance of parallel
%% lattice Boltzmann code,
%% \textit{Parallel Process Lett.} \textit{13} 549--560 (2003).

%% \bibitem{mattila2007}
%% K. Mattila, J. Hyv\"aluoma, T. Rossi, M. Aspn\"as and J. Westerholm,
%% An efficient swap algorithm for the lattice Boltzmann method,
%% \textit{Comput. Phys. Comms.} \textit{176} 200-210 (2007).

%% \bibitem{wittmann2012}
%% M. Wittmann, T. Zeiser, G. Hager, and G. Wellein,
%% Comparison of different propagation steps for lattice Boltzmann methods,
%% \textit{Comput. Math with Appl.} doi:10.1016/j.camwa.2012.05.002 (2012).

%% \bibitem{walshsaar2012}
%% S.D.C. Walsh and M.O. Saar,
%% Developing extensible lattice Boltzmann simulators for general-purpose
%% graphics-processing units,
%% \textit{Comm. Comput. Phys.}, \textbf{13} 867--879 (2013).


%% \bibitem{williams2011}
%% S. Williams, L. Oliker, J. Carter, and J Shalf,
%% Extracting ultra-scale lattice Boltzmann performance via
%% hierarchical and distributed auto-tuning,
%% \textit{Proc. SC2011}.


%% \bibitem{ch14:stratford-jsp2005}
%% K. Stratford, R. Adhikari, I. Pagonabarraga, and J.-C. Desplat,
%% \textit{Lattice Boltzmann for Binary Fluids with Suspended Colloids},
%% J. Stat. Phys. \textbf{121}, 163 (2005).

%% \bibitem{ladd1994}
%% A.J.C. Ladd,
%% Numerical simulations of particle suspensions via a discretized
%% Boltzmann equation. Part 1. Theoretical foundation,
%% \textit{J. Fluid Mech.} \textbf{271} 285--309 (1994);
%% Part II. Numerical results,
%% \textit{ibid.} \textbf{271} 311--339 (1994).
 

%% \bibitem{nguyen2002}
%% N.-Q. Nguyen and A.J.C. Ladd,
%% Lubrication corrections for lattice Boltzmann simulations of particle
%% suspensions,
%% \textit{Phys. Rev. E} \textbf{66} 046708 (2002).

%% \bibitem{ch14:immersed}
%% C.S. Peskin,
%% Flow patterns around heart valves; a numerical method,
%% \textit{J. Comp. Phys.}, \textbf{10}, 252--271 (1972);
%% C.S. Peskin,
%% The immersed boundary method,
%% \textit{Acta Nummerica} \textbf{11} 479--517 (2002).

%% \bibitem{ch14:immersed-lb}
%% Z.-G. Feng and E.E. Michaelides,
%% The immersed boundary-lattice Boltzmann method for solving
%% fluid-particles interaction problem,
%% \textit{J. Comp. Phys.}, \textbf{195} 602--628 (2004).

%% \bibitem{ch14:spm}
%% Y. Nakayama and R. Yammamoto,
%% Simulation method to resolve hydrodynamic interactions in colloidal
%% dispersions,
%% \textit{Phys. Rev. E}, \textbf{71} 036707 (2005).


%\end{thebibliography}







% LocalWords:  CRESTA EPSRC Succi Desplat Pagonabarraga