Multi-core strategies for particle methods
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Williams, John R., Holmes, David, and Tilke, Peter (2010) Multicore strategies for particle methods. ISBN 978-0-9551179-8-5.
Proceedings of the Fifth International Conference on Discrete
Element Methods In: Fifth International Conference on Discrete
Element Methods, 25-26 August 2010, London, UK.
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DISCRETE ELEMENT METHODS
Multi-core Strategies For Particle Methods
John R. Williams, David Holmes and Peter Tilke*
Massachusetts Institute of Technology, *Schlumberger-Doll Research Laboratory
This paper discusses the implementation of particle based numerical methods
on multi-core machines. In contrast to cluster computing, where memory is distributed across machines, multi-core machine can share memory across all
cores. Here general strategies are developed for spatial management of particles
and sub-domains that optimize computation on shared memory machines. In
particular, we extend cell hashing so that cells bundle particles into orthogonal
tasks that can be safely distributed across cores avoiding the use of “memory
locks” while still protecting against race conditions. Adjusting task size provides for optimal load balancing and maximizing cache hits. Additionally, the
way in which tasks are mapped to execution threads has a significant influence
on the memory footprint and it is shown that minimizing memory usage is one
of the most important factors in achieving execution speed and performance on
multi-core. A novel algorithm called H-Dispatch is used to pipeline tasks to
processing cores. The performance is demonstrated in speed-up and efficiency
tests on a smooth particle hydrodynamics (SPH) flow simulator. An efficiency
of over 90% is achieved on a 24-core machine.
INTRODUCTION
Recent trends in computational physics suggest a rapidly growing interest in particle based numerical techniques. While particle based methods allow robust handling of mass advection, this comes at the computational cost of managing the
spatial interactions of particles. For problems involving millions of particles, it is
necessary to develop efficient strategies for parallel implementation. A variety of
authors have reported parallel particle codes on clusters of machines (see for example Nelson et al [1], Morris et al [2], Sbalzarini et al [3], Walther and Sbalzarini [4], Ferrari et al [5]). The key difference between parallel computing on clusters and on multi-core is that memory can be shared across all cores on today’s
multi-core machines. However, there are well known challenges of thread safety
and limited memory bandwidth. A variety of approaches to programming on
multi-core have been proposed to-date. Concurrency tools from traditional cluster
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computing, like MPI [6] and OpenMP [7] have been used to achieve fast take-up
of the new technology. If one runs an MPI process on each core, the operating
system guarantees memory isolation across processes. Communication of information across process boundaries requires MPI messages to be sent, which is relatively slow compared with direct memory sharing.
In response to this, Chrysanthakopoulos and co-workers [7, 8] have implemented multi-core concurrency libraries using Port based abstractions. These
mimic the functionality of a message passing library like MPI, but use shared
memory as the medium for data exchange, rather than exchanging serialized
packets over TCP/IP as is the case for MPI. Such an approach provides extensibility in program structure, while still capitalizing on the speed advantages of shared
memory. In an earlier paper, Holmes et al. [9] have shown that a programming
model developed using such port based techniques provides significant performance advantages over MPI and OpenMP. In this paper, we apply the programming
model proposed in [9] to the parallelization of particle based numerical methods
on multi-core.
PARTICLE BASED NUMERICAL METHODS
Mesh based numerical methods have been the cornerstone of computational physics for decades. Here, integration points are positioned according to some topological connectivity or mesh to ensure compatibility of the numerical interpolation. Examples of Eulerian mesh based methods include finite difference (FD)
and the lattice Boltzmann method (LBM), while Lagrangian examples include the
finite element method (FEM). While powerful for a wide range of problems, mesh
limitations for problems involving large deformation and complex material interfaces has led to significant developments in meshless and particle based methodologies. For such methods, integration points are positioned freely in space, capable of advecting with material in a Lagrangian sense. For methods like molecular
dynamics (MD) and the discrete element method (DEM), such points represent
literal particles (atoms and molecules for MD and discrete grains for DEM), while
for methods like dissipative particle dynamics (DPD) [10], smooth particle hydrodynamics (SPH) [11], wavelets [12], and the reproducing kernel particle method
(RKPM) [13], the particle analogy is largely figurative. For such methods, parti-
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DISCRETE ELEMENT METHODS
cles provide positions at which to enforce a partition of unity. By partitioning
unity across the particles, continuity can be imposed without a defined mesh, allowing such methods to represent a continuum in a generalized way. In Eulerian
mesh based methods, such as FD and LBM, continuity is inherently provided by
the static mesh, while for Lagrangian mesh based approaches like FEM, continuity is enforced through the use of element shape functions. The partition of unity
imposed on mesh-free particle methods can be seen to be a generalization of shape
functions for arbitrary integration point arrangements. From Li and Liu [14]
...meshfree methods are the natural extension of finite element methods, they provide a perfect habitat for a more general and more appealing computational
paradigm - the partition of unity.
The advantage of partition of unity methods is that any expression related to a
field quantity can be imposed on the continuum. Where, for a bounded domain in Eucledean space, a set of nonnegative compactly supported functions, sums to
unity. Correspondingly, the value of some field function can be determined from
its value at all other points via
f ( x )   ak  ( x  k )
k 0 ,1, 2... n
The function f (x) can be related to any physical field expression; hydrodynamic,
mechanical, electrical, chemical, magnetic etc. Such versatility is a key advantage
of particle methods.
Strategies for Programming on Multi-Core By eliminating the need for ghost
regions and high volume to surface ratio sub-domains, shared memory enables a
near arbitrary selection of spatial divisions by which to distribute a numerical
task. Correspondingly, it becomes convenient to repurpose a spatial division strategy already in use in one form or other in most particle based numerical codes.
Spatial hashing techniques have been used widely to achieve algorithmically efficient interaction detection in particle codes (see for example DEM [15, 16]).
SPATIAL HASHING IN PARTICLE METHODS
The central idea behind spatial hashing is to overlay some regular spatial structure
over the randomly positioned particles e.g. an array of equally sized cells. We can
then perform spatial reasoning on the cells rather than on the particles themselves.
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DISCRETE ELEMENT METHODS
Spatial hashing assigns particles to cells or ‘bins’ based on a hash of particle coordinates. The numerical expense of such an algorithm is O (N) where N is particle number [15,16]. A variety of programmatic implementations of hash cell
methods have been used (for example linked list). In this work a dictionary hash
table is used where a generic list of particles is stored for each cell and indexed
based on an integer unique key to that cell, i.e. key  ( k  ny  j )  nx  i where
nx and ny are the total number of cells in the x and y dimensions and i, j and k are
the integer cell coordinates in the x, y and z dimensions.
Managing Thread Safety The cells also provide a means for defining task packages to the cores. By assigning a number of cells to each core we assure “task orthogonality” in that each core is operating on different set of particles. Traditional
software applications for shared memory parallel architectures have utilized locks
to avoid thread contention. We note that a core may “read” the memory of particles belonging to surrounding cells but may not update them. We execute a single
loop in which global memory for both previous and current field values ( vtn and
vt n1 ) is stored for each particle. Gradient terms can then be calculated as functions of values in previous memory, while updates are written to the current value
memory, in the same loop. In parallel, minimizing the frequency of so called synchronization points has advantages for performance and we utilize this “rolling
memory algorithm”. This allows the previous and updated terms to be maintained
without needing to replace the former with the latter at the end of each step and
thus, ensuring a single synchronization per step.
Counter Intuitive Design In a typical SPH simulator, two operations must be
done on particles in each cell per step, separated by a synchronization, the first to
determine the particle number density of each particle, and the second to perform
the field variable updates. Interacting particles must be known for each of these
two stages. Using a standard SPH formulation, the performance of two structure
variations are compared in this case study: A. Interacting particles are determined
in the first stage for all cells and stored in a global list for use in the second, and
B. Interacting particles are determined as needed in each stage for each cell, i.e.
twice per cell per time step. Recalculation of interacting particles means they need
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DISCRETE ELEMENT METHODS
only be kept in a local thread list that is overwritten with each newly dispatched
cell. While differences in execution memory are to be expected of the two code
versions, the differences in execution time are more surprising. For low core
counts (< 10 cores), as would be expected, the single search variant (A) solves
more quickly than the double (B) due to less computations. After this point, however, the double search (B) is shown to provide marked improvements in speed
over (A) (up to 50%). This can be attributed to better cache blocking of the second
approach and the significantly smaller amount of data experiencing latency when
being loaded from RAM to cache. The fact that such performance gains only
manifest when more than 10 cores are used, suggests that for less than 10 cores,
RAM pipeline bandwidth is sufficient to handle a global interaction list.
Fig 1. (a) memory vs. Number of Particles (b) Speed-up vs. Number of Cores
The MIT SPH simulator (see http://geonumerics.mit.edu ) has been validated for
single and multi-phase flow. Figure 2 shows results for two phase RayleighTaylor instability [17].
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Fig 2. Rayleigh-Taylor instability example showing extremely complex fluid
phase interface geometries; clear liquid is water, green liquid is oil. [9].
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