Octree-based, GPU implementation of a continuous cellular automaton for the simulation of complex, evolving surfaces

Abstract

Presently, dynamic surface-based models are required to contain increasingly larger numbers of points and to propagate them over longer time periods. For large numbers of surface points, the octree data structure can be used as a balance between low memory occupation and relatively rapid access to the stored data. For evolution rules that depend on neighborhood states, extended simulation periods can be obtained by using simplified atomistic propagation models, such as the Cellular Automata (CA). This method, however, has an intrinsic parallel updating nature and the corresponding simulations are highly inefficient when performed on classical Central Processing Units (CPUs), which are designed for the sequential execution of tasks. In this paper, a series of guidelines is presented for the efficient adaptation of octree-based, CA simulations of complex, evolving surfaces into massively parallel computing hardware. A Graphics Processing Unit (GPU) is used as a cost-efficient example of the parallel architectures. For the actual simulations, we consider the surface propagation during anisotropic wet chemical etching of silicon as a computationally challenging process with a wide-spread use in microengineering applications. A continuous CA model that is intrinsically parallel in nature is used for the time evolution. Our study strongly indicates that parallel computations of dynamically evolving surfaces simulated using CA methods are significantly benefited by the incorporation of octrees as support data structures, substantially decreasing the overall computational time and memory usage.

An investigation of the efficient implementation of cellular automata on multi-core CPU and GPU hardware

“…The optimised use of further GPGPU data structures are used to minimise the amount of traffic between the CPU and GPGPU, which is known to be a bottleneck. However, Ferrando et al [24] are more keenly interested in carrying out the high resolution of simulation in feasible amounts of time, and so do not directly claim that this approach provides speed ups because their system works as a co-operation of the CPU and GPGPU. In experiments conducted later in this paper it is found that with the GPGPU as a co-processor, a certain amount of instruction passing from the CPU is necessary.…”

“…There are circumstances where a sparse implementation has been implemented on a GPGPU, for example Ferrando et al (2011) [24] have employed an Oct-tree representation which subdivides a 3 dimensional cube of space into 8 smaller cubes at each branch of the tree. Although this does mean that the tree structure must be stored and manipulated using the CPU, a lot of processing can still be carried out on the GPGPU.…”

“…In addition, these key CA parameters cover the majority of variations in CA that might be implemented to simulate a variety of natural systems. Through the intensive study of the multi-core/many-core speed-up available for a  Engineering (wet chemical etching [24], designing hardware(FPGA) to run CA, communication [21])  Mathematics [39]  Hydroinfomatics (fluid dynamics [8] [4] [33], sewer optimisation [40], pluvial flooding modelling [41])  Economics (stock markets) [21] This variety of applications demonstrates the wide applicability of CA models and in many cases illustrates that the discretisation of time and space for use with a CA model is able to provide results of acceptable accuracy with greater efficiency than traditional models. Although the application level details for these CA will be different, they each implement the core principles of CA which is that time and space are discrete, cells will be updated in parallel and interactions between cells will be local.…”

Evolutionary continuous cellular automaton for the simulation of wet etching of quartz

“…Crystal surfaces of any orientation can be created by replicating the orthorhombic unit cell along the three axes. The use of the unit cell in combination with an octree data structure enables a significant reduction in memory use and a dramatic increase in the simulation speed for these models [19].…”

“…To make this large number of simulations feasible, an efficient CCA simulator is needed. We use a fast, GPU-based implementation [19], whereby each hemisphere simulation takes about 3 seconds. The initial hemisphere surface contains 472298 surface atoms and ∼ 1.4 · 10 6 atoms are removed per simulation.…”