Nodal discontinuous Galerkin methods on graphics processors

Abstract

a b s t r a c tDiscontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of accuracy without compromising simulation stability. Lately, another property of DG has been growing in importance: The majority of a DG operator is applied in an element-local way, with weak penalty-based element-to-element coupling.The resulting locality in memory access is one of the factors that enables DG to run on off-the-shelf, massively parallel graphics processors (GPUs). In addition, DG's high-order nature lets it require fewer data points per represented wavelength and hence fewer memory accesses, in exchange for higher arithmetic intensity. Both of these factors work significantly in favor of a GPU implementation of DG.Using a single US$400 Nvidia GTX 280 GPU, we accelerate a solver for Maxwell's equations on a general 3D unstructured grid by a factor of around 50 relative to a serial computation on a current-generation CPU. In many cases, our algorithms exhibit full use of the device's available memory bandwidth. Example computations achieve and surpass 200 gigaflops/s of net application-level floating point work.In this article, we describe and derive the techniques used to reach this level of performance. In addition, we present comprehensive data on the accuracy and runtime behavior of the method.

Accelerated finite element elastodynamic simulations using the GPU

“…This is related to the fact that the algorithm is memory bandwidth limited rather than compute limited. It is well known that the use of linear finite elements with explicit time steps results in relatively few calculations being performed for a given amount of data being loaded from the memory; this is why it has been proposed to use higher order elements on GPUs [35] to increase the number of calculations per node at each time step. In the case of bandwidth limited algorithms such as this, since the amount of data which must be loaded from memory doubles for double precision, the run time will be expected to approximately double.…”

“…Two partitioning schemes are considered in this paper. The first is the simple 'greedy partitioner'; the algorithm follows that of [35]. The second is a more efficient partitioning scheme -the 'aligned partitioner' -which has been developed to subdivide the mesh into neat aligned blocks.…”

“…The resulting 125-node elements fitted well within a block of 128 threads, which could then be efficiently arranged in memory; while this is an elegant solution it is unlikely to be practical for the general case, particularly when considering lower order problems. Klöckner et al [35] recognised the problem for the more general case, where many elements must be allocated to each block; a partitioning scheme was therefore required to divide the mesh into blocks in an efficient manner. It was recognised that general purpose partitioners such as those developed for parallel computing [36] were rarely suitable, since the specific nature of the hardware configuration requires particular limits on size which are challenging to implement.…”

An explicit dynamics GPU structural solver for thin shell finite elements

“…A recent review of applications in FEM (Finite Element Method)-based structural mechanics can be found in [1], but GPUs are being used also in a variety of other contexts. In [2], the authors apply high-order Discontinuos Galerkin (DG) method to the solution of Maxwell's equa-tions. In DG methods most operators are defined locally at element level.…”

“…Hybrid implementations of multiscale finite element approaches, whereby constitutive material computations at Gauss point level are carried out on the GPU, are discussed in [13] and [14] In most of the above-mentioned applications, results are limited to single-precision arithmetic (e.g. [2,9,14,10]). It is important to recall, as suggested in [8], that, concerning NVIDIA GPUs, in all the implementations before CUDA compute capability 1.3, only single-precision floating point operations are supported directly by the hardware.…”