GPU accelerated spectral finite elements on all-hex meshes

Remacle, Jean‐François; Gandham, R.; Warburton, Tim

doi:10.1016/j.jcp.2016.08.005

Cited by 32 publications

(29 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Dealing with small-scale heterogeneities in seismic wave simulation is a difficult task because it usually involves enormous computation costs. To handle them, Graphics Processing Unit (GPU) and/or High Performance Computing (HPC) implementations of well-established numerical techniques such as the Spectral Element method (SEM), the Discontinuous Galerkin method (DGM) and the Finite Difference method (FDM), have been proposed (Komatitsch et al 2010;Peter et al 2011;Weiss & Shragge 2013;Gokhberg & Fichtner 2016;Remacle et al 2016;Rietmann et al 2017). Furthermore, the DGM allows local time-stepping and p-adaptivity (e.g.…”

Section: Discussion a N D C O N C L U S I O N Smentioning

confidence: 99%

Non-periodic homogenization of 3-D elastic media for the seismic wave equation

Cupillard

Capdeville

2018

Geophysical Journal International

View full text Add to dashboard Cite

Because seismic waves have a limited frequency spectrum, the velocity structure of the Earth that can be extracted from seismic records has a limited resolution. As a consequence, one obtains smooth images from waveform inversion, although the Earth holds discontinuities and small scales of various natures. Within the last decade, the non-periodic homogenization method shed light on how seismic waves interact with small geological heterogeneities and 'see' upscaled properties. This theory enables us to compute long-wave equivalent density and elastic coefficients of any media, with no constraint on the size, the shape and the contrast of the heterogeneities. In particular, the homogenization leads to the apparent, structure-induced anisotropy. In this paper, we implement this method in 3-D and show 3-D tests for the very first time. The non-periodic homogenization relies on an asymptotic expansion of the displacement and the stress involved in the elastic wave equation. Limiting ourselves to the order 0, we show that the practical computation of an upscaled elastic tensor basically requires (i) to solve an elastostatic problem and (ii) to low-pass filter the strain and the stress associated with the obtained solution. The elastostatic problem consists in finding the displacements due to local unit strains acting in all directions within the medium to upscale. This is solved using a parallel, highly optimized finite-element code. As for the filtering, we rely on the finiteelement quadrature to perform the convolution in the space domain. We end up with an efficient numerical tool that we apply on various 3-D models to test the accuracy and the benefit of the homogenization. In the case of a finely layered model, our method agrees with results derived from Backus. In a more challenging model composed by a million of small cubes, waveforms computed in the homogenized medium fit reference waveforms very well. Both direct phases and complex diffracted waves are accurately retrieved in the upscaled model, although it is smooth. Finally, our upscaling method is applied to a realistic geological model. The obtained homogenized medium holds structure-induced anisotropy. Moreover, full seismic wavefields in this medium can be simulated with a coarse mesh (no matter what the numerical solver is), which significantly reduces computation costs usually associated with discontinuities and small heterogeneities. These three tests show that the non-periodic homogenization is both accurate and tractable in large 3-D cases, which opens the path to the correct account of the effect of small-scale features on seismic wave propagation for various applications and to a deeper understanding of the apparent anisotropy.

show abstract

Section: Discussion a N D C O N C L U S I O N Smentioning

confidence: 99%

Non-periodic homogenization of 3-D elastic media for the seismic wave equation

Cupillard

Capdeville

2018

Geophysical Journal International

View full text Add to dashboard Cite

show abstract

“…This nearpeak performance is likely due to the loading of multiple geometric change-of-variables factors per node per element and the low arithmetic intensity of tensor-product operations, which are effectively hidden during memory retrievals. It has been noted in [39] that, for vertex-mapped hexahedra, the IO bound nature of the hex volume kernel may be addressed by computing these geometric factors on the fly inside the kernel, loading only vertex positions (regardless of order).…”

Section: Estimated Gflops and Bandwidthmentioning

confidence: 99%

GPU-accelerated discontinuous Galerkin methods on hybrid meshes

Chan

Wang

Modave

et al. 2016

Journal of Computational Physics

Self Cite

View full text Add to dashboard Cite

We present a time-explicit discontinuous Galerkin (DG) solver for the time-domain acoustic wave equation on hybrid meshes containing vertex-mapped hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable formulations are presented for both Gauss-Legendre and Gauss-Legendre-Lobatto (Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions for hybrid meshes are derived by bounding the spectral radius of the DG operator using order-dependent constants in trace and Markov inequalities. Computational efficiency is achieved under a combination of element-specific kernels (including new quadrature-free operators for the pyramid), multi-rate timestepping, and acceleration using Graphics Processing Units.

show abstract

“…Engineering analysis is generally based on finite element computations and requires a mesh. Compared to tetrahedral meshes, hexahedral meshes have important numerical properties: faster assembly [Remacle et al 2016], high accuracy in solid mechanics [Wang et al 2004], or for quasi-incompressible materials [Benzley et al 1995]. Unstructured hexahedral meshing is an active research topic, e.g.…”

Section: Introductionmentioning

confidence: 99%

There are 174 subdivisions of the hexahedron into tetrahedra

2018

View full text Add to dashboard Cite

This article answers an important theoretical question: How many different subdivisions of the hexahedron into tetrahedra are there? It is well known that the cube has five subdivisions into 6 tetrahedra and one subdivision into 5 tetrahedra. However, all hexahedra are not cubes and moving the vertex positions increases the number of subdivisions. Recent hexahedral dominant meshing methods try to take these configurations into account for combining tetrahedra into hexahedra, but fail to enumerate them all: they use only a set of 10 subdivisions among the 174 we found in this article.The enumeration of these 174 subdivisions of the hexahedron into tetrahedra is our combinatorial result. Each of the 174 subdivisions has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling. We further show that exactly 171 of these subdivisions have a geometrical realization, i.e. there exist coordinates of the eight hexahedron vertices in a three-dimensional space such that the geometrical tetrahedral mesh is valid. We exhibit the tetrahedral meshes for these configurations and show in particular subdivisions of hexahedra with 15 tetrahedra that have a strictly positive Jacobian.

show abstract

GPU accelerated spectral finite elements on all-hex meshes

Cited by 32 publications

References 16 publications

Non-periodic homogenization of 3-D elastic media for the seismic wave equation

Non-periodic homogenization of 3-D elastic media for the seismic wave equation

GPU-accelerated discontinuous Galerkin methods on hybrid meshes

There are 174 subdivisions of the hexahedron into tetrahedra

Contact Info

Product

Resources

About