From h to p efficiently: Strategy selection for operator evaluation on hexahedral and tetrahedral elements

Cantwell, Chris D.; Sherwin, Spencer J.; Kirby, Robert M.; Kelly, Paul H. J.

doi:10.1016/j.compfluid.2010.08.012

Cited by 84 publications

(103 citation statements)

References 7 publications

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“…The nature of this construction allows for the sum-factorisation evaluation strategy [10] to be employed, even in the tetrahedral region, which leads to a higher performance in some circumstances [2]. However, to ensure only the minimum necessary degrees of freedom are used in the tetrahedral case (an extension to three dimensions of the triangular case shown in Fig.…”

Section: Spectral/ℎ Element Frameworkmentioning

confidence: 99%

“…As shown in [2] for the three-dimensional case, and in [15] in two dimensions, careful selection of the implementation strategy is essential to achieve the best performance at a given (ℎ, )-discretisation. As well as global operations and local elemental operations the tensor-product construction of the elemental basis modes allows for a third method of implementation known as sum-factorisation [10].…”

Section: Spectral/ℎ Element Frameworkmentioning

confidence: 99%

“…The sum-factorisation approach factorises the fully three-dimensional ( 6 ) elemental operation as a series of three one-dimensional ( 4 ) operations. This provides significantly improved performance with some discretisations [2], reducing both memory storage and computational cost. This is not always the case, and with some linear differential operators with certain discretisations it may provide a significantly lower performance than the local matrix or global strategies.…”

Section: Spectral/ℎ Element Frameworkmentioning

confidence: 99%

“…Cantwell et al [2] examined the runtime performance of a spectral/ℎ implementation in actioning four fundamental operators in three dimensions (including the backward transform and evaluation of the Helmholtz linear differential operator). They found that choosing the correct implementation strategy for a given discretisation was essential when evaluating the different operators.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, one would like to minimise the runtime cost of the calculation. For each (ℎ, )-discretisation there exists an optimal evaluation strategy for each mathematical operator [2] and so we therefore seek the (ℎ, )-discretisation, from the space of those which provide acceptable accuracy, which minimises the runtime cost when using the optimal implementation. In practice, other factors may influence the selection, such as memory limitations, but we discount these for the purposes of this study.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

From h to p Efficiently: Selecting the Optimal Spectral/hpDiscretisation in Three Dimensions

Cantwell

Sherwin

Kirby

2011

Math. Model. Nat. Phenom.

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Abstract. There is a growing interest in high-order finite and spectral/ℎ element methods using continuous and discontinuous Galerkin formulations. In this paper we investigate the effect of ℎ-and -type refinement on the relationship between runtime performance and solution accuracy. The broad spectrum of possible domain discretisations makes establishing a performance-optimal selection non-trivial. Through comparing the runtime of different implementations for evaluating operators over the space of discretisations with a desired solution tolerance, we demonstrate how the optimal discretisation and operator implementation may be selected for a specified problem. Furthermore, this demonstrates the need for codes to support both low-and high-order discretisations.

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Section: Spectral/ℎ Element Frameworkmentioning

confidence: 99%

Section: Spectral/ℎ Element Frameworkmentioning

confidence: 99%

Section: Spectral/ℎ Element Frameworkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

From h to p Efficiently: Selecting the Optimal Spectral/hpDiscretisation in Three Dimensions

Cantwell

Sherwin

Kirby

2011

Math. Model. Nat. Phenom.

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Spectral Element andhpMethods

Kirby

Karniadakis

2017

Encyclopedia of Computational Mechanics Second Edition

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Spectral/ hp element methods provide high‐order discretization, which is essential in the longtime integration of advection–diffusion systems and for capturing dynamic instabilities in solids. In this chapter, we review the main formulations for simulations of incompressible and compressible viscous flows as well as for solid mechanics and present several examples with some emphasis on fluid–structure interactions and interfaces. The first generation of (nodal) spectral elements was limited to relatively simple geometries and smooth solutions. However, the new generation of spectral/ hp elements, consisting of both nodal and modal forms, can handle very complex geometries using unstructured grids and can capture strong shocks by employing discontinuous Galerkin methods. New flexible formulations allow simulations of multiphysics problems including extremely complex geometries and multiphase flows. Several implementation strategies have also been developed on the basis of multilevel parallel algorithms that allow dynamic ‐refinement at constant wall clock time. After three decades of intense developments, spectral element and hp methods are mature and efficient to be used effectively in applications of industrial complexity. They provide the capabilities that standard finite element and finite volume methods do, but, in addition, they exhibit high‐order accuracy and error control.

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Finite element assembly strategies on multi‐core and many‐core architectures

Markall

Slemmer

Ham

et al. 2012

Numerical Methods in Fluids

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SUMMARY We demonstrate that radically differing implementations of finite element methods (FEMs) are needed on multi‐core (CPU) and many‐core (GPU) architectures, if their respective performance potential is to be realised. Our numerical investigations using a finite element advection–diffusion solver show that increased performance on each architecture can only be achieved by committing to specific and diverse algorithmic choices that cut across the high‐level structure of the implementation. Making these commitments to achieve high performance for a single architecture leads to a loss of performance portability. Data structures that include redundant data but enable coalesced memory accesses are faster on many‐core architectures, whereas redundancy‐free data structures that are accessed indirectly are faster on multi‐core architectures. The Addto algorithm for global assembly is optimal on multi‐core architectures, whereas the Local Matrix Approach is optimal on many‐core architectures despite requiring more computation than the Addto algorithm. These results demonstrate the value in making the correct choice of algorithm and data structure when implementing FEMs, spectral element methods and low‐order discontinuous Galerkin methods on modern high‐performance architectures. Copyright © 2012 John Wiley & Sons, Ltd.

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From h to p efficiently: Strategy selection for operator evaluation on hexahedral and tetrahedral elements

Cited by 84 publications

References 7 publications

From h to p Efficiently: Selecting the Optimal Spectral/hpDiscretisation in Three Dimensions

From h to p Efficiently: Selecting the Optimal Spectral/hpDiscretisation in Three Dimensions

Spectral Element andhpMethods

Finite element assembly strategies on multi‐core and many‐core architectures

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