An Analysis of Solution Point Coordinates for Flux Reconstruction Schemes on Triangular Elements

Witherden, Freddie D.; Vincent, Peter

doi:10.1007/s10915-014-9832-2

Cited by 40 publications

(23 citation statements)

References 14 publications

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“…Takex u ei to be a set of standard element solution points associated with a given element type, where 0 ≤ i < N e . Examples of such point distributions are Gauss-Legendre or GaussLobatto points, when N D = 1, Witherden-Vincent points [22] for triangles, when N D = 2, and Shunn-Ham points [23] for tetrahedra, when N D = 3. The set of solution pointsx u ei within the standard element Ω e can be used to define a nodal basis set l ei (x) of order P(N e ) that spans a polynomial space P .…”

Section: Flux Reconstructionmentioning

confidence: 99%

A high-order cross-platform incompressible Navier–Stokes solver via artificial compressibility with application to a turbulent jet

Loppi

Witherden

Jameson

et al. 2018

Computer Physics Communications

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a b s t r a c tModern hardware architectures such as GPUs and manycore processors are characterised by an abundance of compute capability relative to memory bandwidth. This makes them well-suited to solving temporally explicit and spatially compact discretisations of hyperbolic conservation laws. However, classical pressure-projection-based incompressible Navier-Stokes formulations do not fall into this category. One attractive formulation for solving incompressible problems on modern hardware is the method of artificial compressibility. When combined with explicit dual time stepping and a high-order Flux Reconstruction discretisation, the majority of operations can be cast as compute bound matrix-matrix multiplications that are well-suited for GPU acceleration and manycore processing. In this work, we develop a high-order cross-platform incompressible Navier-Stokes solver, via artificial compressibility and dual time stepping, in the PyFR framework. The solver runs on a range of computer architectures, from laptops to the largest supercomputers, via a platform-unified templating approach that can generate/compile CUDA, OpenCL and C/OpenMP code at runtime. The extensibility of the cross-platform templating framework defined within PyFR is clearly demonstrated, as is the utility of P-multigrid for convergence acceleration. The platform independence of the solver is verified on Nvidia Tesla P100 GPUs and Intel Xeon Phi 7210 KNL manycore processors with a 3D Taylor-Green vortex test case. Additionally, the solver is applied to a 3D turbulent jet test case at Re = 10,000, and strong scaling is reported up to 144 GPUs. The new software constitutes the first high-order accurate cross-platform implementation of an incompressible Navier-Stokes solver via artificial compressibility and P-multigrid accelerated dual time stepping to be published in the literature. The technology has applications in a range of sectors, including the maritime and automotive industries. Moreover, due to its cross-platform nature, the technology is well placed to remain relevant in an era of rapidly evolving hardware architectures. Solution method: Artificial compressibility formulation discretised with a high-order Flux Reconstruction approach in space and P-multigrid accelerated dual time stepping in time. Program summary

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Section: Flux Reconstructionmentioning

confidence: 99%

A high-order cross-platform incompressible Navier–Stokes solver via artificial compressibility with application to a turbulent jet

Loppi

Witherden

Jameson

et al. 2018

Computer Physics Communications

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“…Following Witherden and Vincent [12] we will use this property to define the truncation error associated with an N p point rule…”

Section: Basis Polynomialsmentioning

confidence: 99%

“…The traditional objective when constructing quadrature rules is to obtain a rule of strength φ inside of a domain using the fewest number of points. To this end efficient quadrature rules have been derived for a variety of domains: triangles [4][5][6][7][8][9][10][11][12][13], quadrilaterals [11,14,15], tetrahedra [7,9,16,17], prisms [18], pyramids [19], and hexahedra [11,[20][21][22]. For finite element applications it is desirable that (i) points are arranged symmetrically inside of the domain, (ii) all of the points are strictly inside of the domain, and (iii) all of the weights are positive.…”

Section: Introductionmentioning

confidence: 99%

On the identification of symmetric quadrature rules for finite element methods

Witherden

Vincent

2015

Computers & Mathematics with Applications

130

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a b s t r a c tIn this paper we describe a methodology for the identification of symmetric quadrature rules inside of quadrilaterals, triangles, tetrahedra, prisms, pyramids, and hexahedra. The methodology is free from manual intervention and is capable of identifying a set of rules with a given strength and a given number of points. We also present polyquad which is an implementation of our methodology. Using polyquad v1.0 we proceed to derive a complete set of symmetric rules on the aforementioned domains. All rules possess purely positive weights and have all points inside the domain. Many of the rules appear to be new, and an improvement over those tabulated in the literature.

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“…This change gave rise to a marked improvement in both stability and accuracy. These results were subsequently confirmed by Witherden and Vincent [8] in 2014 who analysed the performance triangular quadrature rules when used as solution points. Inside of tetrahedra Williams [7] compared the three dimensional α-optimised points against the tetrahedral quadrature rules of Shunn and Ham [12].…”

Section: Quadrature Rulesmentioning

confidence: 65%

“…It has been demonstrated, both theoretically [5] and empirically [6][7][8] that the degree of aliasing driven instabilities depends upon the location of the solution points inside each element. Specifically, it has been found that placing points at the abscissa of strong Gaussian quadrature rules has a positive impact on their performance.…”

Section: Introductionmentioning

confidence: 99%

An Analysis of Solution Point Coordinates for Flux Reconstruction Schemes on Tetrahedral Elements

2016

Self Cite

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The flux reconstruction (FR) approach offers an efficient route to high-order accuracy on unstructured grids. In this work we study the effect of solution point placement on the stability and accuracy of FR schemes on tetrahedral grids. To accomplish this we generate a large number of solution point candidates that satisfy various criteria at polynomial orders ℘ = 3, 4, 5. We then proceed to assess their properties by using them to solve the non-linear Euler equations on both structured and unstructured meshes. The results demonstrate that the location of the solution points is important in terms of both the stability and accuracy. Across a range of cases it is possible to outperform the solution points of Shunn and Ham for specific problems. However, there appears to be a degree of problem-dependence with regards to the optimal point set, and hence overall it is concluded that the Shunn and Ham points offer a good compromise in terms of practical utility.

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An Analysis of Solution Point Coordinates for Flux Reconstruction Schemes on Triangular Elements

Cited by 40 publications

References 14 publications

A high-order cross-platform incompressible Navier–Stokes solver via artificial compressibility with application to a turbulent jet

A high-order cross-platform incompressible Navier–Stokes solver via artificial compressibility with application to a turbulent jet

On the identification of symmetric quadrature rules for finite element methods

An Analysis of Solution Point Coordinates for Flux Reconstruction Schemes on Tetrahedral Elements

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