Symmetric quadrature rules for tetrahedra based on a cubic close-packed lattice arrangement

Shunn, Lee; Ham, Frank

doi:10.1016/j.cam.2012.03.032

Cited by 41 publications

(35 citation statements)

References 16 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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show abstract

“…Note that the SCP-based quadrature points of Shunn, Ham, and Williams 18,21 produce values for Λ which are larger than those of the α-optimized points. However, the values of Λ are still seen to be much smaller than those associated with the naive choice of placing points at the centers of the spheres in the SCP lattices.…”

Section: Conditioning Of Nodal Setsmentioning

confidence: 98%

“…In order to evaluate the conditioning of each of the point distributions, the Lebesgue constants were computed for p = 1 to p = 5 for the nodal points of Hesthaven and Warburton 10 , Chen and Babushka 22 , and Shunn, Ham, and Williams 18,21 . In addition, in order to establish a reference (or baseline), the Lebesgue constants were computed for nodal sets with points located at the centers of the 2D and 3D spheres in the 2D and 3D SCP lattices.…”

Section: Conditioning Of Nodal Setsmentioning

confidence: 99%

“…Experiments on equations (70) and (71) were performed with the VCJH schemes in conjunction with two 'representative' nodal sets: the nodal set based on the SCP-quadrature points of Shunn, Ham, and Williams 18,21 , and the nodal set based on the α-optimized points of Hesthaven and Warburton 10 . Figure (4) illustrates the locations of the nodal solution points and nodal flux points for each nodal set, on the reference tetrahedron, for the case of p = 3.…”

Section: Aliasing Errors Of Nodal Setsmentioning

confidence: 99%

“…This paper extends these efforts to 3D and attempts to find nodal sets that reduce aliasing errors on tetrahedral elements. In particular, the majority of attention is focused on the recently discovered quadrature points due to Shunn, Ham, and Williams 18,21 . These points, along with more traditional sets of nodal points due to 10,22 are evaluated and their suitability for nonlinear problems is assessed.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nodal Points and the Nonlinear Stability of High-Order Methods for Unsteady Flow Problems on Tetrahedral Meshes

Williams

Jameson

2013

21st AIAA Computational Fluid Dynamics Conference

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High-order methods have the potential to efficiently generate accurate solutions to fluid dynamics problems of practical interest. However, high-order methods are less robust than lower-order methods, as they are less dissipative, making them more susceptible to spurious oscillations and aliasing driven instabilities that arise during simulations of nonlinear phenomena. An effective approach for addressing this issue comes from noting that, for nonlinear problems, the stability of high-order nodal methods is significantly effected by the locations of the nodal points. In fact, in 1D and 2D, it has been shown that placing the nodal points at the locations of quadrature points reduces aliasing errors and improves the robustness of high-order schemes. In this paper, the authors perform an investigation of a particular set of nodal points whose locations coincide with quadrature points. Analysis is performed in order to determine the conditioning of these points and their suitability for interpolation. Thereafter, numerical experiments are performed on several canonical 3D problems in order to show that this set of nodal points is effective in reducing aliasing errors and promoting nonlinear stability.

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High‐order cubature rules for tetrahedra

Jaśkowiec

Sukumar

2020

Numerical Meth Engineering

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In this paper, we construct new high-order numerical integration schemes for tetrahedra, with positive weights and integration points that are in the interior of the domain. The construction of cubature rules is a challenging problem, which requires the solution of strongly nonlinear algebraic (moment) equations with side conditions given by affine inequality constraints. We present a robust algorithm based on a sequence of three modified Newton procedures to solve the constrained minimization problem. In the literature, numerical integration rules for the tetrahedron are available up to order p = 15. We obtain integration rules for the tetrahedron from p = 2 to p = 20, which are computed using multiprecision arithmetic. For p ≤ 15, our approach provides integration rules that have the same or fewer number of integration points than existing rules; for p = 16 to p = 20, our rules are new. Numerical tests are presented that verify the polynomial-precision of the cubature rules. Convergence studies are performed for the integration of exponential, rational, weakly singular and trigonometric test functions over tetrahedra with flat and curved faces. In all tests, improvements in accuracy is realized as p is increased, though in some cases nonmonotonic convergence is observed. K E Y W O R D Sleast squares, numerical integration, polynomial basis, positive weights, strictly interior integration points, tetrahedral domain 1 2418

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Symmetric quadrature rules for tetrahedra based on a cubic close-packed lattice arrangement

Cited by 41 publications

References 16 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

Nodal Points and the Nonlinear Stability of High-Order Methods for Unsteady Flow Problems on Tetrahedral Meshes

High‐order cubature rules for tetrahedra

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