425 research outputs found
On the Identification of Symmetric Quadrature Rules for Finite Element Methods
In 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 an ensemble 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 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.Comment: 17 pages, 6 figures, 1 tabl
An extended range of stable-symmetric-conservative Flux Reconstruction correction functions
The Flux Reconstruction (FR) approach offers an efficient route to achieving high-order accuracy on unstructured grids. Additionally, FR offers a flexible framework for defining a range of numerical schemes in terms of so-called FR correction functions. Recently, a one-parameter family of FR correction functions were identified that lead to stable schemes for 1D linear advection problems. In this study we develop a procedure for identifying an extended range of stable, symmetric, and conservative FR correction functions. The procedure is applied to identify ranges of such correction functions for various orders of accuracy. Numerical experiments are undertaken, and the results found to be in agreement with the theoretical findings
PyFR: An Open Source Framework for Solving Advection-Diffusion Type Problems on Streaming Architectures using the Flux Reconstruction Approach
High-order numerical methods for unstructured grids combine the superior
accuracy of high-order spectral or finite difference methods with the geometric
flexibility of low-order finite volume or finite element schemes. The Flux
Reconstruction (FR) approach unifies various high-order schemes for
unstructured grids within a single framework. Additionally, the FR approach
exhibits a significant degree of element locality, and is thus able to run
efficiently on modern streaming architectures, such as Graphical Processing
Units (GPUs). The aforementioned properties of FR mean it offers a promising
route to performing affordable, and hence industrially relevant,
scale-resolving simulations of hitherto intractable unsteady flows within the
vicinity of real-world engineering geometries. In this paper we present PyFR,
an open-source Python based framework for solving advection-diffusion type
problems on streaming architectures using the FR approach. The framework is
designed to solve a range of governing systems on mixed unstructured grids
containing various element types. It is also designed to target a range of
hardware platforms via use of an in-built domain specific language based on the
Mako templating engine. The current release of PyFR is able to solve the
compressible Euler and Navier-Stokes equations on grids of quadrilateral and
triangular elements in two dimensions, and hexahedral elements in three
dimensions, targeting clusters of CPUs, and NVIDIA GPUs. Results are presented
for various benchmark flow problems, single-node performance is discussed, and
scalability of the code is demonstrated on up to 104 NVIDIA M2090 GPUs. The
software is freely available under a 3-Clause New Style BSD license (see
www.pyfr.org)
Heterogeneous Computing on Mixed Unstructured Grids with PyFR
PyFR is an open-source high-order accurate computational fluid dynamics
solver for mixed unstructured grids that can target a range of hardware
platforms from a single codebase. In this paper we demonstrate the ability of
PyFR to perform high-order accurate unsteady simulations of flow on mixed
unstructured grids using heterogeneous multi-node hardware. Specifically, after
benchmarking single-node performance for various platforms, PyFR v0.2.2 is used
to undertake simulations of unsteady flow over a circular cylinder at Reynolds
number 3 900 using a mixed unstructured grid of prismatic and tetrahedral
elements on a desktop workstation containing an Intel Xeon E5-2697 v2 CPU, an
NVIDIA Tesla K40c GPU, and an AMD FirePro W9100 GPU. Both the performance and
accuracy of PyFR are assessed. PyFR v0.2.2 is freely available under a 3-Clause
New Style BSD license (see www.pyfr.org).Comment: 21 pages, 9 figures, 6 table
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