165 research outputs found
An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs
We extend the entropy stable high order nodal discontinuous Galerkin spectral
element approximation for the non-linear two dimensional shallow water
equations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J.
Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkin
method for the two dimensional shallow water equations on unstructured
curvilinear meshes with discontinuous bathymetry. Journal of Computational
Physics, 340:200-242, 2017] with a shock capturing technique and a positivity
preservation capability to handle dry areas. The scheme preserves the entropy
inequality, is well-balanced and works on unstructured, possibly curved,
quadrilateral meshes. For the shock capturing, we introduce an artificial
viscosity to the equations and prove that the numerical scheme remains entropy
stable. We add a positivity preserving limiter to guarantee non-negative water
heights as long as the mean water height is non-negative. We prove that
non-negative mean water heights are guaranteed under a certain additional time
step restriction for the entropy stable numerical interface flux. We implement
the method on GPU architectures using the abstract language OCCA, a unified
approach to multi-threading languages. We show that the entropy stable scheme
is well suited to GPUs as the necessary extra calculations do not negatively
impact the runtime up to reasonably high polynomial degrees (around ). We
provide numerical examples that challenge the shock capturing and positivity
properties of our scheme to verify our 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
Towards industrial large eddy simulation using the FR/CPR method
NASA’s 2030 CFD Vision calls for the development of accurate and efficient scale-resolving simulations for turbulent flow, such as large eddy simulation (LES) and direct numerical simulation (DNS). This is primarily because the Reynolds-averaged Navier-Stokes (RANS) approach has failed to predict vortex-dominated flow involving large flow separations, e.g., flow through a jet engine or over aircraft near the edge of the flight envelope, i.e., during take-off and landing at high angles of attack. Although the DNS approach resolves all turbulence scales, it is too expensive in the foreseeable future for real world flow problems because of the disparate length and time scales in the flow. LES resolves the energetic large scales while modeling the smaller scales, so it provides a good compromise between accuracy and cost. As a result, LES is widely considered to be the method of choice for next generation CFD design tool. The major obstacle for LES is its considerable computational cost since unsteady 3D simulations need to be performed to obtain the mean flow quantities such as the drag and lift coefficients. In order to resolve the dominant scales in a turbulent flow, numerical methods used for LES should have low dissipation and dispersion errors. This means standard second order finite-volume methods are usually not accurate or efficient enough for LES applications. High-order methods (order of accuracy 2) have demonstrated their potential for LES and DNS in the past decade because of their low embedded numerical dissipation and dispersion errors. In the present study, we develop and demonstrate a recently developed high-order method, called flux reconstruction (FR) or correction procedure via reconstruction (CPR), for industrial LES. A major advantage of the FR/CPR method is its capability to handle unstructured mixed meshes, and its compactness and scalability, which is particularly desired on modern super-computers. We therefore address the following major pacing items in industrial LES in the present study: High-order methods Geometric flexibility Efficient time integration Efficient implementation on modern super computers Demonstration for real world application
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
Implicit High-Order Flux Reconstruction Solver for High-Speed Compressible Flows
The present paper addresses the development and implementation of the first
high-order Flux Reconstruction (FR) solver for high-speed flows within the
open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid
Dynamics) platform. The resulting solver is fully implicit and able to simulate
compressible flow problems governed by either the Euler or the Navier-Stokes
equations in two and three dimensions. Furthermore, it can run in parallel on
multiple CPU-cores and is designed to handle unstructured grids consisting of
both straight and curved edged quadrilateral or hexahedral elements. While most
of the implementation relies on state-of-the-art FR algorithms, an improved and
more case-independent shock capturing scheme has been developed in order to
tackle the first viscous hypersonic simulations using the FR method. Extensive
verification of the FR solver has been performed through the use of
reproducible benchmark test cases with flow speeds ranging from subsonic to
hypersonic, up to Mach 17.6. The obtained results have been favorably compared
to those available in literature. Furthermore, so-called super-accuracy is
retrieved for certain cases when solving the Euler equations. The strengths of
the FR solver in terms of computational accuracy per degree of freedom are also
illustrated. Finally, the influence of the characterizing parameters of the FR
method as well as the the influence of the novel shock capturing scheme on the
accuracy of the developed solver is discussed
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