Commit 244a2415 authored by Mike Kirby's avatar Mike Kirby
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parent e128cfaf
%%% Software
author = "Peter E. J. Vos and Spencer J. Sherwin and Robert M. Kirby",
title = "h-to-p Efficiently: Implementing Finite and Spectral/hp Element Methods to Achieve Optimal Performance for Low- and High-Order Discretisations",
journal = "Journal of Computational Physics",
volume = "229",
issue = "13",
pages = "5161-5181",
year = "2010"
author = "C.D. Cantwell and S.J. Sherwin and R.M. Kirby and P.H.J. Kelly",
title = "From h-to-p Efficiently: Strategy Selection for Operator Evaluation on Hexahedral and Tetrahedral Elements",
journal = "Computers and Fluids",
volume = "43",
issue = "1",
pages = "23-28",
year = "2011"
author = "C.D. Cantwell and S.J. Sherwin and R.M. Kirby and P.H. Kelly",
title = "From h-to-p Efficiently: Selecting the Optimal Spectral/hp Discretisation in Three Dimensions",
journal = "Math. Model. Nat. Phenom.",
volume = "6",
number = "3",
pages = "84-96",
year = "2011"
author = "A. Bolis and C.D. Cantwell and R.M. Kirby and S.J. Sherwin",
title = "h-to-p efficiently: Optimal implementation strategies for explicit time-dependent problems using the spectral/hp element method",
journal = "International Journal for Numerical Methods in Fluids",
volume = "75",
issue = "8",
pages = "591-607",
year = "2014"
author = "C.D. Cantwell and D. Moxey and A. Comerford and A. Bolis and G. Rocco and G. Mengaldo and D. de Grazia and S. Yakovlev and J-E Lombard and D. Ekelschot and B. Jordi and H. Xu and Y. Mohamied and C. Eskilsson and B. Nelson and P. Vos and C. Biotto and R.M. Kirby and S.J. Sherwin",
title = "Nektar++: An open-source spectral/hp element framework",
journal = "Computer Physics Communications",
volume = "192",
pages = "205-219",
year = "2015"
title = {Camellia: {A} software framework for discontinuous {Petrov-Galerkin} methods},
volume = {68},
shorttitle = {Camellia},
abstract = {The discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan minimizes the solution residual in a user-determinable energy norm and offers a built-in mechanism for evaluating error in the energy norm, among other desirable features. However, the methodology also brings with it some additional complexity for researchers who wish to experiment with DPG in their computations. In this paper, we introduce Camellia, a software framework whose central design goal is to enable developers to create efficient hp-adaptive DPG solvers with minimal effort.},
number = {11},
journal = {Computers \& Mathematics with Applications},
author = {Roberts, Nathan V.},
year = {2014},
pages = {1581--1604}
title = {Elvis: {A} system for the accurate and interactive visualization of high-order finite element solutions},
volume = {18},
shorttitle = {Elvis},
abstract = {This paper presents the Element Visualizer (ElVis), a new, open-source scientific visualization system for use with high-order finite element solutions to PDEs in three dimensions. This system is designed to minimize visualization errors of these types of fields by querying the underlying finite element basis functions (e.g., high-order polynomials) directly, leading to pixel-exact representations of solutions and geometry. The system interacts with simulation data through runtime plugins, which only require users to implement a handful of operations fundamental to finite element solvers. The data in turn can be visualized through the use of cut surfaces, contours, isosurfaces, and volume rendering. These visualization algorithms are implemented using NVIDIA's OptiX GPU-based ray-tracing engine, which provides accelerated ray traversal of the high-order geometry, and CUDA, which allows for effective parallel evaluation of the visualization algorithms. The direct interface between ElVis and the underlying data differentiates it from existing visualization tools. Current tools assume the underlying data is composed of linear primitives; high-order data must be interpolated with linear functions as a result. In this work, examples drawn from aerodynamic simulations-high-order discontinuous Galerkin finite element solutions of aerodynamic flows in particular-will demonstrate the superiority of ElVis' pixel-exact approach when compared with traditional linear-interpolation methods. Such methods can introduce a number of inaccuracies in the resulting visualization, making it unclear if visual artifacts are genuine to the solution data or if these artifacts are the result of interpolation errors. Linear methods additionally cannot properly visualize curved geometries (elements or boundaries) which can greatly inhibit developers' debugging efforts. As we will show, pixel-exact visualization exhibits none of these issues, removing the visualization scheme as a source of - ncertainty for engineers using ElVis.},
number = {12},
journal = {Visualization and Computer Graphics, IEEE Transactions on},
author = {Nelson, Blake and Liu, Eric and Kirby, Robert M. and Haimes, Robert},
year = {2012},
pages = {2325--2334}
title = {Gmsh: {A} 3-{D} finite element mesh generator with built-in pre-and post-processing facilities},
volume = {79},
shorttitle = {Gmsh},
abstract = {Gmsh is an open-source 3-D finite element grid generator with a build-in CAD
engine and post-processor. Its design goal is to provide a fast, light and user-friendly
meshing tool with parametric input and advanced visualization capabilities. This paper presents the overall philosophy, the main design choices and some of the original algorithms implemented in Gmsh. Copyright© 2009 John Wiley \& Sons, Ltd.},
number = {11},
journal = {International Journal for Numerical Methods in Engineering},
author = {Geuzaine, Christophe and Remacle, Jean-François},
year = {2009},
pages = {1309--1331},
file = {[PDF] from Support/Zotero/Profiles/7xumqoyz.default/zotero/storage/IASXUV6Z/Geuzaine and Remacle - 2009 - Gmsh A 3-D finite element mesh generator with bui.pdf:application/pdf}
title = "Nek5000 User Manual",
author = "Paul Fischer and James Lottes and Stefan Kerkemeier and Oana Marin and Katherine Heisey and Aleks Obabko and Elia Merzari and Yulia Peet",
publisher = "ANL/MCS-TM-351",
year = "2014"
title = "High-Order Methods for Incompressible Fluid Flow",
author = "M.O. Deville and P.F. Fisher and E.H. Mund",
publisher = "Cambridge University Press",
year = "2002"
author = "J.S. Hesthaven and T.C. Warburton",
title = "Nodal Discontinuous {G}alerkin Methods: Algorithms, Analysis, and Applications",
publisher = "Springer Texts in Applied Mathematics 54. Springer Verlag: New York",
year = "2008"
author = "F.X. Giraldo and J.F. Kelly and E.M. Constantinescu",
title = "Implicit explicit formulations of a three dimensional non-hydrostatic unified model of the atmosphere {(NUMA)}",
journal = "SIAM Journal of Scientific Computing",
volume = "35",
pages = "1162-1194",
year = "2013"
title = "Automated Solution of Differential Equations by the Finite Element Method",
author = "Anders Logg and Kent-Andre Mardal and Garth Wells (editors)",
publisher = "Springer Lecture Notes in Computational Science and Engineering, Volume 84",
year = "2012"
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% Introduction
\caption{Figure 1}
\caption{Figure 2}
\section{Software Implementations and Frameworks}
In the last ten years a collection of software frameworks has been put forward to try to bridge the gap between the
mathematics of high-order methods and their implementation. A major challenge many practitioners have with
spectral/{\em hp\/} elements and high-order methods, in general, is the complexity (in terms of algorithmic design) they
encounter. In this section, we give an incomplete but
representative summary of several of these attempts to overcome this challenge.
FEniCS \cite{FEniCS} is a collaborative project for the development of scientific computing tools, with a particular focus
on the automated solution of differential equations by finite element methods (FEM). Through the use of concepts such as meta-programming,
FEniCS tries to keep the solving of PDEs with FEM, from the application programmers' perspective, as close to the mathematical expressions
as possible without sacrificing computational efficiency.
The current effort, Nektar++ \cite{CantwellMCBRMGYLEJXMENVBKS2015}, is an open-source software framework designed to support the development of
high-performance scalable solvers for partial differential equations using the spectral/{\em hp\/} element method. Nektar++ is an outgrowth of the original {\nektar} effort; its design and principal implementation
being by two former {\nektar} developers. The two software suites are similar in terms of the basis choices available, etc. However, what makes Nektar++ unique is that it is
an initiative to overcome this limitation by encapsulating the mathematical complexities of the underlying method within an efficient C++ framework, making the techniques more accessible to the broader scientific and industrial communities. The software supports a variety of discretization techniques and
implementation strategies \cite{VosSK2010,CantwellSKK2011a,CantwellSKK2011b,BolisCKS2014},
supporting methods research as well as application-focused computation, and the multi-layered structure of the framework allows the user to embrace as much or as little of the complexity as they need. The libraries capture the mathematical constructs of spectral/{\em hp\/} element methods, while the associated collection of pre-written PDE solvers provides out-of-the-box application-level functionality and a template for users who wish to develop solutions for addressing questions in their own scientific domains.
With respect to software infrastructures that tackle specific domain problems,
the nodal hexahedral spectral element code Nek5000 \cite{Nek5000,DevilleFM02},
which is an outgrowth of the original code NEKTON, provides the application
developer with a framework for solving the incompressible Navier-Stokes equations and its extensions in a scalable parallel way.
Similarly the Non-hydrostatic Unified Model of the Atmosphere (NUMA) \cite{GiraldoKC13} is a spectral element framework that employs continuous
and discontinuous Galerkin
strategies for solving a particular problem of interest, but in a way on which others could adopt and build. The discontinuous Galerkin
of Hesthaven and Warburton \cite{HesthavenW08} and a software framework for the discontinuous Petrov-Galerkin method \cite{roberts_camellia:_2014}
provide a similar flexibility to the user-community trying to jump-start their high-order software experience.
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\section{Linear Algebra}
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\section{The Fundamentals Behind LocalRegions}
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\section{The Fundamentals Behind MultiRegions}
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\section{The Fundamentals Behind SpatialDomains}
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\section{The Fundamentals Behind StdRegions}
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% Library Master File
\chapter{Inside the Library: LibUtilities}
In this chapter, we walk the reader through the different components of the LibUtilities Directory.
\chapter{Inside the Library: StdRegions}
In this chapter, we walk the reader through the different components of the StdRegions Directory.
\chapter{Inside the Library: SpatialDomains}
In this chapter, we walk the reader through the different components of the SpatialDomains Directory.
\chapter{Inside the Library: LocalRegions}
In this chapter, we walk the reader through the different components of the LocalRegions Directory.
\chapter{Inside the Library: MultiRegions}
In this chapter, we walk the reader through the different components of the MultiRegions Directory.
% Preface
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