A high-resolution Petrov–Galerkin method for the 1D convection–diffusion–reaction problem

Nadukandi, Prashanth; Oñate, Eugenio; García, Julio Martínez

doi:10.1016/j.cma.2009.10.009

Cited by 29 publications

(40 citation statements)

References 53 publications

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“…The problem can be solved using various stabilization techniques. The one we favor here can be derived using either FIC or OSS approaches, to give an additional nonlinear term

\underset{bold-italicS}{falsemml-underline} ((), \underset{bold-italicv}{falsemml-underline} MathClass-punc, \underset{ϕ}{falsemml-underline})

in the form

{\underset{bold-italicS}{falsemml-underline}}_{IJ} MathClass-punc: MathClass-rel= τ_{II} ((), {MathClass-op\int}_{Ω} MathClass-rel\nabla N_{I} MathClass-bin• ((), bold-italicv MathClass-bin\otimes bold-italicv) MathClass-bin• MathClass-rel\nabla N_{J} d Ω MathClass-bin- {MathClass-op\int}_{Ω} MathClass-rel\nabla N_{I} MathClass-bin• ((), bold-italicv MathClass-bin\otimes bold-italicv) MathClass-bin• N_{J} {\underset{Π}{falsemml-underline}}_{J} d Ω)

where

\underset{Π}{falsemml-underline}

represents the

L^{2}

projection of the gradient of

\underset{ϕ}{falsemml-underline}

onto the FE mesh

{\underset{Π}{falsemml-underline}}_{I} MathClass-punc: MathClass-rel= {\underset{\underset{bold-italicM}{falsemml-underline}}{falsemml-underline}}^{MathClass-bin- 1} {MathClass-op\int}_{Ω} N_{I} MathClass-rel\nabla N_{J} {\underset{ϕ}{falsemml-underline}}_{J} d Ω

and

τ

is a scalar, which we will take as

τ_{II} MathClass-punc: MathClass-rel= ((), \frac{β}{dt} MathClass-bi...)

…”

Section: Edge‐based Finite Element Formulationsmentioning

confidence: 99%

OpenCL‐based implementation of an unstructured edge‐based finite element convection‐diffusion solver on graphics hardware

Mossaiby

Rossi

Dadvand

et al. 2011

Numerical Meth Engineering

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SUMMARYThe solution of problems in computational fluid dynamics (CFD) represents a classical field for the application of advanced numerical methods. Many different approaches were developed over the years to address CFD applications. Good examples are finite volumes, finite differences (FD), and finite elements (FE) but also newer approaches such as the lattice-Boltzmann (LB), smooth particle hydrodynamics or the particle finite element method. FD and LB methods on regular grids are known to be superior in terms of raw computing speed, but using such regular discretization represents an important limitation in dealing with complex geometries. Here, we concentrate on unstructured approaches which are less common in the GPU world. We employ a nonstandard FE approach which leverages an optimized edge-based data structure allowing a highly parallel implementation. Such technique is applied to the 'convection-diffusion' problem, which is often considered as a first step towards CFD because of similarities to the nonconservative form of the Navier-Stokes equations. In this regard, an existing highly optimized parallel OpenMP solver is ported to graphics hardware based on the OpenCL platform. The optimizations performed are discussed in detail. A number of benchmarks prove that the GPU-accelerated OpenCL code consistently outperforms the OpenMP version.

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“…The problem can be solved using various stabilization techniques. The one we favor here can be derived using either FIC or OSS approaches, to give an additional nonlinear term

\underset{bold-italicS}{falsemml-underline} ((), \underset{bold-italicv}{falsemml-underline} MathClass-punc, \underset{ϕ}{falsemml-underline})

in the form

{\underset{bold-italicS}{falsemml-underline}}_{IJ} MathClass-punc: MathClass-rel= τ_{II} ((), {MathClass-op\int}_{Ω} MathClass-rel\nabla N_{I} MathClass-bin• ((), bold-italicv MathClass-bin\otimes bold-italicv) MathClass-bin• MathClass-rel\nabla N_{J} d Ω MathClass-bin- {MathClass-op\int}_{Ω} MathClass-rel\nabla N_{I} MathClass-bin• ((), bold-italicv MathClass-bin\otimes bold-italicv) MathClass-bin• N_{J} {\underset{Π}{falsemml-underline}}_{J} d Ω)

where

\underset{Π}{falsemml-underline}

represents the

L^{2}

projection of the gradient of

\underset{ϕ}{falsemml-underline}

onto the FE mesh

{\underset{Π}{falsemml-underline}}_{I} MathClass-punc: MathClass-rel= {\underset{\underset{bold-italicM}{falsemml-underline}}{falsemml-underline}}^{MathClass-bin- 1} {MathClass-op\int}_{Ω} N_{I} MathClass-rel\nabla N_{J} {\underset{ϕ}{falsemml-underline}}_{J} d Ω

and

τ

is a scalar, which we will take as

τ_{II} MathClass-punc: MathClass-rel= ((), \frac{β}{dt} MathClass-bi...)

…”

Section: Edge‐based Finite Element Formulationsmentioning

confidence: 99%

OpenCL‐based implementation of an unstructured edge‐based finite element convection‐diffusion solver on graphics hardware

Mossaiby

Rossi

Dadvand

et al. 2011

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…We favor here the use of the so-called split-OSS approach [10], which is known to work properly in a wide range of applications. FIC stabilization [11,12] could be used as an alternative since it leads to very similar discrete forms. Since a discussion of the properties of the chosen stabilization method falls outside the scope of this work, we refer the reader to the literature for a detailed description of the properties of such technique.…”

Section: Finite Element Formulationmentioning

confidence: 99%

A portable OpenCL-based unstructured edge-based finite element Navier–Stokes solver on graphics hardware

2013

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The rise of GPUs in modern high-performance systems increases the interest in porting portion of codes to such hardware. The current paper aims to explore the performance of a portable state-of-the-art FE solver on GPU accelerators. Performance evaluation is done by comparing with an existing highly-optimized OpenMP version of the solver.Code portability is ensured by writing the program using the OpenCL 1.1 specifications, while performance portability is sought through an optimization step performed at the beginning of the calculations to find out the optimal parameter set for the solver.The results show that the new implementation can be several times faster than the OpenMP version.

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“…These tests are for models with relatively low element Peclet numbers, defined as Pe :¼ vh D In practice, in many problems of interest, v > > D. Many numerical schemes cannot produce accurate results when element Peclet numbers are large [23,24]. If Pe values are large, then implementation of the LCM scheme, as presented here, may be problematic, because the values of c i calculated using Eqn (1.6) can exceed floating point overflow limits.…”

mentioning

confidence: 99%

A method for numerical modelling of convection–reaction–diffusion using electrical analogues

Shafaq

Kennedy

Delauré

2015

Int J Numerical Modelling

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In this paper, a new method, called the lumped-component circuit method (LCM), is developed for one-dimensioal and two-dimensional convection-reaction-diffusion with low to moderate Peclet numbers, tested for modelling both steady-state and transient problems, and compared with standard finite volume method (FVM) schemes. The method has been developed principally for solving equations with piecewise-constant coefficients using nodes that are not positioned to correspond to the coefficient discontinuities. In such situations, the FVM solutions do not converge consistently as the node spacing is decreased, but LCM solutions do. In general, the LCM method is more accurate than the FVM schemes tested, and, while the computational cost of LCM is higher, results suggest that it can be more efficient. Like the transmission line method (TLM), it is an indirect scheme in which the problem to be solved is first represented by an analogous transmission line (TL). Unlike with TLM, however, the TL is then modelled using a lumped-component circuit, the voltages at nodes within that circuit being calculated.

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A high-resolution Petrov–Galerkin method for the 1D convection–diffusion–reaction problem

Cited by 29 publications

References 53 publications

OpenCL‐based implementation of an unstructured edge‐based finite element convection‐diffusion solver on graphics hardware

OpenCL‐based implementation of an unstructured edge‐based finite element convection‐diffusion solver on graphics hardware

A portable OpenCL-based unstructured edge-based finite element Navier–Stokes solver on graphics hardware

A method for numerical modelling of convection–reaction–diffusion using electrical analogues

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