Analysis and Computation of Symmetry-Breaking Bifurcation and Scaling Laws Using Group-Theoretic Methods

Aston, Philip J.

doi:10.1137/0522012

Cited by 38 publications

(29 citation statements)

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“…Independently, Healey and his students [21][22][23][24], motivated by examples in structural mechanics, and Aston [25,26], motivated by a problem in water waves, developed similar theories but which allowed for infinite groups. An important idea is that, under the action of a group and for an appropriately chosen basis, the Jacobian of a non-linear problem 'block-diagonalizes', with consequent savings for numerical algorithms, and improved theoretical understanding [20,23,26]. Many of the group theoretic ideas can appear complicated, and one aim of this paper is to explain them as simply as possible for the O(2) group, which is relevant for many problems defined on a cylindrical domain.…”

Section: Introductionmentioning

confidence: 93%

O(2)-symmetry breaking bifurcation: with application to the flow past a sphere in a pipe

Cliffe

Spence

Tavener

2000

Int. J. Numer. Meth. Fluids

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Section: Introductionmentioning

confidence: 93%

O(2)-symmetry breaking bifurcation: with application to the flow past a sphere in a pipe

Cliffe

Spence

Tavener

2000

Int. J. Numer. Meth. Fluids

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“…In the sequel we consider the numerical approximation of the corresponding steady state problem and investigate its linear stability. With this in mind, employing the continuity equation (5.2), we rewrite the steady NavierStokes equations in the following divergence form (to facilitate the DG discretization): find u 0 and p 0 such that 6) subject to the boundary conditions outlined in (5.3)-(5.4) above, with u and p replaced by u 0 and p 0 , respectively. Here, for vectors v ∈ R m and w ∈ R n , m, n ≥ 1, the matrix v ⊗ w ∈ R m×n is the standard outer product defined by (v ⊗ w) kl = v k w l .…”

Section: Incompressible Navier-stokes Equationsmentioning

confidence: 99%

“…Aston [6]. Thereby, for u ∈ V s , the Jacobian operator F ′ u (u, λ; ·) has a diagonal block structure; more precisely, the following result holds.…”

mentioning

confidence: 96%

Adaptivity and a Posteriori Error Control for Bifurcation Problems II: Incompressible Fluid Flow in Open Systems with Z 2 Symmetry

et al. 2011

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Access from the University of Nottingham repository:http://eprints.nottingham.ac.uk/1257/1/cliffe_et_al_z2_bifurcation.pdf Copyright and reuse:The Nottingham ePrints service makes this work by researchers of the University of Nottingham available open access under the following conditions. This article is made available under the University of Nottingham End User licence and may be reused according to the conditions of the licence. For more details see: http://eprints.nottingham.ac.uk/end_user_agreement.pdf A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. Abstract. In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork or Hopf bifurcation occurs when the underlying physical system possesses reflectional or Z 2 symmetry. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard DualWeighted-Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented.Key words. Incompressible flows, bifurcation problems, a posteriori error estimation, adaptivity, discontinuous Galerkin methods, Z 2 symmetry 1. Introduction. Due to the highly nonlinear governing equations and complex geometries involved, understanding fluid flow remains one of the fundamental engineering challenges today. Often it is impossible to obtain analytical solutions to problems and numerical methods must instead be exploited. Broadly speaking, there are two numerical approaches to understanding the Navier-Stokes equations: the first involves direct 'simulation' of the time dependent problem, while the second, less commonly used approach focuses on applying nonlinear analysis to compute paths of steady solutions using numerical continuation methods and to determine their stability based on eigenvalue information. In this article we focus on the latter technique, specifically where we seek to understand how the solution structure changes as one parameter of interest is varied; in the case of the incompressible Navier-Stokes equations this parameter is typically the Reynolds number. We are particularly interested in the location of critical parameters at which a bifurcation first occurs; a review of techniques for bifurcation detection can be found in Cliffe et al. [16], for example.Over the past few decades, ...

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“…see Aston [5]. Thereby, for u ∈ V O(2) , our Jacobian operator F ′ u (u, λ; ·) has a diagonal block structure; more precisely the following result holds.…”

mentioning

confidence: 95%

“…It is a standard result, see Aston [5], that, for the O(2) case, there exists a unique orthogonal decomposition of V , namely,…”

mentioning

confidence: 99%

Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry

et al. 2011

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Access from the University of Nottingham repository:http://eprints.nottingham.ac.uk/1437/1/cliffe_et_al_o2_article.pdf Copyright and reuse:The Nottingham ePrints service makes this work by researchers of the University of Nottingham available open access under the following conditions. This article is made available under the University of Nottingham End User licence and may be reused according to the conditions of the licence. For more details see: http://eprints.nottingham.ac.uk/end_user_agreement.pdf A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. Abstract. In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the bifurcation problem associated with the steady incompressible Navier-Stokes equations. Particular attention is given to the reliable error estimation of the critical Reynolds number at which a steady pitchfork bifurcation occurs when the underlying physical system possesses rotational and reflectional or O(2) symmetry. Here, computable a posteriori error bounds are derived based on employing the generalization of the standard Dual Weighted Residual approach, originally developed for the estimation of target functionals of the solution, to bifurcation problems. Numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on adaptively refined computational meshes are presented. Here, particular attention is devoted to the problem of flow through a cylindrical pipe with a sudden expansion, which represents a notoriously difficult computational problem.Key words. Incompressible flows, bifurcation problems, a posteriori error estimation, adaptivity, discontinuous Galerkin methods, O(2) symmetry 1. Introduction. In this article, we study the stability of the three-dimensional incompressible Navier-Stokes equations in the case when the underlying system possesses both rotational and reflectional symmetry, or more precisely, O(2) symmetry. To this end, we are interested in numerically estimating the critical Reynolds number Re, at which a (pitchfork) bifurcation point first occurs; a review of techniques for bifurcation detection can be found in Cliffe et al. [13], for example. The work in this article expands upon our recent work in [12] and [10] to include problems whose geometries exhibit both rotational and reflectional symmetry. The detection of bifurcation points in this setting is now well understood, for example, see Golubitsky and Schaeffer [19]. For the purposes of this article, we assume that a symmetric steady state solution to the incompressible Navier-Stokes equations undergoes a steady pitchfork bifurcation at a critical value of the Reynolds number. Estimation of the critical...

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Analysis and Computation of Symmetry-Breaking Bifurcation and Scaling Laws Using Group-Theoretic Methods

Cited by 38 publications

References 32 publications

O(2)-symmetry breaking bifurcation: with application to the flow past a sphere in a pipe

O(2)-symmetry breaking bifurcation: with application to the flow past a sphere in a pipe

Adaptivity and a Posteriori Error Control for Bifurcation Problems II: Incompressible Fluid Flow in Open Systems with Z 2 Symmetry

Adaptivity and a Posteriori Error Control for Bifurcation Problems III: Incompressible Fluid Flow in Open Systems with O(2) Symmetry

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