Méthode asymptotique numérique

Cochelin, De Bruno; Damil, Noureddine; Potier‐Ferry, Michel

doi:10.1080/17797179.2008.9737353

Cited by 79 publications

(168 citation statements)

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“…In the case of thin elastic shell equilibrium equations, the unknowns are U = (u, S) and λ; u is the displacement and S is the second Piola Kirchhoff stress tensor and λ is the load parameter. In this paper, we limit ourselves to the quadratic framework (2.1), more difficult problems can be found in [2]. The ANM has proved to be an efficient method to compute solution path of (2.1).…”

Section: Local Parameterizations In the Anmmentioning

confidence: 99%

“…Using the evaluation of the series at the end of the interval of validity, we obtain a new starting point and we define, in this way, the ANM continuation procedure. This continuation method, based on ANM, has proved to be an efficient method to compute solution of non-linear partial differential equations [2]. The step lengths depend on the definition of the path parameter.…”

Section: Introductionmentioning

confidence: 99%

“…The Asymptotic Numerical Method (ANM) is a family of algorithms for path following problems [2]. The principle is simply to expand the unknown in power series with respect to a path parameter.…”

Section: Introductionmentioning

confidence: 99%

“…The step lengths depend on the definition of the path parameter. In [2], we have used the pseudo arc-length parameterization. Another choice, the so-called local parameterization [6,7] has been examined by J. J. Gervais et al [4] in the framework of the ANM.…”

Section: Introductionmentioning

confidence: 99%

“…The Asymptotic Numerical Method (ANM) is a family of algorithms, based on computation of truncated vectorial series, for path following problems [2]. In this paper, we present and discuss some techniques to define local parameterization [4,6,7] in the ANM.…”

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confidence: 99%

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Local Parameterization and the Asymptotic Numerical Method

Mottaqui

Braikat

Damil

2010

Math. Model. Nat. Phenom.

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Abstract. The Asymptotic Numerical Method (ANM) is a family of algorithms, based on computation of truncated vectorial series, for path following problems [2]. In this paper, we present and discuss some techniques to define local parameterization [4,6,7] in the ANM. We give some numerical comparisons of pseudo arc-length parameterization and local parameterization on nonlinear elastic shells problems.

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Section: Local Parameterizations In the Anmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Local Parameterization and the Asymptotic Numerical Method

Mottaqui

Braikat

Damil

2010

Math. Model. Nat. Phenom.

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Numerical bifurcation analysis for 3‐dimensional sudden expansion fluid dynamic problem

Guevel

Allain

Girault

et al. 2017

Numerical Methods in Fluids

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Summary This paper deals with bifurcation analysis methods based on the asymptotic‐numerical method. It is used to investigate 3‐dimensional (3D) instabilities in a sudden expansion. To do so, high‐performance computing is implemented in ELMER, ie, an open‐source multiphysical software. In this work, velocity‐pressure mixed vectors are used with asymptotic‐numerical method–based methods, remarks are made for the branch‐switching method in the case of symmetry‐breaking bifurcation, and new 3D instability results are presented for the sudden expansion ratio, ie, E=3. Critical Reynolds numbers for primary bifurcations are studied with the evolution of a geometric parameter. New values are computed, which reveal new trends that complete a previous work. Several kinds of bifurcation are depicted and tracked with the evolution of the spanwise aspect ratio. One of these relies on a fully 3D effect as it breaks both spanwise and top‐bottom symmetries. This bifurcation is found for smaller aspect ratios than expected. Furthermore, a critical Reynolds number is found for the aspect ratio, ie, Ai=1, which was not previously reported. Finally, primary and secondary bifurcations are efficiently detected and all post‐bifurcated branches are followed. This makes it possible to plot a complete bifurcation diagram for this 3D case.

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Bifurcation points and bifurcated branches in fluids mechanics by high‐order mesh‐free geometric progression algorithms

Rammane

Mesmoudi

Tri

et al. 2020

Numerical Methods in Fluids

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In this article, we propose to investigate numerically the steady bifurcation points and bifurcated branches in fluid mechanics by employing high‐order mesh‐free geometric progression algorithms. These algorithms are based on the use of the geometric progression (GP) in a high‐order mesh‐free approach. The first proposed algorithm is applied on a strong formulation using the moving least squares (MLS) approximation coupled with GP (HO‐MLS‐GPM). While the second proposed algorithm is applied on a weak formulation using the element‐free Galerkin (EFG) coupled also with GP (HO‐EFG‐GPM). The incompressibility condition is taken by introducing the penalty technique to transform the stationary Navier–Stokes equations verified by the pressure and velocity into ones verified by only the velocity. The high‐order mesh‐free algorithm permits to transform this nonlinear equations into a succession of linear ones. The GP allows to detect with precision the bifurcation points and the Lyapunov–Schmidt reduction is coupled with HO‐MLS‐GPM and HO‐EFG‐GPM as a continuation procedure to follow the many bifurcated branches. The aim of this resolution strategy concerns the treatment of the bifurcation phenomena for a fluid flow through an expansion in several geometries, where the steady flow becomes unstable after a critical Reynolds value. The obtained results are compared with those presented in literature and with those computed using the high‐order finite element algorithm coupled with GP.

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Méthode asymptotique numérique

Cited by 79 publications

References 0 publications

Local Parameterization and the Asymptotic Numerical Method

Local Parameterization and the Asymptotic Numerical Method

Numerical bifurcation analysis for 3‐dimensional sudden expansion fluid dynamic problem

Bifurcation points and bifurcated branches in fluids mechanics by high‐order mesh‐free geometric progression algorithms

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