Local Parameterization and the Asymptotic Numerical Method

Mottaqui, H.; Braikat, Bouazza; Damil, Noureddine

doi:10.1051/mmnp/20105703

Cited by 19 publications

(9 citation statements)

References 5 publications

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“…In addition, we adopt the so‐called penalty method, which provides a convenient way to satisfy the incompressibility constraint by eliminating the pressure term . We also present the computation of the solution from a weak formulation using the high order with the finite element method (HO‐FEM) for comparison . To demonstrate the effectiveness and the robustness of the presented algorithm in comparison with an iterative algorithm based on the Newton‐Raphson method with MLS and HO‐FEM, three examples of an incompressible fluid flows are studied.…”

Section: Introductionmentioning

confidence: 99%

Solving the incompressible fluid flows by a high‐order mesh‐free approach

Rammane

Mesmoudi

Tri

et al. 2019

Numerical Methods in Fluids

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Summary In this paper, we propose for the first time to extend the application field of the high‐order mesh‐free approach to the stationary incompressible Navier‐Stokes equations. This approach is based on a high‐order algorithm, which combines a Taylor series expansion, a continuation technique, and a moving least squares (MLS) method. The Taylor series expansion permits to transform the nonlinear problem into a succession of continuous linear ones with the same tangent operator. The MLS method is used to transform the succession of continuous linear problems into discrete ones. The continuation technique allows to compute step‐by‐step the whole solution of the discrete problems. This mesh‐free approach is tested on three examples: a flow around a cylindrical obstacle, a flow in a sudden expansion, and the standard benchmark lid‐driven cavity flow. A comparison of the obtained results with those computed by the Newton‐Raphson method with MLS, the high‐order continuation with finite element method, and those of literature is presented.

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Section: Introductionmentioning

confidence: 99%

Solving the incompressible fluid flows by a high‐order mesh‐free approach

Rammane

Mesmoudi

Tri

et al. 2019

Numerical Methods in Fluids

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“…a.1 Perturbation technique. To solve the nonlinear problem (55), the perturbation method is used and "a" is considered as a perturbation parameter [25,24]. The unknowns ∆r g are sought in the form of an integro-power series with respect to homotopy parameter "a" starting from order 1 and truncated at order p as follows:…”

Section: The Homotopy Transformationsmentioning

confidence: 99%

“…Introducing the vector {θ} defined in (25), one can express the strain vector {γ} in terms of {θ} by [22]:…”

mentioning

confidence: 99%

Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings

Kaimbillah¹,

Bourihane²,

Braikat³

et al. 2019

Discrete &Amp; Continuous Dynamical Systems - S

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This paper aims to investigate, in large displacement and torsion context, the nonlinear dynamic behavior of thin-walled beams with open cross section subjected to various loadings by high-order implicit solvers. These homotopy transformations consist to modify the nonlinear discretized dynamic problem by introducing an arbitrary invertible pre-conditioner [K ] and an arbitrary path following parameter. The nonlinear strongly coupled equations of these structures are derived by using a 3D nonlinear dynamic model which accounts for large displacements and large torsion without any assumption on torsion angle amplitude. Coupling complex structural phenomena such that warping, bending-bending, and flexural-torsion are taken into account. Two examples of great practical interest of nonlinear dynamic problems of various thin-walled beams with open section are presented to validate the Key words and phrases. Thin-walled beam, open cross section, large displacement, large torsion, forced nonlinear dynamic, finite element, homotopy, power series expansion, Asymptotic Numerical Method (ANM).

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“…The objective of this work is to propose a high-order algorithm based on the asymptotic numerical method (ANM) [12][13][14][15][16][17] to calculate the displacements and rotations at a point of the middle line of a helical structure. The originality of our numerical approach is to calculate these unknowns without neglecting any nonlinear term of the strain.…”

Section: Introductionmentioning

confidence: 99%

Numerical High-Order Model for the Nonlinear Elastic Computation of Helical Structures

Boussaoui

Lahmam

Braikat

2021

Modelling and Simulation in Engineering

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In this work, we propose a high-order algorithm based on the asymptotic numerical method (ANM) for the nonlinear elastic computation of helical structures without neglecting any nonlinear term. The nonlinearity considered in the following study will be a geometric type, and the kinematics adopted in this numerical modeling takes into account the hypotheses of Timoshenko and de Saint-Venant. The finite element used in the discretization of the middle line of this structure is curvilinear with twelve degrees of freedom. Using a simple example, we show the efficiency of the algorithm which was carried out in this context and which resides in the reduction of the number of inversions of the tangent matrix compared to the incremental iterative algorithm of Newton-Raphson.

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Local Parameterization and the Asymptotic Numerical Method

Cited by 19 publications

References 5 publications

Solving the incompressible fluid flows by a high‐order mesh‐free approach

Solving the incompressible fluid flows by a high‐order mesh‐free approach

Efficient high-order implicit solvers for the dynamic of thin-walled beams with open cross section under external arbitrary loadings

Numerical High-Order Model for the Nonlinear Elastic Computation of Helical Structures

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