High-order prediction–correction algorithms for unilateral contact problems

Aggoune, Woihida; Zahrouni, Hamid; Potier‐Ferry, Michel

doi:10.1016/j.cam.2003.05.015

Cited by 18 publications

(18 citation statements)

References 9 publications

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“…Let us mention that the efficiency of Asymptotic Numerical Method for contact problem has been assessed in several previous papers, see for instance [12] [17] and this will not be repeated here. Few numerical results are presented in Figures 3, 4, 5 and 6.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…One sees many slope discontinuities, due to appearance or disappearance of new nodes in contact. To get the response curve of Figure 3 with an ANM order equals 20 and the accuracy parameter δ = 10 −6 , 95 ANM steps have been necessary, see [12] or [17]. Note that the number of steps could be reduced by using Padé approximants [31] or by changing the accuracy parameter δ or the regularization parameters, η or α.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Automatic solver for non‐linear partial differential equations with implicit local laws: Application to unilateral contact

Lejeune

Boudaoud

Potier‐Ferry

et al. 2013

Numerical Meth Engineering

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In general, nonlinear continuum mechanics combine global balance equations and local constitutive laws. In this work, frictionless contact between a rigid tool and a thin elastic shell is considered. This class of boundary value problems involves two nonlinear algebraic laws: the first one gives explicitly the stress field as a function of the strain throughout the continuum part, while the second one is a nonlinear equation relating the contact forces and the displacement at the boundary.Given the fact that classical computational approaches sometimes require significant effort in implementation of complex nonlinear problems, a computation technique based on Automatic Differentiation of constitutive laws is presented in this paper. The procedure enables to compute automatically the higher-order derivatives of these constitutive laws and thereafter to define Taylor series that are the basis of the continuation technique called Asymptotic Numerical Method. The algorithm is about the same with an explicit or an implicit constitutive relation. In the modeling of forming processes, many tool shapes can be encountered. The presented computational technique permits an easy implementation of these complex surfaces, for instance in a finite element code: the user is only required to define the tool geometry and the computer is able to get the higher-order derivatives.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Numerical Resultsmentioning

confidence: 99%

Automatic solver for non‐linear partial differential equations with implicit local laws: Application to unilateral contact

Lejeune

Boudaoud

Potier‐Ferry

et al. 2013

Numerical Meth Engineering

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“…One can also insert correction phases at the beginning of some steps. Especially, ANM‐corrective algorithms are available that are based on homotopy transformations . In this manner, one defines a continuation technique allowing the determination of the whole solution branch.…”

Section: Branch Solutions For Nonlinear Poisson Problemsmentioning

confidence: 99%

Bifurcation indicator based on meshless and asymptotic numerical methods for nonlinear poisson problems

Tri

Zahrouni

Potier‐Ferry

2014

Numerical Methods Partial

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We propose in this work new algorithms associating asymptotic numerical method and meshless discretization (MFS‐MPS: Method of fundamental solutions‐Method of particular solutions) to compute branch solutions of nonlinear Poisson problems. To detect singular points on these branches, geometrical indicator, Padé approximants, and analytical bifurcation indicator are proposed. Numerical applications show the robustness and the effectiveness of the proposed algorithms. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 978–993, 2014

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“…Several numerical tests of solid and fluid mechanics show that the representation of the solution by rational fractions permits one reducing about 50% the step number in comparison with the power series solution. It is also possible to perform iterative procedures using a high-order algorithm for the prediction and a high-order algorithm for the correction (see [3,4,29]). Moreover, efficient algorithms have been established to detect bifurcation points and to compute post-buckling responses of thin structures [9,10,51].…”

Section: Introductionmentioning

confidence: 99%

Solving hyperelastic material problems by asymptotic numerical method

2010

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This paper presents a numerical algorithm based on a perturbation technique named asymptotic numerical method (ANM) to solve nonlinear problems with hyperelastic constitutive behaviors. The main advantages of this technique compared to Newton-Raphson are: (a) a large reduction of the number of tangent matrix decompositions; (b) in presence of instabilities or limit points no special treatment such as arc-length algorithms is necessary. The ANM uses high order series approximation with auto-adaptive step length and without need of any iteration. Introduction of this expansion into the set of nonlinear equations results into a sequence of linear problems with the same linear operator. The present work aims at providing algorithms for applying the ANM to the special case of compressible and incompressible hyperelastic materials. The efficiency and accuracy of the method are examined by comparing this algorithm with Newton-Raphson method for problems involving hyperelastic structures with large strains and instabilities.

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High-order prediction–correction algorithms for unilateral contact problems

Cited by 18 publications

References 9 publications

Automatic solver for non‐linear partial differential equations with implicit local laws: Application to unilateral contact

Automatic solver for non‐linear partial differential equations with implicit local laws: Application to unilateral contact

Bifurcation indicator based on meshless and asymptotic numerical methods for nonlinear poisson problems

Solving hyperelastic material problems by asymptotic numerical method

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