A numerical continuation method based on Padé approximants

Elhage-Hussein, Ahmad; Potier‐Ferry, Michel; Damil, N.

doi:10.1016/s0020-7683(99)00323-6

Cited by 102 publications

(99 citation statements)

References 24 publications

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“…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. In Figure 4, the evaluation of three contact forces according to λ is plotted and compared to Abaqus results (reference curves).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…If 1/α = 0 the penetration is prevented and the relation (15) leads to a barrier function as in [30]. The barrier contact law (1/α = 0) has been used in many papers about ANM, for instance [31] [13]. By combining (15) with the projection equations of subsection 2.1.2, the contact normal pressure R N can be considered as an implicit function of the displacement u.…”

Section: Contact Modeling : the Mechanical Partmentioning

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: Contact Modeling : the Mechanical Partmentioning

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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“…The same remark is observed for the other orders. The use of Padé representation reduces the number of steps as has been observed in [3].…”

Section: Numerical Comparison and Discussionmentioning

confidence: 99%

“…The Padé representation allows to increase the range of validity of the series representation (2.2) [2,3]. This rational representation can be written in the form: …”

Section: Local Parameterizations In the Anmmentioning

confidence: 99%

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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“…The study of anelastic problems by ANM has begun with deformation plasticity [7,8]. Unilateral frictionless contact is the simplest example of non smooth mechanics and ANM has proved to be efficient in this case [9,10].…”

Section: Introductionmentioning

confidence: 99%

Solving fracture problems using an asymptotic numerical method

Daridon

Wattrisse

Chrysochoos

et al. 2011

Computers & Structures

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The present work deals with the use of asymptotic numerical methods (ANM) to manage crack onset and crack growth in the framework of Continuum Damage Mechanics (CDM). More specifically, an application of regularization techniques to a 1D cohesive model is proposed. The standard "triangle" damageable elastic model, often used in finite element codes to describe fracture of brittle materials, was chosen. Results associated with load-unload cycle showed that ANM is convenient to take numerically this specific non regular behaviour into account.Moreover, the present paper also shows that the chosen damageable interface model can be introduced in the generalized standard material formalism which unables us to define a complete energy balance associated with the damage process. In such a framework, the new damage state variable is a displacement. Finally, a 1D finite element application to a simple elastic damageable structure is shown to emphasize the potentialities of such an approach.

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A numerical continuation method based on Padé approximants

Cited by 102 publications

References 24 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

Local Parameterization and the Asymptotic Numerical Method

Solving fracture problems using an asymptotic numerical method

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