A generic approach for the solution of nonlinear residual equations. Part I: The Diamant toolbox

Koutsawa, Yao; Charpentier, Isabelle; Daya, El Mostafa; Cherkaoui, Mohammed

doi:10.1016/j.cma.2008.09.003

Cited by 45 publications

(43 citation statements)

References 17 publications

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“…This paper exploits the Diamant abilities for the solution of complex nonlinear eigenvalue problems. To that end, we revise, combine and extend theoretical arguments discussed in [8,17]. On theoretical aspects, a new Diamant-based ANM driver is designed for nonlinear problems involving a homotopy, that is a deformation of the original problem into a simpler one whose solutions are known.…”

Section: Introductionmentioning

confidence: 99%

A generic approach for the solution of nonlinear residual equations. Part II: Homotopy and complex nonlinear eigenvalue method

Bilasse

Charpentier

Daya

et al. 2009

Computer Methods in Applied Mechanics and Engineering

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Section: Introductionmentioning

confidence: 99%

A generic approach for the solution of nonlinear residual equations. Part II: Homotopy and complex nonlinear eigenvalue method

Bilasse

Charpentier

Daya

et al. 2009

Computer Methods in Applied Mechanics and Engineering

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“…With the chosen material parameters for the interface, we observe that there is no localisation of the damage with mesh resizing. In further development, we then propose to tackle the complete crack propagation problem with a FE implementation of the ANM based on Automatic Differentiation technique [30,31]. …”

Section: Numerical Resultsmentioning

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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“…Recently, DIAMANT approach has been proposed and applied to several kinds of academic problems [32], [21], [27]. Right-hand side terms and tangent matrix occurring in ANM are obtained by application of AD technique.…”

Section: Diamant Approachmentioning

confidence: 99%

“…Finally, combination of (28), (29) and (30) and the obvious k th -derivative of the problem (27) leads, at a given order k, to the linear set of equations satisfied by the mixed unknown u k , R N k and the k th component λ k of the load parameter:…”

Section: Unilateral Contact With An Elastic 3d Bodymentioning

confidence: 99%

“…This allows for the automation of the implementation of various complex tool shapes, tool motions or gap-pressure relations. There have been many works about Automatic Differentiation [20] [21] and about its applications to partial differential equations using or not finite element method ( see for instance [22], [23], [24], [25], [26],... ), but only few papers have described coupling of AD and ANM [27] [28] [21] [29].…”

Section: Introductionmentioning

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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A generic approach for the solution of nonlinear residual equations. Part I: The Diamant toolbox

Cited by 45 publications

References 17 publications

A generic approach for the solution of nonlinear residual equations. Part II: Homotopy and complex nonlinear eigenvalue method

A generic approach for the solution of nonlinear residual equations. Part II: Homotopy and complex nonlinear eigenvalue method

Solving fracture problems using an asymptotic numerical method

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

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