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

Abstract : In general, non-linear 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 non-linear algebraic laws: the first one gives explicitly the stress field as a function of the strain throughout the continuum part, whereas the second one is a non-linear 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 non-linear 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 the Taylor series that are the basis of the continuation technique called asymptotic numerical method. The algorithm is about the same with an explicit or implicit constitutive relation. In the modelling 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 obtain the higher-order derivatives.
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Arnaud Lejeune, Hakim Boudaoud, Michel Potier-Ferry, Isabelle Charpentier, Hamid Zahrouni. Automatic solver for non-linear partial differential equations with implicit local laws: Application to unilateral contact. International Journal for Numerical Methods in Engineering, Wiley, 2013, 94, pp.850 - 867. ⟨10.1002/nme.4483⟩. ⟨hal-00982813⟩

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