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

International audienceIn 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

Object-oriented design to automate a high order non-linear solver based on asymptotic numerical method

International audienceThe Manitoo library is devoted to the resolution of analytical non-linear problems using a high order method called asymptotic numerical method. We describe here the Object Oriented design of this library and especially the choices made to obtain a quite generic and flexible numerical solver. Through classical examples, we present a comparison with some existing tools implemented in Matlab and Fortran 77