9/11/2022 Friday 11h30 to 11h55

**Pascal VENTURA**, Michel Potier-Ferry

Laboratoire LEM3

Université de Lorraine

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*How to accelerate and improve the continuation algorithm called Asymptotic Numerical Method using FreeFem++*

Abstract :

The Asymptotic Numerical Method (ANM) [1] is a robust continuation algorithm for solving non-linear problems that depends on a parameter. It has been applied to various fields in solids and fluid mechanics, in physics, in finance, …

We will first explain why FreeFem++ is a very efficient numerical tool to implement the ANM algorithm in order to solve non-linear solid mechanical problems.

Then, new encouraging results obtained regarding convergence acceleration techniques of ANM applied to film/substrate systems will be given.

Recently, Henderson et alt. [2] have shown that real solutions of real continuation of non-linear problems frequently have complex solutions bifurcating from then. These complex solutions might connect later to new real solutions which cannot been discovered using the real continuation algorithm. We will show our first attempt to generalize the ANM algorithm for complex continuation, and the first results obtained for a film/substrate thermo-mechanical problem.

[1] B. Cochelin, N. Damil, M. Potier-Ferry, ″Méthode asymptotique numérique″, Hermes, Lavoisier 2007.

[2] M. E. Henderson, H. B. Keller, ″Complex bifurcation from real paths, SIAM Journal on Applied Mathematics″, 50(2), 460-482.