An efficient implementation of ANM with FEM for solving nonlinear solid thermo-elastic problems in the FreeFEM++ language.
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Abstract: Asymptotic Numerical Method (ANM), based on a perturbation technique, is known to be a robust continuation method for solving nonlinear problems dependent on a loading parameter [1]. ANM is often associated with the Finite Element Method to analyze nonlinear solid or fluid mechanical problems. In this paper, we will show the implementation of ANM in the FreeFEM++ language in the case of elastic or thermo elastic nonlinear problems. FreeFEM++ uses a high-level language that makes it easy to derive a variational formulation and implement the ANM algorithm. A detailed description of the ANM algorithm and the key features of its FreeFEM++ implementation will be presented in the case of a film/substrate systems subjected to thermal shrinkage of its core. Thanks to the coupling between FreeFEM++ and efficient numerical libraries like MUMPS, PETsc, and others, first attempts to parallelize the ANM/FEM algorithm written in FreeFEM++ using a multigrid method will be presented. References: [1] B. Cochelin, N. Damil, M. Potier-Ferry, “Méthode Asymptotique Numérique ”, Hermès Lavoisier: Paris, France, 2007. [2] P. Ventura, F. Hecht, M. Potier-Ferry, H. Zahrouni, F. Xu, H. Azzayani, M. Brun, A-K Chau, “Mathematical Aspects of ANM/FEM Numerical Model, Applied to Nonlinear Elastic, and Thermo Elastic Analysis of Wrinkles in Film/Substrate Systems, and a New Implementation in the FreeFEM++ Language”, Mathematics 2025, 13(19), 3063; https://doi.org/10.3390/math13193063.
