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Tuning effective dynamical properties of periodic media by FFT‐accelerated topological optimization
Author(s) -
Cornaggia Rémi,
Bellis Cédric
Publication year - 2020
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.6352
Subject(s) - homogenization (climate) , topology optimization , fourier transform , scalar (mathematics) , fast fourier transform , dispersion (optics) , wave propagation , mathematics , mathematical optimization , asymptotic homogenization , anisotropy , computer science , mathematical analysis , topology (electrical circuits) , algorithm , physics , optics , geometry , finite element method , biodiversity , ecology , combinatorics , composite number , biology , thermodynamics
Summary This works concerns the propagation of waves in periodic media, whose microstructure is optimized to obtain specific dynamical properties (typically, to maximize the dispersion in given directions). The present study, focusing on scalar waves in two dimensions, for example, antiplane shear waves, aims at setting a generic optimization framework. The proposed optimization procedure relies on a number of mathematical and numerical tools. First, the two‐scale asymptotic homogenization method is deployed up to second‐order to provide an effective dispersive model. Simple dispersion indicators and cost functionals are then considered on the basis of this model. Then, the minimization of these functionals is performed thanks to an algorithm that relies on the concept of topological derivative to iteratively perform phase changes in the unit cell characterizing the material. Finally, fast Fourier transform‐accelerated solvers are extensively used to solve the cell problems underlying the homogenized model. To illustrate the proposed approach, the resulting procedure is applied to the design of anisotropic media with maximal dispersion in specific directions, and to the reconstruction of unknown microstructures from effective phase velocity data.