Optimization of dispersive coefficients in the homogenization of the wave equation in periodic structures

Allaire, Grégoire; Yamada, Takayuki

doi:10.1007/s00211-018-0972-4

Cited by 20 publications

(23 citation statements)

References 53 publications

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“…Remark The ability of the second‐order homogenized model to predict the effective dispersion due to the microstructure is supported by the comparison with the Bloch‐wave homogenization method in the case ρ=1, see, for example, References . It is notably proven that the second‐order expansion coincides with the one obtained when (ω, k ) are the eigenfrequency and wavenumber associated with the first Bloch mode of the unit cell.…”

Section: Problem Setting and Topological Optimizationmentioning

confidence: 72%

“…This approach is commonly used in static to optimize the stiffness properties of elastic composites, for example, in References . It was recently extended to the optimization of the low‐frequency dynamics of regular and highly contrasted composites, and also to higher frequency regimes (in this latter case, both Bloch eigenvalue problems and asymptotic homogenization are needed to describe the wave motion) …”

Section: Introductionmentioning

confidence: 99%

“…In the present article, as in Reference , we focus on the low‐frequency dispersive effects of periodic materials, using the effective description provided by the second‐order asymptotic homogenization to set the optimization problem. The effective wave equation is then enriched with additional microstiffness and microinertia terms featuring higher order derivatives of the wavefield, similarly to the so‐called gradient elasticity models .…”

Section: Introductionmentioning

confidence: 99%

“…Efficient topological optimization procedures are therefore needed to provide optimal microstructures in more general settings, and the present article aims at providing a simple and efficient approach. It differs from Reference as (i) it permits nonconstant density (ii) it uses an optimization algorithm based on the topological derivatives of the effective coefficients rather than their shape derivatives, and (iii) it takes advantage of a fast Fourier transform (FFT)‐accelerated method to solve cell problems, instead of finite element methods.…”

Section: Introductionmentioning

confidence: 99%

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Tuning effective dynamical properties of periodic media by FFT‐accelerated topological optimization

Cornaggia

Bellis

2020

Numerical Meth Engineering

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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. K E Y W O R D Sdispersive waves, phononic crystals, second-order homogenization, topological derivatives 1 3178

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Section: Problem Setting and Topological Optimizationmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Tuning effective dynamical properties of periodic media by FFT‐accelerated topological optimization

Cornaggia

Bellis

2020

Numerical Meth Engineering

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“…Here, we focus on the similarity of the geometrical features of dispersive coefficient in periodic homogenization. As proofed in Allaire and Yamada's work [1], dispersive coefficient values in k times scaled unit cell are equivalent to k 2 times values of the original dispersive coefficient. Based on this feature and the similarity, the fictitious physical model is formulated as follows:…”

Section: Formulation Of the Fictitious Physical Modelmentioning

confidence: 87%

Thickness Constraints for Topology Optimization Using the Fictitious Physical Model

Yamada

2018

EngOpt 2018 Proceedings of the 6th International Conference on Engineering Optimization

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Thickness constraint is an important geometrical constraint in topology optimization methods. I present a novel approach of the thickness constraint based on the Fictitious Physical Model (FPM). The FPM is formulated using the similarity of the dispersive coefficient in high order homogenization. The thickness constraint is represented using the solutions of the linear partial deferential equation system. Its design sensitivity is derived using the adjoint variable method. Numerical example is shown to confirm the validity and utility of the proposed method using the level set-based topology optimization method. The main advantage of the proposed method is the allowance of thickness constraint violations during the optimization procedure. Furthermore, the thickness is computed without computing minimum distances from the boundaries of target shape.

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Symbolic homogenization and structure optimization for a periodically perforated cylindrical shell

et al. 2021

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The main focus of this paper lies in the drastic model reduction for a complex multi‐scale problem of linear elasticity. The core of the work lies in the structural optimization of periodic perforated cylindrical shells under a given load on a small portion of the surface. Periodic structure of the shell is a frame of beams. Algorithm presented in the paper utilizes two our recent analysis works on homogenization and dimension reduction of the problem in the perforated 3D shell to two‐dimensional homogenized shell, and then dimension reduction w.r.t. the beam thickness in auxiliary cell‐problems to problem on a frame of beams, by asymptotic method. In this paper, we found the analytical solution for the orthotropic cylindrical shell, obtained in the limit, by variable separation and Fourier analysis. The solution depends explicitly on the effective properties, which are computed symbolically, and on the design variables of the structure. Cell‐problems are solved by symbolic means and the structural design for this type of the loading has been optimized. Moreover, this yields a semi‐analytic optimization problem and the practical usage of the underlying theoretical derivation. We stress that the homogenization and dimension reduction of a shell with holes and the analytic solution to the corresponding macroscopic problem are new.

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Optimization of dispersive coefficients in the homogenization of the wave equation in periodic structures

Cited by 20 publications

References 53 publications

Tuning effective dynamical properties of periodic media by FFT‐accelerated topological optimization

Tuning effective dynamical properties of periodic media by FFT‐accelerated topological optimization

Thickness Constraints for Topology Optimization Using the Fictitious Physical Model

Symbolic homogenization and structure optimization for a periodically perforated cylindrical shell

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