Bloch wave approach to almost periodic homogenization and approximations of effective coefficients

Sista Sivaji Ganesh; Vivek Tewary

doi:10.3934/dcdsb.2021119

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2022, Volume 27, Issue 4: 1989-2024. Doi: 10.3934/dcdsb.2021119

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Bloch wave approach to almost periodic homogenization and approximations of effective coefficients

Sista Sivaji Ganesh and
Vivek Tewary^,

Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

^* Corresponding author: Vivek Tewary, Present Address: TIFR Centre for Applicable Mathematics, Bengaluru, India
^* Corresponding author: Vivek Tewary, Present Address: TIFR Centre for Applicable Mathematics, Bengaluru, India

Received: August 2020

Revised: February 2021

Early access: April 23, 2021

Published online: April 23, 2021

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Abstract

Bloch wave homogenization is a spectral method for obtaining effective coefficients for periodically heterogeneous media. This method hinges on the direct integral decomposition of periodic operators, which is not available in a suitable form for almost periodic operators. In particular, the notion of Bloch eigenvalues and eigenvectors does not exist for almost periodic operators. However, we are able to recover the almost periodic homogenization result by employing a sequence of periodic approximations to almost periodic operators. We also establish a rate of convergence for approximations of homogenized tensors for a class of almost periodic media. The results are supported by a numerical study.

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Mathematics Subject Classification: Primary: 47A55, 35J15, 35B27, 34C27.

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Figure 1. The error $ |A^{R,*}-A^*| $ for approximations to homogenized tensor using periodic correctors in log-log scale for the functions $ A_1 $ and $ A_2 $ with respect to $ R $.

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Figure 2. The error $ |A^{R,*}-A^*| $ for approximations to homogenized tensor using periodic correctors in log-log scale for the function $ A_3 $ with respect to $ R $.

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Figure 3. The error $ |A^{R,D,*}-A^*| $ for Dirichlet approximations in log-log scale for the functions $ A_1 $ and $ A_2 $ with respect to $ R $.

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Figure 4. The error $ |A^{R,D,*}-A^*| $ for Dirichlet approximations in log-log scale for the function $ A_3 $ with respect to $ R $.

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Figure 5. The averaged $ L^2 $ norm of the difference of the gradients $ E(R) = \left(\mathit{{\rlap{-} \smallint }}_{Y_{R}}|\nabla w^{R,D,e_1}(y)-\nabla w^{R,e_1}(y)|^2\,dy\right)^{1/2} $ in log-log scale for the correctors corresponding to the periodic matrices $ A_1 $ and $ A_2 $ plotted as a function of $ R $.

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Figure 6. The averaged $ L^2 $ norm of the difference of the gradients $ E(R) = \left(\mathit{{\rlap{-} \smallint }}_{Y_{R}}|\nabla w^{R,D,e_1}(y)-\nabla w^{R,e_1}(y)|^2\,dy\right)^{1/2} $ in log-log scale for the correctors corresponding to the quasiperiodic matrix $ A_3 $ plotted as a function of $ R $.

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Figure 7. The absolute error $ |A^{R,D,*}-A^{R,*}| $ in log-log scale for the periodic matrices $ A_1 $ and $ A_2 $ plotted as a function of $ R $.

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Figure 8. The absolute error $ |A^{R,D,*}-A^{R,*}| $ in log-log scale for the quasiperiodic matrix $ A_3 $ plotted as a function of $ R $.

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