Rapid computation of effective conductivity of 2D composites by equivalent circuit and spectral methods
This work presents models for homogenizing or finding the effective transport or mechanical properties of microscale composites formed from highly contrasting phases described on a grid. The methods developed here are intended for engineering applications where speed and geometrical flexibility are...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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AIMS Press
2022-05-01
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| Series: | Mathematics in Engineering |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/mine.2022020?viewType=HTML |
| _version_ | 1818554588645556224 |
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| author | Ben J. Ransom Dean R. Wheeler |
| author_facet | Ben J. Ransom Dean R. Wheeler |
| author_sort | Ben J. Ransom |
| collection | DOAJ |
| description | This work presents models for homogenizing or finding the effective transport or mechanical properties of microscale composites formed from highly contrasting phases described on a grid. The methods developed here are intended for engineering applications where speed and geometrical flexibility are a premium. A canonical case that is mathematically challenging and yet can be applied to many realistic materials is a 4-phase 2-dimensional periodic checkerboard or tiling. While analytic solutions for calculating effective properties exist for some cases, versatile methods are needed to handle anisotropic and non-square grids. A reinterpretation and extension of an existing analytic solution that utilizes equivalent circuits is developed. The resulting closed-form expressions for effective conductivity are shown to be accurate within a few percent or better for multiple cases of interest. Secondly a versatile and efficient spectral method is presented as a solution to the 4-phase primitive cell with a variety of external boundaries. The spectral method expresses the solution to effective conductivity in terms of analytically derived eigenfunctions and numerically determined spectral coefficients. The method is validated by comparing to known solutions and can allow extensions to cases with no current analytic solution. |
| first_indexed | 2024-12-12T09:41:57Z |
| format | Article |
| id | doaj.art-cac5d12d444d4fe48b6ac2714aca3e39 |
| institution | Directory Open Access Journal |
| issn | 2640-3501 |
| language | English |
| last_indexed | 2024-12-12T09:41:57Z |
| publishDate | 2022-05-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematics in Engineering |
| spelling | doaj.art-cac5d12d444d4fe48b6ac2714aca3e392022-12-22T00:28:33ZengAIMS PressMathematics in Engineering2640-35012022-05-014312410.3934/mine.2022020Rapid computation of effective conductivity of 2D composites by equivalent circuit and spectral methodsBen J. Ransom0Dean R. Wheeler1Department of Chemical Engineering, Brigham Young University, Provo, Utah 84602, USADepartment of Chemical Engineering, Brigham Young University, Provo, Utah 84602, USAThis work presents models for homogenizing or finding the effective transport or mechanical properties of microscale composites formed from highly contrasting phases described on a grid. The methods developed here are intended for engineering applications where speed and geometrical flexibility are a premium. A canonical case that is mathematically challenging and yet can be applied to many realistic materials is a 4-phase 2-dimensional periodic checkerboard or tiling. While analytic solutions for calculating effective properties exist for some cases, versatile methods are needed to handle anisotropic and non-square grids. A reinterpretation and extension of an existing analytic solution that utilizes equivalent circuits is developed. The resulting closed-form expressions for effective conductivity are shown to be accurate within a few percent or better for multiple cases of interest. Secondly a versatile and efficient spectral method is presented as a solution to the 4-phase primitive cell with a variety of external boundaries. The spectral method expresses the solution to effective conductivity in terms of analytically derived eigenfunctions and numerically determined spectral coefficients. The method is validated by comparing to known solutions and can allow extensions to cases with no current analytic solution.https://www.aimspress.com/article/doi/10.3934/mine.2022020?viewType=HTMLcoarse grainingcomposite materialsconductivitycheckerboardpercolationrenormalization |
| spellingShingle | Ben J. Ransom Dean R. Wheeler Rapid computation of effective conductivity of 2D composites by equivalent circuit and spectral methods Mathematics in Engineering coarse graining composite materials conductivity checkerboard percolation renormalization |
| title | Rapid computation of effective conductivity of 2D composites by equivalent circuit and spectral methods |
| title_full | Rapid computation of effective conductivity of 2D composites by equivalent circuit and spectral methods |
| title_fullStr | Rapid computation of effective conductivity of 2D composites by equivalent circuit and spectral methods |
| title_full_unstemmed | Rapid computation of effective conductivity of 2D composites by equivalent circuit and spectral methods |
| title_short | Rapid computation of effective conductivity of 2D composites by equivalent circuit and spectral methods |
| title_sort | rapid computation of effective conductivity of 2d composites by equivalent circuit and spectral methods |
| topic | coarse graining composite materials conductivity checkerboard percolation renormalization |
| url | https://www.aimspress.com/article/doi/10.3934/mine.2022020?viewType=HTML |
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