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.
Citation: Ben J. Ransom, Dean R. Wheeler. Rapid computation of effective conductivity of 2D composites by equivalent circuit and spectral methods[J]. Mathematics in Engineering, 2022, 4(3): 1-24. doi: 10.3934/mine.2022020
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.
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Ben J. Ransom, Dean R. Wheeler. Rapid computation of effective conductivity of 2D composites by equivalent circuit and spectral methods[J]. Mathematics in Engineering, 2022, 4(3): 1-24. doi: 10.3934/mine.2022020
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