Abstract
BackgroundIn the framework of periodic homogenization, the conduction problem can be formulated as an integral equation whose solution can be represented by a Neumann series. From the theory, many efficient numerical computation methods and analytical estimations have been proposed to compute the effective conductivity of composites.MethodsWe combine a Fast Fourier Transform (FFT) numerical method based on the Neumann series and analytical estimation based on the integral equation to solve the problem. Specifically, the analytical approximation is used to estimate the remainder of the series.ResultsFrom some numerical examples, the coupling method have shown to improve significantly the original FFT iteration scheme and results are also superior to the analytical estimation.ConclusionsWe have proposed a new efficient computation method to determine the effective conductivity of composites. This method combines the advantages of the FFT numerical methods and the analytical estimation based on integral equation.
Highlights
In the framework of periodic homogenization, the conduction problem can be formulated as an integral equation whose solution can be represented by a Neumann series
Results and discussion we study the results coming from the implementation of the coupled method for the case of a simple cubic system
1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Volume fraction f periodic problem with prescribed temperature gradient E is solved by three approaches: the NIH approximation (31), the conventional primal iterative scheme (PIS), and the coupled method
Summary
In the framework of periodic homogenization, the conduction problem can be formulated as an integral equation whose solution can be represented by a Neumann series. Many efficient numerical computation methods and analytical estimations have been proposed to compute the effective conductivity of composites. Composite materials can exist in nature or be fabricated by purpose. Due to their technological importance, micromechanical approaches are developed to determine the overall behavior of composites from the properties of their constituents. Finite element method (FEM) and boundary element method (BEM) are widely used for homogenization problems. These methods have been reported in numerous works [8,9,10,11,12]. A more recent method, introduced in the 1990s and described thereafter, uses extensively the Fourier transform and the introduction of a ‘reference material’
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