Multiscale Asymptotic Analysis And Numerical Simulation For Composite Structures

In recent years, composite materials have been widely used for their excellent per-formance and outstanding designability. The broad application of composite materials inengineering calls for higher requirements for the comprehensive performance of compositematerials. Therefore, it is necessary to study the microstructure of composite materials.The research on composite materials’ physics-mathematical models, performance predic-tion and optimization design under certain conditions has become an interdisciplinary sub-ject. In the recent100years, aiming at the processing of composite materials and theirproduct structures, numerous methods have been proposed in mechanics, mathematics andengineering. Besides macro mechanical method and micro mechanical method on the as-pect of mechanics, from the mathematical point of view, there are homogenization theory,multiscale asymptotic analysis method and so on. The multiscale method based on the ho-mogenization theory can not only be applied to forecast the macro-micro performance ofcomposite materials efciently, but also quantitatively describes the micro structural char-acteristics and efectively captures the local response of materials.In this paper, in order to study the efect of arrangement style of cell on the macro-scopic properties of materials, we will discuss multiscale asymptotic analysis for one classof staggered arrangement structure, which is diferent to the traditional periodic structure.The details are presented as follows:To discuss staggered arrangement structure, the concept of staggered continuationfunction is introduced in the first part. Using coordinate transformation, we prove the rele-vant fundamental theorems of multiscale asymptotic analysis method for staggered arrange-ment structure. Then, taking a kind of elliptic equations as an example, we make multiscaleasymptotic analysis and obtain the first order and the second order multiscale asymptotic ex-pansion solutions. By introducing error function and analyzing its concrete form, we provetheoretically the multiscale convergence rate with the orderε12under appropriate regular-ity hypothesis regularity hypothesis. The analysis shows that the periodic cell problem ofstaggered arrangement structure requires special boundary conditions, which is apparentlydiferent from the traditional periodic structure.In the second part, on the basis of theoretical analysis and finite element technique,two kinds of multiscale finite element algorithms are given for solving elliptic equationwith staggered arrangement structure. And we demonstrate the equivalence of the twoalgorithms by the theory of multiscale asymptotic analysis. Numerical simulations arecarried out to validate the multiscale finite element method algorithm. Moreover, the efectof diferent staggered arrangement on the equivalent performance of materials is discussed. The result indicates that the staggered arrangement is superior to the periodic arrangementin thermal insulation structure.