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The equations that govern elastostatic equilibrium between a prescribed force field and an unknown displacement field for materials with periodic skeletal micro-structures are studied. It is shown that as the size of the micro-structure tends to zero, the displacement field will converge to the solution of a constant coefficient partial differential equation. This equation is shown to be either a classical or a micro-polar continuum elasticity equation, depending on the micro-structural geometry and the nature of the external load field. Convergence is proved for representative model problems in Sobolev energy norms and in the maximum norm. In addition, it is shown that by considering pseudo-differential homogenized equations, any order of convergence can be achieved. [ABSTRACT FROM AUTHOR]

The classical problem of homogenization deals with elliptic operators in periodically oscillating media of small period ℇ>0 and the asymptotic behavior of solution uℇ of boundary value problems associated with such operators. In a previous work [Conca et al., SIAM J. Math. Anal. 33 (2002), 1166–1198], the authors introduced what is called Bloch approximation which provided energy norm approximation for the solution in $\mathbb{R}^{N}$. This paper continues with the above development and proposes a modified Bloch approximation. This function takes into account boundary effects. Its connection with the first order classical correctors is also established with the corresponding error estimate. All the proofs are worked out entirely in the Fourier–Bloch space. [ABSTRACT FROM AUTHOR]

This paper presents two methods for the numerical solution of the classical homogenization problem of elliptic operators with periodically oscillating coefficients. The numerical solution of such problems is difficult because of the presence of rapidly oscillating coefficients. The first method based on the classical one which consists of the homogenized solution, the first- and second-order correctors, whereas the second one is based on the Bloch wave approach. Further, for the calculation of the homogenized coefficients and some auxiliary functions involved in this method, we applied both methods and compared their accuracies. The Bloch approximation consists in determining an oscillating integral, numerically. The Bloch method provides a better approximation to the exact solution than the classical first-order corrector term in the smooth coefficients case. Moreover, we provided Taylor approximations for the Bloch approximation function and implemented it numerically. In order to show the efficiency of these methods, exhaustive numerical examples in both one and two-dimensional cases are presented. [Copyright &y& Elsevier]