This special issue is devoted to the homogenization technique and the associated topics. Traditionally, the homogenization deals with the heterogeneous media, described by the partial differential equations with rapidly oscillating coefficients. The composite materials, the porous media, the lattice-like structures, periodic trusses, gridworks, frameworks provide a non-exhaustive list of examples of heterogeneous media. The small parameter ε in such models is the ratio of the microscopic and the macroscopic scales. The coefficients of the partial differential equations depend on the so-called fast variable, that is, the spatial variable x divided by the small parameter ε. The homogenization is an asymptotic method justifying the passage from the micro- to the macro-scale. The macroscopic model is normally a boundary-value problem for an equation with constant or smooth coefficients set for the leading term of the asymptotic expansion of the solution of the equation describing the microscopic behaviour. The same approach could be applied in the case of so-called thin structures: bars, plates, channels, tubes and the ‘bundles’ (unions) of these structural elements. The small parameter is the ratio of the thickness and the length of the structural elements. In this case, the limit macroscopic models are normally models of the reduced dimension. So, the homogenization plays the part of the dimension reduction method, justifying the passage from the three-dimensional models to the two-dimensional or one-dimensional models, or from the two-dimensional to one-dimensional models. The homogenization can be applied to the discrete (atomistic) models justifying the passage to the continuous models. All these aspects of the homogenization are presented in this selected set of papers.
The special issue starts with the paper by Dag Lukkassen and Annette Meidel on effective (macroscopic) properties of some self-similar chessboard-type structures with infinitely many microscales. It is well-known Dykhne formula in which the clasical chessboard periodic structure the effective conductivity is the geometric mean of the conductivities of ‘white’ and ‘black’ materials. The authors show that in the general case this formula does not work, but it is possible to construct sequences of approximations to the effective coefficients. The next paper by R. Hochmuth is devoted to the homogenization of a nonlocal problem: the Laplace equation in a periodically perforated domain with an integral operator on the boundaries of the holes. This problem models the heat transfer in a tissue with blood vessels. The paper by Grigory Panasenko and Natalia Pshenitsyna deals with the problem of homogenization of the Burgers equation with the integral ‘relaxation’ term in the right-hand side. The coefficients of this equation oscillate not necessarily periodically. This equation has not been studied earlier mathematically. The macroscopic model is constructed and the estimate of the difference between the exact and asymptotic solution is proved. Further, Uldis Raitums and Juris Roberts Kalnin contribute a short paper on the homogenization of composite materials with non-standard interface conditions between the phases. Making a special change of unknown functions they reduce it to a problem which can be homogenized by means of the G-convergence. The new approach to the homogenization technique via blow-up is developed in the paper by Andrea Brides, Michail Maslennikov and Laura Sigalotti. In particular, the authors apply this new techniques for nonlinear problems and obtain the convergence results. The spectral problem for the Laplace-Beltrami operator on Riemannian manifold is considered in the paper by Andrii Khrabustovskyi. The manifold has a microscopic structure: it contains a great number of small holes glued to some spherical segments of a diameter of order ϵ. The spectrum of the Laplace-Beltrami operator converges to the spectrum of some Laplace type operator as ϵ tends to zero.
The next two papers deal with thin structures. The paper by Catherine Choquet and Andro Mikelić studies a time-dependent reactive flow with the dominant Peclet number through a thin channel. Starting from a two-dimensional in space model they pass to the one-dimensional in space limit problem and obtain the estimates of the difference between the exact and asymptotic solution. The article by Grigory Panasenko and Marie-Claude Viallon discusses the finite volume implementation of the method of partial asymptotic domain decomposition for thin domains. The estimates of the error are proved for the difference between the exact solution and the numerical one. It depends on the small parameter ε and the mesh step h.
The homogenization technique for the discrete or discretized models has been applied in the next two papers. Marthe Betoue Etoughe and Grigory Panasenko introduce the hybrid semi-discrete and semi-continuous models as approximations of the discrete models in the case when the coefficients have a singular behaviour in some small domain close to the boundary. The estimates between the solutions of the discrete and semi-discrete models justifies this approach. The paper by Gregory A. Chechkin, Dag Lukkassen and Annette Meidell justify the Sapondzhyan-Babuska paradox for the elasticity equations in a polygonal plate as the length of the side of polygon tends to zero. To this end, an asymptotic solution of the elasticity equations in the polygonal plate is constructed.
The paper by J. Casado-Diaz, J. Couce-Calvo, M. Luna-Laynez and J.D. Martin-Gomez provides a numerical analysis of an optimal design problem for a non-linear cost in the gradient. The homogenization technique is applied in order to get the relaxed formulation. Unfortunately, this relaxed problem is not explicitely known. The paper shows that in the discrete approximation one can replace this functional by one of its bounds.
This special issue is intended to stimulate the discussion and further development of the homogenization and asymptotic analysis of thin structures. I would like to express my appreciation to all the reviewers for their critical and encouraging remarks and to the authors for their participation. Special thanks to Professor Robert Gilbert, the Editor in Chief, for his guidance and encouragement.