Homogenization of reticulated structures
Development of the homogenization theory of lattice-type structures began in early 80th in papers by G. P. Panasenko; many examples and applications were presented in the monograph by [\textit{N. S. Bakhvalov} and \textit{G. P. Panasenko}, Averaging processes in periodic media, Nauka, Moskva (1984; Zbl 0607.73009) (Russian) and Kluwer, Dordrecht (1998; Zbl 0692.73012)]. D. Cioronescu and J. Saint Jean Paulin began in 1986 the investigation of the same structures with a different approach. Their results (as well as results of other authors) are collected in this interesting and from the point of view of applications important book, with a comprehensive and unified presentation. The book consists of six chapters. In Chapter 1 an introduction to homogenization of perforated media, including multi-scale and Tartar's method is given. In Chapter 2 reticulated structures, gridworks (networks), tall, honeycomb and reinforced structures are considered. All these cases are characterized by a small thickness of the material. Because of that fact it is possible to find out the homogenized coefficients explicitly. The authors developed a general method for the treatment of structures with very complicated geometry. Chapter 3 deals with thermal problem for gridworks, consisting of regular arrays of thin wires. Here periodicity occurs only in two dimensions and the third dimension appears as a new small parameter. In Chapter 4 the linear elasticity problem for the same structure is considered; these are previously unpublished topics. In Chapter 5 and Chapter 6 the thermal and elasticity problem, respectively, for thin tall structures is treated; in this case the thickness of the structure is a new parameter. As the authors pointed out in the Preface, the book is written from the point of view of applied mathematicians, attention being paid to the mathematical rigour, convergence results and error estimates.
scientific article; zbMATH DE number 1313468