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In recent years the traditional subject of continuum mechanics has grown rapidly and many new techniques have emerged. This text provides a rigorous, yet accessible introduction to the basic concepts of the network approximation method and provides a unified approach for solving a wide variety of applied problems. As a unifying theme, the authors discuss in detail the transport problem in a system of bodies. They solve the problem of closely placed bodies using the new method of the network approximation for partial differential equations with discontinuous coefficients. Intended for graduate students in applied mathematics and related fields such as physics, chemistry and engineering, the book is also a useful overview of the topic for researchers in these areas. Encyclopedia of Mathematics and Its Applications This series is devoted to significant topics or themes that have wide application in mathematics or mathematical science and for which a detailed development of the abstract theory is less important than a thorough and concrete exploration of the implications and applications. Books in the Encyclopedia of Mathematics and Its Applications cover their subjects comprehensively. Less important results may be summarized as exercises at the ends of chapters. For technicalities, readers can be referred to the bibliography, which is expected to be comprehensive. As a result, volumes are encyclopedic references or manageable guides to major subjects.

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Closed-form solutions based on a general homogenization composite shell model are obtained for the effective stiffness moduli of the high-stiffness fiber-reinforced angle-ply composite shell. The design problem for the fiber-reinforced shell having the required set of effective stiffnesses is formulated and solved. The set of prescribed stiffnesses for which the design problem is solvable is described, and the effective method of the design parameters calculation based on convex analysis is developed. The minimum number of reinforcing layers required for the design of the fiber-reinforced angleply shell with the prescribed stiffnesses is determined. The solution of design problem is generalized on account of minimization of the fiber volume content.