- Modifies known asymptotic procedures
- Constructs new asymptotic procedures that account for the contrast of components
- Presents methods initiated by real-world problems such as effective permittivity and dielectric loss
- Provides results of numerical computations

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*Is it possible to apply a network model to composites with conical inclusions?*

*How does the energy pass through contrast composites?*

Devoted to the analysis of transport problems for systems of densely packed, high-contrast composite materials, **Capacity and Transport in Contrast Composite Structures: Asymptotic Analysis and Applications** answers questions such as these and presents new and modified asymptotic methods for real-world applications in composite materials development.

A mathematical discussion of phenomena related to natural sciences and engineering, this book covers historical developments and new progress in mathematical calculations, computer techniques, finite element computer programs, and presentation of results of numerical computations.

The "transport problem"—which is described with scalar linear elliptic equations—implies problems of thermoconductivity, diffusion, and electrostatics. To address this "problem," the authors cover asymptotic analysis of partial differential equations, material science, and the analysis of effective properties of electroceramics. Providing numerical calculations of modern composite materials that take into account nonlinear effects, the book also:

- Presents results of numerical analysis, demonstrating specific properties of distributions of local fields in high-contrast composite structures and systems of closely placed bodies
- Assesses whether total flux, energy, and capacity exhaust characteristics of the original continuum model
- Illustrates the expansion of the method for systems of bodies to highly filled contrast composites

This text addresses the problem of loss of high-contrast composites, as well as transport and elastic properties of thin layers that cover or join solid bodies. The material presented will be particularly useful for applied mathematicians interested in new methods, and engineers dealing with prospective materials and design methods.

**IDEAS AND METHODS OF ASYMPTOTIC ANALYSIS AS APPLIED TO TRANSPORT IN COMPOSITE STRUCTURES**

Effective properties of composite materials and the homogenization theory

Transport properties of periodic arrays of densely packed bodies

Disordered media with piecewise characteristics and random collections of bodies

Capacity of a system of bodies

**NUMERICAL ANALYSIS OF LOCAL FIELDS IN A SYSTEM OF CLOSELY PLACED BODIES**

Numerical analysis of two-dimensional periodic problem

Numerical analysis of three-dimensional periodic problem

The energy concentration and energy localization phenomena

Which physical field demonstrates localization most strongly?

Numerical analysis of potential of bodies in a system of closely placed bodies with finite element method and network model

Energy channels in non-periodic systems of disks

**ASYMPTOTIC BEHAVIOR OF CAPACITY OF A SYSTEM OF CLOSELY PLACED BODIES. TAMM SHIELDING. NETWORK APPROXIMATION**

Problem of capacity of a system of bodies

Formulation of the problem and definitions

Heuristic network model

Proof of the principle theorems

Completion of proof of the theorems

Some consequences of the theorems about NL zones and network approximation

Capacity of a pair of bodies dependent on shape

**NETWORK APPROXIMATION FOR POTENTIALS OF CLOSELY PLACED BODIES**

Formulation of the problem of approximation of potentials of bodies

Proof of the network approximation theorem for potentials

The speed of convergence of potentials for a system of circular disks

**ANALYSIS OF TRANSPORT PROPERTIES OF HIGHLY FILLED CONTRAST COMPOSITES USING THE NETWORK APPROXIMATION METHOD**

Modification of the network approximation method as applied to particle-filled composite materials

Numerical analysis of transport properties of highly filled disordered composite material with network model

**EFFECTIVE TUNABILITY OF HIGH-CONTRAST COMPOSITES**

Nonlinear characteristics of composite materials

Homogenization procedure for nonlinear electrostatic problem

Tunability of laminated composite

Tunability amplification factor of composite

Numerical design of composites possessing high tunability amplification factor

The problem of maximum value for the homogenized tunability amplification factor

What determines the effective characteristics of composites?

The difference between design problems of tunable composites in the cases of weak and strong fields

Numerical analysis of tunability of composite in strong fields

**EFFECTIVE LOSS OF HIGH-CONTRAST COMPOSITES**

Effective loss of particle-filled composite

Effective loss of laminated composite material

**TRANSPORT AND ELASTIC PROPERTIES OF THIN LAYERS**

Asymptotic of first boundary-value problem for elliptic equation in a region with a thin cover

Elastic bodies with thin underbodies layer (glued bodies)

**A.A. Kolpakov** works in the Department of Mathematics and Mechanics at Novosibirsk State University, Russia and Université de Fribourg, Fribourg Pérolles, Switzerland. **A.G. Kolpakov** works as Marie Curie Fellow at Università degli Studi di Cassino, Italy and Siberian State University of Telecommunications and Informatics, Russia.

... deals with interesting questions of strongly heterogeneous media, such as the analysis of capacities and transport properties. ... The book is intended to be self-contained and it is of interest to researchers in the fields of "homogenization theory" and "asymptotic analysis" in the areas of applied mathematics, physics and engineering sciences. More specifically, it may be of interest to students and researchers in mathematical models related to diffusion, electricity, magnetism, mechanics, new materials and design methods. It is written in terms of electrostatics, and it pays special attention to the so-called (by the authors) "Tamm screening effect" or "Tamm shielding effect" and the problems of the "effective tunability" and "effective loss" of composite materials. These effects/terms (and some others) arising in physics and engineering are in the reviewer’s opinion rarely considered in the literature of applied mathematics, and the authors provide a mathematical interpretation in this book .... Many figures in the book are important for the understanding of the corresponding problems and the results obtained.

—Eugenia Pérez, in *Mathematical Reviews*, 2012a