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Introduction to the Network Approximation Method for Materials Modeling

Book Description

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 network approximation for PDE with discontinuous coefficients, developed in the 2000s by applied mathematicians in the USA and Russia. 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.

Table of Contents

  1. Cover
  2. Half title
  3. Series
  4. Title
  5. Copyright
  6. Dedication
  7. Table of Contents
  8. Preface
  9. 1 Review of mathematical notions used in the analysis of transport problems in densely-packed composite materials
    1. 1.1 Graphs
      1. 1.1.1 General information about graphs (networks)
      2. 1.1.2 Delaunay–Voronoi graphs in modeling of disordered structures
    2. 1.2 Functional spaces and weak solutions of partial differential equations
      1. 1.2.1 Distributions and distributional derivatives
      2. 1.2.2 Sobolev functional spaces
      3. 1.2.3 Traces of functions from H[sup(m)](Q)
      4. 1.2.4 Weak solutions of partial differential equations with discontinuous coefficients
      5. 1.2.5 Variational form of boundary value problems
    3. 1.3 Duality of functional spaces and functionals
      1. 1.3.1 Legendre transform
      2. 1.3.2 The minimax problem
    4. 1.4 Differentiation in functional spaces
    5. 1.5 Introduction to elliptic function theory
      1. 1.5.1 Weierstrass p-function
      2. 1.5.2 Weierstrass ζ-function
      3. 1.5.3 Weierstrass σ-function sectitle
      4. 1.5.4 γ-function
      5. 1.5.5 Eisenstein–Rayleigh sums
      6. 1.5.6 Eisenstein series
    6. 1.6 Kirszbraun’s theorem
  10. 2 Background and motivation for the introduction of network models
    1. 2.1 Examples of real-world problems leading to discrete network models
      1. 2.1.1 Optimal design of electrical capacitors
      2. 2.1.2 Thermal management in the electronics industry
      3. 2.1.3 Suspensions
    2. 2.2 Examples of network models
      1. 2.2.1 Resistor network models
      2. 2.2.2 Spring network models
      3. 2.2.3 Beam network models
      4. 2.2.4 Network models in physics
      5. 2.2.5 Network models in materials science, electrostatics and hydrodynamics
    3. 2.3 Rigorous mathematical approaches
    4. 2.4 When does network modeling work?
      1. 2.4.1 Constitutive equation of scalar transport models
      2. 2.4.2 Periodic and disordered structures
      3. 2.4.3 Media with continuously distributed characteristics. Kozlov model of high-contrast media
      4. 2.4.4 Media with piecewise constant characteristics and systems of bodies. The Maxwell–Keller model
    5. 2.5 History of the mathematical investigation of overall properties of high-contrast materials and arrays of bodies
      1. 2.5.1 The problem of computation of overall properties of a periodic system of bodies
      2. 2.5.2 Homogenization theory for periodic and random structures
      3. 2.5.3 Keller’s analysis of the conductivity of a medium containing a periodic array of perfectly conducting spheres or cylinders
    6. 2.6 Berryman–Borcea–Papanicolaou analysis of the Kozlov model
      1. 2.6.1 Direct and dual problems
      2. 2.6.2 Trial functions
    7. 2.7 Numerical analysis of the Maxwell–Keller model
      1. 2.7.1 Numerical verification of energy localization in the periodic case
      2. 2.7.2 Numerics for the non-periodic problem
      3. 2.7.3 Difference between localization effect and concentration effect
    8. 2.8 Percolation in disordered systems
    9. 2.9 Summary
  11. 3 Network approximation for boundary-value problems with discontinuous coefficients and a finite number of inclusions
    1. 3.1 Variational principles and duality. Two-sided bounds
      1. 3.1.1 The variational formulation of the problem
      2. 3.1.2 Formulas for the effective conductivity
      3. 3.1.3 The effective conductivity of the composite material
    2. 3.2 Composite material with homogeneous matrix
      1. 3.2.1 The dimensionless problem
      2. 3.2.2 Extremal form of the problem. The direct problem
      3. 3.2.3 Existence and uniqueness of solution of the problem (3.2.3)–(3.2.7)
      4. 3.2.4 The dual problem
      5. 3.2.5 Modeling of particle-filled composite materials using the Delaunay–Voronoi method. The notion of pseudo-particles
    3. 3.3 Trial functions and the accuracy of two-sided bounds. Construction of trial functions for high-contrast densely-packed composite materials
      1. 3.3.1 Trial functions and the accuracy of two-sided bounds
      2. 3.3.2 Construction of trial functions for high-contrast densely-packed composite materials
    4. 3.4 Construction of a heuristic network model. Two-dimensional transport problem for a high-contrast composite material filled with densely packed particles
    5. 3.5 Asymptotically matching bounds
    6. 3.6 Proof of the network approximation theorem
      1. 3.6.1 The refined lower-sided bound
      2. 3.6.2 Construction of a trial function for the dual problem
      3. 3.6.3 Construction of the function ζ[sub(ij)](x), which determines the trial function
      4. 3.6.4 Calculation of the double integral (3.6.18) from (3.6.1)
      5. 3.6.5 Calculation of the boundary integral in (3.6.1)
      6. 3.6.6 The refined upper-sided bound
      7. 3.6.7 Step A. Construction of the trial function in the neck
      8. 3.6.8 Step B. Construction of the trial function in the triangle
      9. 3.6.9 Step C. Construction of the trial function in the curvilinear triangle
      10. 3.6.10 Step D. Construction of the trial function for the near boundary disks
      11. 3.6.11 Step E. Evaluation of the Dirichlet integral in the whole domain
    7. 3.7 Close-packing systems of bodies
      1. 3.7.1 Some definitions for close-packing systems of bodies
      2. 3.7.2 Modeling of close-packing systems of bodies
    8. 3.8 Finish of the proof of the network approximation theorem
    9. 3.9 The pseudo-disk method and Robin boundary conditions
  12. 4 Numerics for percolation and polydispersity via network models
    1. 4.1 Computation of flux between two closely spaced disks different radii using the Keller method
    2. 4.2 Concept of neighbors using characteristic distances
    3. 4.3 Numerical implementation of the discrete network approximation and fluxes in the network
    4. 4.4 Property of the self-similarity problem (3.2.4)–(3.2.7)
    5. 4.5 Numerical simulations for monodispersed composite materials. The percolation phenomenon
      1. 4.5.1 Description of the numerical procedure
      2. 4.5.2 Verification of the computer program
    6. 4.6 Polydispersed densely-packed composite materials
      1. 4.6.1 The influence of polydispersity on the effective transport properties of a composite material
      2. 4.6.2 Numerical analysis of effective transport properties of a polydispersed composite material
  13. 5 The network approximation theorem for an infinite number of bodies
    1. 5.1 Formulation of the mathematical model
    2. 5.2 Triangle–neck partition and discrete network
      1. 5.2.1 Triangle–neck partition
      2. 5.2.2 Asymptotic equivalence of fluxes given by formulas (2.5.8) and (5.2.7)
      3. 5.2.3 The discrete network
    3. 5.3 Perturbed network models
    4. 5.4 δ-N connectedness and δ-subgraphs
    5. 5.5 Properties of the discrete network
    6. 5.6 Variational error estimates
    7. 5.7 The refined lower-sided bound
    8. 5.8 The refined upper-sided bound
    9. 5.9 Construction of trial function for the upper-sided bound
      1. 5.9.1 Construction of the trial function and estimation of the Dirichlet integral in the necks
      2. 5.9.2 Construction of the trial function and estimation of the Dirichlet integral in the triangles
      3. 5.9.3 Upper bounds on the relative half-neck widths
    10. 5.10 The network approximation theorem with an error estimate independent of the total number of particles
    11. 5.11 Estimation of the constant in the network approximation theorem
    12. 5.12 A posteriori numerical error
  14. 6 Network method for nonlinear composites
    1. 6.1 Formulation of the mathematical model
    2. 6.2 A two-step construction of the network
      1. 6.2.1 The domain partitioning step
      2. 6.2.2 The asymptotic step
    3. 6.3 Proofs for the domain partitioning step
      1. 6.3.1 Iterative minimization lemma
      2. 6.3.2 Boundary gradient estimates for the perforated medium
      3. 6.3.3 Optimal Lipschitz extensions
    4. 6.4 Proofs for the asymptotic step
      1. 6.4.1 Asymptotics in the necks
      2. 6.4.2 Connectivity of networks
  15. 7 Network approximation for potentials of bodies
    1. 7.1 Formulation of the problem of approximation of potentials of bodies
    2. 7.2 Network approximation theorem for potentials
      1. 7.2.1 Step A. An auxiliary boundary-value problem
      2. 7.2.2 Step B. An auxiliary estimate for the energies
      3. 7.2.3 Step C. Estimation of the difference between the solutions to the original and network problems
  16. 8 Application of the method of complex variables
    1. 8.1 R-linear problem and functional equations
      1. 8.1.1 R-linear problem
      2. 8.1.2 Functional equations
      3. 8.1.3 Flux between almost touching disks
      4. 8.1.4 Combination of functional equations and structural approximation
    2. 8.2 Doubly-periodic problems
      1. 8.2.1 Formulation of the boundary-value problem
      2. 8.2.2 The complex variable formula for effective conductivity
    3. 8.3 Optimal design problem for monodispersed composites
      1. 8.3.1 “Shaking model” for a random composite
      2. 8.3.2 Optimal design problem for a deterministic monodisperced composite
    4. 8.4 Random polydispersed composite
      1. 8.4.1 Well-separated model (identical shaking parameters)
      2. 8.4.2 Bumping model (two different shaking parameters)
  17. References
  18. Index