Computational Analysis of Structured Media presents a systematical approach to analytical formulae for the effective properties of deterministic and random composites. Schwarz’s method and functional equations yield for use in symbolic-numeric computations relevant to the effective properties. The work is primarily concerned with constructive topics of boundary value problems, complex analysis, and their applications to composites. Symbolic-numerical computations are widely used to deduce new formulae interesting for applied mathematicians and engineers. The main line of presentation is the investigation of two-phase 2D composites with non-overlapping inclusions randomly embedded in matrices.
1. Chapter 1: Introduction - p. 1 2. Chapter 2: Complex Potentials and R-linear problem - p. 7 3. Chapter 3: Constructive homogenization - p. 41 4. Abstract - p. 41 5. 1 Introduction - p. 41 6. 2 Deterministic and stochastic approaches - p. 44 7. 3 Series expansions for the local fields and effective tensors. Traditional approach - p. 56 8. 4 Schwarz’s method - p. 59 9. 5 Remark on asymptotic methods - p. 74 10. Chapter 4: From Basic Sums to effective conductivity and RVE) - p. 79 11. Chapter 5: Introduction to the method of self-similar approximants - p. 103 12. Abstract - p. 103 13. 1 Brief introduction to extrapolation - p. 103 14. 2 Algebraic renormalization and self-similar bootstrap - p. 106 15. 3 Extrapolation problem and self-similar approximants - p. 111 16. 4 Corrected Padé approximants for indeterminate problem - p. 123 17. 5 Calculation of critical exponents - p. 138 18. 6 Interpolation with self-similar root approximants - p. 149 19. Chapter 6: Conductivity of regular composite. Square lattice - p. 161 20. Chapter 7: Conductivity of regular composite. Hexagonal array - p. 191 21. Chapter 8: Effective Conductivity of 3D regular composites - p. 221 22. Abstract - p. 221 23. 1 Modified Dirichlet problem. Finite number of balls - p. 221 24. 2 3D periodic problems - p. 226 25. 3 Triply periodic functions - p. 232 26. 4 Functional equations on periodic functions - p. 233 27. 5 Analytical formulae for the effective conductivity. Discussion and overview of the known results - p. 235 28. 6 Non-conducting inclusions embedded in an conducting matrix. FCC lattice - p. 242 29. 7 Non-conducting inclusions embedded in an conducting matrix. SC and BCC lattices - p. 244 30. Chapter 9: Random 2D composites - p. 249 31. Chapter 10: Elastic problem - p. 283 32. 11: Table of the main analytical formulae - p. 305 33. Index - p. 307
Computational Analysis of Structured Media presents a systematical approach to analytical formulae for the effective properties of deterministic and random composites. Schwarz’s method and functional equations yield for use in symbolic-numeric computations relevant to the effective properties. The work is primarily concerned with constructive topics of boundary value problems, complex analysis, and their applications to composites. Symbolic-numerical computations are widely used to deduce new formulae interesting for applied mathematicians and engineers. The main line of presentation is the investigation of two-phase 2D composites with non-overlapping inclusions randomly embedded in matrices.
1. Chapter 1: Introduction - p. 1 2. Chapter 2: Complex Potentials and R-linear problem - p. 7 3. Chapter 3: Constructive homogenization - p. 41 4. Abstract - p. 41 5. 1 Introduction - p. 41 6. 2 Deterministic and stochastic approaches - p. 44 7. 3 Series expansions for the local fields and effective tensors. Traditional approach - p. 56 8. 4 Schwarz’s method - p. 59 9. 5 Remark on asymptotic methods - p. 74 10. Chapter 4: From Basic Sums to effective conductivity and RVE) - p. 79 11. Chapter 5: Introduction to the method of self-similar approximants - p. 103 12. Abstract - p. 103 13. 1 Brief introduction to extrapolation - p. 103 14. 2 Algebraic renormalization and self-similar bootstrap - p. 106 15. 3 Extrapolation problem and self-similar approximants - p. 111 16. 4 Corrected Padé approximants for indeterminate problem - p. 123 17. 5 Calculation of critical exponents - p. 138 18. 6 Interpolation with self-similar root approximants - p. 149 19. Chapter 6: Conductivity of regular composite. Square lattice - p. 161 20. Chapter 7: Conductivity of regular composite. Hexagonal array - p. 191 21. Chapter 8: Effective Conductivity of 3D regular composites - p. 221 22. Abstract - p. 221 23. 1 Modified Dirichlet problem. Finite number of balls - p. 221 24. 2 3D periodic problems - p. 226 25. 3 Triply periodic functions - p. 232 26. 4 Functional equations on periodic functions - p. 233 27. 5 Analytical formulae for the effective conductivity. Discussion and overview of the known results - p. 235 28. 6 Non-conducting inclusions embedded in an conducting matrix. FCC lattice - p. 242 29. 7 Non-conducting inclusions embedded in an conducting matrix. SC and BCC lattices - p. 244 30. Chapter 9: Random 2D composites - p. 249 31. Chapter 10: Elastic problem - p. 283 32. 11: Table of the main analytical formulae - p. 305 33. Index - p. 307
