Computational Analysis of Structured Media

By Simon Gluzman, Vladimir Mityushev and Wojciech Nawalaniec

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.

• Computational methodology for main classes of problems in structured media
• Theory of Representative Volume Element
• Combines exact results, Monte-Carlo simulations and Resummation techniques under one umbrella
• Contains new analytical formulae obtained in the last ten years and it combines different asymptotic methods with the corresponding computer implementations

• Language: English
• Publisher: Academic Press
• Release date: Sep 20, 2017
• ISBN: 9780128110478

Simon Gluzman

Simon Gluzman is presently an Independent Researcher (Toronto, Canada) and formerly a Research Associate at PSU in Applied Mathematics. He is interested in Re-summation methods in theory of random and regular composites and the method of self-similar and rational approximants.

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Computational Analysis of Structured Media

Series Editor

Themistocles M. Rassias

Simon Gluzman

Bathurst St. 3000, Apt 606, ON M6B3B4 Toronto, Canada

Vladimir Mityushev

Pedagogical University, Faculty of Mathematics, Physics and Technical Science Institute of Computer Sciences, ul. Podchorazych 2, 30-084 Krakow, Poland

Wojciech Nawalaniec

Pedagogical University, Faculty of Mathematics, Physics and Technical Science Institute of Computer Sciences, ul. Podchorazych 2, 30-084 Krakow, Poland

Mathematical Analysis and its Applications Series

Table of Contents

Cover

Title page

Copyright

Dedication

Acknowledgment

Preface

Nomenclature

Chapter 1: Introduction

Abstract

Chapter 2: Complex Potentials and R-linear problem

Abstract

1 Complex potentials

2 R-linear problem

3 Metod of functional equations

Chapter 3: Constructive homogenization

Abstract

1 Introduction

2 Deterministic and stochastic approaches

3 Series expansions for the local fields and effective tensors. Traditional approach

4 Schwarz’s method

5 Remark on asymptotic methods

Chapter 4: From Basic Sums to effective conductivity and RVE)

Abstract

1 Basic Sums

2 Identical circular inclusions.

3 Representative volume element

4 Method of Rayleigh

Chapter 5: Introduction to the method of self-similar approximants

Abstract

1 Brief introduction to extrapolation

2 Algebraic renormalization and self-similar bootstrap

3 Extrapolation problem and self-similar approximants

4 Corrected Padé approximants for indeterminate problem

5 Calculation of critical exponents

6 Interpolation with self-similar root approximants

Chapter 6: Conductivity of regular composite. Square lattice

Abstract

1 Introduction

2 Critical point, square array

3 Critical Index s

4 Crossover formula for all concentrations

5 Expansion near the threshold

6 Additive ansatz. Critical amplitude and formula for all concentrations

7 Interpolation with high-concentration Padé approximants

8 Comment on contrast parameter

Chapter 7: Conductivity of regular composite. Hexagonal array

Abstract

1 Effective conductivity and critical properties of a hexagonal array of superconducting cylinders

2 Series for hexagonal array of superconducting cylinders

3 Critical Point

4 Critical index and amplitude

5 Critical amplitude and formula for all concentrations

6 Interpolation with high-concentration Padé approximants

7 Discussion of the ansatz (7.5.30)

8 Square and hexagonal united

9 Dependence on contrast parameter

Chapter 8: Effective Conductivity of 3D regular composites

Abstract

1 Modified Dirichlet problem. Finite number of balls

2 3D periodic problems

3 Triply periodic functions

4 Functional equations on periodic functions

5 Analytical formulae for the effective conductivity. Discussion and overview of the known results

6 Non-conducting inclusions embedded in an conducting matrix. FCC lattice

7 Non-conducting inclusions embedded in an conducting matrix. SC and BCC lattices

Chapter 9: Random 2D composites

Abstract

1 Critical properties of an ideally conducting composite materials

2 Random composite: stirred or shaken?

3 2D Conductivity. Dependence on contrast parameter

Chapter 10: Elastic problem

Abstract

1 Introduction

2 Method of functional equations for local fields

3 Averaged fields in finite composites

4 Roadmap to composites represented by RVE

5 Effective constants

11: Table of the main analytical formulae

Abstract

Index

Copyright

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Dedication

For Our Families

Acknowledgment

Simon Gluzman; Vladimir Mityushev; Wojciech Nawalaniec April 2017

We thank our friends and colleagues, Pierre Adler, Igor Andrianov, Leonid Berlyand, Piotr Drygaś, Barbara Gambin, Leonid Filshtinsky, Alexander G. Kolpakov, Pawel Kurtyka, Natalia Rylko, Ryszard Wojnar, Vyacheslav Yukalov, for all their support and useful discussions. We are also indebted to the whole Elsevier team for their help and guidance throughout the publishing process. We are grateful to the editor for carefully reviewing the book during this process. The authors (V.M. and W.N.) were partially supported by Grant of the National Centre for Research and Development 2016/21/B/ST8/01181.

Preface

Four legs good, two legs bad

George Orwell, Animal farm

The present book may be considered as an answer to the question associated to the picture on the last front matter page. Why does James Bond prefer shaken, not stirred martini with ice?¹ Highly accurate computational analysis of structural media allows us to explain the difference between various types of random composite structures.

We are primarily concerned here with the effective properties of deterministic and random composites. The analysis is based on accurate analytical solutions to the problems considered by respected specialists as impossible to find their exact solutions. Consensual opinions can be summarized as follows: "It is important to realize that solution to partial differential equations, of even linear material models, at infinitesimal strains, describing the response of small bodies containing a few heterogeneities are still open problem. In short, complete solutions are virtually impossible" (see [5, p. 1]).

Of course, it is impossible to resolve all the problems of micromechanics and their analogs, but certain classes such as boundary value problems for Laplace’s equation and bi-harmonic two-dimensional (2D) elasticity equations can be solved in analytical form.

At least for an arbitrary 2D multiply connected domain with circular inclusions our methods yield analytical formulae for most of the important effective properties, such as conductivity, permeability, effective shear modulus and effective viscosity. Randomness in such problems is introduced through random locations of non-overlapping disks. It is worth noting that any domain can be approximated by special configurations of packed circular disks.

Many respectful authors apply various self-consistent methods (SCM) such as effective medium approximation, differential scheme, Mori–Tanaka approach, etc. [2]. They claim that such methods give general analytical formula for the effective properties. Careful analysis though shows their restriction to the first- or second- order approximations in concentration.

Actually SCM perform elaborated variations on the theme of the celebrated Maxwell formula, Clausius–Mossotti approximation, and so forth [3]. All of them are justified rigorously only for a dilute composites when interactions among inclusions are neglected. In the same time, exact and high-order formulae for special regular composites which go beyond SCM were derived.

Despite a considerable progress made in the theory of disordered media, the main tools for studying such systems remain numerical simulations and questionable designs to extend SCM to high concentrations. These approaches are sustained by unlimited belief in numerics and equal underestimation of constructive analytical and asymptotic methods. They have to be drastically reconsidered and refined. In our opinion there are three major developments which warrant such radical change of view.

–linear problems for multiply connected domains, see Chapter 1, [4].²

2. Significant progress in symbolic computations (see MATHEMATICA®, MAPLE®, MATLAB® and others) greatly extends our computational capacities. Symbolic computations operate on the meta-level of numerical computing. They transform pure analytical constructive formulae into computable objects. Such an approach results in symbolic algorithms which often require optimization and detailed analysis from the computational point of view. Moreover, symbolic and numeric computations do integrate harmoniously [1].

But we can not declare a victory just yet, because even long power series in concentration and contrast parameters are not sufficient because they won’t allow to cover the high-concentration regime. Sometimes the series are short, in other cases they do not converge fast enough, or even diverge in the most interesting regime. Your typical answer to the challenges is to apply an additional methods powerful enough to extract information from the series. But in addition to a traditional Padé approximants applied in such cases, we would need a

3. New post-Padé approximants for analysis of the divergent or poorly convergent series, including different asymptotic regimes discussed in Chapter 5.

In the present book, we demonstrate that the theoretical results [4] can be effectively implemented in symbolic form that yields long power series. Accurate analytical formulae for deterministic and random composites and porous media can be derived employing approximants, when the low-concentration series are supplemented with information on the high-concentration regime where the problems we encounter are characterized by power laws.

As to the engineering needs we recognize the need for an additional fourth step. The engineer would like to have a convenient formula but also to incorporate in it all available information on the system, with a particular attention to the results of numerical simulations or known experimental values.

Method of regression on approximants consists in applying different multivariate regression techniques to the results of interpolation and (or) extrapolation with various approximants within the framework of supervised learning paradigm. In this case approximants by themselves are treated as new variables and experimental, exact or numerically exact data, or else training data set, are used to construct regression on approximants. Some statistical learning procedure should be included in order to select the best regression and to make sure it is better indeed than choosing the best performer among the approximants.

Consider a non-dimensional effective property ϑ(f). Here f stands for the concentration of particles. Let the asymptotic expansion of ϑ(f) in the weak-coupling limit be

   (0.0.1)

In addition to the expansion, let us consider as available to us, exact numerical values ϑ(fi) (or labels), for the effective property for some typical values of f = fi, i = 1, 2, …, K.

This information to be incorporated into the algorithm:

(a) construct all possible approximants for such expansion, such as Apj*(f), j = 1, 2, …, M. The approximants are also assumed to incorporate the information from the high-concentration regime whenever such information is available.

(b) make predictions, i.e., calculate with all constructed approximants for all fi the values of Apj* (fi).

Let us consider all predictions as a j-dimensional vector Ap*(fi) and organize Ap*(fi) and ϑ(fi) into pairs {Ap∗(fi), ϑ(fi)} , i = 1 , 2 , … , K, thus creating a training data set.

(c) based on information contained in the training data set, we attempt to learn the (multivariate) mapping ϑ∗ = F(Ap∗) (regression model), allowing to make predictions for arbitrary f (since Ap* depends on f);

(d) most simple regression models, such as multivariate linear regression and k-Nearest Neighbors could be used.

(e) for learning, i.e., selection of the best regression based on prediction error incurred by particular regression within a given training data set, one can use, e.g., a jackknife (or leave-one-out cross-validation) error estimates. Starting from the whole training data set, the jackknife begins with throwing away the first label, leaving a re-sampled data set, which is used to construct regression and prediction is made for the missing label. In turn, such obtained prediction is compared with the true label. Resampling is continued till predictions and comparison are performed for each and every label from the training data set.

(f) as a cumulative measure for the prediction error one can take mean absolute percentage error calculated over all labels.

Such approach is amenable to automation and true predictions of the sought physical quantity for any f can be generated.

Simon Gluzman; Vladimir Mityushev; Wojciech Nawalaniec, Kraków, Poland, April 2017

Reference

1. Czapla R., Nawalaniec W., Mityushev V. Effective conductivity of random two-dimensional composites with circular non-overlapping inclusions. Comput. Mat. Sci. 2012;63:118–126.

2. Kanaun S.K., Levin V.M. Self-consistent methods for composites. Dordrecht: Springer-Verlag; 2008.

3. Landauer R. Electrical conductivity in inhomogeneous media. In: Garland J.C., Tanner D.B., eds. Electrical Transport and Optical Properties of Inhomogeneous Media. 1978.

4. Mityushev V.V., Rogosin S.V. Constructive methods to linear and non-linear boundary value problems of the analytic function. Theory and applications. Boca Raton etc: Chapman & Hall/CRC; 1999/2000 [Chapter 4].

5. Zohdi T.I., Wriggers P. An introduction to Computational mechanics. Berlin etc: Springer-Verlag; 2008 [Chapter 1].

---

¹ The complete answer on the question is yet to be found, and most likely after many experiments. But the mathematical answer is given on 268.

² Poisson’s type formula for an arbitrary circular multiply connected domain is one of its particular cases.

Nomenclature

Geometry

ak 

 stands for circular multiply connected domain

D 

Dk 

∂ D stands for boundary of D

Ek(z) stands for Eisenstein function of order k

em1 , m2 , m3 … , mn stands for basic sum of the multi-order m = (m1, …, mn)

f stands for volume fraction (concentration) of the inclusions

n = (n1, n2) normal vector to a smooth oriented curve L

n = n1 + in2 complex form of the normal vector n = (n1, n2)

s = (− n2, n1) tangent vector

 stands for the Euclidean space

Sm stands for Eisenstein–Rayleigh lattice sums

 stands for unit disk on plane

BCC means body-centered cubic lattice

FCC means face-centered cubic lattice

HCP means hexagonal close-packed lattice

SC means simple cubic lattice

ω1 and ω2 stand for the fundamental translation vectors on the complex plane

Conductivity

σ(x) stands for scalar local conductivity

σk is used for conductivity of the kth inclusion

σ is used for conductivity of inclusions of two-phase composites when the conductivity of matrix is normalized to unity

σe stands for effective conductivity tensor

σij stands for components of the effective conductivity tensor

σe stands for scalar effective conductivity of the macroscopically isotropic composites

σ*, σe,n(f), σadd, σ1ad etc stands for various approximations of σe

 stands for contrast parameter between media with the conductivities σ+ and σ

s stands for the critical index for superconductivity

t stands for the critical index for conductivity

Elasticity

μ stands for shear modulus and viscosity

k stands for bulk modulus

ν stands for Poisson’s ration

κ stands for Muskhelishvili’s constant

σxx, σxy = σyx, σyy the components of the stress tensor

ϵxx, ϵxy = ϵyx, σyy the components of the strain tensor

Other symbols and abbreviations

~ indicates asymptotic equivalence between functions, i.e., indicates that functions are similar, of the same order

≃ indicates that the two functions are similar or equal asymptotically

A and B stand for the critical amplitudes

 stands for the complex numbers

C stands for the operator of complex conjugation

error measures the error as a ratio of the exact value

 stands for functions Holder continuous on a simple smooth curve L

 stands for functions analytic in D and Hölder continuous in its closure

PadeApproximant[F[z], n, m] stands for the Padé (n, m)-approximant of the function F(z)

 stands for the real numbers

-function

ζ(z) stands for the Weierstrass ζ-function

1D, 2D, and 3D mean one-, two-, and three- dimensional

LHS and RHS mean the expressions left-hand side and right-hand side (of an equation), respectively

EMA means effective medium approximations

RS means random shaking

RSA means Random Sequential Addition

RVE stands for representative volume element

RW means random walks

SCM means self-consistent methods

Chapter 1

Introduction

Abstract

In this chapter, we discuss what the terms exact, approximate, and numerical solutions of the problem does mean.

Keywords

Asymptotic methods; Constructive solution; Numerical solution; Riemann-Hilbert problem; Schottky-Klein prime function; Series method; Truncation method

Man’s quest for knowledge is an expanding series whose limit is infinity, but philosophy seeks to attain that limit at one blow, by a short circuit providing the certainty of complete and in-alterable truth. Science meanwhile advances at its gradual pace, often slowing to a crawl, and for periods it even walks in place, but eventually it reaches the various ultimate trenches dug by philosophical thought, and, quite heedless of the fact that it is not supposed to be able to cross those final barriers to the intellect, goes right on.

Stanislav Lem, His Master’s Voice

The terms closed form solution, analytical solution, etc., are frequently used in literature in various contexts. We introduce below some definitions for various solutions with different levels of exactness.

Consider an equation Ax = b. One can write its solution as x = A− 1b where A− 1 is an inverse operator to A. Let a space where the operator A acts be defined and existence of the inverse operator in this space be established. Then, a mathematician working on the level of functional analysis can say that the deal is done. This is a standard question of the existence and uniqueness in theoretical mathematics. In the present book, we consider problems when the existence and uniqueness take place and we are interested in a constructive form of the expression x = A− 1b.

1. We say that x = A− 1b is a closed form solution if the expression A− 1b consists of a finite number of elementary and special functions, arithmetic operations, compositions, integrals and derivatives.

2. A solution is an analytical form solution if the expression A− 1b consists of a finite number of elementary and special functions, arithmetic operations, compositions, integrals, derivatives and series.

The difference between items 1 and 2 lies with the usage of the infinite series for a analytical solution. We use the term constructive solution for items 1 and 2 following the book [20]. In the second case, we have to cut the considered infinite series to get an expression convenient for symbolic analytical and numerical analysis. This book steadfastly adheres to the analysis of obtained truncated series. In particular, to constructive investigation of their behaviour near divergence points when the physical percolation effects occur. The main feature of such a constructive study is in retention of the fundamental physical parameters, concentration for instance, in symbolic form. The term analytical solution means not only closed form solution, but also a solution constructed from asymptotic approximations.

Asymptotic methods[1, 2, 3, 5, 15] are assigned to analytical methods when solutions are investigated near the critical values of the geometrical and physical parameters. Hence, asymptotic formulae can be considered as analytical approximations.

In order to distinguish our results from others, we proceed to classify different types of solutions.

3. Numerical solution means here the expression x = A− 1b which can be treated only numerically. Integral equation methods based on the potentials of single and double layers usually give such a solution. An integral in such a method has to be approximated by a cubature formula. This makes it pure numerical, since cubature formulae require numerical data in kernels and fixed domains of integrations. Fredholm’s and singular integral equations frequently arise in applications [6, 20, 22, 23]. Effective numerical methods were developed and systematically described in the books [6, 10, 11, 12, 23] and many others. Nevertheless, some integral and integro-differential equations can be solved in a closed form, e.g., [25].

Central for our study, series method arises when an unknown element x is expanded into a series x = ∑k = 1∞ ckxk on the basis {xk}k = 1∞ with undetermined constants ck. Substitution of the series into equation can lead to an infinite system of equations on ck. In order to get a numerical solution, this system is cut short and a finite system of equations arise, say of order n. Let the solution of the finite system tend to a solution of the infinite system, as n → ∞. Then, the infinite system is called regular and can be solved by the described truncation method. This method was justified for some classes of equations in the fundamental book [14]. The series method can be applied to general equation Ax = b in a discrete space in the form of infinite system with infinite number of unknowns. In particular, Fredholm’s alternative and the Hilbert–Schmidt theory of compact operators can be applied [14]. So in general, the series method belongs to numerical methods. In the field of composite materials, the series truncation method was systematically used by Guz et al. [13], Kushch [17], and others.

The special structure of the composite systems or application of a low-order truncation can lead to an approximate solution in symbolic form. Par excellence examples of such solutions are due to Rayleigh [26] and to McPhedran’s et al, [18, 24] where classical but incomplete analytical approximate formulae for the effective properties of composites were deduced.

It is worth noting that methods of integral equations and methods of series are often well developed computationally. So that many mathematicians would stop on a formula which includes numerically computable objects without actually computing. For instance, the Riemann–Hilbert problem for a simply connected domain D was solved in a closed form and up to a conformal mapping . The solution of the Riemann–Hilbert problem can be written as a composition of Poisson’s type integral and the mapping φ. Construction of the map φ is considered as a separate computational problem.

Analogous remark concerns the Riemann–Hilbert problem for a multiply connected domain. In accordance with [27] this problem is solved in terms of the principal functional on the Riemann surface which could be found from integral equations. So, realization of the theory [27] could be purely numerical, but again remains undeveloped into concrete algorithm.

was solved analytically in [20], more precisely in terms of the uniformly convergent Poincaré series [19]. Therefore, it is solved for any multiply connected domain D up to a conformal mapping φ(see demonstration of its existence in [9]).

Similar remark concerns the Schottky–Klein prime function [4]. This function was constructively expressed in terms of the Poincaré αin a symbolic form.

4. Discrete numerical solution refers to applications of the finite elements and difference methods. These methods are powerful and their application is reasonable when the geometries and the physical parameters are fixed. In this case the researcher can be fully satisfied with numerical solution to various boundary value problems. Many specialists perceive a pristine computational block (package) as an exact formula: just substitute data and get the result! They have a good case if such a block allows to investigate symbolic dependencies and, perhaps by fitting methods to understand a process or an object. However, a sackful of numbers is not as useful as an analytical formulae. Pure numerical procedures can fail as a rule for the critical parameters and analytical matching with asymptotic solutions can be useful even for the numerical computations.

Moreover, numerical packages sometimes are presented as a remedy from all deceases. It is worth noting again that numerical solutions are useful if we are interested in a fixed geometry and fixed set of parameters for engineering purposes. Different geometry or parameters would require to relaunch the numerical procedure every time properly generating random data and conducting Monte-Carlo experiment [16].

This book deals with constructive analytical solutions. In other words with exact and approximate analytical solutions, when the resulting formulae contain the main physical and geometrical parameters in symbolic form. The obtained truncated series, actually are considered as polynomials. They are supposed to remember their infinite expansions, so that with a help of some additional re-summation procedure one can extrapolate to the whole series. A significant part of the book is devoted to restoration of these infinite series by means of special constructive forms called approximants, expressions that are asymptotically equivalent to the truncated series. But the approximants are richer in a sense that they also suggest an infinite additional number of the coefficients {ck}k = n + 1∞. The