\chapter{Background and literature review}
\section{Homogenization Methods}
\paragraph
The effective properties of macroscopic homogeneous composite materials can be derived from the microscopic heterogeneous material structures using homogenization techniques. Homogenization is a technique for macro micro-scale transition. In a two-scale method two spatial variables are introduced: $x$ is a macroscopic spatial coordinate and $y = x/\varepsilon$ is a microscopic one. The variable $y$ is associated with the small length scale of the inclusions or heterogeneities. The two-scales process introduced in the partial differential equations of the problem produces equations in x, y and both variables. Generally speaking, equations in y are "solvable" if the microscopic structure is periodic, and this leads to a "rigorous" deduction of the macroscopic equations (in x) for the global behavior.
\paragraph
In most problems, a mathematical proof of the convergence of solutions to the "homogenized solutions" is available when $\epsilon\rightarrow 0$. It should be noticed that the "homogenized coefficients" only depend on the local (or microscopic) structure of the medium, and may be obtained by numerical solution of some boundary value problems in a period of the structure, the boundary conditions being mostly of the periodic type. In fact, homogenization gives relevant information on the relation between the local and global behaviors; for most problems in mechanics the micro and macro-processes are of very different nature.
\paragraph
The macroscopic mechanical properties of composite material is described by its microstructure behavior which is exemplified by the interaction between the constituents. Many heterogeneous materials have regular microstructure which makes it possible to consider only one small periodic element of the structure -- representative volume element (RVE). (see Fig \ref{RVE}).
\begin{figure}[htb]
\begin{center}
\begin{tabular}[3]{ccc}
\includegraphics[width=30mm]{RVE_1-1.eps} &
\includegraphics[width=30mm]{RVE_1-2.eps} &
\includegraphics[width=30mm]{RVE_1-3.eps} \\
(a) & (b) & (c) \\
\end{tabular}
\caption{Representative Volume Element Selection. }
\label{RVE}
\end{center}
\end{figure}
All computations are performed over RVE, and then, they are extended to the whole material. As a general procedure it could be said that in most homogenization methods a local problem of a single inclusion is solved to get the approximation of the local field behavior,(as it was done by Eshelby in 1957 for elastic fields of an ellipsoidal inclusion) then, by averaging these local fields, we could find global properties. This is called basic mean field homogenization method. Hashin Shtrikman type bounds, Mori-Tanaka type models and classical self schemes are the most famous homogenization methods of this type.
\paragraph
In all of these methods, composite is restricted to the matrix-inclusion type with perfect interfacial bonds between inclusions and their immediate surrounding matrix and differences between methods related to the assumption which is made to treat with these interaction. The Mori-Tanaka method approximates the interaction between the phases by assuming that each inclusion is embedded in an infinite matrix which is remotely loaded by the average matrix strain or average matrix stress.
\paragraph
Hashin and Shtrikman uses variational bounding techniques to obtain more useful estimate of modula for isotropic heterogenous materials. They assumed that the particles are spherical and the action of whole heterogenous material on any inclusion is transmitted through a spherical shell, which lies wholly in the matrix \cite{S.Kurukuri}. According to \cite{Klusemann_2010}, it can be seen that the upper Hashin-Shtrikman bound corresponds to the Mori-Tanaka result. The upper bound can also be obtained with the Mori-Tanaka method just by interchanging matrix and inclusion material.
\paragraph
Lielens method is another homogenization method which uses the advantages of Mori-Tanaka method for a two phase material. This method is a properly chosen interpolation between the Mori-Tanaka and inverse Mori-Tanaka method and also between the Hashin-Shtrikman bounds, respectively. The self-consistent method approximates the interaction between phases by assuming that each phase is an inclusion embedded in a homogenous medium that has the overall properties of the composite. Therefore the equation will be implicit and nonlinear. In general, the self-consistent method gives a sufficient prediction of the behavior of polycrystal but it is less accurate in the case of two-phase composites\cite{Klusemann_2010}.
\paragraph
Also, the "Effective Self-Consistent Scheme (ESCS)" and "Interaction Direct Derivative (IDD)" are the most recent homogenization method, proposed by Zheng and Du(2001) based on three-phase model.
In the three-phase model, the inclusion is embedded in a matrix which itself embedded in an unbound, initially unknown effective medium. The ESCS overcomes the restrictions on spherical and cylindrical inclusions but still has a complicated structure. Interaction Direct Derivative (IDD)method is a simplified and explicit version of the ESCS method and has a very simple structure with clear physical meaning of the single constituent parts\cite{Klusemann_2010}.
\paragraph
\subsection{Asymptotic Expansion homogenization method (AEH)}
\paragraph
Asymptotic expansion homogenization (AEH) method is another approach for multi-scale problems. This method is a very good methodology to model physical phenomena in heterogenous material with periodic microstructure. The method of asymptotic homogenization proceeds by introducing the fast variable $\xi=x/\epsilon$ and posing a formal expansion in $\epsilon$ :
\begin{equation}
\label{asymptotic}
u(x)=u(x,\xi)=u_0(x,\xi)+\epsilon u_1(x,\xi)+\epsilon^2 u_2(x,\xi)+ O(\epsilon^3)
\end{equation}
According to \cite{Peter.W}, we can introduce a recipe for this type of homogenization as following:
\paragraph
\begin{itemize}
\item Step 1. Modeling: the scale is modeled with any Simulation software (mostly FEM) and define the $X^e$ as global and $Y^e$ as local variables.
\item Step 2. Introduce the asymptotic series approximation in $\epsilon$: AEH is a perturbation technique based on an asymptotic series expansion in $\epsilon$, a scale parameter, of a primary variable such as displacement. The first two terms of the series, represent the sum of global terms and oscillating small scale term. As mentioned earlier, $\epsilon$ is a scale parameter which is a kind of ratio between micro and macro scale. We could refer to smooth global term as $u^0$ and the small local term as $u^1$.
\item Step 3. Derivation of hierarchical equations: this step is problem dependant. In other words it can be different for elasticity, plasticity or heat conduction problems.
\item Step 4. Micro equation: derivation equation in step 4 , results in partial differential equation in $u^1$ to $u^0$. Instead of solving the equation directly for $u^1$ a characteristic function will be used. This function can be obtained by solving an auxiliary equation with any numerical method (FEM for example). The characteristic function which could also be called corrector, is used to relate $u^1$ to $(\partial u)^0 /\partial x $ either through a numerical or an analytical solution.
\item Step 5. Homogenization: the gradient of the corrector is used to define a homogenized property tensor in $Y^e$.
\item Step 6. Solve the global boundary value problem: by the use of any method such as FEM and characteristic function from step 4, the field equation can be solved in $X^e$. This gives the globally smooth solution $u^0$.
\item Step 7. Localization: at last the localization equations will be derived indirectly from step 3.
\end{itemize}
\paragraph
Up to now, different methods of homogenization were introduced. There are more mathematical details about these methods that could be found in \cite{Klusemann_2010} and \cite{S.Kurukuri} and \cite{Peter.W}. Also, there are bunch of papers about application of these methods in structural analysis which used constitutive equations for their calculation which are out of scope of this research so they are ignored. In this research, we are considering homogenized thermal conductivity of the heterogeneous composite material. Unlike solid mechanic problems in composite materials, only a few works have been done in thermo-fluid area which in the following some of them are mentioned.
\paragraph
In \cite{Jan.Vorel} Mori-Tanaka method is applied for finding effective thermal conductivity of the composite media reinforced with ellipsoidal inclusions, then extended to account for random orientation of particles and particle size distribution. Comparison of experimental and numerical results demonstrated that Mori-Tanaka method is still applicable for these complex systems.
\paragraph
In \cite{James.K.Carson} authors used both Hashin-Shtrikman bounds and Effective Medium Theory(EMT) to find thermal conductivity of porous materials. According to them isotropic materials which presented as 'porous' may be divided into two classes: internal porosity material in which the optimal heat transfer pathway is through the continuous phase like granular or particulate materials, in which the void volume may be occupied by either liquid or gaseous components and external porosity materials are those in which the optimal heat transfer pathway is through the dispersed phase. It may refer to a material having a continuous solid matrix that contains pores/bubbles, which may be isolated or interconnected. They believed that a model that accurately predicts the effective thermal conductivity of internal porosity will not necessarily be applicable to external porosity materials or vice versa. Their proposed bounds support conclusions from previous studies that suggested there was inherently greater uncertainty involved with predicting the effective thermal conductivity of external porosity materials than there is with internal porosity materials.
\paragraph
Among different homogenization methods, Asymptotic Expansion is more often used for modeling the thermal conductivity in heterogenous material. Young Seok Song and Jae Ryoun Youn examined the effective thermal conductivity tensor of carbon-nano-tubes (CNT) filled composites by using Asymptotic Expansion Homogenization method(AEH) \cite{Young.Seok}. According to them this method is able to perform both localization and homogenization for the heterogenous medium. In multi-scale approach, the homogenization and the localization are the main concerns: the former yields smeared material properties used in the macroscopic field equations and the latter provides estimation of the microscopic material behaviour based on the macroscopic solution and this is one of the superiorities over other methods. Also contribution of complex geometries and anisotropic material properties of fillers can be precisely calculated through AEH method which can not be handled by other analytic models. Yasser M.Shabana and Naotake Noda in \cite{M.Shabana} used AEH to evaluate the thermomechanical effective properties of another group of heterogenous material which is called Functionally Graded Material (FGM) . We will also use Asymptotic Expansion Homogenization method to predict the thermal conductivity of the composite ceramic materials.
\section{Meshfree method with distance field}
\paragraph
In all of the aformentioned homogenization methods, one thing was in common and it was the usage of Finite Element Method (FEM) in discritizing the domain.The Finite Element Method (FEM) has become one of the most popular and powerful analytical tools for studying the behavior of a wide range of engineering and physical problems. Quick development of several general- purpose finite element software packages which verified and calibrated over the years made them available almost to everyone who asks and pays for them\cite{Soheil}.
\paragraph
In applied mathematics, finite element method (FEM) is a general mathematical tool for obtaining approximate solutions to boundary value problems. It uses variational methods (the Calculus of variation) to minimize an error function and produce a stable solution.
\paragraph
Same as the idea of approximating the larger circle by many tiny straight lines, FEM also approximates a complex equation on a large domain by many simple equations on smaller sub-domain called finite elements. Applying finite element method dates back to long times ago. For instance, finding the circumference of a circle by approximating it by the perimeter of polygon by ancient mathematicians was among first FEM applications \cite{Rao} The basic ideas of the finite element method as known today were presented in the papers of Turner et al., \cite{Martin} and Argyris et al.,\cite{Argiris}.
\paragraph
Application of finite element method was quickened by super-fast development of high speed digital computers. The book by Przemieniecki \cite{perzem} presents the finite element method as applied to the solution stress analysis problem. Zienkiewich et al.,\cite{cheung} presented the broad interpretation of the method and its applicability to any general field problem.
\paragraph
Although FEM based methods were considered as a revolution in computational analysis, but it has some shortcomings in finding the physical properties from the realistic geometry of materials. In this regard, it is required to construct a spatial grid that must conform to the shape of the geometric model which is taken from the micro graphs. It will be very difficult due to the fact that meshing could not be done precisely for this kind of geometries. As a result, to avoid difficulties of mesh construction, meshfree method is applied.
\paragraph
At the core of the meshfree method of analysis with distances field is representation of a physical field by the Taylor series expansion, originally proposed by Kantorovich\cite{Kantorovich} and developed by Rvachev \cite{Rvachev_1} and \cite{Rvachev_2}.
\begin{equation}
\label{general_structure}
u\left( w \right)= u(0)+ \sum \limits_{k=1}^m \frac{1}{k!} u_k(0) \omega^k + \omega^{m+1} \Phi
\end{equation}
This representation is a straightforward generalization of a classical Taylor series, where the term $|x-x_0|$ measuring the distance to the point $x_0$, is replaced by $\omega$ measuring the distance to a set of points. Similarly, the $ k^{th} $ order derivatives of the function $u$ in the classical Taylor series are replaced by coefficients $u_k$ that are $k^{th}$ order derivatives of the function $u$ in the direction $n$ normal to the boundary of a geometric domain. In contrast to classical Taylor series, where the coefficients are constants, $ u_k(x,y,z)$ in the expression (\ref{general_structure}) may be arbitrary functions. This also holds when $\omega$ represents approximate distance to the geometric boundary.
\paragraph
Taylor series (\ref{general_structure}) provides connection between the values of a physical field at any spatial point and values of the field and its normal derivatives prescribed on the boundary of a geometric domain. In the context of engineering analysis this means that the function $u$ given by expression (\ref{general_structure}) satisfies specified boundary conditions exactly. The remainder term $\omega^{m+1}\phi$ assures completeness of the Taylor series (\ref{general_structure}), and it can be used to satisfy additional constraints imposed on $u$, which are usually formulated in the form of differential equations, integral equations, or variational principles.
\paragraph
To find a function $u$ that satisfies both boundary conditions and additional constraints one needs to determine the function $\phi$. In most cases, this problem has no exact solution. Thus, $ \phi $ is approximated by linear combination of basis functions:
\begin{equation}
\label{shape function}
\phi = \sum \limits_{i=1}^N C_i \chi_i.
\end{equation}
Now, the solution of the original problem is transformed into determining the numerical values of the coefficients $C_i$ in expression (\ref{shape function}) by any standard numerical method. The basis functions $ \chi_i $ in the last expression can be chosen from any sufficiently complete system of linearly independent functions: polynomials, radial basis functions, B-splines or even finite elements.
Representing physical fields by Taylor series (\ref{general_structure}) reveals two salient features of our meshfree method: exact treatment of boundary conditions (this is the only meshfree method which allows exact treatment of different types of boundary conditions), and clean and modular separation of geometric and analytic information \cite{VShapiro}.
\paragraph
The shape of the geometric domain is completely described by distance $\omega$ to the boundary; and the basis functions can be defined on a grid that does not conform to the geometric input. Since a physical field u represented by expression (\ref{general_structure}) satisfies the prescribed boundary conditions exactly, the solution procedure needs to determine numerical values of the coefficients $C_i$ in the remainder term (\ref{shape function}) such that u gives the best approximation to the differential equation of the problem.
\paragraph
A typical solution procedure includes construction of distance fields to the boundaries where boundary conditions are specified, differentiation of the functions in the Taylor series (\ref{general_structure}) with respect to spatial coordinates, integration over the un-meshed geometric domain and its boundary, solution of an algebraic problem, and visualization of the analysis results.
\subsection{Scan and Solve Approach}
\paragraph
The use of distance fields derived from sampled data makes it possible to implement a ‘scan-and-solve’ approach to modeling of physical fields, which is particularly effective when physical fields need to be modeled and analyzed in existing artifacts for which traditional geometric models may not exist. Reverse engineering of geometric models for such parts is a difficult and time consuming process fraught with difficulties due to inaccuracies, wear, deformations, and imprecision of both natural and engineered objects. The key observation is that the traditional reverse engineering and meshing pipeline may be bypassed if an object model and its boundary are described by an approximate distance field.
\paragraph
Such fields may be often constructed directly from sampled data generated by 3D laser scanners or other scanning technology. Once the points on the surface of the part have been generated, a variety of methods can be used to compute an approximate distance field. In our examples the exact distance field was randomly sampled throughout the volume, and then represented by B-splines with coefficients computed using the least square method. The ‘scan and solve’ approach is depicted in the flowchart in the Fig. \ref{scan and solve} with operands depicted as ovals, and operations depicted as rectangles. At the outset, the geometry is scanned to produce a 2D or 3D image. Here, image refers to a regular grid of pixels or voxels for 2D or 3D geometry, respectively. The image is then segmented using image processing techniques to produce a binary image with foreground and background only. A Euclidean distance transform is then applied to compute a distance value for each image element. Samples of the distance image are taken at randomly distributed points and a set of basis functions is fit to these samples to produce an approximate distance field. The approximate distance field is then used to support meshfree simulation of stress or other physical quantities. \cite{scan}.
\paragraph
\begin{figure}[htb]
\begin{center}
\includegraphics[width=100mm]{scan.eps}
\end{center}
\caption{ "Scan and Solve" Flowchart .}
\label{scan and solve}
\end{figure}
\paragraph
As a demonstration of "Scan and Solve" approach, Fig \ref{David_analysis} shows Scan\&Solve stress analysis performed from a SEM micrograph of fracture surface of TaC sample. The first picture in the analysis pipeline shows a SEM micrograph of fracture surface of TaC sample. The last three pictures illustrates the distributions of the components of the displacement vector and normal stress $\sigma_y$ (shown on the scale from 0 to 16 GPa). Segmentation of the SEM micrograph results in a binary image in which white color depicts material and black color corresponds to the pores. Samples of the Euclidean distance to the boundary are computed using Euclidean distance transformation. These samples are used to construct a smooth approximate distance field whose zero set describes the geometry of the boundary of a cross section. Distance fields to the fixed portions of the boundary are constructed by trimming\cite{sm99}. Meshfree analysis combines distance fields, boundary conditions and basis functions to compute the displacement field and stresses.
\paragraph
\begin{figure}[htb]
\begin{center}
\includegraphics[width=165mm]{SnS.eps}
\end{center}
\vspace{-4mm}
\caption{Scan\&Solve stress analysis performed from a SEM micrograph of fracture surface of TaC sample.}
\label{David_analysis}
\end{figure}