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\title{Stochastic Finite Element Analysis for Transport Phenomena in Geomechanics using Polynomial Chaos}
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             \author[1]{S.  Drakos}

             \affil[1]{  International Center for Computational Engineering, Rhodes, Greece.}

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\date{\small \em Received: 10 June 2015 Accepted: 5 July 2015 Published: 15 July 2015}

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\begin{abstract}
        


Transport Phenomena in Geomechanics occur under uncertain conditions and their parameters dominated by spatial randomness. The prediction of the progress of these phenomena is a stochastic problem rather than a deterministic. To solve the problem a procedure of conducting Stochastic Finite Element Analysis using Polynomial Chaos is presented. It eliminates the need for a large number of Monte Carlo simulations thus reducing computational time and making stochastic analysis of practical problems feasible. This is achieved by polynomial chaos expansion of the concentration. An example of a pollution development in a soil is presented and the results are compared to those obtained from Random Finite Element Analysis. A close matching of the two is observed.

\end{abstract}


\keywords{The prediction of the progress, a large number of Monte Carlo simulations.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
he Tranposrt phenomena in geomechanics is a complex problem and cover a wide range of application. Problem as the contamination of land in many countries or recovery of oil in reservoir e.t.c, are dominated from the uncertainty and spatial variability of the properties of soil materials. Various forms of uncertainties arise which depend on the nature of geological formation, the extent of site investigation, the type and the accuracy of design calculations etc. In recent years there has been considerable interest amongst engineers and researchers in the issues related to quantification of uncertainty as it affects safety, design as well as the cost of projects.\par
Zhang, P. and Hu, L. (2014) presented in their study, a generalized model of preferential flow paths based on a dual-domain model which reflects contaminant transport and the dynamic transfer between two domains, and quantitatively analyzes the difference between advection-dispersion model (ADM) and dual-domain mass transfer model (DDM) in pore scale. Basic idea of this paper was the complexity of contaminant transport through highly heterogeneous soil on predicting pollution of the soil and groundwater system. Mousavi et all (2013) presented the development and the application of a numerical model for simulation of advective and diffusive-dispersive contaminant transport using a stochastic finite-element approach. Employing the stochastic finite-element method proposed in this study, the response variability was reproduced with a high accuracy. Johnson et all (2010) introduced an approach which has been incorporated into a spreadsheet model which uses a one-dimensional solution to the advection dispersion equation, which readily lends itself to Monte Carlo applications. The scope of this work was the definition of contaminant source release of contaminant mass transport simulation in the saturated zone as many sites the release history is unknown.  {\ref Nadim et all (2004)} proposed a three-dimensional stochastic model for the generation and transport of LFG in order to quantify the uncertainties. Using Monte Carlo simulations, multiple realizations of key input parameters was generated. For each realization, LFG transport was simulated and then used to evaluate probabilistically the rates and efficiency of energy recovery. Wörman, A. and Xu, S.  {\ref (2001)} developed a new modeling framework for internal erosion in heterogeneous stratified soils by combining existing methods used to study sediment transport in canals, filtration phenomena, and stochastic processes. The framework was used to study the erosion process in sediments caused by a flow of water through a covering filter layer that deviates from geometrical filter rules. By the use of spectral analysis and Laplace transforms, mean value solutions to the transport rate and the variance about the mean was derived for a transport constraint at the upstream boundary and a constant initial transport.\par
In this paper we present a general SFEM for transport phenomena in geomechanics using the method of Generalized Polynomial Chaos (GPC). In the first part of the paper a new algorithm based on RFEM using the Circulant Embedding method (Lord et all 2014) is presented in order to generate the random fields. In the second part, development of SFEM based on the Karhunen-Loeve Expansion for stochastic process discretisation and GPC is described. Finally in the last part of the paper, the problem of contamination of soil due to a surface source is solved by the two methods and the results are compared. and characterized by specific distribution. The Gaussian distribution is the default distribution in most application of probabilistic engineering mechanics. In our case the Gaussian distribution is inappropriate because of coefficients' nature. None of them cannot take negative value. Thus in order to cover all the values of them greater than zero a long normal distribution is adopted.\par
The expected value of a quantity of the problem is given by the following norm:??? ?? 2 ???,?? 2 (??)? = ? ? |?| 2 (??, ð??"ð??")??????? ?? ?? = ð?"¼ð?"¼???? ?? 2 (D) ?\textbf{(2)}\par
In essence the solution of the problem is a function of the form ?? ?? ? ?? × ?? ? ? for every fixed t, i.e. a random field and is not a deterministic function. \par
II. 
\section[{Random Finite Element}]{Random Finite Element}\par
The most common way to solve this problem is to create a random field of the soil properties which is mapped to a grid of finite element and then for different every time realization of the fields \{?? ???? ( ?? , ??) : ? ?? × ??\} \{??( ?? , ??) : ? ?? × ??\} and the source or sink \{??( ?? , ??) : ? ?? × ??\} to solve an ordinary boundary value problem using the Monte Carlo method. Assuming the same randomness for all leads to computational efficiency and is not far from the reality.\par
In the current work the random field is generated by the Circulant Embedding method (Lord et all 2014) using the Fast Fourier Algorithm resulting, as will be shown in the following sections, an exact simulation of stochastic processes.\par
Following the random field generation the displacement field ?? ?? (??) for each realization is taken place. At the end of all running the statistical moment based on the Monte Carlo method is calculated. The problem for each realization of ??? ???? ( ?? , ??) : ? ?? × ??? and the source or sink \{??( ?? , ??) : ? ?? × ??\} is: ? ???? ?? (??,\par
And the expected values at the end:ð?"¼ð?"¼(??(??)) = 1 ?? ? ?? ?? (??) ?? ??=1\textbf{(6)}\par
And the variance??????(??(??)) = 1 ?? ? 1 ?(?? ?? (??) ? ð?"¼ð?"¼(??(??)) 2 ?? ??=1\textbf{(7)}\par
III. 
\section[{Random Field Generation}]{Random Field Generation}\par
The Circulant Embedding method (Lord et all 2014) is a technique used for the generation of realizations of Gaussian stochastic processes. This technique has two main advantages among others. The first is that the statistical properties of the generated process are exactly the same process that we aim. The second advantage arises from the Fast Fourier Transform Algorithm which significantly reduces the computational cost. This method seems to have initially been studied in problems of one dimension by \hyperref[b5]{Davies \& Harte (1987)} and more systematically by \hyperref[b6]{Dembo et al. (1989)}, \hyperref[b7]{Dietrich \& Newsam (1993}\hyperref[b8]{, 1997)}, \hyperref[b9]{Gneiting (2000)}, \hyperref[b11]{Stein (2001)},  {\ref Craigmile (2003), and}\hyperref[b10]{Percival (2006)}. Extension of the method to a multi-parameter problems studied by  {\ref Wood (1999)}, Helgasonetal (2011) whereas in random fields by \hyperref[b7]{Dietrich \& Newsam (1993)}, \hyperref[b14]{Wood \& Chan (1994}\hyperref[b8]{, 1997)}, \hyperref[b12]{Stein (2002}\hyperref[b13]{Stein ( , 2012))}, \hyperref[b10]{Gneiting et al. (2006)}. Considerable work was recently featured by Lord et all (2014) whose principles are followed in this work. According to the method, in the case where the samples are uniformly distributed in space in a twodimensional problem, then the covariance matrix C is Toeplitz (Appendix A) and has as elements Toeplitz blocks (block Toeplitz with Toeplitz blocks (BTTB)). The covariance matrix can be described by the Fast Fourier Algorithm:?? = ?????? ??\textbf{(8)}\par
Where:?? ??=?? 1 ?? 2 ×?? 1 ?? 2 = ?? 1 ??? 2 (9)\par
The F 1 , F 2 are the Fourier matrices with dimension of n 1 × n 1 and n 1 × n 2 respectively and the diagonal matrix D includes the eigenvalues of the covariance matrix. For the application of the method the covariance matrix has to be circulant and for that reason using the BTTB matrix invokes a new circulant ((Appendix A)) matrix is created with ?? 2 blocks of circulant matrices ?? 1 × ?? 1 which is represented uniquely by the reduced matrix ?? ð??"ð??"ð??"ð??"ð??"ð??" = [?? 0 , ? , ?? n 2 ?1 ]. The latter can be replaced uniquely by the vector ?? ð??"ð??"ð??"ð??"ð??"ð??" ? ? n 1 n 2 .\par
Following the process described and provided that the covariance matrix has non-negative and real eigenvalues we get: Where:?? = ?? 1/2 ?? ???? ??\textasciitilde ??(0,2?? ?? )\textbf{(10)}\par
??\textasciitilde ??(0, ??) ??\textasciitilde ??(0, ??)\par
Considering the covariance matrix:??(??, ??) = ?? 2 exp ?? ??? ?? ??? ?? ? ?? ?? ? ??? ?? ??? ?? ? ?? ?? ?\textbf{(12)}\par
In Figures \hyperref[fig_6]{3 to 6}, examples of random fields realization for different correlation lengths of covariance above matrix are presented.  
\section[{a) Karhunen-Loeve Expansion}]{a) Karhunen-Loeve Expansion}\par
One of the major points of the SFEM is the separation of deterministic part from the stochastic part of the formulation. Thus the method has two types of discretization, the ordinary FEM discretization of geometry and the stochastic discretization of random fields. In the current paper in order to reach in these results the Karhunen-Loeve expansion has been used which is the most efficient method for the discretization of a random field, requiring the smallest number of random variables to represent the field within a given level of accuracy. Based on that the stochastic process of the Diffusion and convection coefficients over the spatial domain with the known mean values ?? ? ???? and ??(x) and covariance matrix ??????(x 1 , x 2 ) is given by:?? ???? ???, ??(ð??"ð??")? = exp (?? ???? ? (??) + ? ??? ?? ?? ?? ?? ?? (ð??"ð??") ? ??=1 ) ?????, ??(ð??"ð??")? = exp (??(??) + ? ??? ?? ?? ?? ?? ?? (ð??"ð??") ? ??=1 )\textbf{(13)}\par
In practice, calculations were carried out over a finite number of summations (for example 1-5) and the approximate stochastic representation is given by the trancuated part of expansion. The number of truncated is coming from the Karhunen-Loeve property where the eigenvalues ?? ?? decay as the k increase:?? ???? ???, ??(ð??"ð??")? = exp (?? ???? ? (??) + ? ??? ?? ?? ?? ?? ?? (ð??"ð??") ?? ??=1 ) ?????, ??(ð??"ð??")? = exp (??(??) + ? ??? ?? ?? ?? ?? ?? (ð??"ð??") ?? ??=1 )\textbf{(14)}\par
Where: The rate of eigenvalue decay is inversely proportional to the correlation length. Thus for high correlation length (strong correlation) there is fast decay of the eigenvalues. For small correlation length (weak correlation) we have low decay. For zero correlation length there is not correlation and there is not decay of the eigenvalues. As the correlation length increases the decay rate increasing. If the correlation length is very small i.e correlation length =0.01 then the decay rate is not noticeable.\par
Implemented the Mercem theorem the first four eigenfunction are presented in figure \hyperref[fig_6]{6} where in figures     The Karhunen-Loeve expansion method enables to replace the calculating procedure for the expected value using instead of the abstract space ?? of random fields ? their figures and finally to solve a deterministic problem in space ?? × ?? ? ? ?? instead of space ?? × ??. By performing such replacements in fact a deterministic problem is solved, in contrast to the case of Monte Carlo where a large number of problems carried out. According that the test function of the weak form determined by ?? ? ?? ?? 2 (??, ?? 0 1 (??)) while the solution of the problems in the general form of the boundaries conditions is a function ?? ? ? ?? = ?? ?? 2 (??, ?? ð??"ð??" 1 (??)) which is satisfied the equation: \par
? ?? ? ?? ?? 2 (??, ?? 0 1 (??))\par
The expected value of each side assuming that the behavior of source or sink is deterministic and constant: \par
Where:\par
?? ? Î?" ? ? is the ? joint density of independent random variables ?\par
In order to solve the problem according to the finite element method in the current paper we consider a triangle K with nodes ?? ?? ? ?? (??) , ?? (??) ?, ?? = 1,2,3 . To each node N i there is a hat function ?? ?? associated, which takes the value 1 at node ?? ?? and 0 at the other two nodes. Each hat function is a linear function on K so it has the form:?? i = ?? ?? + ?? ?? ?? + ?? ?? ??\textbf{(19)}\par
The test ?? function belongs to the space:V h = ????????\{?? 1 , ?? 2 , ? , ?? ?? \} ? ? 0 1 (D)\textbf{(20)}\par
Any type of higher order shape functions can be used although it will increase the computational cost.\par
In order to solve the problem of equation 1 we have to create the new space L p 2 (Î?", ? 0 1 (D)). For that reason the subspace S k ? L p 2 (Î?") is considered as (Lord et all 2014).?? ?? = ????????\{ð??"ð??" 1 , ð??"ð??" 2 , ? , ð??"ð??" ? \} (21)\par
Using the dyadic product of the space V h , ?? ?? the space L p 2 (Î?", ? 0 1 (D)) created. Thus?? ??? = ?? ? ??? ?? = ????????\{?? ?? ð??"ð??" ?? , i = 1 ? ??, j = 1, ? ??\} (22)\par
The space ?? ??? has dimension QN and regards the test function v. In the case where exists ?? ?? finite element supported by boundaries condition then the subspace of solution belongs is: W hk = V hk ?????????\{?? ??+1 , ?? ??+2 , ? , ?? ??+???? \} (23)?? ?? ?? = ??? ?? ?? ?? (?? ?? ), ?? ?? = 1,2, ? ???, ?? = 1, ? ??\textbf{(24)}\par
The tensor product of the M ?? ?? ?? subspace results the space of the Generalized Polynomial Chaos:?? ?? = ?? 1 ??? 2 ? ??? ??\textbf{(25)}\par
Xiu \& Karniadakis  {\ref (2003)} show the application of the method for different kind of orthonormal polynomials and in the current paper the Hermite polynomial was used with the following characteristics:?? 0 = 1, < ?? ?? >= 0, ?? > 0 (\textbf{26})\par
Where:\par
?? ?? =< ?? ?? 2 > : are the normalization factors.\par
?? ???? is the Kronecker delta??(??) = 1 ?2?? ?? ? ?? 2 is the density function (27)\par
And:?? ?? = (?1) ?? ?? ?? 2 ?? ?? ???? ?? ?? ? ?? 2 (28)\par
The function ?? ?? ? ?? ??? can be written as the summation of ?? ?? polynomials base as?? ?? (??, ??) = ? ?? ?? (??)ð??"ð??" ?? (??) ?? ??=1 (29)\par
According that and using the inner product of the weak form equation on each polynomial of the ?? ?? base and get: < ??????,\par
And let calculate these two integrals:?? 1 =< ? ?? ???? (??, ??)???(??, ??, ??) ? ???(??)????, ð??"ð??" ?? > ?? = < ? ?? ???? (??, ??) ? ?? ?? (??, ??, ??) nnode i ? ??? ?? (??)??? ?? (??)????, ð??"ð??" ?? > ?? = < ? ?? ?? (??, ??, ??) nnode i ? ?? ???? (??, ??) ? ??? ?? (??)??? ?? (??)????, ð??"ð??" ?? > ?? = < ? ? ?? ?? (??, ??)?? ?? (??) P k nnode i ? ?? ???? (??, ??) ? ??? ?? (??)??? ?? (??)????, ð??"ð??" ?? > ?? = < ? ? ?? ?? (??, ??)?? ?? (??) P k nnode i ? ?? ?? ???? ? (??)+? ??? ?? ?? ?? ?? ?? (??) ?? ??=1 ? ??? ?? (??)??? ?? (??)????, ð??"ð??" ?? > ?? = < ? ? ?? ???? (??, ??)?? ?? (??) P k nnode i ?? ? ??? ?? ?? ?? ?? ?? (??) ?? ??=1 ? ?? ?? ???? ? (??) ? ??? ?? (??)??? ?? (??)????, ð??"ð??" ?? > ?? = ? ? ?? ???? (??, ??) ? ??(??)?? ? ??? ?? ?? ?? ?? ?? (??) ?? ??=1\par
ð??"ð??" ?? (??)ð??"ð??" ?? (??)?? P k nnode i ? ?? ?? ???? ? (??) ? ??? ?? (??)??? ?? (??)???????? ?? = ?? ?< ?? ??(??) ð??"ð??" ?? (??)ð??"ð??" ?? (??) > ?? ?(??, ??)\textbf{(36)}\par
Where?? = ? ?? ?? ???? ? (??) ? ??? ?? (??)??? ?? (??)???? ??\textbf{(37)}\par
< ?? ??(??) ð??"ð??" ?? (??)ð??"ð??" ?? (??) > = ? ??(??)?? ? ??? ?? ?? ?? ?? ?? (??) \par
V. 
\section[{Numerical Example}]{Numerical Example}\par
Considering a point source of pollution and the need to estimate its progress due to diffusion and advection phenomena. The problem and the geometry of the finite elements used presented in figure \hyperref[fig_4]{10}. The advection coefficient for simplicity is assumed to be deterministic and constant where the diffusion coefficient present spatial randomness and it is simulated as a random field (figure \hyperref[fig_12]{11}).\par
To solve the problem the application of the numerical algorithms described in the previous paragraphs is presented and results are compared to those obtained from RFEM using Monte Carlo simulations. The dimensionless input data of the problem is the random field diffusion coefficient with a constant average value equal to 0.1 and a fixed advection coefficient equal to 0.1. An initial source of point pollution equal to 2 is applied as described in figure \hyperref[fig_4]{10}.\par
In the figures 12-14 the two methods results of the expected values and standard deviation of concentration are shown for variation coefficient ?? ?? = 0.4 and number of Monte Carle samples for the RFEM equal to 1000.\par
The problem then was solved for three different number of Monte Carle samples for the RFEM and 5, 50, 500 simulations are executed creating 10,100, and 1000 realisation while for SFEM were used one dimensional Hermite GPC with order 3  {\ref (} In figures 14, 15 the results for the various calculation are presented. It is observed that as the number of sample increase the results of the Monte Carlo convergence to GPC method and for number of samples equal to 1000 a great accuracy is presented. In figures 17 and 18 this reduction is shown for a variation coefficient ?? ?? = 0.5 .    
\section[{Conclusions}]{Conclusions}\par
A procedure of conducting a Stochastic Finite Element Analysis of Transport phenomena in Geomechanics where uncertainty arises due to spatial variability of mechanical parameters of soil/rock has been presented. Two different approaches in order to quantifying uncertainty are discussed. The first approach involves generating a random field based on Circulant embedding method and the second Stochastic Finite Element using Polynomial Chaos. A problem of a point source of pollution its progress due to diffusion and advection phenomena used to show the application of the methods.\par
It is shown that the results of SFEM using polynomial chaos compare well with those obtained from Random Finite Element Method. The main advantage in using the proposed methodology is that a large number of realisations which have to be made for RFEM are avoided, thus making the procedure viable for realistic practical problems. 0.00\% \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{13}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Figure 1 : 3 )}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}Stochastic(}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{21031045}\includegraphics[]{image-5.png}
\caption{\label{fig_3}Figure 2 : 10 Figure 3 : 10 Figure 4 :Figure 5 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-6.png}
\caption{\label{fig_4}1 ??,}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-7.png}
\caption{\label{fig_5}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{6}\includegraphics[]{image-8.png}
\caption{\label{fig_6}Figure 6 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{7}\includegraphics[]{image-9.png}
\caption{\label{fig_7}Figure 7 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{8}\includegraphics[]{image-10.png}
\caption{\label{fig_8}Figure 8 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{910}\includegraphics[]{image-11.png}
\caption{\label{fig_9}Figure 9 :Figure 10 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-12.png}
\caption{\label{fig_10}?}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-13.png}
\caption{\label{fig_11}<<}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{11}\includegraphics[]{image-14.png}
\caption{\label{fig_12}Thus?? 1 = 1 +=}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-15.png}
\caption{\label{fig_13}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1011}\includegraphics[]{image-16.png}
\caption{\label{fig_14}Figure 10 :Figure 11 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1213}\includegraphics[]{image-17.png}
\caption{\label{fig_15}Figure 12 :Figure 13 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{142015151617518}\includegraphics[]{image-18.png}
\caption{\label{fig_16}Figure 14 : 2015 EFigure 15 :Figure 16 :Figure 17 : 5 0Figure 18 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.3386973180076628\textwidth}P{0.23448275862068965\textwidth}P{0.03582375478927203\textwidth}P{0.032567049808429116\textwidth}P{0.20842911877394635\textwidth}}
\multicolumn{5}{l}{c) Generalized Polynomial of chaos and stochastic}\\
\tabcellsep Galerkin solution\tabcellsep \tabcellsep \tabcellsep \\
\tabcellsep \multicolumn{4}{l}{Assuming that the ?? ?? ?? represents a space}\\
of\tabcellsep univariate\tabcellsep orthonormal\tabcellsep polynomial\tabcellsep of\\
\multicolumn{4}{l}{variable ?? ?? ? ?? ?? ? ? with order k or lower and:}\tabcellsep \\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep ( ) Volume XV Issue\\
\tabcellsep \tabcellsep \tabcellsep \tabcellsep Global Journal of Researches in Engineering\end{longtable} \par
 
\caption{\label{tab_1}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.09588086185044359\textwidth}P{0.006463878326996198\textwidth}P{0.6259188846641318\textwidth}P{0.06463878326996197\textwidth}P{0.03124207858048162\textwidth}P{0.0075411913814955635\textwidth}P{0.014005069708491762\textwidth}P{0.004309252217997465\textwidth}}
\tabcellsep \tabcellsep \tabcellsep \tabcellsep \tabcellsep ?\\
\tabcellsep \tabcellsep ??\tabcellsep \tabcellsep \\
\tabcellsep \tabcellsep ?< ? ??(??, ??)???(??, ??, ??)\tabcellsep \multicolumn{3}{l}{? ??(??)????, ð??"ð??" ?? > ? ? ??(??, ??) ? ??(??)????}\tabcellsep (33)\\
\tabcellsep \tabcellsep ??\tabcellsep \tabcellsep \tabcellsep ??\\
\tabcellsep \tabcellsep \tabcellsep Set\tabcellsep \\
\tabcellsep \tabcellsep \multicolumn{4}{l}{?? 1 =< ? ?? ???? (??, ??)???(??, ??, ??) ? ???(??)????, ð??"ð??" ?? >}\tabcellsep (34)\\
\tabcellsep \tabcellsep ??\tabcellsep \tabcellsep \\
Year 2015\tabcellsep \tabcellsep \multicolumn{2}{l}{?? ?? 2 < ? ??(??, ??)???(??, ??, ??)}\tabcellsep \multicolumn{2}{l}{? ??(??)????, ð??"ð??" ?? >}\\
40\tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
I\tabcellsep \tabcellsep \tabcellsep \tabcellsep \\
Global Journal of Researches in Engineering E ( ) Volum e XV Issue II Version\tabcellsep Where:\tabcellsep \multicolumn{4}{l}{ð??"ð??" ?? >=< ? ?? < ? ???? ?? ? ???? ??=1 ?? ???? (? ?? ?? ?? ? ?? ?? ? ?? ?? ???? ? ? < ? ?????????? ???? ???? (??, ??) ???? ?? ?????????? ??=1 ? ? ???? ???? (??, ??) ???? ?? ? ??(??)ð??"ð??" ?? (??)ð??"ð??" ?? (??) ? ?? ?? (??)????, ð??"ð??" ?? >= ?? ?? ??=1 ? ?? ?? (??) ? ?? ?? (??)???????? ?? ?????????? ??=1 ????(??, ??) ???? ? ??? < ð??"ð??" ?? ð??"ð??" ?? > ??=1 ? ?? ?? ??????????}\tabcellsep =\tabcellsep (30)\\
\tabcellsep \tabcellsep \multicolumn{3}{l}{?? = ? ?? ?? (??) ? ?? ?? (??)}\tabcellsep ????\tabcellsep (31)\\
\tabcellsep \tabcellsep \multicolumn{2}{l}{??}\tabcellsep \end{longtable} \par
  {\small\itshape [Note: ?? (??) ? ?? ?? (??)????, ð??"ð??" ?? >= ???? (??, ??)ð??"ð??" ?? (??) ?? ??=1 ) ? ?? ?? (??) ? ?? ?? (??)????, ð??"ð??" ?? >= ?? ð??"ð??" ?? (??) ?? ??=1 ? ?? ?? (??) ? ?? ?? (??)????, ð??"ð??" ?? >= < ð??"ð??" ?? ð??"ð??" ?? >= ? ??(??) ?? ð??"ð??" ?? (??)ð??"ð??" ?? (??)???? (32) And the RHS of the weak form: © 2015 Global Journals Inc. (US)]} 
\caption{\label{tab_2}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.85\textwidth}}
16.00\%\\
14.00\%\\
12.00\%\\
10.00\%\\
8.00\%\\
6.00\%\\
2.00\% 4.00\% Î?"std[uGlobal Journal of Researches in Engineering 1 10 100 1000 ( ) Volume XV Issue II Version I\end{longtable} \par
 
\caption{\label{tab_3}] of the two methods Number of realisations}\end{figure}
 			\footnote{© 2015 Global Journals Inc. (US)} 		 		\backmatter  			 \par
Appendix A Definition 1. A Toeplitz matrix is an n × n matrix ?? ?? = [?? ??,?? : ??, ?? = 0, 1 ? , ?? ? 1] where ?? ??,?? = ?? ????? , i.e., a matrix of the form:\par
When every row of the matrix is a right cyclic shift of the row above it so that ?? ?? = ?? ?(?????) for ?? = 1,2 ? , ?? ? 1. In this case the matrix is called Circulant and is equal to:\par
Toeplitz matrix then the ?? × ?? matrix with ?? = ?? 1 ?? 2 and the form:\par
is called Block Toeplitz matrix with Toeplitz Blocks (BTTB).			 			  				\begin{bibitemlist}{1}
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