STOCHASTIC FINITE ELEMENTS IN A STOCHASTIC FINITE ELEMENTS IN A COMPLEX SYSTEM: VIBRATINGCOMPLEX SYSTEM: VIBRATING

NON STATIONARY FLUID FLOW, MASS NON STATIONARY FLUID FLOW, MASS TRANSPORT, HEAT CONDUCTION.TRANSPORT, HEAT CONDUCTION.

Skender Osmani, Margarita QirkoPolytechnic University ,Tirana, ALBANIA

E-mail s_osmani@yahoo.com

AbstractAbstract

The paper presents a view on applications of stochastic finite The paper presents a view on applications of stochastic finite elements (S.F.E.) elements (S.F.E.) RR33 in one complex system, considering a in one complex system, considering a differential equation with random parameters,differential equation with random parameters,

where,where,xx, , yy, , zz are the 3D spatial coordinates,. are the 3D spatial coordinates,. tt- the processes time- the processes time is a process function( temperature, pressure etc)is a process function( temperature, pressure etc)Kij- the conductivity tensor.In general case it is:Kij- the conductivity tensor.In general case it is:

( 12)( 12)

d- capacity coefficient (function),g- mass coefficient (function),Q- d- capacity coefficient (function),g- mass coefficient (function),Q- density of the volume flux,density of the volume flux,

V –velocity vector.V –velocity vector.

zzzxzx

zzyyyx

xzxyxx

KKK

KKK

KKK

K

Qt

gt

dVz

Vy

Vx

kzy

kyx

kx zyxzyx

2

)()(

This equation in specific conditions goes to:This equation in specific conditions goes to:- non stationary fluid flow in porous medium (or / and) , - non stationary fluid flow in porous medium (or / and) , - mass transport (or /and) - mass transport (or /and) - heat transfer (or/and) - heat transfer (or/and) - vibrating system etc.- vibrating system etc.The paper contains:The paper contains:1-An approach of the mean value estimation of the parameter distributions, 1-An approach of the mean value estimation of the parameter distributions, using S.F.E..using S.F.E..The next is defined as a block vi, with the random function Z(x), where The next is defined as a block vi, with the random function Z(x), where xxvvii. is a random variable i.e. the value at point x determines the . is a random variable i.e. the value at point x determines the respective probability distribution p(x) .After the estimation of the mean respective probability distribution p(x) .After the estimation of the mean value over the domain vi , is calculated by value over the domain vi , is calculated by zzvivi = (1) = (1)2.-Development of the numerical model using S.F.E . applying a mixed 2.-Development of the numerical model using S.F.E . applying a mixed algorithm at the , i.e. it is applied . the Galerkin,s approach not “as a whole” algorithm at the , i.e. it is applied . the Galerkin,s approach not “as a whole” as it is often happened in the literature[3][15] , but partly , combining it with as it is often happened in the literature[3][15] , but partly , combining it with other numerical procedure as Runge-Kutta of the fourth order etc.other numerical procedure as Runge-Kutta of the fourth order etc.In this treatment, the initial and boundary conditions have been supposed to In this treatment, the initial and boundary conditions have been supposed to be treated specifically according by the given problem.be treated specifically according by the given problem. 3.- Several simple examples as particular case of the mentioned equation.3.- Several simple examples as particular case of the mentioned equation.4.Conclusions, the good things of the S.F.E in stationary flow, mass 4.Conclusions, the good things of the S.F.E in stationary flow, mass transport, heat transfer, vibrating , subsidence, waving, deformations, transport, heat transfer, vibrating , subsidence, waving, deformations, consolidations, earthquakes and other phenomenon’s.consolidations, earthquakes and other phenomenon’s.

ivi

dxxZv

)(1

2. Stochastic Finite Element2. Stochastic Finite Element2.1 Mean value estimation2.1 Mean value estimation Let’s consider: a zone V Let’s consider: a zone V RR33 and a random function Z(x) , x and a random function Z(x) , xV; the zone V is partitioned into blocks vi by a V; the zone V is partitioned into blocks vi by a parallelepiped grid :parallelepiped grid : V= vV= v ii (2) (2) where vi is a parallelepiped element with eight nodes. In each node Z(x) is known i.e the probability density on its point where vi is a parallelepiped element with eight nodes. In each node Z(x) is known i.e the probability density on its point support as it is shown in Fig(1) .support as it is shown in Fig(1) .

Fig. 1. Parallelpiped element.Fig. 1. Parallelpiped element.Let, suppose ,it is required Let, suppose ,it is required 88::-the distribution p.d.f in whatever point x-the distribution p.d.f in whatever point xV V -the estimation of the mean value -the estimation of the mean value zzvivi = 1/v = 1/v ii Z(x)dx (3) Z(x)dx (3)over the domain vover the domain v ii . .The stochastic finite element is defined as a block vi, with the random function Z(x), where xThe stochastic finite element is defined as a block vi, with the random function Z(x), where xvvii. is a random variable . is a random variable i.e. the value at point x determines the respective probability distribution p(x).i.e. the value at point x determines the respective probability distribution p(x).Let us consider a reference element wLet us consider a reference element w ii in the coordinate system s in the coordinate system s11 s s22 s s33 22. If we choose an incomplete base: . If we choose an incomplete base: PP(s)=(s)=1 s1 s11 s s22 s s33 s s11ss22 s s22ss33 s s33ss11 s s11ss22ss33 (4) (4)

v

Therefore Therefore ZZvivi= = Z(x Z(x11(s(s11 s s22 s s33 ), x ), x22(s(s11 s s22 s s33 ), x ), x33(s(s11 s s22 s s33 )) det J ds )) det J ds11 ds ds22 ds ds33== < N (s) >{ Z< N (s) >{ Zss

88 } det } det J J dsds11dsds22dsds33 = H = HiiZZii(x) (x) (7) (7) where where JJ is the Jacobian of the transformation , H is the Jacobian of the transformation , H ii are the are the distribution weights depending only on the node coordinates and in distribution weights depending only on the node coordinates and in other words they make the weighted average of the given other words they make the weighted average of the given distributions at the nodes. Furthermore, if we consider the distributions at the nodes. Furthermore, if we consider the expectation:expectation: E{Zv(x)}=E(1/vE{Zv(x)}=E(1/vii (H (HiiZZii(x) dx(x) dx (8) (8)under the hypothesis:under the hypothesis: E (ZE (Zii(x) )= m(x) )= mii m = const , m = const , I = 1,8 I = 1,8 (9) (9)then then E{ Zv(x)}= HE{ Zv(x)}= Hii E(Z E(Zii(x)) = H(x)) = Hiim =m , m =m , i=1,8i=1,8 (10) (10)

Thus,the stochastic estimator of chose finite element is a linear Thus,the stochastic estimator of chose finite element is a linear interpolator, related to the distributions given at its nodes.interpolator, related to the distributions given at its nodes.

1

1

1

1

1

1

8

1

8

1

8

1i

8

1i

Fig 2 Node density distributionFig 2 Node density distribution Fig 3 Node cumulative functionFig 3 Node cumulative function

Fig 4 Mean value density Fig 5 Mean value cumulativeFig 4 Mean value density Fig 5 Mean value cumulative

distribution functiondistribution function

0

20

40

0

0.5

1

1.5

0 1 2 3 4 5

0

0.5

12 13 14 15 16 17012

12 13 14 15 16 17

2.2 Numerical model2.2 Numerical model

Let’s briefly consider some differential equations [12] in a zone Let’s briefly consider some differential equations [12] in a zone RR33

1-1-Heat equation Heat equation

Q (11)Q (11)

where,where,

x, y, z are the 3D spatial coordinates. x, y, z are the 3D spatial coordinates.

T-the temperature function: T = T(x, y, y, z, t)T-the temperature function: T = T(x, y, y, z, t)

t- the processes time, d- capacity coefficient (function), g- mass t- the processes time, d- capacity coefficient (function), g- mass coefficient (function), Q- density of the volume fluxcoefficient (function), Q- density of the volume flux

2

2

t

Tg

t

Td

z

Tk

yx

Tk

x yx

zzzxzx

zzyyyx

xzxyxx

KKK

KKK

KKK

K

22-Solute transport equation-Solute transport equation

QKCt

CRcV

ycV

x

cD

zy

cD

yx

cD

x yxzyx

)()(

c- concentration in the liquid phase as mass of contaminant per unit volume of solutionDx, Dy, Dz, - directional hydrodynamic dispersion coefficient.,R- retardation factor depending on other factors [15][16], k- the overall first order decay rate,Vx, Vy, Vz, -directional (seepage) velocity components

3.Vibrating (system) equation: (14)

m-mass of the system, d-damping coefficient (function), k-elasticity coefficient (function), f-external force[14]As the applied SFE mechanism is analogues to above equations, below we are considering another equation in a given zone R3

Fkut

ud

t

um

2

2

Qt

gt

dVz

Vy

Vx

kzy

kyx

kx zyxzyx

2

)()(

According to the Galerkin’s approach we haveAccording to the Galerkin’s approach we have::

So in these circumstances the velocities, Vx, Vy, Vz will be So in these circumstances the velocities, Vx, Vy, Vz will be considered known. Being so, further we will again simplify the idea, considered known. Being so, further we will again simplify the idea, considering only the differential equations system.considering only the differential equations system.

(31)(31)

thenthen

(36)(36)

Basing on the mentioned algorithm with three dimension SFE it is Basing on the mentioned algorithm with three dimension SFE it is compiled a C++ code for the general equation(15).compiled a C++ code for the general equation(15).

0(

0(

0(

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dy

NhCNVN

dx

NhCNVN

zz

eyy

exx

e

e

e

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dt

dDK eeee

2

2

][

EKXEDQEGdt

Xd

33..Illustrative examplesIllustrative examples

3.13.1.Vibrating system:.Vibrating system:In one SFE structure are considered only two nodes of a vibrating In one SFE structure are considered only two nodes of a vibrating system[13][14] , characterized of the following differential system[13][14] , characterized of the following differential equations:equations:

dd22u/dxu/dx2 2 - 3du/dx + 4u = 6sin(2t)- 3du/dx + 4u = 6sin(2t) dd22v/dxv/dx2 2 - 3dv/dx + 2v = exp(2t)*sin(t) (37)- 3dv/dx + 2v = exp(2t)*sin(t) (37)

The solution is asked at the process time t[0, 1].In order to obtain it, The solution is asked at the process time t[0, 1].In order to obtain it, we take into account the Runge Kutta approach and transform the we take into account the Runge Kutta approach and transform the system as bellow: system as bellow:

dudu11/dt = u/dt = u22 = f = f11(t, u(t, u11, u, u22,u,u3, 3, uu44) (39)) (39)

dudu22/dt = 6sin(2t)+3u/dt = 6sin(2t)+3u22-4u-4u1 1 = f= f22(t,u(t,u11,u,u22,u,u33, u, u44) )

dudu33/dt = u/dt = u44 = f = f33(t, u(t, u11, u, u22,u,u33, u, u44) )

dudu22/dt = exp(2t)*sin(t)+2u/dt = exp(2t)*sin(t)+2u44-2u-2u33= f= f44(t, u(t, u11, u, u22, u, u33, u, u44) )

For different time : For different time : t=0.1, 0.2, 0.3, 0.4, 0.5t=0.1, 0.2, 0.3, 0.4, 0.5

(dividing the given segment into elementary ones with n=4000),are (dividing the given segment into elementary ones with n=4000),are obtained the following results:obtained the following results:U[0.1]=0.9801 U[0.2]= 0.9212 U[0.3]= 0.8256 U[0.4]= 0.6971 U[0.1]=0.9801 U[0.2]= 0.9212 U[0.3]= 0.8256 U[0.4]= 0.6971 U[0.5]= 0.5408 U[0.5]= 0.5408V[0.1]=-0.4617 V[0.2]=-0.5256 V[0.3]=-0.5886 V[0.4]=-0.6466 V[0.1]=-0.4617 V[0.2]=-0.5256 V[0.3]=-0.5886 V[0.4]=-0.6466 V[0.5]=-0.6936 V[0.5]=-0.6936Taking into account the exact solution of the given system:Taking into account the exact solution of the given system: u=cos(2*t);u=cos(2*t); v=0.2(exp(2t)*(sin(t)-2*cos(t))v=0.2(exp(2t)*(sin(t)-2*cos(t))the calculation error of the solution results generally smaller than the calculation error of the solution results generally smaller than 0.0001. 0.0001.

3.2.3.2. Non stationary heat transfer:Non stationary heat transfer:

As a application of heat transfer let’s consider the plane problem : As a application of heat transfer let’s consider the plane problem : cp*(dcp*(d22U/dxU/dx22+k*d+k*d22U/dyU/dy22)+cd*dU/dt +cg*d)+cd*dU/dt +cg*d22U/dtU/dt22 = Q(x,y,t) (39 ) = Q(x,y,t) (39 )

cp=1,cd=1,cg=1;cp=1,cd=1,cg=1;

in one SFE face in which three nodes having their coordinates as: in one SFE face in which three nodes having their coordinates as: 1 x=0,y=1; 2 x=0,y=0; 3. x=1,y=0 1 x=0,y=1; 2 x=0,y=0; 3. x=1,y=0

(39.1)(39.1)t=10t=10U1[1]=-29.000001 U2[1]=-3U1[1]=-29.000001 U2[1]=-3U1[2]=-28.866354 U2[2]=-3.0382929U1[2]=-28.866354 U2[2]=-3.0382929U1[3]=-29.000001 U2[3]=-3U1[3]=-29.000001 U2[3]=-3The exact solution of this problem is:The exact solution of this problem is:

U=xU=x22 + y + y22 - 3*t. - 3*t. Comparing theoretical results to those obtained , the error of heat Comparing theoretical results to those obtained , the error of heat transfer gets to be smaller for transfer gets to be smaller for tt >10. >10.

3.33.3 Heat general equation Heat general equation

Now, in one reference [ ] SFE , let’s consider the Now, in one reference [ ] SFE , let’s consider the general equation of the heat transfer :general equation of the heat transfer :

cpcp11*d*d22u/dxu/dx22 + cp + cp22*d*d22u/dyu/dy22+cp+cp33*d*d22u/dzu/dz22 + cg*d + cg*d22u/dtu/dt22+cd*du/dt +cd*du/dt = Q(x,y,z,t)= Q(x,y,z,t)

cpcp11= cp= cp22= cp= cp33 = cd = cg = 1; = cd = cg = 1; t=0.30t=0.30

U1[1]=2.2224547 U2[1]=-2.2224547U1[1]=2.2224547 U2[1]=-2.2224547U1[2]=2.2224547 U2[2]=-2.2224547U1[2]=2.2224547 U2[2]=-2.2224547U1[3]=2.2224547 U2[3]=-2.2224547 etc.U1[3]=2.2224547 U2[3]=-2.2224547 etc.The exact solution of the above differential equation is The exact solution of the above differential equation is U=(x2+y2+z2)exp(-t), so the solution has a spheric U=(x2+y2+z2)exp(-t), so the solution has a spheric symmetry .This fact is accomplished only for t=0.1 , while symmetry .This fact is accomplished only for t=0.1 , while for t=0.3 that is not. for t=0.3 that is not.

4. Conclusions4. Conclusions The study of complex phenomenon’s could be done using SFE. Their The study of complex phenomenon’s could be done using SFE. Their applications are essential in the geodynamics, energy[8] and water applications are essential in the geodynamics, energy[8] and water resources, reservoir engineering( exploration, exploitation, oil, gas water resources, reservoir engineering( exploration, exploitation, oil, gas water flow[6], well testing[1] ) , hydrology, environment, geostatistics, risk flow[6], well testing[1] ) , hydrology, environment, geostatistics, risk analysis[11] etcanalysis[11] etcSFE algorithm could be used not only in the solution of the mentioned SFE algorithm could be used not only in the solution of the mentioned equations ,but also in geostatistics (Kriking estimation),plane and space equations ,but also in geostatistics (Kriking estimation),plane and space stress, bending etc [12],in the local estimation of different proprieties, stress, bending etc [12],in the local estimation of different proprieties, solving the inverse local problems etc .The geostatistics methods solving the inverse local problems etc .The geostatistics methods could be simultaneously used with other dynamic ones for parameter could be simultaneously used with other dynamic ones for parameter estimation , reserve calculation[7] [9] etc - SFE could be coded using estimation , reserve calculation[7] [9] etc - SFE could be coded using Object – Oriented Languages as C++[5], independently of (excessive) Object – Oriented Languages as C++[5], independently of (excessive) data, complex and difficulties programs, which in the future will need to data, complex and difficulties programs, which in the future will need to be modified , extended etc as it was the case analyzed etc. be modified , extended etc as it was the case analyzed etc. In the view of resources management [10], the SFE approach could In the view of resources management [10], the SFE approach could and should be used as an important strategy in the exploration, and should be used as an important strategy in the exploration, exploitation, using , monitoring, reserve estimation, optimal placement exploitation, using , monitoring, reserve estimation, optimal placement and location etc of energy, oil, water etc. resources in different and location etc of energy, oil, water etc. resources in different engineering projects, shortly say to determine and decide of optimally engineering projects, shortly say to determine and decide of optimally economical solution, the best of the contingency alternatives .economical solution, the best of the contingency alternatives .