Computational Mechanics of Composite Materials part 2 ppsx

The second part of the simulation procedure is a statistical estimation procedure [29], which enables approximation of probabilistic moments and the relevant coefficients for the given s

( )

2 2 1 2 1

1

2 2

2

1

y

y arctg

,(

),

2 2 1 2 2

2 1

2 1

2 2 1 2 2 1 1 1

y y y

y

x x

y x y x y x y x

since it is a product of the functions of y and 2 y separately Finally, we obtain 1

each y is returned as the independent Gaussian variable

The second part of the simulation procedure is a statistical estimation procedure [29], which enables approximation of probabilistic moments and the relevant coefficients for the given series of output variables and for the specified number of random trials The equations listed below are implemented in the statistical

estimation procedure to compute the probabilistic moments with respect to M,

which denotes here the total number of Monte Carlo random trials

Statistical estimation theory is devoted to determination and verification of statistical estimators computed on a basis of the random trials sets These estimators are necessary for efficient approximation of the analysed random variable and they are introduced for the random variables, fields and processes to assure their stochastic convergence

then the series of random variables X stochastically converges to X Let us note n

that the consistent, unbiased, most effective and asymptotically most effective estimators are available in statistical estimation theory

)(

X E

1 1

X Var

1

2 1

X Var

1

2 1

It can be demonstrated that

( ) ( )

m

1 1

Definition

The estimator of the kth order central probabilistic moment is defined as

( ) ( ω ) [ ( ( )ω ) ( ( )ω ) ]

Any central moments of odd order are equal to 0 in case of the normalized

Gaussian PDF N(m,σ), while the first three even moments are given below

Definition

The estimator of the second order central moment is equal to

( ) ( ) 2

6 6

15

m

Using the proposed estimators of the central moments of the random variable X(ω)

valid for the n-element random event, the following probabilistic coefficients are usually calculated:

Definition

The coefficient of variation for X(ω) is equal to

( ) ( ) ( [ ( ) ( )ω ] )

ωσωα

X E

n i i

1 1 1

Definition

The coefficient of correlation for two variables X( ω) and Y(ω) in two dimensional

n-element random event is equal to

( ) ( )

( ) ( ω ) ( ( )ω )

ωωρ

Y Var X Var

Y X Cov

Equations (1.101) and (1.102) are very useful together with the relevant PDF estimator in recognising of the probabilistic distribution function type for the output variables – using the Central Limit Theorem the Gaussian variables can be found This is very important aspect considering the fact that theoretical considerations in this subject are rather complicated and not always possible

1.3 Stochastic Second Moment Perturbation

Approach

1.3.1 Transient Heat Transfer Problems

The main concept of stochastic second order perturbation technique [263] applied in the next chapters to various transient heat transfer computations can be explained on the example of the following equation [135]:

Q T K T

where K, C are some linear stochastic operators equivalent to the heat conductivity and capacity matrices, T is the random thermal response vector for the structure

with T& representing its time derivative, while Q is the admissible heat flux (due to

the boundary conditions) applied on the system To introduce a precise definition

of K, for instance, let us consider the Hilbert space H defined on a real domain D

and the probability space (Ω,σ,P), where x∈ ,D θ∈Ω and Θ :Ω→R Then, the operator K(x;ω) is some stochastic operator defined on H×Θ, which means that it is a differential operator with the coefficients varying randomly with respect

to one or more independent design random variables of the system; the operator C

can be defined analogously As is known, the analytical solutions to such a class of partial differential equations are available for some specific cases and that is why quite different approximating numerical methods are used (simulation, perturbation

or spectral methods as well)

Further, let us denote the vector of random variables of a problem as {b r(x;θ)}

and its probability density functions as ( r)

b

g and ( r s)

b b

with its covariance in the form

b

Next, all material and physical parameters of Ω as well as their state functions being random fields are extended by the use of stochastic expansion via the Taylor series as follows:

n

n r

r n

n

b x

K n x

K x

E and K(n)( )x;θ represents the nth

order partial derivatives with respect to the random variables determined at the expected values The variable θ represents here the random event belonging to the corresponding probability space of admissible events (nonnegative, for instance) The second order perturbation approach is now analysed and then the random operator K( )x;θ is expanded as

s r rs

r r

b b x K b

x K x K

x

K( ;θ)= 0( ;θ)+ε , ( ;θ)∆ +12ε2 , ( ;θ)∆ ∆ (1.109)

It can be noted that the second order equation is obtained by multiplying the

R-variate probability density function p R(b1,b2, ,b R) by the ε2

-terms and by integrating over the domain of b( )x;θ Assuming that the small parameter ε of the expansion is equal to 1 and applying the stochastic second order perturbation methodology to the fundamental deterministic equation (1.105), we find

• zeroth order equations:

)

;()

;()

;()

;()

;()

;()

;()

;(

)

;()

;()

;()

;

(

, ,

0 0

,

, 0

0 ,

θθ

θθ

θ

θθ

θθ

x Q x T x K x T x K

x T x C x T

x

C

r r

r

r r

=

⋅+

⋅+

⋅+

(1.111)

• second order equations (for r,s=1,…,R):

;()

;(2)

;()

;

(

)

;()

;()

;()

;(2)

;(

, ,

0 ,

, 0

, ,

0

,

θ

θθ

θθ

θθ

θθ

θθ

θθ

x

Q

x T x K x T x K x

T

x

K

x T x C x T x C x T

x

C

rs

rs s

r rs

rs s

r rs

=

⋅+

⋅+

⋅

+

⋅+

⋅+

(1.112)

It is clear that coefficients for the products of K, C and T are the successive

orders of the initial basic deterministic eqn (1.110) and they are taken from the well-known Pascal triangle As far as the nth order partial differential

perturbation-based approach is concerned, then the general statement can be written out using the Leibniz differentiation rule in the following form:

)

;()

;()

;(

)

;()

;()

;()

;(1

)

;()

;(1

)

;()

;(0)

;()

;(

)

;()

;(1

)

;()

;(

0

) ( )

( )

0

(

) ( )

0 ( )

1 ( )

1 (

) 1 ( )

1 (

0 )

) ( )

0

(

) 1 ( )

1 ( 0

)

θθ

θ

θθ

θθ

θθ

θθ

θθ

θθ

θθ

x Q x T x K

n

n

x T x K n

n x T x K

n

n

x T x K

n

x T x K

n x T x C

n

n

x T x C

n x T x

C

n

n n

n n

n

n n

n n

Furthermore, it can be noted that system (1.111) is rewritten for all random

parameters of the problem indexed by r =1,…,R (R equations), while system (1.112) gives us generally R2 equations The unnecessary equations are eliminated here through multiplying both sides of the highest order equation by the appropriate covariance matrix of input random parameters There holds

• zeroth order equations:

)

;()

;()

;()

;()

;()

;()

;()

;()

;(

)

;()

;()

;()

;

(

, ,

0 0

,

, 0

0 ,

θθ

θθ

θ

θθ

θθ

x Q x T x K x T x K

x T x C x T

x

C

r r

r

r r

=

⋅+

⋅+

⋅+

rs

s r

rs rs

b b Cov x T x C x

T x

C

x T x K x

T x K x

Q

x T x K x T

x

C

,)

;()

;(2)

;()

;

(

)

;()

;(2)

;()

;()

;

(

)

;()

;()

;()

;

(

, ,

0 ,

, ,

0 ,

,

) 2 ( 0

) 2 ( 0

θθ

θθ

θθ

θθ

θ

θθ

θθ

&

&

&

⋅+

⋅

−

⋅+

⋅

−

=

⋅+

⋅

(1.116)

It is observed that solving for the nth order perturbation equations system, the closure of the entire hierarchical system is obtained by nth order correlation of input random vector components br and bs, respectively; for this purpose nth order

statistical information about input random variables is however necessary To obtain the probabilistic solution for the analysed heat flow problem, eqn (1.114) is solved for T0, eqn (1.115) for first order terms T,r

and, finally, eqn (1.116) for

d p

b b x T

b T

T

R s r rs

r r

θθ

θθ

1

, 0

∆

∆+

∆+

∆+

b x b x

x x

b x b x

x b x b x

d p

b b

T

d p

b T

d p

T

R s

r rs

R r

r R

θθ

θθ

θθ

θθ

S T

T T

2 1

Now, using the perturbation approach, both spatial and temporal covariances can be determined separately There holds for spatial cross-covariance computed at the specific time moment τ

b

x x x

x b x

x

b

d p

T E T

T E T

S T

T

Cov

R

ij T

θτ

θτ

θ

τθτ

θ

ττ

θτ

2 ( )

2

(

) 1 ( ) 1 ( )

1 ( ) 1

(

) 2 ( ) 1 ( )

2 ( ) 2 ( )

1 ( )

x b

x x

x b x x b

d p

T E T

T E T

S T

T

Cov

R

ij T

θτ

θτ

θ

τθτ

θ

τττ

θτ

1 1

2 1 2

1.3.2 Elastodynamics with Random Parameters

Generally, the following problem is solved now [56,181,198]:

f Ku u C u

various mathematical approaches to the solution of that problem are reported and presented in [233,249,324,326] However the second order perturbation second central probabilistic moment approach is documented below

The stochastic second order Taylor series based extension [208] of the basic deterministic equation (1.124) of the problem leads by equating of the same order terms for τ∈[0,∞) to

• zeroth order equations:

τ

ττ

, 0 0 0 , 0

0

0 0 0 , 0 0 0 , 0 0 0

,

b

f

b u b K b u b C b u b

M

b u b K b u b C b u

b

M

r

r r

r

r r

r

=

++

+

++

τ

ττ

τ

ττ

;2

;2

0

0

0 , 0 , 0

, 0 , 0

, 0

,

0 0 0 , 0 0 0 , 0 0

b

M

b u b K b

u b C b

u b

M

b u b K b u b C b u

b

M

rs

rs rs

rs

s r s

r s

r

rs rs

rs

=

++

+

++

+

++

k n

n

k

k k

n

n

k

k k

n

x b f x

b u x b K

k

n

x b u x b C

k

n

x b u x b M

k

n

0

0 , 0

, 0

, 0

0 , 0

, 0

0 , 0

,

);

;();

;()

;(

);

;()

;(

);

;()

;(

τθτ

θθ

τθθ

τθθ

&

&&

(1.128)

where the operators M,n,C,n,K,n denote nth order partial derivatives of mass,

damping and stiffness matrices with respect to the input random variables determined at the expected values of these variables, respectively The vectors ( )0;τ

represent analogous nth order partial

derivatives of external excitation, accelerations, velocities as well as displacements

of the system

Let us note that the stochastic hierarchical equations of motion for desired

perturbation order m can be obtained from eqn (1.128) by successive expansion and substitution of n by the natural numbers 0,1,…,m, which returns the system of (m+1) equations Then zeroth order solution is obtained from the first equation;

then, inserting the zeroth order solution into the second equation (of the first order), the first order solution can be determined An analogous procedure is repeated to determine all orders of the structural response, which are finally used in the calculation of the response probabilistic moments

Assuming that higher than second order perturbations can be neglected, this equation system constitutes the equilibrium problem The detailed convergence studies should be carried out in further extensions of the model with respect to perturbation order, parameter θ and coefficient of variation of input random variables If higher than the second probabilistic moment approach is considered, then the coefficients of assymetry, concentration, etc., also influence final effectiveness of the perturbation-based solution

Analogously to the stochastic expansion of (1.105), the first and second order equations are modified and it is found that

• zeroth order equations:

( ) ( ) ( ) ( ) ( ) ( ) ( )0 0 0;τ 0 0 0 0;τ 0 0 0 0;τ 0 0;τ

0

b f b u b K b u b C b

,

0 , 0 0 0 , 0 0 0

,

0

0

b u b K b u b C b u b M b

f

b u b K b u b C b

u

b

M

r r

r r

r r

r

++

−

=

++

rs rs

b b Cov b

u b K b

u b C b

u

b

M

b u b K b u b C b u b M

b

f

b u b K b u b C b

;2

;2

, 0 , 0

,

0

,

0 0 0 , 0 0 0 , 0 0 0 , 0

,

0 ) 2 ( 0 0 0 ) 2 ( 0 0 0

τ

ττ

ττ

ττ

Let us observe that looking for the nth order perturbation approach, the closure

of hierarchical equations is obtained by the nth order correlation of input random process components br and bs, respectively; nth order statistical information about

input random variables is however necessary for this purpose

To obtain the probabilistic solution for the considered equilibrium problem, (1.129) is solved for 0

u (and its time derivatives 0

2 1 0

)()

()

b b Cov t u t u t u t u

which completes the two-moment characterization of the perturbation-based solution for the dynamic equilibrium problem (1.124) The entire solution simplifies in the case of free vibrations when the following equations are to be solved:

0]

)

( α

Ω and Φ are the eigenvalues and eigenvectors, respectively and α=1, ,N

denotes the total number of degrees of freedom of a structure The second order expansion leads to the following equation system:

0]

0 , 0 ) ( 0 , ) ( , , 0 0 ) (

M M

K M

{

s r r

r

rs s

r rs

rs

b b Cov M

M K

M M

M K

M K

,]

[

2

]2

[

][

, , 0 ) ( 0 , ) ( ,

0 , 0 ) ( , , ) ( 0 , ) ( ,

) 2 ( 0 0 ) ( 0

ΦΩ

−Ω

−

−

ΦΩ

−Ω

−Ω

−

−

=ΦΩ

−

α α

α α

Equation (1.138) is transformed for this purpose by multiplying by the transposed zeroth order eigenvector, which gives

T

M K

M M

M

Let us observe that Ω,r

is diagonal and therefore

0 0 0 , ) ( 0 0 , ) ( 0

ΦΦΩ

=ΦΩ

which finally results in

[ 0 , ] 0

) ( , 0 , )

s r rs T

T T

b b Cov M

M K

b b Cov M

M K

M M

K

,

,2

, , 0 ) ( 0 , ) ( , 0

0 , 0 ) ( , , ) ( , 0

0 0 ) 2 ( ) ( 0 ) 2 ( 0 0 ) ( 0 0

ΦΩ

−Ω

−Φ

−

ΦΩ

−Ω

−Φ

−

=

ΦΩΦ

−ΦΩ

−Φ

α α

α α

α α

(1.144)

which finally implies

[ r r r] s ( )r s T

s r rs

s r rs T

b b Cov M

M K

b b Cov M

M K

,2

,2

, , 0 ) ( 0 , ) ( , 0

0 , 0 ) ( , , ) ( , 0 )

2

(

ΦΩ

−Ω

−Φ

+

ΦΩ

−Ω

−Φ

=

Φ

α α

α

The next problem is to determine the first and second order derivatives of the eigenvectors Basically, the eigenvector derivative is expressed as a linear combination of all the eigenvectors in the original system Equations describing the

coefficients of the linear combination are formed using the M-orthonormality and

K-orthogonality conditions Starting from (1.138), the αth eigenpair is determined

as

) ( , 0 ) ( 0 , ) ( , ,

) ( 0 0

) (

0

α α

α α

Ω

M M

K M

and (1.143) in the following form:

) ( , 0 ) ( , 0 ) ( , )

)

M M

M K

K

Further, it is assumed that the αth first order eigenvector ,r

) ( α

Φ can be expressed

as a linear combination of all the zeroth order eigenvectors as

r r

a( )

0 , )

ααω

ω

α α

α α

α α α

ˆ,

21

ˆ,

0 ) ( , 0 ) (

0 ) 0 ) (

) ( 0 )

) (

for M

for R

ααω

ω

α α α α

α α

α α

α α α

ˆ

,2

2

1

ˆ,

) ( ) ( , ) ( , 0 ) ( 0 ) ( , 0 ) (

0 ) ( 0 ) (

) 2 ( ) (

0 ) (

M

for R

Finally, the first two probabilistic moments of the eigenvalues and eigenvectors are found in a typical way, which completes the solution of the second order second central probabilistic moment eigenvalue and eigenvector problem

Summing up the applications of the stochastic perturbation methodology it should be pointed out that the main disadvantage is dependence between the assumed order of the expansion, interrelations between input probabilistic characteristics and overall precision of such a computational methodology The method found its numerous applications in structural engineering [88,208,237], in homogenisation [162,164,192] as well as in fluid dynamics computations [184] Computational implementation in conjunction with Finite Element Method both in displacement [208] and stress versions [186], Boundary Element Method [51,185]

as well as with Finite Difference Method [187,198] are available now, whereas the scaled the Boundary-Finite Element Method has no such extension [369]

Nevertheless, the perturbation method can be very useful after successful implementation in symbolic computations programs, which will enable automatic perturbation-based extension of up to nth order [178] for any variational equation

[25,297] as well as ordinary or partial differential equations solutions [68,90] The application of the perturbation method in stochastic processes [319,326] modelling needs its essential improvements, because now the randomness of an input cannot

be introduced both in space and time