Stochastic dynamics and global analysis of wind turbines ... · x X d px tdxd dt W´W ò = Q qq...

Stochastic dynamics and global reliability analysis of wind turbines (II): numerical methods and applications

Tongji University

August 7, 2013

Jianbing Chen, Jie Li

2nd International Summer School on Stochastic Dynamics of Wind Turbines and Wave Energy Absorbers, Aalborg, Denmark, August 6-8, 2014.

Contents

Partition of Probability Space and Point Selection

Finite Difference Method for GDEE Engineering Applications Concluding Remarks

Physical equation (1):

Physical equation (2):

GDEE:

Marginalization:

Generalized Density Evolution Equation (GDEE)

0 0( , , ), ( )t t= =X f X X x Q

0 0[ , ], ( )t= =Z X X Z z Z

1

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

n

jj j

p t p tZ t

t z=

¶ ¶+ =

¶ ¶åZ

Z Zz zQ Qq qq

( , ) ( , , )p t p t dW

= òZ Zz zQ

Q q q

2( )tX

1( )tX

xi

The random event at time instant t1

The same random event at time instant t2

qW

,i iVq

,j jVq

A certain random event

1tW

2tW

Li J, Chen J, Sun W, Peng Y. Advances of probability density evolution method for nonlinear stochastic systems. Probabilistic Engineering Mechanics, 2012, 28: 132‐142.

0( , , ) ( , , )

( , )X Xp x t p x t

d X t dt xW W

¶ ¶+ =

¶ ¶ò ò Q Qq qq q q

0( , , )x

X

dp x t dxd

dt W W=ò Q q q

Partition of Probability‐assigned space, Point Evolution and Ensemble Evolution

0( , , ) ( , , )

( , )X Xp x t p x t

X t dt xqW

æ ö¶ ¶ ÷ç ÷+ =ç ÷ç ÷÷ç ¶ ¶è øò Q Qq q

q q

Note that the following equation

hold for any arbitrary . Then it should of course hold for a partition of , i.e.

qW ÎWQ

WQ

,i iVq

,j jVq

1

for

n

qq

q rq rf

=W = W

W W = " ¹ Q

and thus

0( , , ) ( , , )

( , )q

X Xp x t p x t

X t dt xW

æ ö¶ ¶ ÷ç ÷+ =ç ÷ç ÷÷ç ¶ ¶è øò Q Qq q

q q

hold for .1 2, , ,q n=


We have, thus

which becomes

,i iVq

,j jVq

0 for 1 2( , , ) ( , , )

( , ) , , , ,q

X Xp x t p x t

X t d q nt xW

æ ö¶ ¶ ÷ç ÷+ = =ç ÷ç ÷÷ç ¶ ¶è øò Q Qq q

q q

for .1 2, , ,q n=

0 for 1 2( , , ) ( , ) ( , , ) , , , ,q q

X Xp x t d X t p x t d q n

t xW W

¶ ¶+ = =

¶ ¶ò ò Q Qq q q q q

Denote , we have ( , ) ( , , )q

q Xp x t p x t d

W= ò Q q q

0( , )

( , ) ( , , )q

qX

p x tX t p x t d

t x W

¶ ¶+ =

¶ ¶ ò Qq q q


If we want to solve equations

,i iVq

,j jVq

( , ) ( , , )q

q Xp x t p x t d

W= ò Q q q

0 for 1 2( , )

( , ) ( , , ) , , , ,q

q

X

p x tX t p x t d q n

t x W

¶ ¶+ = =

¶ ¶ ò Qq q q

where , then we need to know all the

information of for all . In this case, we call it the

approach of Ensemble Evolution.

( , )X t qq

ÎWq

Alternatively, we could use some pointto represent the information in the partitioned domain, e.g. , thus we have

q qÎWq

for ( , ) ( , ),q q

X t X t= ÎWq q q

0 for 1 2( , )

( , ) ( , , ) , , , ,q

q

q X


t x W

¶ ¶+ = =

¶ ¶ ò Qq q q


If we want to solve equations

,i iVq

,j jVq

Again note that , we have( , ) ( , , )q

q Xp x t p x t d

W= ò Q q q

0 for 1 2( , )

( , ) ( , , ) , , , ,q

q

q X


t x W

¶ ¶+ = =

¶ ¶ ò Qq q q

0 for 1 2( , ) ( , )

( , ) , , , ,q qq

p x t p x tX t q n

t x

¶ ¶+ = =

¶ ¶ q

i.e, we need to select a set of representative

points , and then get the

information of , and then

solve the above equations. The could be called

the approach of Point Evolution.

1 2, , , ,q

q n= q

for 1 2( , ), , , ,q

X t q n= q


Then in the point evolution approach, we solve the following equations:

,i iVq

,j jVq

where .

After solving the above equations, we have

( , ) ( , , )q

q Xp x t p x t d

W= ò Q q q

0 for 1 2( , ) ( , )

( , ) , , , ,q qq

p x t p x tX t q n

t x

¶ ¶+ = =

¶ ¶ q

1

1

1

( , ) ( , , ) ( , , )

( , , )

( , )

nq q

q

X X

n

Xqn

qq

p x t p x t d p x t d

p x t d

p x t

=W W

W=

=

= =

=

=

ò ò

åò

å

Q Q

Q

q q q q

q q

Q

Because

we could let , which could be called

Assigned Probability. It is also the probability over . Clearly, .

( , ) ( , , ) ( , , ) ( )q q q

q X Xp x t dx p x t d dx p x t dx d p d

¥ ¥ ¥

-¥ -¥ W W -¥ W

æ ö æ ö÷ ÷ç ç= = =÷ ÷ç ç÷ ÷ç÷ç è øè øò ò ò ò ò òQ Q Qq q q q q q

( , ) ( )q

q qP p x t dx p d

¥

-¥ W= =ò ò Q q q

qW

1

1n

qq

P=

=å


Now let , then we can solve the equivalent equations

,i iVq

,j jVq

0 for 1 2( , ) ( , )

( , ) , , , ,q qq

p x t p x tX t q n

t x

¶ ¶+ = =

¶ ¶ q

1 1

( , ) ( , ) ( , )n n

q q qq q

p x t p x t P p x t= =

= = ⋅å å

Thus, the assigned probabilities play roles of

weights in a way in the approach of point

evolution.


( , ) ( , )q q q

p x t p x t P=

and give the resulting PDF by

We should note that the points are related closely to the partition of

the space. One of the methods to connect them is the adoption of

Voronoi cells.

(a) Packing radius (b) Covering radius (c) Voronoi cells

Voronoi cell:

( ) { : for all }sq q q jV V j= Î - £ -x x x q q q


Point selection and partition of spaceGeneralized F‐Discrepancy (GF‐discrepancy)

( , )( , ) sup ([ , ])

s

Nn

nÎ= - 0

n

nnV

1

1n

2n1

We could define the Discrepancy (star‐discrepancy) of a point set as (Hua & Wang, 1981; Niederreiter, 1992; Dick & Pillichshammer, 2010)

1

1( ) ( ) TV( ) ( , )

s

n

kk

f d f f nn =

- £åò x x x

Koksma‐Hlawka inequality (1942/43, 1961):

12

MCS( log log )O n n

-= 1NTM

( )O n e- +=

1

1

1

1

1

UGP( )sO n

-=

Generalized F‐Discrepancy (GF‐discrepancy)

( , )( , ) sup ([ , ])

s

Nn

nÎ= - 0

n

nnV

1

1n

2n1

Discrepancy (star‐discrepancy) (Hua & Wang, 1981; Niederreiter, 1992; Dick & Pillichshammer, 2010)

1

1( ) ( ) TV( ) ( , )

s

n

kk

f d f f nn =

- £åò x x x


• Non‐uniform distribution?• Assigned Probability?• Computational complexity?

12

MCS( log log )O n n

-=1

NTM( )O n e- +=

Point selection and partition of space

From discrepancy to F‐discrepancy

xo

( )x1

( )n x

F‐Discrepancy

(Fang & Wang, 1994)

( , ) sup ( ) ( )s

nnÎ

= -x

x x

1

1( ) { }

n

n qq

In =

= £åx x xE‐CDF:



( , )( , ) sup ([ , ])

s

Nn

nÎ= - 0

n

nnV

1

1n

2n1

( , )( , ) su (p Pr [ , ])

s

Nn

nÎ

= - 0

n

n n

xo

( )x1

( )n x

F‐Discrepancy

(Fang & Wang, 1994)

( , ) sup ( ) ( )s

nnÎ

= -x

x x

1

1( ) { }

n

n qq

In =

= £åx x xE‐CDF:

,i iVq

,j jVq

1

( ) { }n

n q qq

P I=

= ⋅ £åx x xModified E‐CDF:

Point selection and partition of spaceFrom discrepancy to F‐discrepancy


1

1n

2n1

( , ) sup ( ) ( )s

EF nn

Î

= -x

x x

xo

( )x1

( )n x

F‐Discrepancy

(Fang & Wang, 1994)

( , ) sup ( ) ( )s

nnÎ

= -x

x x

1

1( ) { }

n

n qq

In =

= £åx x xE‐CDF:

,i iVq

,j jVq

1

( ) { }n

n q qq

P I=

= ⋅ £åx x xModified E‐CDF:

1

( ) ( ) TV( ) ,1

( )s

n

kk

fn

d f f n=

- £åò x x x


1

( ) ( ) TV( ) ( , )s

n

kk Ek

Ff d f fP n

=

- £åò x x x

Extension of Koksma‐Hlawka inequality (Proinov , 1985; Chen et al, 2013):

Point selection and partition of spaceFrom discrepancy to F‐discrepancy


NP hard problem for computation of EF‐discprency

y = 0.2556x2 + 0.3271xR² = 0.9996

0

20

40

60

80

100

120

140

0 5 10 15 20 25


1

( ) ( ) TV( ) ( , )s

n

kk Ek

Ff d f fP n

=

- £åò x x x



From EF‐discrepancy to GF‐discrepancy

,1max sup ( ) ( )

GF n i ii d xD F x F x

£ £ Î

é ù= -ê ú

ê úë û

{ }, ,1

( )n

n i q i qq

F x P I x x=

= ⋅ £å

——Maximum of marginal discrepancy

{ }, ,1

1( )

n

n i i qq

F x I x xn=

= ⋅ £å

-4 -2 0 2 40

0.2

0.4

0.6

0.8

1

-4-2

02

400.20.40.60.81

,i iVq

,j jVq

Point selection and partition of space• Non‐uniform distribution?• Assigned Probability?• Computational complexity?

,1max sup ( ) ( )

GF n i ii d xD F x F x

£ £ Î

é ù= -ê ú

ê úë û

{ }, ,1

( )n

n i q i qq

F x P I x x=

= ⋅ £å

{ }, ,1

1( )

n

n i i qq

F x I x xn=

= ⋅ £å

1

( ) ( ) ( , )TV( )s

n

k k GFk

f d P f n f=

- £åò x x x

?


From EF‐discrepancy to GF‐discrepancy


1

( ) ( ) TV( ) ( , )s

n

kk Ek

Ff d f fP n

=

- £åò x x x

10-2 10-1

10-2

10-1

DEF

DG

F

10-2 10-1

10-2

10-1

DEF

DG

F

10-2 10-1

10-2

10-1

DEF

DG

F

2‐d case 3‐d case 4‐d case


Optimal partition of probability‐assigned space

Application: Rotational Quasi‐Symmetric Point Set

Q‐SPM (Victoir, 2004; Xiu, 2009)

x

y

x

y

0 20 40 60 80 1000.045

0.05

0.055

0.06

0.065

0.07

0.075

Number of generations

Gen

eral

ized

F-d

iscr

epan

cy

-4 -2 0 2 40

0.2

0.4

0.6

0.8

1

CD

F

PDEM(RQ-SPM)PDEM(Q-SPM)Original CDF

Xu J, Chen JB & Li J. Computational Mechanics, 2012, 50(1): 135–156.Chen JB, Zhang SH, 2013, SIAM Journal on Scientific Computing, in press.

Algebraic accuracy = 50 0.05 0.1 0.15 0.2 0.25

0

0.02

0.04

0.06

0.08

0.1

0.12

Generalized F-discrepancy

Error of Std. DError of Mean

0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1


Error of Std. DError of Mean


Application: Rotational Quasi‐Symmetric Point Set

0 2 4 6 8 10 12 14 16 18 200

0.02

0.04

0.06

Time (s)

Std.

D (m

)

PDEM(Q-SPM) PDEM(RQ-SPM) MCS(9999points)

Time (sec)16.6 16.8 17 17.2 17.4

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-0.05 0 0.05 0.10

10

20

30

40

50

Displacement (m)

PDF

PDF at 7.00 secPDF at 11.25 secPDF at 17.00 sec

-0.05 0 0.05 0.10

0.2

0.4

0.6

0.8

1

Displacement (m)

CD

F

PDEM(RQ-SPM)PDEM(Q-SPM)MCS(9999 points)

-0.05 0 0.05 0.10

0.2

0.4

0.6

0.8

1

Displacement (m)

CD

F


-0.05 0 0.050

0.2

0.4

0.6

0.8

1

Displacement (m)C

DF


nm

( )gx t

to

1k

2k

1nk-

nk

1nm -

2m

1m

-0.1 -0.05 0 0.05 0.1-4000

-2000

0

2000

4000

Inter-story Drift (m)

Res

torin

g fo

rce

(kN

)


0 0.02 0.04 0.06 0.08 0.10

0.02

0.04

0.06


Erro

r of m

ean

MCSRQ-SPM

0 0.02 0.04 0.06 0.08 0.10

0.02

0.04

0.06

0.08

0.1


Erro

r of s

tand

ard

devi

atio

n

MCSRQ-SPM

-0.2 -0.1 0 0.1 0.2 0.30

5

10

15

Displacement (m)

PD

F

PDEM(RQ-SPM)NormalLognormalEVD-I

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

Displacement (m)

CD

F


0.1 0.12 0.14 0.16 0.18 0.20

0.02

0.04

0.06

0.08

Coefficient of variation of stiffness

Err

or o

f mea

n

Q-SPMRQ-SPM

0.1 0.12 0.14 0.16 0.18 0.20

0.02

0.04

0.06

0.08

0.1

0.12

Coefficient of variation of stiffness

Erro

r of s

tand

ard

devi

atio

n

Q-SPMRQ-SPM


Contents

Partition of probability space and point selection


I. Partition probability-assigned space:

II. Carry out deterministic dynamic analysis :

III. Solve GDEE:

IV. Synthesize:

: qW Q q

0( , , ) ( , , )

( , )Z Zp z t p z t

Z tt z

¶ ¶+ =

¶ ¶Q Qq qq

( , ) ( , , )Z Z qq

p z t p z t=å Q q

( ) ( , , ) ( , )q q q

t+ = M Y f Y Y Fq q q

Generalized Density Evolution Equation (GDEE)

Some clues to analytical solution

( , ) ( , )0

p x t p x ta

t x¶ ¶

+ =¶ ¶

Analytical solution: ( , ) ( )p x t f x at= -

x at c x at c- = = +Characteristic line:

t

x

p

o

Some clues to analytical solutionSome clues to analytical solution

( , ) ( , )0

p x t p x ta

t x

¶ ¶+ =

¶ ¶

Analytical solution: ( , ) ( )p x t f x at= -

x at c x at c- = = +Characteristic line:

Without loss of generality, we first deal with the equation in the form of GDEE which is renumbered here for convenience of reference

7.1.1 The Finite Difference Method

It is a hyperbolic partial differential equation. The basic idea of the FDM is to discretize the partial differential equation into an algebraic equation.

( , ) ( , )( ) 0

p x t p x ta t

t x¶ ¶

+ =¶ ¶

To use the FDM, the plane will be meshed by two families of lines

x t

such that a uniform grid with time step and spatial mesh size is determined. For notational convenience, denote the value at the point by .

tx

,j k

p x t ,j k

x j x t k t k

jp

, ; 0; 1, 2, ; 0,1,2,j kx x t t j k= = = =

t

x

o

t

p x

o

and therefore the partial differentiation in terms of can be approximated by

x

then we can have

or

where is the ratio of the time step to the special mesh size. The schematic illustration of this FDM is shown in Figure 7.1.

( ) ( )( )1

1

k kkj j

j

p ppx x

-

-

-é ù¶ê úê ú¶ Dë û

( 1) ( ) ( ) ( ) ( )1[ ]k k k k k

j j j jp p a p pl+-= - -

( 1) ( ) ( ) ( ) ( )1(1 )k k k k k

j j jp a p a pl l+-= - +

t xl = D D

( )kjp

( )1

kjp -

( 1)kjp +

Figure 7.1 One sided scheme

x

to1t 2t kt

1x

2x

1x-

0t

jx

tD

xD 1l

1a

Characteristic linex

to1t 2t kt

1x-

0x

2x-

0t

jx

tDxD 1l

1a

Characteristic line

( ) 0a a> ( ) 0b a<

Figure 7.2 Characteristics and propagation of probability

( 1) ( ) ( ) ( ) ( )1(1 )k k k k k

j j jp a p a pl l+-= - +

it is noticed that the non-zero points at the time instants are limited in the shaded area in Figure 7.2(a), denoted by the small hollow circles. In the case , the actual propagation of the probability is along the characteristic line from the origin on which the points at are denoted by the black points. The condition of stability is

1 2, ,t t

0a >

1 2, ,t t

0a <On the other hand, if , the practical propagation of probability is along the characteristic line shown in Figure 7.2(b). Because now , thus that means this scheme is unstable. Therefore, in this case, the scheme should be modified such that the propagation direction should be the shaded area in Figure 7.2(b). Then we can have

1al <

0a < 1 1al- >

( ) ( )1( , ) k k

j jp pp x tx x

+ -¶¶ D

( 1) ( ) ( ) ( ) ( )1[ ]k k k k k

j j j jp p a p pl++= - -

Likewise, to guarantee the stability of the scheme it is requires that

( )1 0kal- £ £

The schematic illustration of the scheme is shown in Figure 7.3( )

1k

jp +

( )kjp ( 1)k

jp +

Figure 7.3 One-sided scheme

It should be stressed that according to the above analysis, the appropriate scheme should be chosen according to the sign of such that the propagation direction of the numerical solution coincides with the propagation of the real solution, which is determined by the characteristic curves. The schemes for and can also be written in a unified way by

( )a t

0a > 0a <

( ) ( ) ( )

( )

( 1) ( ) ( ) ( )1 1

( 1) ( ) ( ) ( )1 1

1 11

2 2

( ) 1 ( )

k k k kj j j j

k k k kj j j j

p a p a a p a a p

or

p ap u h a p ap u h

l l l l l

l l l

++ -

++ -

= - + - + +

= - - + - +

The Courant-Friedrichs-Lewy(CFL conditions) condition now become

( ) 1kal £

Using the difference to approximate the partial differentiation with accuracy up to second order, should be represented by a central difference, namely

Simultaneously, the second-order partial differentiation can be approximated by a second order difference

2 22

2 2( )

p pa t

t x¶ ¶

=¶ ¶

( )( ) 2 2( 1) ( ) 2 2

2

( )( ) ( )

2

kkk k

j jj j

a tp pp p a t t t o t

x x+

é ùé ù¶ ¶ê ú= - D + D + Dê ú ê úê ú¶ ¶ë û ë û

( )k

jp xé ù¶ ¶ê úë û

( ) ( )( )1 1 2( )2

k kkj j

j

p ppo x

x x+ --é ù¶

= + Dê úê ú¶ Dë û

This is the widely-used Lax-Wendroff scheme, of which the schematic illustration is shown in Figure 7.4.

( ) ( ) ( ) ( )21 1 2

2 2

2( )

k k k kj j j

j

p p ppo x

x x+ -é ù + -¶ê ú = + D

ê ú¶ Dë û

2 2( 1) ( ) ( ) ( ) ( ) ( ) ( )

1 1 1 1[ ] [ 2 ]2 2

k k k k k k kj j j j j j j

a ap p p p p p p

l l++ - + -= - - + + -

( 1) 2 2 ( ) 2 2 ( ) 2 2 ( )1 1

1 1(1 ) ( ) ( )

2 2k k k k

j j j jp a p a a p a a pl l l l l++ -= - + - + +

Figure 7.4 Two-sided scheme

Similarly, the numerical flux is now

( )1

kjp +

( )kjp ( 1)k

jp +

( )1

kjp -

( ),LW ( ) 2 ( ) 2 ( )1

1 1( ) ( )

2 2k k k k

j j j jF F a a p a a pl l+= - + +

This indicates that the numerical flux of the Lax-Wendroff scheme can be regarded as the numerical flux of the one-sided scheme plus a correction second order term.The propagation of the non-zero points at the time instants, is shown in the shaded area in Figure 7.5, where the characteristic line when is a constant is also plotted. Clearly, in contrast to the one-sided schemes, it is seen that the scheme works both for and . The CFL condition for the Lax-Wendroff scheme is

12

( ),LW ( ),One-sided 2 ( )1( )

2k k k

j j jF F a a pl += + - D

1 2, ,t t

a

0a > 0a <

( )1 1ka or al l£ £

x

to1t 2t kt

1x

1x-

0t

jx tD

xD

1l

1a

2x-

Characteristic line

Figure 7.5 The Lax-Wendroff scheme

7.1.4 Dissipation, Dispersion and TVD Schemes

The features of the difference schemes play one of thecentral roles in judging if the numerical solution is a physically reasonable solution. From Figure 7.6, it is seen that if the one-sided difference scheme is employed the numerical solution is greatly smoothed in the vicinity of the left-hand side discontinuity point while tiny high-frequency oscillation occurs in the vicinity of the right-hand side discontinuity point. If the Lax-Wendroff scheme is employed, on the other hand, the numerical solution is closer to the exact solution in the vicinity of the left-hand side discontinuity point. However, severe high-frequency oscillation occurs to the left of the two discontinuity points.

(a) One-sided scheme (b) Lax-Wendroff scheme

Figure 7.6 Numerical solutions computed by different schemes

The effect of smoothing is related to dissipation while the high-frequency oscillation is due to dispersion.

Then we have

The above equation is the modified equation of the difference scheme.Similar treatment works for the Lax-Wendroff scheme. In this case, we can have

2 32 3

2 3

( , ) ( , ) ( , )1 1( , ) ( , )

2 6j k j k j k

j k j k

p x t p x t p x tp x x t p x t x x x

x x x

¶ ¶ ¶-D = - D + D - D +

¶ ¶ ¶

2 2 3 32 3

2 2 3 3,

p p p pa a

t x t x¶ ¶ ¶ ¶

= = -¶ ¶ ¶ ¶

2 2 22 3

2 3

(1 ) ( 1)2 6

a a x a a xp p p pa

t x x xl l- D - D¶ ¶ ¶ ¶

+ = +¶ ¶ ¶ ¶

The one-sided difference scheme is non-negativeness preserving and the numerical results are usually smooth, but the dissipation is too large. On the other hand, the two-sided scheme, e.g. the Lax-Wendroff scheme, is much less dissipative, however, much more dispersive especially in the vicinity of discontinuity. Can we have a balance between them by some of hybrid schemes? This is possible by constructing a TVD scheme. Compared to the real solution, the numerical solution by the Lax-Wendroff scheme is more irregular because of the high-frequency oscillation. This can be measured by a quantity called total variation of a function defined by

discretization

( , )TV[ (, )]

p x tp t dx

x

¥

-¥

¶⋅ =

¶ò

It can be proved that the real solution of partial differentiation equation satisfies

This attribute of the solution function is total variation diminishing (TVD for brevity). Any TVD scheme is monotonicity preserving. We can rewrite the Lax-Wendroff scheme to a flux-difference form

( ) ( ) ( )1TV( )k k k

j jj

p p p¥

⋅ +=-¥

= -å

2 1 0 2 1 0TV[ (, )] TV[ (, )] TV[ (, )], for p t p t p t t t t⋅ £ ⋅ £ ⋅ > >

1 12 2

( 1) ( ) ( ),LW ( ),LW1

( ) ( ),One-sided ( ),One-sided 2 2 ( ) ( )1

( )

1 = ( ) ( )( )

2

k k k kj j j j

k k k k kj j j j j

p p F F

p F F a a p p

l

l l l

+-

- + -

= - -

- - - - D -D

where can be regarded as the difference of numerical flux. Here it is clearly seen that a second order correction term is imposed on the one-sided scheme to construct the Lax-Wendroff scheme. The most intuitive approach to balance between the one-sided and the Lax-Wendroff scheme is to construct a hybrid scheme with the numerical flux as combination of numerical flux of the Lax-Wendroff scheme and that of the one-sided scheme, namely, for

To retain the second order accuracy, must be a nonlinear factor dependent on the data, denoted by , and the above equation is rewritten as

b

1 12 2

( ) ( ) ( ) ( ) ( ) ( )1 1

,k k k k k kj j j jj j

p p p p p p+ -+ -D = - D = -

( ),LWkjF ( ),One-sidedk

jF

12

( ),Hybrid ( ),One-sided ( ),LW

( ),One-sided 2 2 ( )

(1 )

1 ( )

2

k k kj j j

k kj j

F F F

F a a p

b b

b l l +

= - +

= + - D

12jy +

One-sided scheme Lax-Wendroff scheme

Therefore, we require

1 12 2

( ),Hybrid ( ),One-sided 2 2 ( )1( )

2k k k

j j j jF F a a pl l y + += + - D

1 1 1 12 2 2 2

( 1) ( ) ( ),Hybrid ( ),Hybrid1

( ) ( ),One-sided ( ),One-sided 2 2 ( ) ( )1

( )

1 ( ) ( )( )

2

k k k kj j j j

k k k k kj j j j j j j

p p F F

p F F a a p p

l

l l l y y

+-

- + + - -

= - -

= - - - - D - D

1 12 2

1 1 1 12 2 2 2

( 1) ( ) ( ) ( )

2 2 ( ) ( )

1 1( ) ( )

2 21

( )( )2

k k k kj j j j

k kj j j j

p p a a p a a p

a a p p

l l l l

l l y y

++ -

+ + - -

= - - D - + D

- - D - D

1 12 2

0j jy y+ -º º 1 12 2

1j jy y+ -º º

Investigations proved that away from the extreme value the TVD scheme is second order accurate while near the extreme value it is first order accurate. The TVD scheme does work well in the probability density evolution analysis of most problems.

Example 7.1 Comparison between Lax-Wendroff and TVD schemesAgain we study the SDOF system with uncertain natural frequency in Example 6.2. The PDF of the displacement is solved respectively by the Lax-Wendroff scheme and TVD scheme. Shown in Figure 7.7(a) and (b) are the PDFs at 1.00 sec computed respectively by the two schemes, in Figure 7.7 (c) and (d) are the PDFs evolving against time during 0.9 through 1.1 sec. These figures show that the Lax-Wendroff scheme can capter the exact results in most places but not work well in the vicinity of discontinuity because of dispersion, whereas in the TVD scheme the accuracy is high even in the vicinity of discontinuity and the high-frequency oscillation disappears.

PDF compared with the closed form solution

PDF at typical time instants

is a random variable.

SDOF system:

Formal solutions

(Lagrangian description):

20 0 00 0; ( ) , ( )X X X t x X tw+ = = =

0

0

( ) cos( ),

( ) sin( )

X t x t

X t x t

ww w

=

= -w

Numerical examples: SDOF system

Evolution of the PDF The mean and the standard deviation

Numerical examples: SDOF system

is a random variable.

SDOF system:

Formal solutions

(Lagrangian description):

20 0 00 0; ( ) , ( )X X X t x X tw+ = = =

0

0

( ) cos( ),

( ) sin( )

X t x t

X t x t

ww w

=

= -w

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x 106

Inter-floor Displacement (m)

Res

torin

g fo

rce

(N)

1h

h

h

h

h

h

h

h

1m

8m

7m

6m

5m

4m

3m

2m

Jie Li, Jian-Bing Chen. The probability density evolution method for dynamic responseanalysis of non-linear stochastic structures. International Journal for Numerical Methodsin Engineering, 2006; 65: 882-903.

Numerical examples: MDOF system

Joint probability density function

Numerical examples: MDOF system

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Displacement (m)

PD

F

FE-PDEE at 0.5sFE-PDEE at 1sFE-PDEE at 2.5sFE-PDEE at 6sThe exact stationary solution

A one‐dimensional nonlinear system (Di Paola & Sofi, 2002)

Dimension‐reduction of FPK equation: examples

Chen JB, Yuan SR. Equivalent of flux and dimension‐reduction of FPK equation. Journal of Engineering Mechanics, 2014, DOI: 10.1061/(ASCE)EM.1943‐7889.0000804.

A 50‐DOF system ‐linear

(compared with exact solution)

-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

10-15

10-10

10-5

100

Velocity (m/s)

PD

F

The exact solution at 0.08 secThe exact solution at 0.5 secThe exact solution at 6.2secPE-PDEE at 0.08 secPE-PDEEat 0.5 secPE-PDEE at 6.2 sec

0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

0.25

Time (s)

Std

.D (m

/s)

The exact solutionPDEMFE-PDEE

Dimension‐reduction of FPK equation: examples

Dynamic reliability evaluation: Example

A 50‐DOF nonlinear system

(compared with 800,000 times of MCS)

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Velocity (m/s)

CD

F

MCS at 0.04 secMCS at 0.6 secMCS at 23.7 secPDEM at 0.04 secPDEM at 0.6 secPDEM at 23.7 secFE-PDEE at 0.04 secFE-PDEEat 0.6 secFE-PDEE at 23.7 sec

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.410-8

10-6

10-4

10-2

100

Velocity (m/s)

CD

F

MCS at 0.04 secMCS at 0.6 secMCS at 23.7 secPDEM at 0.04 secPDEM at 0.6 secPDEM at 23.7 secFE-PDEE at 0.04 secFE-PDEEat 0.6 secFE-PDEE at 23.7 sec

-0.02 -0.015 -0.01 -0.005 0 0.005 0.01-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 104

Inter-story Drift (m)

Res

torin

g fo

rce

(kN

)

LinearNonlinear

Chen JB, Yuan SR. PDEM‐based Dimension‐Reduction of FPK Equation for Additively Excited Hysteretic Nonlinear Systems. Probabilistic Engineering Mechanics, 2014, http://dx.doi.org/10.1016/j.probengmech.2014.05.002.

Contents

Partition of Probability Space and Point Selection


First-passage reliability:

Dynamic reliability evaluation: absorbing boundary

Once the criterion is violated, the corresponding probability could not return to the safety domain.

Dynamic reliability evaluation: absorbing boundary

First-passage reliability:

Physical equation:

GDEE:

Absorbing condition:

Reliability:

Dynamic reliability evaluation: Example

0 2 4 6 8 10 12-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time [s]

Failu

re P

roba

bilit

yThreshold = MaxSigma

Fflux-based Eq, (N=500)PDEM (N=500)MCS (N=10000 c.o.v=1.26%)MCS (N=500)

0 2 4 6 8 10 12-0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

Time [s]

Failu

re P

roba

bilit

y

Threshold = 2.5MaxSigma

Flux-based Eq. (N=500)PDEM (N=500)MCS (N=200000 c.o.v=1.39%)MCS (N=500)

0 2 4 6 8 10 12-1

0

1

2

3

4

5

6

7x 10-3

Time [s]

Failu

re P

roab

abilit

y

Threshold = 3*MaxSigma

Flux-based Eq. (N=500)PDEM (N=500)MCS (N=100000 c.o.v=4.19%)

Applications (1): World Financial Center（Shanghai, 492 m）

Applications (2)：Large Egg-shaped digester

0 0.2 0.4 0.6 0.8 1.0 1.2-0.1

0

0.1

0.2

0.3

Time (sec)

Mea

n (M

Pa)

LWTVD

0 0.2 0.4 0.6 0.8 1.0 1.20

0.05

0.10

0.15

0.20

Time (sec)

Std

.D (M

Pa)

LWTVD

Jie Li, Huaming Chen, Jianbing Chen. Studies on seismic performances of theprestressed egg‐shaped digester with shaking table test. Engineering Structures,2007, 29(4): 552‐599.

Applications (3)：Base-isolated structure in a hospital

Jianbing Chen, Weiqing Liu, Yongbo Peng, Jie Li. Stochastic seismic response and reliability analysis of base‐isolated structures. Journal of Earthquake Engineering, 2007, 11(6): 903‐924.

0.994 0.995 0.996 0.997 0.998 0.999 10

50

100

150

可靠度

楼数

层层

R = 0.9948Reliability

Num

ber o

f floor

Applications (4)：Shanghai Center Building

Reliability and control of offshore wind turbines

China is now the country with largest installed capacity of wind turbines.

In 2012 the economic loss due to stop of normal operation is 10 billion RMB Yuan, accounted for 30% of the whole capacity!

Applications (5)：Offshore wind turbines

Physical stochastic model for fluctuating wind

Model of wind turbines and loadings



Fluctuating wind speed spectrum Contours of PDF of tip response

10-3

10-2

10-1

100

101

10-3

10-2

10-1

100

101

102

Frequency(Hz)

Ran

dom

Fou

rier S

pect

rum

((m/s

)2 )

PDEMMCM

Time (s)

Dis

plac

emen

t (m

)

120 122 124 126 128 130 132 134 136 138 1400

0.05

0.1

0.15

0.2

0.25



Thank you for your kind attention！

Jianbing ChenSchool of Civil EngineeringTongji UniversityNo.1239, Siping Road, Shanghai 200092, ChinaE‐mail: [email protected] http://dpcetj.org/quntichengyuan/chenjb.htm

Stochastic dynamics and global analysis of wind turbines ... · x X d px tdxd dt W´W ò = Q qq...

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