Andrzej Pownuk Naveen Kumar Goud Ramunigari The University...

Andrzej PownukNaveen Kumar Goud Ramunigari

The University of Texas at El Paso

Problems with uncertain geometry

Modeling of uncertainty

Sensitivity analysis

Topological derivative

Optimization based on topological derivative

Examples of application

Conclusion

: ( )nf R M x f x R →

( ) { ( ) : }f f x x =

inf ( )lf f=

sup ( )uf f=

if (x<y) then f(x)>f(y)

if (x<y) then f(x)<f(y)

Interval numbers

Interval functions

Interval sets

Interval ???

, { : }l u l ux x x X x x x =

( ) ( )f g x dx

=

( ) ( )f F g x dx

=

1 2

[0, ]

( ) ( )( ) ( ) ( )

L

dN x dN xf E x A x dx

dx dx =

( )

( )( )

xd x

fd x

=

..., ( ) ( ),... ..., ( ) ( ),...K g x d x u Q g x d x

=

..., ( ) ( ),... (..., ,...)

e

eu u g x d x u

= =

Displacement vector u is complicated function of many sets

e

inf ( ) : ,l l uf f =

sup ( ) : ,u l uf f =

( ) ( )A B A B

( ) ( ) ( , )f f df+ −

( ) ( ) ( )f g x d x

=

( , ) ( )df g x

For example

2 2( ) ( )zI x y d

= +

2 2( , , , ) ( )zdI x y d x y d = +

( ) ( )( ) 0 ( )f f f f+ − +

( ) ( ) ( , )f f df+ −

If ( , ) 0df then

( )( )f f+

( ) 0

( ( )) ( )lim

( ) ( )x

f x f df

x d x →

+ − =

2 2 0( , )

zdIx y

d x y= +

2 2( ) ( )zI x y d

= +

2 2( , , , ) ( )zdI x y d x y d = +

2 2 0( , )

zdIx y

d x y= +

+

( ) ( )z zI I+

( ) ( ) ( )f F g x d x

=

( , ) ' ( ) ( ) ( )df F g x d x g x

=

( ) ( ) ( )f F g x d x

=

' ( ) ( ) ( )( )

dfF g x d x g x

d x

=

( )

xd

fd

=

2

( ) ( )( )

( )

d dxd d xd d

d x d xdf

d xd

−

=

2

( )

( )

x d xddf

d xd

−

=

( )( ) ( )K u Q =

( )( , ) ( ) ( ) ( , ) ,dK d u K du d dQ d + =

( , )du d

can be used in optimization problems.

( )( ) ( , ) , ( , ) ( )K du d dQ d dK d u = −

( ), ( , )u du d - Solution of the system of equation

( )

( )

( )

,

,

K

dQ d

dK d

( )( ) ( )K u Q =

- Can be calculated directly

( ) ( ) ( )( ) ( )

( ) ( ) ( )

du dQ dKK u

d x d x d x

= −

http://andrzej.pownuk.com/

( , ), ( , ), , 0

dy xF x y x

dx

=

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

F F ddy x d dy x d dF x y d

y dx

+ + =

where( , )

v .dy x

dx

=

From this equation it is possible to calculate dyand then use this result in the sensitivity analysis.

( , ), ( , ), , 0

dy xF x y x

dx

=

0( ) v ( ) ( )

F dy F d dy F

y d x dx d x x

+ + =

Using presented approach it is possible to avoid approximations errors.

( )22 2

2 2

ji ii

j jj i j

uu uF

x x x t

+ + + =

( )2 2 2

2 2

( , )( , ) ( , )

( ) ( ) ( )

ji i

j jj i j

u x tu x t u x t

x y x x y t y

+ + =

( )22 2

2 2

ji i

j jj i jx x x t

+ + =

( , )

( )

j

j

u x t

y

=

( , )0

d du x AE A n

dx dx

+ =

( )( , )

( , ) 0u x A

E u x A E Ax x x A

+ =

0

( , )

LP

u x A dxE A

=

( ) ( )2 2 2

1 1 1 1| | 1

| |A

A A AA A A

= − = − = −

( )A

A dA =

( ) ( ) 2

0 0

( , )0

L Lu x A P P

dx dxA A E A E A

= = −

( )l uu u A=

( )u lu u A=

2

( )0

( )

x d xddf

d xd

−

= =

xd

xd

=

( , )g x x d xd

= −

( )( )

( )

dg dx d xd

d x d x

= − =

1 0x x= − =

Interesting property

max( )u

c Cx x=

0x d xd

−

Numerical examples use code which is written in Java

( , )0

( )

C idx x

d x

1) Create a list of boxes ( , )i ix

remove the box form the list.

2) Calculate the area .ii

=

3) If

4) Calculate new center of gravity .new

Cx

5) If new old

C Cx x − then, stop.

6) Go to point 2.

( )L L

W W L Pdx Qdy = = + =

[ , ]l uL L L

min : ( , ) 0 uncertainL x L dW x dL=

max : ( , ) 0 uncertainL x L dW x dL=

( , ) ( )dW x dL dr Pdx Qdy= = +

min( )lW W L=

max( )uW W L=

( , )dW x L P x Q y +

dL dr=

( , ) ( )

( ) ( )

dW x r r

Pdx r Qdy r P x Q y

= =

= + = +

( , ) ( )

( ) ( )

dW x r r

dx dyPdx r Qdy r P t Q t

dt dt

= =

= + = +

( )

dW dx dyP Q

dL t dt dt= +

( )S S

f S Pdy dz Qdz dx Rdx dy = + + =

1 2( ) ( ) ( , )df S S r r = =

Pdy dz Qdz dx Rdx dy = + +

dS

( , , )x y z

1 2( ) ( , )S r r =

Divide the S into

Check the sign of

Depending on the sign of define

.iS

( ).iS

( )iS min ,S max .Smin max( ), ( ).l uf f S f f S= =

( , ) ( , ) ( , )

( , ) ( , ) ( , )

y z z x y zPdy dz Qdz dx Rdx dy P Q R ds dt

s t s t s t

= + + = + +

( )( , ) ( , ) ( , )

, ( , ) ( , ) ( , )

d y z z x y zP Q R

dS s t s t s t s t

= + +

2 2 2( )L L L

f L fdl f dx dy dz = = + + =

2 2 2( )df dr f dx dy dz= + +

2 2 2

( )

df dx dy dzf

dL t dt dt dt

= + +

( ) ( ) ( )2 2 2

( )L L S

f S fdS f dy dz dz dx dx dy = = + + =

( ) ( ) ( )2 2 2

( , )df S dS f dy dz dz dx dx dy= + +

( )

2 2 2

( , ) ( , ) ( , )

, ( , ) ( , ) ( , )

df y z z x y zf

dS s t s t s t s t

= + +

( )M

f M =

( )( )df dM dM=

Let us consider volume integral

If the volume is regular enough then

( )V

f V gdx dy dz=

( , )

0

( )

xy

z x y

D

f V gdz dxdy =

{( , , ) : 0 ( , ), ( , ) }xyV x y z z z x y x y D=

( , )

0

( )

xy

z x y

xy

D

df dV gdz D g zdxdy g z + +

I would like to thank Dr Behzad Rouhaniand Dr Vladik Kreinovichfor their useful comments.

Sensitivity analysis can be applied to thesolution of problems with uncertaingeometry.

If it is not possible to apply topologicalderivative then the sign of differential can beused.