andrzej pownuk naveen kumar goud ramunigari the university...
TRANSCRIPT
![Page 1: Andrzej Pownuk Naveen Kumar Goud Ramunigari The University ...andrzej.pownuk.com/cv/html/Pownuk-Naveen-Scan-2008.pdf · Andrzej Pownuk Naveen Kumar Goud Ramunigari The University](https://reader035.vdocuments.net/reader035/viewer/2022081517/5fc249b81acaa0748a5475a8/html5/thumbnails/1.jpg)
Andrzej PownukNaveen Kumar Goud Ramunigari
The University of Texas at El Paso
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Problems with uncertain geometry
Modeling of uncertainty
Sensitivity analysis
Topological derivative
Optimization based on topological derivative
Examples of application
Conclusion
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: ( )nf R M x f x R →
( ) { ( ) : }f f x x =
inf ( )lf f=
sup ( )uf f=
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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 =
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( ) ( )f g x dx
=
( ) ( )f F g x dx
=
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1 2
[0, ]
( ) ( )( ) ( ) ( )
L
dN x dN xf E x A x dx
dx dx =
( )
( )( )
xd x
fd x
=
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..., ( ) ( ),... ..., ( ) ( ),...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
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inf ( ) : ,l l uf f =
sup ( ) : ,u l uf f =
( ) ( )A B A B
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( ) ( ) ( , )f f df+ −
( ) ( ) ( )f g x d x
=
( , ) ( )df g x
For example
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2 2( ) ( )zI x y d
= +
2 2( , , , ) ( )zdI x y d x y d = +
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( ) ( )( ) 0 ( )f f f f+ − +
( ) ( ) ( , )f f df+ −
If ( , ) 0df then
( )( )f f+
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( ) 0
( ( )) ( )lim
( ) ( )x
f x f df
x d x →
+ − =
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2 2 0( , )
zdIx y
d x y= +
2 2( ) ( )zI x y d
= +
2 2( , , , ) ( )zdI x y d x y d = +
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2 2 0( , )
zdIx y
d x y= +
+
( ) ( )z zI I+
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( ) ( ) ( )f F g x d x
=
( , ) ' ( ) ( ) ( )df F g x d x g x
=
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( ) ( ) ( )f F g x d x
=
' ( ) ( ) ( )( )
dfF g x d x g x
d x
=
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( )
xd
fd
=
2
( ) ( )( )
( )
d dxd d xd d
d x d xdf
d xd
−
=
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2
( )
( )
x d xddf
d xd
−
=
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( )( ) ( )K u Q =
( )( , ) ( ) ( ) ( , ) ,dK d u K du d dQ d + =
( , )du d
can be used in optimization problems.
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( )( ) ( , ) , ( , ) ( )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
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( ) ( ) ( )( ) ( )
( ) ( ) ( )
du dQ dKK u
d x d x d x
= −
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http://andrzej.pownuk.com/
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( , ), ( , ), , 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.
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( , ), ( , ), , 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.
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( )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
=
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( , )0
d du x AE A n
dx dx
+ =
( )( , )
( , ) 0u x A
E u x A E Ax x x A
+ =
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0
( , )
LP
u x A dxE A
=
( ) ( )2 2 2
1 1 1 1| | 1
| |A
A A AA A A
= − = − = −
( )A
A dA =
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( ) ( ) 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=
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2
( )0
( )
x d xddf
d xd
−
= =
xd
xd
=
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( , )g x x d xd
= −
( )( )
( )
dg dx d xd
d x d x
= − =
1 0x x= − =
Interesting property
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max( )u
c Cx x=
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0x d xd
−
Numerical examples use code which is written in Java
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( , )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.
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( )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=
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( , ) ( )dW x dL dr Pdx Qdy= = +
min( )lW W L=
max( )uW W L=
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( , )dW x L P x Q y +
dL dr=
( , ) ( )
( ) ( )
dW x r r
Pdx r Qdy r P x Q y
= =
= + = +
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( , ) ( )
( ) ( )
dW x r r
dx dyPdx r Qdy r P t Q t
dt dt
= =
= + = +
( )
dW dx dyP Q
dL t dt dt= +
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( )S S
f S Pdy dz Qdz dx Rdx dy = + + =
1 2( ) ( ) ( , )df S S r r = =
Pdy dz Qdz dx Rdx dy = + +
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dS
( , , )x y z
1 2( ) ( , )S r r =
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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= =
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( , ) ( , ) ( , )
( , ) ( , ) ( , )
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
= + +
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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
= + +
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( ) ( ) ( )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
= + +
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( )M
f M =
( )( )df dM dM=
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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 =
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{( , , ) : 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 + +
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I would like to thank Dr Behzad Rouhaniand Dr Vladik Kreinovichfor their useful comments.
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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.