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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.
