Optimal topologies in case of probabilistic
loading
János Lógó
Department of Structural Mechanics
Budapest University of Technology and Economics
Hungary
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

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Introduction, Motivation
Mathematical background
Assumptions, Mechanical models
Parametric Study
Conclusions
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Introduction, Motivation
K. Marti “Stochastic Optimization Methods”, Springer-Verlag, Berlin-Heidelberg,
2005.
K. Marti, “Reliability Analysis of Technical Systems/Structures by means of
polyhedral Approximation of the Safe/Unsafe Domain”, GAMM-Mitteilungen,
30, 2, 211-254, 2007.
G. Kharmada, N. Olhoff, A. Mohamed, M. Lemaire “Reliability-based Topology
Optimization”, Structural and Multidisplinary Optimization, 26, 295-307, 2004.
A. Prékopa ”Stochastic Programming”, Akadémia Kiadó and Kluwer, Budapest,
Dordrecht, 1995.
J. Logo „New Type of Optimality Criteria Method in Case of Probabilistic Loading
Conditions”, Mechanics Based Design of Structures and Machines, 35(2), 147162, 2007.
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Mathematical background
1,2 ,...,n
Joint normal distribution
P( x11  x22  ...  xnn  0 )  q,
x n
(1)
Prekopa (1995) -Kataoka (1963):
n
-1
T
x

+

q
x
 i i   K ov x  0
i=1
where
i  E i  ; (i  1, 2,..., n)
K ov : covariance matrix
 -1  q  : probit function
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
 2
Mechanical models, Assumptions
G
1
p
g g
W    g A t  min!
g 1
(3.a)
subject to
u T F  C  0;

t g  tmin  0;

t g  tmax  0;
 for g  1,..., G  ,
 for g  1,..., G  .
(3.b-d)
P(uT F  C  0 )  q
(4.a)
Stochastically linearized form:
uT F  2uT F  uT F
P(2uT F  uT F  C  0 )  q
(4.b)
(4.c)
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Probabilistic compliance constraint


P 2  u1 f1  u2 f 2  ...  un f n    u1 f1  u2 f 2  ...  un f n   C  0  q
FT   f1 , f 2 ,..., f n , f n1  , fi  E  fi  ,
ij ; i  1,..., n  ,
i  1,..., n ;
f n1  E( f n1)  1;
 j  1,..., n : given and n1,i  0; i,n1  0;
xT   x1 , x2 ,..., xn1 
xi  ui ,  i  1,..., n  ; xn 1    u1 f1  u2 f 2  ...  un f n  C  / 2
n 1
P(2 xi fi  0 )  q
(4.d)
i 1
Prekopa model:
n+1
2 xi fi +2 -1  q  xT K ov x  0
(4.e)
i=1
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
 4.c 
Mechanical model
G
1
p
g g
W    g A t  min!
g 1
(5.a)
subject to


 P 2  u1 f1  u2 f 2  ...  un f n    u1 f1  u2 f 2  ...  un f n   C  0  q,


 t g  tmin  0;  for g  1,..., G  ,

t g  tmax  0;  for g  1,..., G  .
n+1
2 xi fi +2 -1  q  xT K ov x  0
i=1
n+1
x f
i i
 uT Ku  C
i=1
uT Ku  C +2-1  q  xT K ov x  0
5.e 
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
(5.b-d)
Minimum weight design with
stochastically calculated compliance
G
1
p
g g
W    g A t  min!
g 1
(6.a)
subject to
uT Ku  C +2 -1  q  xT K x  0;
ov

t g  tmin  0;  for g  1,..., G  ,

t g  tmax  0;  for g  1,..., G  .
(6.b-d)
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Iterative formulation
Determination of the active and passive sets
  p  Rg  Bg  

tg  


Ag  g


if
if
tmin
p
p 1
  p  Rg  Bg  




Ag  g


tmax
if
tmin  t g  tmax
gA
p
p 1
  p  Rg  Bg  




Ag  g


p
p 1

t g  tmin
g P

t g  tmax
g P
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Calculation of the Lagrange-multiplier n
uT Ku  C +2-1  q  xT K ov x =0, C  2 -1  q  xT K ov x 
C  2 -1  q  xT K ov x 

gP
Rg
tg


gA
Rg
  p  Rg  Bg  




Ag  g


p
p 1

gP
Rg
tg


g A
Rg
tg
,
.
p


p 1


Ag  g




R

g




p
R

B
 g g
g A



 
Rg 
-1
T
 C  2  q  x K ov x  

gP t g 





p 1
p
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
,
(for A  0).
Example 1.
42
f1=50
30
f2=50
30
30
30
20160 FEs, Poisson’s ratio is 0. The compliance limit is C=410000. q=0.9
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Stochastic optimal topology with covariances: 0.1, 0.1, 0, 0
and expected probability value 0.9
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Stochastic optimal topology with covariances: 0.1, 0., 0, 0
and expected probability value 0.9
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Stochastic optimal topology with covariances: 0, 0.1, 0, 0
and expected probability value 0.9
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Stochastic optimal topology with covariances: 0.5, 0.5, 0.0, 0.0
and expected probability value 0.9
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Stochastic optimal topology with covariances: 5, 5, 0, 0
and expected probability value 0.9
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Probabilistic topologies with variable expected probability
with fix mean and covariance values: 0.4, 0.4, 0.01, 0.01
expected probability value: 0.60
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Probabilistic topologies with variable expected probability
with fix mean and covariance values: 0.4, 0.4, 0.01, 0.01
expected probability value: 0.65
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Probabilistic topologies with variable expected probability
with fix mean and covariance values: 0.4, 0.4, 0.01, 0.01
expected probability value: 0.70
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Probabilistic topologies with variable expected probability
with fix mean and covariance values: 0.4, 0.4, 0.01, 0.01
expected probability value: 0.75
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Probabilistic topologies with variable expected probability
with fix mean and covariance values: 0.4, 0.4, 0.01, 0.01
expected probability value: 0.80
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Probabilistic topologies with variable expected probability
with fix mean and covariance values: 0.4, 0.4, 0.01, 0.01
expected probability value: 0.85
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Probabilistic topologies with variable expected probability
with fix mean and covariance values: 0.4, 0.4, 0.01, 0.01
expected probability value: 0.90
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Probabilistic topologies with variable expected probability
with fix mean and covariance values: 0.4, 0.4, 0.01, 0.01
expected probability value: 0.95
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Minimum volumes in function of the
expected probability value
volume-probability
2200
2150
volume
2100
2050
2000
1950
1900
1850
0,6
0,65
0,7
0,75
0,8
0,85
0,9
0,95
expexted probability
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
1
Example 2. Cantilever with two forces
40
f1=50
f2=50
40
40
24200 FEs, Poisson’s ratio is 0. The compliance limit is C=320000 q=0.9
The covariances:
2
1,12  0.0 ,  2,2
 0.4
,
2
1,2
 0.0
,
2
 2,1
 0.0
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Probabilistic topologies with variable expected probability
expected probability value: 0.75
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Probabilistic topologies with variable expected probability
expected probability value: 0.95
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Minimum volumes in function of the
expected probability value
1950
1900
1850
volume
1800
1750
1700
1650
1600
1550
0,6
0,65
0,7
0,75
0,8
0,85
0,9
0,95
expexted probability
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
1
Conclusions
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
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The probabilistically constrained topology
optimization problem was solved
The introduced algorithm provides an iterative tool
which allows to use thousands of design variables
The algorithm is rather stable and provides the
convergence to reach the optimum.
Needs rather simple computer programming
The covariance values have significant effect for
the optimal topology
IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria