18th inter-institute seminar, 23-25 september 2011, budapest, hungary 1 j. lógó, d. b. merczel and...
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18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 11
J. Lógó, D. B. Merczel and L. Nagy
Department of Structural Mechanics
Budapest University of Technology and Economics
Hungary
Robust Optimal Topology Design with Uncertain Load Positions
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 22
Introduction, Motivation Assumptions, Mechanical Model Numerical Examples
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 33
Introduction, Motivation
Lógó J., Ghaemi M. and. Vásárhelyi A., Stochastic compliance constrained topology
optimization based on optimality criteria method, Periodica Polytechnica-Civil Engineering, 51, 2, pp. 5-10, 2007.
Lógó J., New Type of Optimality Criteria Method in Case of Probabilistic Loading Conditions, Mechanics Based Design of Structures and Machines, Vol.35, No.2, 2007, pp.147-162 .
Lógó, J., Ghaemi, M. and Movahedi Rad M., Optimal topologies in case of probabilistic loading: The influence of load correlation, Mechanics Based Design of Structures and Machines, 37, 3, pp. 327-348, 2009.
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 44
Mechanical Models, Assumptions
1
1
min!G
pg g g
g
W A t
min
max
0;
0; 1,..., ,
0; 1,..., .
g
g
C
t t for g G
t t for g G
Tu F
(3.a)
subject to
(3.b-d)
P(2 0 ) (4.c)T T C q u F u F
Stochastically linearized form:
2 (4.b)T T T u F u F u F
P( 0 ) (4.a)T C q u F
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 55
Probabilistic Compliance Constraint
Deterministic constraint:
1 2 1 1 1, ,..., , , , 1,..., ; ( ) 1;n n i i n nf f f f f E f i n f E f TF
1, , 1; 1,..., , 1,..., : given and 0; 0;ij n i i ni n j n
1 2 1, ,..., nx x x Tx
1 1 2 2 1 1 2 2P 2 ... ... 0 4.cn n n nu f u f u f u f u f u f C q
n+1
-1ov
i=1
2 +2 0 (4.e)Ti ix f q x K x
1 1 1 2 2, 1,..., ; ... / 2i i n n nx u i n x u f u f u f C 1
1
P(2 0 ) (4.d)n
i ii
x f q
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 66
Minimum Weight Design with Stochastically Calculated Compliance
1
1
min!G
pg g g
g
W A t
(6.a)
subject to
(6.b-d)
-1ov
min
max
+2 0;
0; 1,..., ,
0; 1,..., .
T T
g
g
C q
t t for g G
t t for g G
u Ku x K x
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 77
if 1
p
pg g
gg g
p R Bt
A
Determination of the Active and Passive Sets
min maxgt t t
1
min
p
pg g
g g
p R Bt
A
if
if
g A
g P
g P
1
max
p
pg g
g g
p R Bt
A
mingt t
maxgt t
Iterative Formula
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 88
Probabilistic Compliance Design in the Case of Uncertain Loading Positions: Adjoint Design
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 99
1 1 1
, n n n
i iy i i y i
i i i
x xA f f B f
w w
2
2 2 2
1
ni
A B ii
f
w
Adjoint Design
1 1 1 2 2 2 1 1 1 2 2 2
2 2 2 2 21 1 1 2 2 2 1 1 2 2
M a b a b a M b a M b
D a b a b a D a D
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 1010
Adjoint Design
Table 1: Data of the symmetric case
Original structure adjoint structure
M(x_1) M(x_2) %1 %2 D(x_1) D(x_2) F1 F2 w M(A) D(A) M(B) D(B)
30 90 0 0 0 0 100 100 120 100 0 100 0
30 90 5 5 1,5 1,5 100 100 120 100 0,88388 100 0,88388
30 90 10 10 3 3 100 100 120 100 1,76776 100 1,76776
Table 2: Data of the asymmetric case
Original structure adjoint structure
M(x_1) M(x_2) %1 %2 D(x_1) D(x_2) F1 F2 w M(A) D(A) M(B) D(B)
30 90 0 0 0 0
100 50 120 87,5 0 62,5 0
30 90 5 5 1,5 1,5
100 50 120 87,5 1,39754 62,5 1,39754
30 90 10 10 3 3
100 50 120 87,5 2,79508 62,5 2,79508
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 1111
Probabilistic Compliance Design in the Case of Uncertain Loading Positions: Load Distribution
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 1212
Load Distribution
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 1313
Numerical Example
20160 FEs, Poisson’s ratio is 0. The compliance limit is C=410000. q=0.9
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 1414
Analytical Solution of the Deterministic Problem
Rozvany, Lewinski
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 1515
Numerically Obtained Optimal Topology of the Deterministic Problem
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 1616
Probabilistic Topologies by Adjoint Design
2 21,1 0.88A 2 2
2,2 0.88B 21,2 0.0 2
2,1 0.0
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 1717
Probabilistic Topologies by Adjoint Design
22,1 0.0
2 21,1 1,76A 2 2
2,2 1.76B
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 1818
Probabilistic Topologies by Adjoint Design
21,2 0.0 2
2,1 0.0
2 21,1 3,53A 2 2
2,2 3.53B
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 1919
Numerically Obtained Optimal Topology of the Deterministic Problem in the Case of Asymmetrical Magnitudes
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 2020
Numerically Obtained Optimal Topology of the Stochastic Problem in the Case of Asymmetrical Magnitudes
21,2 0.0 2
2,1 0.0
2 21,1 1,39A 2 2
2,2 1,39B
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 2121
Numerically Obtained Optimal Topology of the Stochastic Problem in the Case of Asymmetrical Magnitudes
21,2 0.0 2
2,1 0.0
2 21,1 2,79A 2 2
2,2 2,79B
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 2222
Numerical Example II.
25600 FEs, Poisson’s ratio is 0. The compliance limit is C=220000. q=0.75
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 2323
Analytical Solution
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 2424
Numerical Solution for Deterministic Design
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 2525
Adjoint Design
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 2626
Distributed Load
18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 2727
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 The asymmetry of the deviations is lost
Conclusions
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