18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 1 J. Lógó, D. B. Merczel and L. Nagy Department of Structural Mechanics Budapest

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


Introduction, Motivation Assumptions, Mechanical Model Numerical Examples


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.


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


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


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


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


Probabilistic Compliance Design in the Case of Uncertain Loading Positions: Adjoint Design


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


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


Probabilistic Compliance Design in the Case of Uncertain Loading Positions: Load Distribution


Load Distribution


Numerical Example

20160 FEs, Poisson’s ratio is 0. The compliance limit is C=410000. q=0.9


Analytical Solution of the Deterministic Problem

Rozvany, Lewinski


Numerically Obtained Optimal Topology of the Deterministic Problem


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



22,1 0.0

2 21,1 1,76A 2 2

2,2 1.76B



21,2 0.0 2

2,1 0.0

2 21,1 3,53A 2 2

2,2 3.53B


Numerically Obtained Optimal Topology of the Deterministic Problem in the Case of Asymmetrical Magnitudes


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


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


Numerical Example II.

25600 FEs, Poisson’s ratio is 0. The compliance limit is C=220000. q=0.75


Analytical Solution


Numerical Solution for Deterministic Design


Adjoint Design


Distributed Load


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

Documents

18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 1 J. Lógó, D. B. Merczel and L. Nagy Department of Structural Mechanics Budapest