1. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 1 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

2. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 2 Introduction, Motivation Assumptions, Mechanical Model Numerical Examples

3. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 3 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.

4. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 4 Mechanical Models, Assumptions (3.a) subject to (3.b-d) Stochastically linearized form:

5. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 5 Probabilistic Compliance Constraint Deterministic constraint:

6. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 6 Minimum Weight Design with Stochastically Calculated Compliance (6.a) subject to (6.b-d)

7. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 7 if Determination of the Active and Passive Sets if Iterative Formula  

8. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 8 Probabilistic Compliance Design in the Case of Uncertain Loading Positions: Adjoint Design

9. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 9 Adjoint Design

10. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 10 Adjoint Design Table 1: Data of the symmetric case Original structureadjointstructure M(x_1)M(x_2)%1%2D(x_1)D(x_2)F1F2wM(A)D(A)M(B)D(B) 30900000100 1201000 0 3090551,5 100 1201000,883881000,88388 309010 33100 1201001,767761001,76776 Table 2: Data of the asymmetric case Originalstructureadjointstructure M(x_1)M(x_2)%1%2D(x_1)D(x_2)F1F2wM(A)D(A)M(B)D(B) 30900000 1005012087,5062,50 3090551,5 1005012087,51,3975462,51,39754 309010 33 1005012087,52,7950862,52,79508

11. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 11 Probabilistic Compliance Design in the Case of Uncertain Loading Positions: Load Distribution

12. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 12 Load Distribution

13. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 13 Numerical Example 20160 FEs, Poisson’s ratio is 0. The compliance limit is C=410000. q=0.9

14. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 14 Analytical Solution of the Deterministic Problem Rozvany, Lewinski

15. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 15 Numerically Obtained Optimal Topology of the Deterministic Problem

16. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 16 Probabilistic Topologies by Adjoint Design

17. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 17 Probabilistic Topologies by Adjoint Design

18. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 18 Probabilistic Topologies by Adjoint Design

19. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 19 Numerically Obtained Optimal Topology of the Deterministic Problem in the Case of Asymmetrical Magnitudes

20. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 20 Numerically Obtained Optimal Topology of the Stochastic Problem in the Case of Asymmetrical Magnitudes

21. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 21 Numerically Obtained Optimal Topology of the Stochastic Problem in the Case of Asymmetrical Magnitudes

22. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 22 Numerical Example II. 25600 FEs, Poisson’s ratio is 0. The compliance limit is C=220000. q=0.75

23. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 23 Analytical Solution

24. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 24 Numerical Solution for Deterministic Design

25. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 25 Adjoint Design

26. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 26 Distributed Load

27. 18th Inter-Institute Seminar, 23-25 September 2011, Budapest, Hungary 27 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