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János Lógó Department of Structural Mechanics Budapest University of Technology and Economics

János Lógó Department of Structural Mechanics Budapest University of Technology and Economics Hungary. Optimal topologies in case of probabilistic loading. Introduction , Motivation Mathematical background Assumptions , Mechanical models Parametric Study Conclusions.

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János Lógó Department of Structural Mechanics Budapest University of Technology and Economics

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  1. János Lógó Department of Structural Mechanics Budapest University of Technology and Economics Hungary Optimal topologies in case of probabilisticloading IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  2. Introduction, Motivation • Mathematical background • Assumptions, Mechanical models • Parametric Study • Conclusions IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  3. 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), 147-162, 2007. IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  4. Mathematical background Joint normal distribution Prekopa (1995) -Kataoka (1963): where IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  5. Mechanical models, Assumptions (3.a) subject to (3.b-d) Stochastically linearized form: IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  6. Probabilistic compliance constraint Prekopa model: IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  7. Mechanical model (5.a) subject to (5.b-d) IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  8. Minimum weight design withstochastically calculated compliance (6.a) subject to (6.b-d) IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  9. Iterative formulation Determination of the active and passive sets if  if  if IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  10. Calculation of the Lagrange-multipliern IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  11. 42 f1=50 f2=50 30 30 30 30 Example 1. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. Stochastic optimal topology with covariances: 5, 5, 0, 0 and expected probability value 0.9 IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. Minimum volumes in function of the expected probability value IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  26. 40 f1=50 f2=50 40 40 Example 2. Cantilever with two forces 24200 FEs, Poisson’s ratio is 0. The compliance limit is C=320000q=0.9 The covariances: , , , IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  27. Probabilistic topologies with variable expected probability expected probability value:0.75 IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  28. Probabilistic topologies with variable expected probability expected probability value:0.95 IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  29. Minimum volumes in function of the expected probability value IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria

  30. Conclusions • 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

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