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Optimal topologies in case of probabilistic loading

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Presentation on theme: "Optimal topologies in case of probabilistic loading"— Presentation transcript:

1 Optimal topologies in case of probabilistic loading
János Lógó Department of Structural Mechanics Budapest University of Technology and Economics Hungary IFIP/IIASA/GAMM, Dec, 2007, Laxenburg, Austria

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

3 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, , 2007. G. Kharmada, N. Olhoff, A. Mohamed, M. Lemaire “Reliability-based Topology Optimization”, Structural and Multidisplinary Optimization, 26, , 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), , 2007. IFIP/IIASA/GAMM, Dec, 2007, Laxenburg, Austria

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

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

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

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

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

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

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

11 IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
Example 1. 42 f1= f2=50 20160 FEs, Poisson’s ratio is 0. The compliance limit is C= q=0.9 IFIP/IIASA/GAMM, Dec, 2007, Laxenburg, Austria

12 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, Dec, 2007, Laxenburg, Austria

13 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, Dec, 2007, Laxenburg, Austria

14 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, Dec, 2007, Laxenburg, Austria

15 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, Dec, 2007, Laxenburg, Austria

16 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, Dec, 2007, Laxenburg, Austria

17 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, Dec, 2007, Laxenburg, Austria

18 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, Dec, 2007, Laxenburg, Austria

19 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, Dec, 2007, Laxenburg, Austria

20 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, Dec, 2007, Laxenburg, Austria

21 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, Dec, 2007, Laxenburg, Austria

22 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, Dec, 2007, Laxenburg, Austria

23 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, Dec, 2007, Laxenburg, Austria

24 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, Dec, 2007, Laxenburg, Austria

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

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

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

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

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

30 IFIP/IIASA/GAMM, 10-12 Dec, 2007, Laxenburg, Austria
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, Dec, 2007, Laxenburg, Austria

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