Copyright © MSC.Software Corporation, All rights reserved. Chaos and Uncertainty: Fighting the Stagnation in CAE J. Marczyk Ph.D. MSC Software Managing Director & Chief Scientist Stochastic Simulation J. Marczyk Ph.D. MSC Software Managing Director & Chief Scientist Stochastic Simulation

Copyright © MSC.Software Corporation, All rights reserved. CONTENT  Introduction  Have Computers Killed Physics?  Is Optimization Really Possible?  Risk Analysis: A Must for Complex Systems  Conclusions

Copyright © MSC.Software Corporation, All rights reserved. Crash: Where is the physics?  Is crash deterministic or stochastic?  Is crash predictable?  Is crash optimizable?  Does it make sense to speak of precision in crash simulations?  Do we need to increase the number of elements in our crash models? What is the reasonable limit?  What is the future of computer-based crash analysis?  Is crash a chaotic phenomenon?

Copyright © MSC.Software Corporation, All rights reserved. Example of Measured Acceleration Signal  A series of tests for chaos are performed with this signal.

Copyright © MSC.Software Corporation, All rights reserved. Log-linear Power Law  Systems that exhibit a log-linear Power Spectrum are potentially chaotic.

Copyright © MSC.Software Corporation, All rights reserved. Typical Tests for Chaos  Hausdorff (Capacity dimension). Signal has fractal dimension (1.8).  Log-linear Power Spectrum (yes).  Correlation dimension (5).  Lyapunov Characteristic Exponents (+0.4).  Poincare’ sections or Return Maps (check for structure).  According to these tests, the measured crash signal possesses a clear chaotic flavour. This explains why each crash is a unique event and cannot be optimised.

Copyright © MSC.Software Corporation, All rights reserved. Crash = Chaos  Chaos can be described by closed-form deterministic equations. Chaos does NOT mean random.  Chaos is characterised by extreme sensitivity to initial conditions.  “Memory” of initial conditions is quickly lost in chaotic phenomena (“butterfly effect”).  Examples of chaotic phenomena:  Tornados (weather in general)  Stock market evolution, economy  Crash, impacts, etc.  Earthquakes  Avalanches  Combustion/turbulence  EEG (alpha-waves in brain)  Duffing, Van der Pol, Lorenz oscillators, etc.

Copyright © MSC.Software Corporation, All rights reserved. The Logistic Map  X(n+1) = k X(n)(1-X(n)) Shows astonishingly complex behaviour:  0 < k < 1, Extinction regime  1 < k < 3, Convergence regime  3 < k < 3.57, Bifurcation regime  3.57 < k < 4, Chaotic regime  4 < k, Second chaotic regime

Copyright © MSC.Software Corporation, All rights reserved.  Phenomena that are chaotic, are unpredictable (nonrepeatable). The main reason is extreme sensitivity to initial conditions.  Phenomena that are unpredictable, cannot be optimized. They must be treated statistically.  All that can be done with chaotic phenomena is increase our understanding of their nature, properties, patterns, structure, main features, quantify the associated risks.  Models for Risk Analysis must be realistic to be of any use. Chaos and Predictability

Copyright © MSC.Software Corporation, All rights reserved. Understanding Risk  Essentially, risk is associated with the existence of outliers Outlier: - warranty - recall - lawsuit Most likely response (highest density) Note: DOE and Response Surface techniques cannot capture outliers }

Copyright © MSC.Software Corporation, All rights reserved. What is Risk and Uncertainty Management?  Understand and remove outliers  Shift entire distribution is safe fails Improved design Initial design Outlier Outliers: unfortunate combinations of operating conditions and design variables that lead to unexpected behaviour.

Copyright © MSC.Software Corporation, All rights reserved. Example of Robust Design: MIR Space Station  Robustness = survivability in the face of unexpected changes in environment (exo) or within the system (endo)

Copyright © MSC.Software Corporation, All rights reserved. Example of Optimal Design  M. Alboreto dies (Le Mans, April 2001) due to slight loss of pressure in left rear tire. The system was extremely sensitive to boundary conditions (was optimal, and therefore very very fragile!).

Copyright © MSC.Software Corporation, All rights reserved. First order RS Second order RS Optimum? Different theories can be shown to fit the same set of observed data. The more complex a theory, the more credible it appears! Optimization: a Dangerous Game

Copyright © MSC.Software Corporation, All rights reserved. Some Lessons  Boundary conditions are most important  Small effects can have macroscopic consequences (watch out for chaos, even in small doses!)  Precision is not everything!  Optimal components don’t give an optimal whole  Optimality = fragility  Robust is the opposite to optimal

Copyright © MSC.Software Corporation, All rights reserved. Conclusions  Phenomena that possess a chaotic component cannot be optimized, but can be improved in statistical sense.  With such systems, it is possible to address:  Risk Analysis  Design for robustness  Increase understanding  Realistic models necessitate:  continuous 3D random fields (geometry)  discrete random field (spotwelds, joints)  randomization of ALL material properties  randomization of ALL thicknesses  variations of boundary/initial conditions

Copyright © MSC.Software Corporation, All rights reserved. Conclusions (2)  Computer models, to be of any use, must be  realistic  validated (not just verified with ONE test!)  Today, HPC resources are often being used in the wrong direction:  accuracy,  precision,  optimality,  “more elements than physics”,  analysis, NOT simulation,  automation, NOT innovation.  Risk originates from fragility. Fragility emanates from optimality.

Copyright © MSC.Software Corporation, All rights reserved. Thought An educated mind is distinguished by the fact that it is content with the degree of accuracy which the nature of things permits, and by the fact that it does not seek exactness where only approximation is possible. Aristotle, Nikomachean Ethics