Chaos and Uncertainty: Fighting the Stagnation in CAE

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Chaos and Uncertainty: Fighting the Stagnation in

CAE

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


CONTENTCONTENT

IntroductionHave Computers Killed Physics?Is Optimization Really Possible?Risk Analysis: A Must for Complex

SystemsConclusions


Crash: Where is the physics?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?


Example of Measured Acceleration SignalExample of Measured Acceleration Signal

A series of tests for chaos are performed with this signal.


Log-linear Power LawLog-linear Power Law

Systems that exhibit a log-linear Power Spectrum are potentially chaotic.


Typical Tests for ChaosTypical 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.


Crash = ChaosCrash = 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.


The Logistic MapThe Logistic Map

X(n+1) = k X(n)(1-X(n))

Shows astonishingly complex behaviour:

0 < k < 1, Extinction regime1 < k < 3, Convergence regime3 < k < 3.57, Bifurcation regime3.57 < k < 4, Chaotic regime4 < k, Second chaotic regime


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 PredictabilityChaos and Predictability


Understanding RiskUnderstanding Risk

Essentially, risk is associated with the existence of outliers

Outlier:- warranty- recall- lawsuit

Most likelyresponse(highest density)

Note: DOE and Response Surface techniques cannot capture outliers

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What is Risk and Uncertainty Management?What is Risk and Uncertainty Management?

Understand and remove outliers Shift entire distribution

is safe fails

Improved designInitial design

Outlier

Outliers:unfortunate combinationsof operating conditions and design variables that lead to unexpected behaviour.


Example of Robust Design: MIR Space StationExample of Robust Design: MIR Space Station

Robustness = survivability in the face of unexpected changes in environment (exo) or within the system (endo)


Example of Optimal DesignExample 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!).


First order RS

Second order RS

Optimum?

Different theories can be shown to fit the same set ofobserved data. The more complex a theory, the morecredible it appears!

Optimization: a Dangerous GameOptimization: a Dangerous Game


Some LessonsSome 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


ConclusionsConclusions 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


Conclusions (2)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.


ThoughtThought

An educated mind is distinguished by the fact that it is content with the degree of accuracywhich the nature of things permits, and by the fact that it does not seek exactness whereonly approximation is possible.

Aristotle, Nikomachean Ethics

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Chaos and Uncertainty: Fighting the Stagnation in CAE