3.13 Nonsmooth - nonconvex analysis, optimalization and mechanics (short version)

Stavroulakis G., Greece

Nonsmooth-nonconvex analysis, optimization and

mechanicsGeorgios E. Stavroulakis,

Institute of Computational Mechanics and Optimizationwww.comeco.tuc.gr

Technical University of Crete, Chania, Greece and Dept. of Civil Engineering,

T.U. Braunschweig, Germany

IFER Workshop, Barcelona, July 4-5, 2010

Motivation: unilateral contact problem

• Nonpenetration relation

• Only compressive stresses are allowed (contact pressure)

• Complementarity relation

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G.E. Stavroulakis – IFER Barcelona July 2010

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Motivation: Frictional Stick-Slip problem(Tangential direction of the interfaces)

Coulomb friction model

• Two contacting surfaces start sliding when- Stick conditions: No sliding when τ < τcr

- τcr: Critical shear stress- µ: Friction coefficient- tn: Contact pressure

• Graphical representation

G.E. Stavroulakis – IFER Barcelona July 2010

Complementarity Problems

Linear Complementarity

Nonlinear Complementarity

G.E. Stavroulakis – IFER Barcelona July 2010

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Unilateral Contact Interface

Linear Complementarity Interface, Variational Inequalities, Nonsmooth Mechanics G.E. Stavroulakis – IFER Barcelona July 2010

Complementarity Problems(other applications)

Flow in Networks (e.g. transportation)

Equilibrium in the sense of Nash – Walrasch

Financial Mathematics

Price analysis of derivatives and options

Flow in porous media

Multiphase flow, salt water front analysis etc

G.E. Stavroulakis – IFER Barcelona July 2010

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Monotone Laws and Convex Nonsmooth Superpotentials

G.E. Stavroulakis – IFER Barcelona July 2010

Monotone Laws

G.E. Stavroulakis – IFER Barcelona July 2010

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Monotone Laws

G.E. Stavroulakis – IFER Barcelona July 2010

Monotone Laws

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Monotone Laws

G.E. Stavroulakis – IFER Barcelona July 2010

Conclusions

• Nonsmooth-nonconvex potentials in mechanics and thermomechanics allow us extend classical approaches

• Algorithms are expanded with the help of nonsmooth – nonconvex optimization

• Inverse and parameter identification problems can be solved by classical optimization or soft computing.

G.E. Stavroulakis – IFER Barcelona July 2010

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