Introduction: Nonquasiconvex Variational Problems: Analysis of Problems that do not have Solutions Dynamic of phase transformation

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IntroductionNonquasiconvex Variational
Problems Analysis of Problems that do not have
SolutionsDynamic of phase transformation

• Andrej Cherkaev
• Department of Mathematics
• University of Utah
• cherk_at_math.utah.edu

The work supported by NSF and ARO
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Plan

• Non-quasiconvex Lagrangian
• Motivations and applications
• Specifics of multivariable problems
• Developments
• Bounds (Variational formulation of several design
  problems)
• Minimizing sequences
• Detection of instabilities (Variational
  conditions) and Detection of zones of instability
  and sorting of structures
• Suboptimal projects
• Dynamics

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Why do structures appear in Nature and
Engineering?

• When the morphology spontaneously becomes more
  complex, there must be an underlying reason,

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Energy of equilibriumand constitutive relations

• Equilibrium in an elastic body corresponds to
  solution of a variational problem
• corresponding constitutive relations
  (Euler-Lagrange eqns) are
• Here
• W is the energy density,
• w is displacement vector,
• q is an external load

If the BVP is elliptic, the Lagrangian W is
(quasi)convex.
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Convexity of the Lagrangian

• In classical (unstructured) materials,
  Lagrangian W(A) is quasiconvex
• The constitutive relations are elliptic.
• The solution w(x) is regular with respect to a
  variation of the domain O and load q.
• However, problems of optimal design, composites,
  natural polymorphic materials (martensites),
  polycrystals, smart materials, biomaterials,
  etc. yield to non(quasi)convex variational
  problems.
• In the region of nonconvexity,
• The Euler equation loses ellipticity,
• The minimizing sequence tends to an
  infinitely-fast-oscillating limit.

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Optimal design and multiwell Lagrangians

• Problem Find a layout c(x) that minimizes the
  total energy of an elastic body with the
  constraint on the used amount of materials.
• An optimal layout adapts itself on the applied
  stress.

Energy
cost
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Examples of Optimal Design Optimal layout is a
fine-scale structure
Thermal lens A structure that optimally
concentrates the current. Optimal structure is an
inhomogeneous laminate that directs the current.
Concentration of the good conductor is variable
to attract the current or to repulse it.
Optimal wheel Structure maximizes the
stiffness against a pair of forces, applied in
the hub and the felly. Optimal geometry radial
spokes and/or two twin systems of spirals.
A.Ch, L.Gibiansky, K.Lurie, 1986
A.Ch, Elena Cherkaev, 1998
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Structural Optimization

• Particularly, the problem of structural
  optimization asks for an optimal mixture of a
  material and void.

Here, g is the cost and C is the stiffness of
material.
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Structures perfected by EvolutionA leaf A
Dinosaur bone
Dragonflys wing
Durers rhino
The structures are known, the goal functional is
unknown!
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Polymorphic materials

• Smart materials, martensite alloys, polycrystals
  and similar materials can exist in several forms
  ( phases). The Gibbs principle states that the
  phase with minimal energy is realized.

Optimality nonconvexity structured materials
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Martensite twins
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Alloys and Minerals
A martensite alloy with twin monocrystals
Polycrystals of granulate
Coal
Steel
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All good things are structured!Mozzarella
cheese Chocolate
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Nonmonotone constitutive relations Instabilities

• Nonconvex energy leads to nonmonotone
  constitutive relations
• and to nonuniqueness of constitutive relations.
• Variational principle selects the solution with
  the least energy.

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Oscillatory solutions and relaxation (1D)(from
optimal control theory)
Young, Gamkrelidge, Warga,. from1960s
Convex envelope Definition
Relaxation of the variational problem
replacement the Lagrangian with its convex
envelope
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Example
Relaxation
Euler equations for an extremal
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Optimal oscillatory solutions in one- and
multidimensional problems

1. When the solution is smooth/oscillatory? The
   Lagrangian is convex/nonconvex function of w.
2. What are the pointwise values (supporting points)
   of optimal solution? They belong to common
   boundary of the Lagrangian and its convex
   envelope
3. What are minimizing sequences? Alternation
   of supporting points. (Trivial in 1d)
4. How to compute or bound the Lagrangian on the
   oscillating solutions? Replace the Lagrangian
   with its convex envelope with respect to. w

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Analysis of multivariable nonconvex variational
problems
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What is special in the multivariable case?

• Formal difference is subject
  to differential constraints
• while in 1D case w is an arbitrary Lp
  function.
• Generally,

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In 1D, the derivative w is an integrable
discontinuous functions

• In a one-dimensional problem,
• The strain in a stretched composed bar is
  discontinuous

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Example of an impossibility of oscillatory
sequence

• In a multidimensional problem,
• the tangential components of the strain are to be
  continuous.
• If the only mode of deformation of an elastic
  medium is the uniform contraction (Material from
  Hoberman spheres), then
• No discontinuities of the strain field are
  possible

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Variational problem (again)

• .

Example
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Quasiconvex envelope

• Minimum over all minimizers with allowed
  discontinuities is called the quasiconvex
  envelope. Quasiconvex envelope is the minimal
  energy of oscillating sequences.

Without this constraint, the definition becomes
the definition for the convex envelope
Murray, Ball, Lurie, Kohn, Strang, Gibiansky,Murat
, Tartar, Dacorogna, Miller, Kinderlehrer, Pedreg
al.
Here O is a cube in
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Minimizers w is a scalar Two
phases -wells

• In this case
• Quasiconvex envelope coincides with the convex
  envelope.
• Field is constant within each phase.
• Minimizing sequences are specified as properly
  oriented laminates.

Continuity constraint serves to define tangent
t to layers
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Minimizers w is a scalar, more than two phases

• Quasiconvex envelope coincides with the convex
  envelope.
• Fields are constant within each phase.
• Minimizing sequences laminates of N-1-th rank.

Remark Optimal structure is not unique For
instance, a permutation of materials is possible.
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General case

• If
• Then the minimizing field is constant in each
  phase, but the structure is specified.
• If
• Then the field is not constant within each phase
  because of too many continuity conditions.
• The quasiconvex envelope is not smaller than the
  convex envelope
• But it is still not larger than the function
  itself

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Questions about optimal oscillatory solutions

• What are minimizing sequences?
• What are the fields in optimal structures?
• How to compute or bound the quasiconvex envelope?
• When the solutions are smooth/oscillatory?
• How to obtain or evaluate suboptimal solutions?

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Atomistic models and Dynamics

• In collaboration with Leonid Slepyan, Elena
  Cherkaev, Alexander Balk, 2001-2004

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Dynamic problems for multiwell energies

• Formulation Lagrangian for a continuous medium
• If W is (quasi)convex
• If W is not quasiconvex
• Questions
• There are infinitely many local minima each
  corresponds to an equilibrium.
• How to choose the right one ?
• The realization of a particular local minimum
  depends on the existence of a path to it. What
  are initial conditions that lead to a particular
  local minimum?
• How to account for dissipation and radiation?

Radiation and other losses
Dynamic homogenization
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Tao of Damage
Tao -- the process of nature by which all things
change and which is to be followed for a life of
harmony. Webster

• Damage happens!
• Dispersed damage absorbs energy concentrated
  damage destroys.
• Design is the Art of Damage Scattering

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Waves in a chain
Twinkling phase
Chaotic phase
Under a smooth excitation, the chain develops
intensive oscillations and waves.
Sonic wave
Wave of phase transition