Nonquasiconvex Variational Problems: Analysis of Problems that do not have Solutions

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Nonquasiconvex Variational Problems Analysis
of Problems that do not have Solutions

• 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?
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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 elliptiticity,
• 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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Structures perfected by EvolutionA leaf A
Dynosaur 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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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 1D problems

1. When the solutions are smooth/oscillatory? (The
   Lagrangian is convex/nonconvex function of w)
2. What are minimizing sequences? (Trivial in 1D --
   alternation)
3. What are the pointwise values of optimal
   solution? (They belong to the common boundary of
   the Lagrangian and its convex envelope)
4. How to compute or bound the Lagrangian on the
   oscillating solutions? Replace the Lagrangian
   with its convex envelope w.r.t. w
5. How to obtain or evaluate suboptimal solutions?
   By forcing a finite scale of oscillations or
   requiring an additional smoothness of minimizers.

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What is special in the multivariable
caseIntegrability conditions
-

Magnitudes of jumps depend on the normal n
Therefore the properties depend on the structure.
n
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Consequences of integrability conditions

• In a one-dimensional problem, w -- the strain in
  a stretched composed bar -- is discontinuous
• In a multidimensional problem, the tangential
  components of the strain must be continuous.
• If the only mode of deformation is the uniform
  contraction (Material made from Hoberman
  spheres), then
• No discontinuities of the strain field are
  possible

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Quasiconvex envelope

• Minimum over all periodic trial minimizers with
  allowed discontinuities is called the quasiconvex
  envelope.
• Quasiconvex envelope equals to the minimal energy
  of periodic oscillating sequences, it is a
  pointwise transform of the Lagrangian

Without this constraint, the definition
coincides With the definition of the convex
envelope
Murray (1956), Ball, Lurie, Kohn, Strang, Ch,
Milton, Gibiansky,Murat, Francfort,Tartar,
Dacorogna, Miller, Kinderlehrer, Pedregal.
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Approaches to calculation of the Quasiconvex
envelope

• Sufficient conditions (Translation bounds)
  replace the variational problem with a minorant
  finite-dimensional optimization problem (analog
  of Lyapunov function). Generally, they are
  better that the lower bound by the convex
  envelope.
• Minimizing sequences correspond to special
  multiscale fractal-type partitions. Generally,
  the optimal nesting partitions (microstructures)
  are not unique and based on a priori conjectures.
  
• Structural variations is a variational method
  that analyses pointwise values of the minimizer
  or the fields in optimal structures It provides
  an upper bound of the quasiconvex envelope stable
  to a class of variations

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Sufficient conditions Translation method

• Sufficient conditions use
• Constancy of the potentials
• Periodicity of the fields
• in the definition of the quasiconvex envelope

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Duality bound of a constrained problem
.

Lurie, Cherkaev, Kohn, Strang, Tartar,
Murat,Milton, Francfort, Gibiansky,Torquato,..
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Calculation of PF(v) for a piece-wise quadratic
Lagrangian
Observe that the first term in PF(v) is a
homogeneous second order function of v but not a
quadratic form of v.
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Translation bound for effective properties
The energy of a heterogeneous mixture equals to
the energy W of the effective
medium quasiconvex envelope QF is the lower
bound of all effective energies
Comparing, we obtain
Inequalities are observed
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Development of translation-type bounds

• The bounds for effective properties are applied
  to various problems
• Optimal conducting structures .(Lurie,Cherkaev,
  1982,84, Murat,Tartar,1985)
• Optimal elastic structures .(Cherkaev, Gibiansky,
  1985, 87, Arraire, Kohn 1987)
• Complex conductivity and viscous-elasticity.(Cherk
  aev, Gibiansky, 1996, ) .(Milton, Gibiansky,
  Berryman )
• Minimization of the sum of energies in all
  directions.
• (Avellaneda, Milton, 1993 (2d), Francfort,Murat,
  Tartar,1998 3d)
• Minimization of the sum of the stiffness and
  compliance.(Ch.,1999)
• Expansion tensor (eigenstrain).(Ch.,Sigmund,
  Vinogradov, 2004)
• Multiphase mixtures (Nesi bounds, 1997),

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Application Bounds for effective conductivity
tensor
The bounds of the set of all effective tensors
with prescribed volume fractions is found from a
variational problem of the layout that minimizes
the energy. The result
Lurie,Cherkaev, 1982,84, Tartar, Murat, 1985
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Improvement of classical Hashin-Shtrikman
boundsfor elasticity (2d)
Gibiansky Ch, 1994

• Hashin and Shtikman in 1963 suggested a bond for
  isotropic elastic moduli of a
  composite
• Translation method allowed to establish the
  coupled bounds between these quantities and the
  bounds for the moduli of anisotropic composites.

Set of possible pairs of the moduli
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Bounds for effective expansion tensor
(eigenstrain)

• Energy (Lagrangian) for each phase
• Effective energy for a mixture
• Problem Bounds for
• Using the Legende transform and the technique of
  polyconvex envelope, we obtain the bounds for
• where
• Cherkaev,Sigmund, Vinogradov 2004 (in progress)

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Minimizing sequences
Bruggeman, Hashin, Shtrickman, Milton, Lurie,
Cherkaev, Gibiansky, Noris, Avellaneda, Murat,
Tartar, Francfort, Bendsoe, Kikuchi,Sigmund.

• Algebra of laminates
• Lego of laminate structures

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Minimizers w is a scalar Two
phases -wells

• As in one-dimensional case, the fields are
  constant at each phase
• Every two fields can be neighbors, if layouts are
  properly oriented laminates.
• Quasiconvex envelope coincides with the convex
  envelope.

Continuity constraint v t0 serves to define
tangent t to layers
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Minimizers more than two wells (phases)

• Again, the quasiconvex envelope coincides with
  the convex envelope.
• Fields are constant within each phase.
• Minimizing sequences Laminates of (N-1)-th rank.

Optimal structure is not unique For instance, a
permutation of materials is possible.
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Properties

• the minimizing field is constant in each
  phase, the structure is a laminate.
• the field is not constant within each phase
  because of too many continuity conditions.
• The quasiconvex envelope is not smaller than the
  convex envelope
• and not larger than the function itself

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Laminates of a rank

• Properties of laminates are explicit functions
  of mixed materials S, their fractions c, and the
  normal to layers n.

Si are the properties tensors, ci are the volume
fractions, and p(n) is the projection on the
subspace of discontinuous fields components
Next step laminates of a rank (repeated
laminates) They are optimal if the fields inside
the structure are either constant within each
material, or a projection of the fields is
constant.
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Differential scheme and control of fractal layouts

• Consider the process of adding a new material in
  infinitesimal quantity and obtain the
  differential equation of evolution of the
  effective properties

A robust scheme, applicable to large class of
problem. Constraints on geometry.
The control problem Choose the order,
orientation and structure of added materials, in
order to maximize the objective at the end of the
process.
The energy of laminate (or other specialized)
structures is an upper bound of the quasiconvex
envelope.
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Optimal fractal geometry Elastic polycrystal
with extreme properties
Sometimes, the nesting sequence is complicated,
it can be found as a stable point of a set of
transforms. The minimizing layouts are
generally not unique.
Avellaneda, Cherkaev, Gibiansky, Milton,
Rudelson, 1997
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ExampleOptimal structures of conducting
composites

• Minimize a functional
• of a conductivity potential

Using the conjugate variables and the Legendre
transform, the functional can be transformed to
the form
j
Elasticity Laminates of a rank are optimal in
an asymptotic case
Conductivity Laminates are always optimal
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Structural variations Analysis of the fields in
optimal structures

• Based on classical Weierstrass test and Eshelby
  approach
• First version was suggested by K.Lurie, 1972
  Sokolovsky, Telega,Fedorov, Ch,
• Presented version- Cherkaev 2000.

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Structural variations

• Aim Description of the discontinuous minimizer
  in the multiwell nonconvex problems, or Fields
  in an optimal structure.
• Consider an infinitesimal variation of the
  layout
• Place an infinitesimal elliptical inclusion of
  one of the admissible phases Sg into the tested
  phase Sh
• d S (Sg Sh )cincl (x)
• Compute the perturbation of minimizer caused by
  this variation and the increment of the
  functional.

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Weierstrass-type test

• Increment of the field depends on the shape of
  the trial ellipse.
• The increment is maximized by choosing
• Shape
• Orientation
• Composition (for multi-material mixtures)
• of the trial ellipse, finding the most dangereos
  variation

Solving we
obtain the region of optimality of the tested
material
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Example Minimization of the energy of a layout
of linear elastic materials.

• The perturbed fields can be explicitly calculated
  if the energy is piece-wise quadratic materials
  are linear.
• To compute the increment we may either use
  modified Eshelby formulas or simply compute
  effective properties of a matrix-laminate.
  structure, when send the volume fraction of the
  inclusions to zero.

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Two-well problem Permitted regions of the fields
Cherkaev, Kucuk, 2004
Forbidden

• A norm of the stress in the weak and cheap
  material in an optimal structure is bounded from
  above

Weak
are the eigenvalues of the stress tensor
Strong

• A norm of the stress in the strong and expensive
  material in an optimal structure is bounded from
  below

There is a region where the NONE of materials is
optimal. If the applied filed belongs to this
region, the structure appears and the point-wise
fields in the materials are sent away from the
forbidden region. This phenomenon explains the
appearance of composites.
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Optimal fields and optimal structures
Cherkaev, Kucuk, 2004

• The jump over forbidden region is only possible
  if the composite has a special microstructure.
• The necessary conditions are examined together
  with the conditions on the boundary between
  materials
• The field in the nuclei is hydrostatic and
  constant.
• The field in the inner layer of the envelope are
  on the boundary of the regimes
• The fields in the external layer lies on the
  straight component of the boundary
• The optimal structures are not unique.

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Interpretation of the optimality conditions

• In order to keep the fields on the boundary of
  the permitted regions, the design forms a
  microstructure that adjusts itself to the loading
  conditions
• In the zone of nonquasiconvexity, a norm of the
  field is each phase is constant everywhere no
  matter what are the external conditions. This
  feature extends the known engineering principle
  of equally stressed designs to the tensor of
  stresses.

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Suboptimal projects
Cherkaev, Kucuk, 2004

• Checking the fields in a design, we can find out
• how close these fields are to the boundaries
• of the permitted domains of optimal field

Color shows the distance from the boundary of
optimality. Remark Similar coloring is used in
the ANSYS to warn about closeness to limits of
carrying capacity
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3D problem Permitted regions
Cherkaev, Kucuk, 2004
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Permitted range of fields in a three-material
mixture

• The field in the intermediate material
  is constrained from zero and infinity.This
  implies that
• the three materials in an optimal structure
  cannot meet in a singular point.
• Intermediate and the weakest material do not have
  a common boundary

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The use of necessary. conditionsOptimal
mixtures of three materials Large fraction of
the best material Small fraction
of the best material

• No contacts
• points between
• the three phases

The set of contact points is dense
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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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Set of unstrained configurations

• The geometrical problem of description of all
  possible unstrained configuration is still
  unsolved.
• Some sophisticated configurations can be found.
• Because of nonuniqueness, the expansion problem
  requires dynamic consideration.

Random lattices Nothing known
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Waves in active materials

• Links store additional energy remaining stable.
  Particles are inertial.
• When an instability develops, the excessive
  energy is transmitted to the next particle,
  originating the wave.
• Kinetic energy of excited waves takes away the
  energy, the transition looks like a domino or an
  explosion.
• Active materials Kinetic energy is bounded from
  below
• Homogenization Accounting for radiation and the
  energy of high-frequency modes is needed.

Extra energy
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Unstable reversible links
Force

• Each link consists of two parallel elastic rods,
  one of which is longer.
• Initially, only the longer road resists the load.
  
• If the load is larger than a critical
  (buckling)value
• The longer bar looses stability (buckling), and
• the shorter bar assumes the load.
• The process is reversible.

Elongation
H is the Heaviside function
No parameters!
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Chain dynamics. Generation of a spontaneous
transition wave
x
0
Initial position (linear regime, close to the
critical point)
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Observed spontaneous 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
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Tao of Damage
Application Structures that can withstand an
impact

• Damage happens!
• Uncontrolled, damage concentrates and destroys
• Dispersed damage absorbs energy
• Design is the art of scattering the damage

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Every variational problem has a solution provided
that the word solution is properly understood.
Conclusion

• David Hilbert