Nonquasiconvex Variational Problems:

Analysis of Problems that do not have Solutions

Andrej CherkaevDepartment of Mathematics

University of Utahcherk@math.utah.edu

The work supported by NSF and ARO

UConn, April 2004

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 projectsDynamics

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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

dswqdxwWJOO

w ∫∫∂

+∇= )(min

, on

, in0),()(

Oqn

OwWw

∂=⋅

=⋅∇∇∇∂∂

=

σ

σσ

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

UConn, April 2004

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 χ(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.

( ) dxwFwdxwWw

dxwWwJ

Oii

ii

N

ii

iii

iN

∫∫ ∑

∫ ∑∑

∇∇=⎟⎠

⎞⎜⎝

⎛ +∇∇=

⎟⎠

⎞⎜⎝

⎛ +∇∇=

)(inf)()...(mininf

)(min)...(inf

0 1

01

βχχχ

βχχχχ

Energy cost

)( wF ∇⎩⎨⎧

∉∈

=i

ii Ox

Oxx

if 0 if 1

)(χ

1O

2O

n Lagrangiamultiwellnonconvex a is })({min)( where,..1 iiNi

wWwF β+∇=∇=

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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, Elena Cherkaev, 1998 A.Ch, L.Gibiansky, K.Lurie, 1986

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Structures perfected by EvolutionA leaf A Dynosaur bone

Dragonfly’s wingDűrer’s 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.

)()...(min)( 1wWwF i

ii

N∇=∇ ∑χχχ

Optimality + nonconvexity =structured materials

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Alloys and Minerals

A martensite alloy with“twin” monocrystals

Polycrystals of granulate

CoalSteelUConn, April 2004

All good things are structured!

Mozzarella cheese Chocolate

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Nonmonotone constitutive relations: Instabilities

w∂∇Φ∂=σ

σ

w∇

w∇

F• Nonconvex energy leads to

nonmonotone constitutive relations

and to nonuniqueness of constitutive relations.

• Variational principle selects the solution with the least energy.

0, =⋅∇∂∇∂= σσ wF

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Oscillatory solutions and relaxation (1D)(from optimal control theory)

{

')1( and

)},,()1(),,({min),,',,( where

)),,,,(min

,,,

1

0,,,

0

wbcac

bwxFcawxcFbawwxCF

dxcbawxCFJ

cbaw

cbaw

=−+

−+=

= ∫

ξ

∫=1

0

)',,(inf dxwwxFJw

CF(w’)

F(w’)Young, Gamkrelidge, Warga,…. from1960s

Convex envelope: Definition

.0periodic,is

),,(||||1inf),,,( 0

=−

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

∫

∫

O

Oz

dxO

dxzwxFOzwxFC

ξξ

ξξξ

Relaxation of the variational problem – replacement the Lagrangian with its convex envelope:

UConn, April 2004

Example

[ ]{ }22

)',(

1

0

2

0)1(,1)0(:)(

)1(,)1(min)(

)'(min

+−=

+= ∫==zzzf

dxwfwJ

wwL

wwxw

44 844 76

Relaxation

⎪⎩

⎪⎨

⎧

≤≤−−=

+=

11- if 0 1 if )1(

1 if )1()(

)'()',(

2

2

2

zzzzz

zCf

wCfwwwCL

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x

ww(x)

f

w’

w’(x)

1

-1

Euler equations for an extremal

⎩⎨⎧

±=≈≤=−

1',01|'| if ,0''

:)(ww

wwwxw

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 case:Integrability conditions

function. - arbitrary an is case 1D in while,0 :sconstraint aldifferenti

subject to is problems, blemultivaria In

∞

=×∇∇=

Lw’v

wv

dxwwFJO

w ∫ ∇= ),(min

[ ] pinvapi

xv

avvVVFF

kjijk

k

jijkn

..1,0 :conditions Cont.

..1,0),..(),( ,Generally 1

==

==∂∂

==

+−

n+ -Magnitudes of jumps depend on the normal n;

Therefore the properties depend on the structure.

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Consequences of integrability conditions

w’(x)

x• 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

[ ] 0=⋅∇ +−tw

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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

.0

,0periodic,is

),,()(1inf),,(

=∂∂

=−

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

∫

∫

kjijk

O

O

xa

dxO

dxAwxFOmeasAwxQW

ξ

ξξ

ξξ

Without this constraint, the definition coincides With the definition of the convex envelope Here O is a cube in

nR Murray (1956), Ball, Lurie, Kohn, Strang, Ch, Milton, Gibiansky,Murat,Francfort,Tartar, Dacorogna,Miller, Kinderlehrer, Pedregal.UConn, April 2004

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 pointwisevalues 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 potentialsPeriodicity of the fields

in the definition of the quasiconvex envelope

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Duality bound of a constrained problem

,||||

1,0:,0),()(:

,)(inf)(

∫∫ ΟΟ

Ξ∈

=⎭⎬⎫

⎩⎨⎧

=∂∂

=∇=+==Ξ

+=

dxZO

Zx

aAdxpxx

vFvQF

k

jijk

ξξξξξξ

ξξ

The differential constraint is replaced with a weaker (integral) constraint )( )( vv Φ≥Φ

dss

wwdsnRdxRdxv

vv

vv

vvvv

v

wwvvvv

OOoo ∂∂

=⋅=⋅∇=Φ

=⎟⎟⎠

⎞⎜⎜⎝

⎛×∇=⎟⎟

⎠

⎞⎜⎜⎝

⎛×∇⎟⎟

⎠

⎞⎜⎜⎝

⎛=

∇==ΦΦ

∫∫∫∫∂∂

21

22

12

21

11

2221

1211

21

)(

0 ,

),(),det()(:divergencea is )( :Example

.v

xv

av:

vvv

k

jijk arbitrary for not but 0for

holds )( )( inequality The :ora translat is )(

=∂∂

Φ≥Φ

Φ

{ }envelope polyconvex thecalled is

)()()())()((max)()()(

PF

vCFvPFvtvtvFCvPFvPFvQF vt ≥Φ−Φ+=≥

Lurie, Cherkaev, Kohn, Strang, Tartar, Murat,Milton, Francfort, Gibiansky,Torquato,..

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Calculation of PF(v) for a piece-wise quadratic Lagrangian:

{ }

{ }

⎟⎠⎞⎜

⎝⎛=+⎟

⎠

⎞⎜⎝

⎛−⋅=

+⋅=

=≥−=Θ

+⋅+⎭⎬⎫

⎩⎨⎧

−⋅∑∑

=

−−

Θ∈

==Θ∈

∑

∑

∑∑

|||| and )(

where

,)(max)( : themexclude and optimal Compute

,...,10:

;)(minmax)(

maxmax

11

00:

001,0

vvtttTtTCmC

mvtCvvPFv

NitTCt

mTvtvvtTCvmvPF

iii

T

iii

Tt

i

i

iii

iiiiimvmvvt iiii

γ

γ

r) translato(the )(,)(set We TvvvvC v(v)W iiii ⋅=Φ+⋅= χγχ

Observe that the first term in PF(v) is a homogeneous second order function of vbut 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

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Comparing, we obtain vvvCC T ∀⎟⎠

⎞⎜⎝⎛≥ ,||||*

{ } ),,()(max)( is of boundlower the

),,(,

000:

00*0*

vmQFmvtCvvPF QW(v)

vmQFmvCv F

iiii

Tt

iiii

≤+⋅=

≥+⋅=

∑

∑

Θ∈γ

γ

( )∑∑

=≥

=≥−−

iiAT

iiHT

CmCC

CmCC 11Inequalities are observed

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.(Cherkaev, 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

.0

operator the toscorrespond 0)( property Effective periodic. - is arbitrary, is Structure

,0)(0and given are and fractions , and tiesConductivi

*

*

121

2121

=∇⋅∇

→∇

==∇⋅∇∞≤≤≤

w

wmw

mm

κεκ

εχχχκκκ

κκ

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:

( )

( )

2211*

121

121

2

12*

111

112

2

11*

)(Tr )(Tr 1)(Tr

)(Tr )(Tr 1)(Tr

κκκ

κκκκκ

κκκκκ

mm

dmm

dmm

+≤

+−≤−

+−≥−

−−−

−−−

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. *

**

*

Kba

Kba

KKK

+≤≤

+

≤≤

+

+

−

−

+−

µ

µ,K +−+−

≤≤≤≤µµµ *

* KKK

−µ

+µ

−Wµ+Wµ Set of possible

pairs of the moduli

−K +K

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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)

kkkk zCW ++= εαεε ::21

**** ::21 zCW ++= εαεε

**,αC

bbb hP ≤−− )(:)( ** αααα

( ),,,,,),,,(),,,( dpkkkkkkbbkkkbb DDCmPPCmhhCm ααααα ===,* UL CCC ≤≤

**,αC

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

Minimizing sequences

Algebra of laminates:Lego of laminate structures

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Minimizers: w is a scalar: Two phases -“wells”:

,wv ∇=

• 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.

( ) 021 =⋅− tvv

Continuity constraint [v] t=0 serves to define

tangent t to layers

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Minimizers: more than two wells (phases),wv ∇=

Again, the quasiconvex envelope coincides with the convex envelope.

• Fields are constant within each phase.

• Minimizing sequences: Laminates of (N-1)-th rank.

( )( ) 21122312

121

)1(,0,0

vccvvvvvv

−+==⋅−

⋅=⋅−

tt

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 envelopeand not larger than the function itself:

( ) CWQWdva jijk =< ,rank If

( ) WQWCWdva jijk ≤≤= ,rank If

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Laminates of a rank

( ) ( )( )( ) )(,

;1

2112

212121

2211

nppppCmCmppN

CCNCCmmCmCmC

TT

lam

=+=

−−−+=

−

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.

)()( 1211 eqeqTT =

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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:

( ) ( )nDDDdcdc composcompos ,,Ψ= 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 boundof 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 generallynot unique.

Avellaneda, Cherkaev, Gibiansky, Milton, Rudelson, 1997

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Example:Optimal structures of conducting composites:

Minimize a functional of a conductivity potential

Ω=∇⋅∇Φ= ∫Ω

infuCdxuJ /)(minC

Using the conjugate variables and the Legendre transform, the functional can be transformed to the form

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Conductivity:Laminates are always optimal

Elasticity: Laminates of a rank are optimal in an asymptotic case

dxjCjvCvdxCuJRR

][minmaxminmin 10j vj,v,C,uC

−

=⋅⋅+∇⋅⇒∇⋅∇= ∫∫ λλ

u∇

λ∇

v

j

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 :δ S= (Sg – Sh )χincl (x)

• Compute the perturbation of minimizer caused by this variation and the increment of the functional.

0|),,(),,(

)],,(),,([

≤=∇−+∇+∇≈

∇−+∇+∇=

−−−−−−−

Ω∫

xtrialxxSwwFSSwwwF

dxSwwFSSwwwFJ

δδ

δδδ

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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”

( ) { } 0),,,(, h,

h max ≤Γ∇=∇ΨΓ

structureggSstructure

SSwJSw δ

Solving we obtain the region of optimality of the tested material

( ) wSw ∇≤∇Ψ for 0, h

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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.

σσσ

βχσχχχ σσ

ii

ii

iii

iN

SW

dxWJ

21)(

,)(min)...(inf0

0:1

=

⎟⎠

⎞⎜⎝

⎛ += ∫ ∑∑=⋅∇

( ) 11 ),()(,::

−− Γ+−=

+−=∆

structhhg

hg

STSSD

DJ σσββ

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Two-well problem: Permitted regions of the fields

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

2 21 2 1 2 1

12 22 1 1 2 1

A B CS

A B C

σ σ σ σ γσ

σ σ σ σ γ

⎫+ + ≤ ⎪ ∈⎬+ + ≤ ⎪⎭

are the eigenvalues of the stress tensor21,σσ

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

2 2 11 2 1 2 2

2

2 21 2 2 2

1

2 2 12 1 1 2 2

2

if 0

1(| | | |) if

if 0

A B C

D S

A B C

σσ σ σ σ γ ασσσ σ γ α σσ ασσ σ σ σ γ ασ

⎫+ + ≤ ≤ ≤ ⎪

⎪⎪

+ ≤ ≤ ≤ ∈⎬⎪⎪

+ + ≤ ≤ ≤ ⎪⎭

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.

Strong

Weak

Forbidden

Cherkaev, Kucuk, 2004

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Optimal fields and optimal structures• 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.

0:)::(0)(:)(:)(

=−=+−=−

ttSStnntnn

BBAA

BABA

σσσσσσ

Cherkaev, Kucuk, 2004

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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Cherkaev, Kucuk, 2004

Suboptimal projects

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

Color shows the distance from the boundary of optimality.

RemarkSimilar 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

The use of necessary. conditions:Optimal mixtures of three materials

Large fraction of the best material Small fraction of the best material

The set of contactpoints is dense

No contacts points betweenthe three phases

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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 mediumIf 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?

)(21)( 2 uWuuL ∇−= &ρ

),,()(21)( 2 uuuuWHuuL D ∇Θ−∇−= &&ρ

Radiation and other

losses

Dynamic homogenization

???)(21)( 2 uQWuuL ∇−= &ρ

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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 knownUConn, April 2004

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

• 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.

Force

Elongation

)1()( −−= xHxxfH is the Heaviside function

No parameters!

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Chain dynamics. Generation of a spontaneous transition wave

1 1

1 1

linear chain

1 1

nonlinear impacts

( ) ( ) ( ) ( )( ) 2

( ) ( ) ( )

k k k k k k k

k k k k

k k k k k

mX f X X f X X L X N XL X X X X

N X H X X a H X X a

− +

− +

− +

= − − − = += − +

= − − − − −

&&

144424443

14444444244444443

x

0

Initial position:

(linear regime, close to the critical point)

kaxk )1( ε−−=

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Observed spontaneous waves in a chain

Under a smooth excitation, the chain develops intensive oscillations and waves.

Sonic wave

“Twinkling” phase “Chaotic” phase

Wave of phase transition

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Application:Structures that can withstand an impact

Tao of Damage

o Damage happens! o Uncontrolled, damage concentrates and destroys o Dispersed damage absorbs energy

o Design is the art of scattering the damage

Conclusion

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

David Hilbert

UConn, April 2004