UConn, April 2004

Introduction:

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

Dynamic of phase transformation

Andrej Cherkaev

Department of Mathematics

University of Utah

cherk@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 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

dswqdxwWJOO

w

)(min

, on

, in0),()(

Oqn

OwWw

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 ellipticity,

– The minimizing sequence tends to an infinitely-fast-oscillating limit.

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Optimal design and multiwell Lagrangians

Problem: Find a layout (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.

dxwFw

dxwWw

dxwWw

J

O

iii

iN

ii

iii

iN

)(inf)()...(

mininf

)(min)...(

inf

0 1

01

Energy cost

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

iiNi

wWwF

)( wF

i

ii Ox

Oxx

if 0

if 1)(

1O

2O

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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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Structural Optimization

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

0if:

,0if0)(

C

Here, is the cost and C is the stiffness of material.

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Structures perfected by EvolutionA leaf A Dinosaur bone

Dűrer’s rhinoDragonfly’s wing

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)(1

wWwF ii

iN

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

CoalSteel

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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.

0, wFw

w

w

F

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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

1

0

)',,(inf dxwwxFJw

CF(w’)

F(w’)

Convex envelope: Definition

Young, Gamkrelidge, Warga,…. from1960s

.0periodic,is

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

O

O

z

dxO

dxzwxFOzwxFC

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

UConn, April 2004

Example

1',0

1|'| if ,0'':)(

ww

wwwxw

Euler equations for an extremal

22

)',(

1

0

2

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

)1(,)1(min)(

)'(min

zzzf

dxwfwJ

wwL

wwxw

Relaxation

11- if 0

1 if )1(

1 if )1(

)(

)'()',(

2

2

2

z

zz

zz

zCf

wCfwwwCL

x

ww(x)

f

w’

w’(x)

1

-1

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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

dxwqwFIO

w ])([min

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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,

we

,0 e

dxwqwFIO

w ])([min

pixv

avvVVFFk

jijkn ..1,0),..(),( 1

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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)

.

.0,)(:

,)(inf

kjijk

A

xAagALA

dxAFJ

Example:

0

,

21

21

AA

AAA

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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.

.0

,0periodic,is

)()(1inf)(

kjijk

O

O

xa

dxO

dxAFOmeasAQW

Here O is a cube in

Without this constraint, the definition becomes the definition for the convex envelope

Murray, Ball, Lurie,Kohn, Strang,Gibiansky,Murat, Tartar, Dacorogna,Miller, Kinderlehrer,Pedregal.

nR

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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

021 tee

,we

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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.

RemarkOptimal structure is not unique: For instance, a permutation of materials is possible.

.0

,0

2312

121

tee

tee

,we

UConn, April 2004

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:CWQW

FQW

dva jijk rank

dva jijk rank

UConn, April 2004

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?

UConn, April 2004

Atomistic models and Dynamics

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

UConn, April 2004

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?

)(21)( 2 uWuuL

),,()(21)( 2 uuuuWHuuL D

Radiation and other

losses

Dynamic homogenization

???)(21)( 2 uQWuuL

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Tao of Damage

o Damage happens! o Dispersed damage absorbs energy; concentrated

damage destroys.

o Design is the Art of Damage Scattering

Tao -- the process of nature by which all things change and which is to be followed for a life of harmony. Webster

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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