Introduction:
Nonquasiconvex Variational Problems:
Analysis of Problems that do not have Solutions
Dynamic of phase transformation
Andrej Cherkaev
Department of Mathematics
University of Utah
[email protected]
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 equilibrium
and constitutive relations
Equilibrium in an elastic body corresponds to solution of a variational problem
J  min  W (w)dx 
w
O
 wqds
O
corresponding constitutive relations (Euler-Lagrange eqns) are

 
W (w),     0 in O,
 (w)
  n  q on O,
If the BVP is elliptic, the Lagrangian
Here:
W is the energy density,
w is displacement vector,
q is an external load
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.
1 if x  Oi
0 if x  Oi
O1
O2
c i ( x)  
An optimal layout adapts itself on the applied stress.
J  inf
 inf
( c 1 ... c N )
w
 min
0
min


c
W
(

w
)

c



i i
i i dx
w   
i
i

0
Energy
cost


F (w)dx
  c i Wi (w)   i dx  inf

( c 1 ... c N )  i

w

O
where F (w)  min {Wi (w)  i } is a nonconvex multiwell Lagrangia n
i 1,.. N
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F ( w )
Examples of Optimal Design:
Optimal layout is a fine-scale structure
Thermal lens:
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 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.
A.Ch, Elena Cherkaev, 1998
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A.Ch, L.Gibiansky, K.Lurie, 1986
Structural Optimization
• Particularly, the problem of
structural optimization asks for
an optimal mixture of a
material and void.
if   0,
0
( )  
    : C if   0
Here,  is the cost and C is
the stiffness of material.
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Structures perfected by Evolution
A leaf
A Dinosaur bone
Dragonfly’s wing
Dűrer’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.
F (w)  min
( c1...c N ) 
i
c iWi (w)
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
F
•
  F w ,     0
   w
and to nonuniqueness of
constitutive relations.
w

Nonconvex energy leads to
nonmonotone constitutive relations
•
Variational principle selects the
solution with the least energy.
w
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Oscillatory solutions and relaxation (1D)
(from optimal control theory)
Young, Gamkrelidge, Warga,…. from1960s
1
J  inf
F(w’)
w
 F ( x, w, w' )dx
0
Convex envelope: Definition
CF(w’)



C z F ( x, w, z,  0 )  inf  1
F
(
x
,
w
,
z


)
dx

 || O || O


 is O  periodic,
  dx  0.
O
Relaxation of the variational problem – replacement the Lagrangian
with its convex envelope:
1
J  min
w ,a ,b ,c
 CF ( x, w, a, b, c))dx
0
where CF ( x, w, w' , a
, b)  min {cF ( x, w, a )  (1  c ) F ( x, w, b)}
0
and
ac  (1  c )b  w'
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w ,a ,b ,c
Example
L ( w , w ')


 w
1
J 
min
w ( x ): w ( 0 ) 1, w (1) 0
2

Relaxation
 f ( w' ) dx
0
f ( z )  min  ( z  1) 2 , ( z  1) 2 
CL( w, w' )  w 2  Cf ( w' )
 ( z  1) 2 if z  1

Cf ( z )   ( z  1) 2 if z  1
 0
if - 1  z  1

f
w(x)
w
w’
Euler equations for an extremal
w' ' w  0, if | w' | 1
w( x ) : 
 w  0, w'  1
1
x
-1
w’(x)
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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
I  min w  [ F (w)  wq]dx
O
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What is special in the multivariable case?
I  min w  [ F (w)  wq]dx
O
• Formal difference:
e  iswsubject to differential constraints:
while in 1D case w’ is an arbitrary Lp function.
  e  0,
• Generally,
 v j

F  F (V ), V  (v1..vn ), aijk 
  0, i  1.. p

x
k 

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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)
.
J  inf
A
 F ( A)dx,

A : L( A)   g , aijk Aj xk  0.
Example:
A  A1  A2 ,
  A1    A2  0
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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.


1

QW ( A)  inf 
F ( A   )dx 

 meas(O)
O


Without this
constraint, the
definition becomes
the definition
for the convex
envelope
 is O  periodic,
  dx  0,
O
aijk  j xk  0.
Here O is a cube in Rn
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Murray, Ball, Lurie,
Kohn, Strang,
Gibiansky,Murat,
Tartar, Dacorogna,
Miller, Kinderlehrer,
Pedregal.
Minimizers:e  w, w is a scalar:
Two phases -“wells”:
e1  e2  t  0
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:e  w,
w is a scalar, more than two phases
e1  e 2   t1  0,
e
12
Quasiconvex envelope coincides
with the convex envelope.
• Fields are constant within each
phase.
• Minimizing sequences: laminates
of N-1-th rank.
 e 3   t 2  0.
Remark
Optimal structure is not unique: For instance, a permutation of
materials is possible.
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General case:


• If rank aijk v j  d
• Then the minimizing field is constant in each phase, but the structure is
specified.
• If
rank aijk v j  d
• 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:
QW  CW
QW  F
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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
•
L(u)  1  u 2  W (u)
2
If W is not quasiconvex
L(u)  1  u 2  QW (u) ???
2
Radiation
and other
losses
L(u)  1  u 2  H DW (u)  (u, u, u)
2
•
Questions:
Dynamic
homogenization
– 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?
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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
o Damage happens!
o Dispersed damage absorbs energy; concentrated damage
destroys.
o 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
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