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Structural optimization

One can obtain stronger, more conducting, or more expanding composites by a proper arrangement of materials in a composite cell.

The picture illustrates concepts of structural optimization.

What kind of structures show optimal effective properties? The answer depends on the feature one wants to optimize.

Here are several examples of optimal conducting materials:

Laminate: maximal conductivity among layers
Laminate: minimal conductivity across layers
Coated spheres: minimal (or maximal) isotropic conductivity
Second-rank laminate: maximal (or minimal) conductivity in a direction if the conductivity in an orthogonal direction is fixed.  (a generalization of coated shperes).


Next Table shows some known optimal elastic structures.



Elasticity: some structural optimization problems, requirements to structures, and the math. formulations
Maximal stiffness  Maximal mean stiffness  Maximal "responce" 
The minimization of the total energy stored in a construction (or the maximization of its integral rigidity) under some fixed loading.

Requires the search for composites which store minimal energy under some given loading. 

The problem is reduced to a non-stable variational problem of the minimization of the energy upon the structures. 


The minimization of the total energy stored in a construction under some uncertain or varying loadings.

Requires the search for composites which store minimal sum of energies under several loadings. 

The problem is reduced to the minimization of the norm of somesome operator  that links the fields and the structures. 


The minimization of other integral (weakly lower-semicontinuous) functionals, like the norm of the displacement, displacement in a point along a certain direction,  etc.

Requires the search for composites which are simultaneously weak with respect to a loading and strong with respect to an orthogonal loading. 

The problem is reduced to minimization of the sum of the energy under a loading and the complementary energy under some other loading. 

Optimal structures: Specially oriented second-rank coated laminates or "coated ellipses" in 2D, or third-rank coated laminates or "coated ellipsoids" in 3D. The orientation of the optimal structures follows the axes of the stress tensor

The structure must be oriented along the principle axes of stress
and adjusted to the ratio of the primciple stresses. Being the stiffest in a given stress fields, this structure has no resistance against shear loadings: it collapses as a deck of cards.

Optimal structures: third-rank coated laminates  in 2D, or six-rank coated laminates in 3D. The orientation of the optimal structures may follow the axes of the stress tensor of stay constant.

 In contrast with the previous structure, it resists all loadings;
the rate of resistance against different loadings must be adjusted by varying of structural parameters.

The whole variety of optimal 
structures is not described yet. 
The optimal structures include herring-bones, 

matrix laminates,  and something else...

Particularly, the structures with maximally negative Poisson ratio and  the matrix lanitates with obliged  branches are among the optimal structures. 


One can also obtain a composite material with an extremal effective property when another property is bounded; for example: A material with the maximal thermal expansion coefficient and a fixed elastic modulus.


Searching for  optimal composites

The extremal effective properties correspond to minimal value of total energy stored in a structure submerged in a number of external fields. The minimization problem requires the variation of the layout. This problem cannot be solved by classical means of Calculus of Variations; it belongs to nonconvex variational problems. Generally, nonconvex variational  problems have nonregular solutions: Optimal structures can be fractals, or infinitely repeated nesting sequence of layers, or similar exotic objects.

Structural Optimization asks for a structure - layout of materials in the composite cell - that maximizes the overall performance of an assemblage. Optimal microstructures yield or to special range of local fields inside them.
The optimization problem has been attacked from different directions. For detailed discussion, see [VMSO].

Most of the approaches to the optimization problem are indirect. The methods include:

 Establishing bounds of the range of the effective properties. The bounds do not depend on the structures, they show the limits of improvements of the effective properties (see section Bounds)
  Modeling microstructures by special geometry, as sequential "laminates of some rank",
    "coated spheres", and by similar special geometry that allow the exact solution of the
    constitutive relation in a periodic cell (see section Modeling).

    Investigating of the fields in optimal structures. Often, the conditions of optimality allow
    for a clear interpretation in terms of requirements to the interior fields inside an optimal
    structure (see section Fields).
  Numerical optimization of a structure, (see section Numerical modelling)

Stability of optimal designs.

Optimal structures are generally unstable. Indeed, they concentrate the resistance capacities in certain directions to stay against a given loading, therefore they are extremely sensitive to the variation of the loadings. We are working on designs that compromise effectiveness and stability.

A simplest example of structural optimization

The next picture shows the crossection of the torsioned bar with maximal rigidity. The bar is assembled from two materials with different elastic moduli, the proportion of the materials in the design is fixed.
It has been proved (Lurie and Cherkaev, 1980) that an optimal project is made from either pure given materials or from laminates.
Here the blue and the red domains denote the originally given strong and weak materials, the color reflects the volume fractions in optimal laminates, lines show their directions. The more is the norm of stresses, the more rigid is the optimal composite.


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