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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.

• The problem of design asks for a layout of several materials within the designed body in order to maximize an integral characteristics.
• As a rule, an optimal design is made from optimal composites: fine scale alternating subdomains of different materials. In order to make optimal constructions one needs to understand properties of composites with extremal properties.
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.

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.