 \newpage
 \subsection{Examples}


\begin{example}
Let us consider the problem of an optimal inhomogeneous conducting
structure that transforms the given boundary potentials to the
desired boundary currents, as  shown in  \fig{f01cond}P.  Suppose
that one has a set of materials of different isotropic
conductivity and  the layout of materials in the designed domain
must be optimized. Clearly, one can control the boundary currents
by varying the materials' layout, because the variation in
conductivity forces the current out of regions of low conductivity
and attracts it into regions of high conductivity. Careful
consideration shows an additional mechanism of control through the
use of anisotropic materials. The current is controlled and sent
in the desired direction by refraction in an anisotropic
composite.  The last mechanism is specific to multidimensional
problems and  has no one-dimensional analogues.  It shows the
usefulness of the anisotropic composite media assembled of
initially isotropic  materials.
\end{example}
\xfgr{f01cond}{The scheme of a composite structure that transforms
the homogeneous boundary potential into an inhomogeneous boundary
current. The horizontal sides are insulated, and the potential on
each vertical side is constant. The current lines are shown.  The
inhomogeneity of the current is caused by the inhomogeneity of the
material layout. The good conductor $ A $ attracts the current,
the bad conductor $ C $ pushes the current away, and the
anisotropic composite $ B $ turns the current in a desired
direction.}{An optimal conducting composite structure}{f01cond}
{0.5}

\begin{figure}
\centerline{\epsffile{f01cond.eps}} \caption{  } \label{fig1}
\end{figure}


The use of anisotropy to control a process in a medium is well
known. Observe a skier on a slope. The skier can control the
direction of his motion because the resistance to sliding along
the ski is much less than the resistance to sliding in the
orthogonal direction.  This mechanism allows the skier to traverse
across the slope and  make turns.  Anisotropy is also used to
steer a sailboat in a direction different from the direction of
the wind. When a current of passive particles moves in a medium
due to an applied force field, the anisotropy of the medium plays
a similar control role:  It generates a current in a direction
different from the direction of the force.


In the optimization of elastic designs, we also find intuitive
reasons for using an anisotropic composite rather than isotropic
materials.
\begin{example}
Let us consider  an elastic material that shows maximal stiffness
under some anisotropic external loading; see \fig{f01elast}P. One
would assume that the larger the stress, the more stiffness is
needed to resist.  Therefore, we anticipate that the structure
tends to be stiffer in the direction of a larger stress, even at
the expense of being weakened in the direction of a smaller
stress. Hence, we expect that anisotropic composites with
controllable degrees of anisotropy are more suitable than
isotropic materials for maximization of the stiffness.
\end{example}



\xfgr{f01elast}{The scheme of an elastic structure with
cavities\index{optimal cavities} showing the maximum stiffness
under a given loading.  The intensity of the loading is
anisotropic, and so is the corresponding optimal structure.
Greater intensity of the loading corresponds to the direction of
greater stiffness.}{ Anisotropy of an optimal elastic
structure}{f01elast}{0.5}
