The text below was written in 2001. Since that time, much has happened in the development of theory, algorithms, and software for free material optimization, in particular in the EU FP6 project PLATO-N. Some of this has been summarized in the following papers.

- M. Kocvara, M. Stingl and J. Zowe. Free
material optimization: recent progress.
*Optimization, 57(1):79-100, 2008*

- J. Haslinger, M. Kocvara,
G. Leugering,
and M. Stingl.
Multidisciplinary Free Material Optimization.
*SIAM J. Appl. Math. 7*0(7):2709-2728, 2010

Optimization of structures is traditionally performed through the
variation of sizing variables (e.g., thicknesses of bars in a truss) and
shape variables (e.g., splines defining the boundary of a body). With the
appearance of composites and other advanced man-made materials it has been
natural to extend this variation to the material choice itself. Recent
years have witnessed increasing development of mathematical methods for
the "generalized'' optimum design problem. These methods, usually called*
topology* or *material optimization* methods, aim at designing
of shape and topology of elastic body as well as its material properties.
This development started in the 1980s by a series of theoretical articles
on relaxation (Allaire-Kohn) and
homogenization (Bendsøe) methods and their
potential application in optimum design. The interest of engineers was
stimulated by the pioneering paper by Bendsøe
and Kikuchi who introduced a numerical technique based on the *homogenization
method*, and showed a practical usefulness of the previously
developed theoretical methods. Since then, many approaches and techniques
have been developed for an overview, see, e.g., Allaire-Kohn
or Bendsøe and the references therein.

Promising start was followed by a slight disillusion, though. The above
methods usually result in (approximation of) an optimal, typically
two-layered material, which may vary from point to point and which is be
difficult to manufacture. Parts of the body, where the material
disappears, are interpreted as voids. The interpretation of the results in
the rest of the body is often unclear, though. These results typically
include an information on some sort of density of the material at each
point. For instance, it isa ratio of the given material and void in a
microscopic cell, i.e., a number varying from zero to one. But there is
more information available, for instance, the way how the material is
organized in this micro-cell, if it forms sort of (micro)layers and under
which angle. This additional information is, however, often ignored. Most
of the resent approaches tend to interpret only the density information in
the zero-one (material-no material) sense.Consequently, these approaches
try to minimize the regions with density values between zero and one,
either directly in the model or by some kind of post-processing. These
techniques go, in our opinion, *against* the spirit of material
optimization: It is known that there* is* no zero-one solution to the
generalized optimum design problem.

*It is therefore important for full utilization of the material
optimization techniques to use the *complete* information obtained
in the results and, consequently, to make a step from conventional to
advanced materials.*

We have developed a collection of tools for conceptual structural
design,called MOPED. The underlying model is the ** free
material
optimization** (FMO) problem, introduced by Bendsøe
et al. and later developed by Kocvara
et
al. and Zowe et al. Our aim is
to optimize not only the distribution of material but also the material
properties themselves. We are thus looking for the ultimately best
structure among all possible elastic continua.

In FMO we minimize the compliance over all positive semidefinite rigidity
tensors **E** (reflecting the Hooke's law) and use the integral over
the domain of some invariant **P(E)** of the rigidity tensor as the
measure of cost. It turns out that this problem can be considerably
simplified.For a particular choice of the cost function **P(E)**, the
reformulation was given, in which we work only with one design variable.
The elements of the optimal rigidity tensor **E** can be recovered
from the optimal value of this auxiliary variable.

The method is supported by powerful optimization and numerical techniques, which allow us to work with bodies of complex initial design and with very fine finite-element meshes, giving thus quite accurate solutions even in "difficult'' parts and for complex geometries.

Rather than curtail our results and interpret them in the zero-one sense, we want to utilize the full information obtained in these results, in order to design an attainable advanced material. This obviously depends on the type of the advanced material and on the manufacturing technology. The composite materials lend themselves to a realization of the computed results. We consider a particular class of composite materials manufactured by the so-called tape-laying technology. In the post-processing phase, we plot curves which indicate the way how to lay these tapes they also show the proposed thickness of the tapes.

MOPED was developed under a BMBF Project *Optimization of Discrete and
Continuous Mechanical Structures *in a cooperation with our
industrial partner Daimler Chrysler Aerospace AG from Munich. For
illustration, we present results of a typical example computed for our
partner. The goal is to design a cross-section of a combat aircraft which
has to carry huge forces coming from the wings when flying in extreme
situations. The initial layout is shown in the next figure.

We solved the problem by MOPED using a mesh of 20000 finite elements. The resulting "density" distribution is presented below.

The final two figures shows the lines of principal stresses which indicate the way how to lay the composite tapes (Druck = compression, Zug = tension)

A. Ben-Tal, M. Kocvara, A. Nemirovski, and J. Zowe. Free material design
via semidefinite programming. The multi-load case with contact conditions.
*SIAMJ. Optimization*, 9(4): 813-832, 1999.

M. P. Bendsøe. *Optimization of Structural
Topology, Shape and Material*. Springer-Verlag, Heidelberg, 1995.

M. P. Bendsøe and N. Kikuchi. Generating
optimal topologies in structural design using a homogenization method.*Comp.
Meth.
Appl. Mechs. Engrg.*, 71:197-224, 1988.

M. P. Bendsøe, J. M. Guades, R. B. Haber, P.
Pedersen, and J. E. Taylor. An analytical model to predict optimal
material properties in the context of optimal structural design. *J.
Applied Mechanics*, 61:930-937, 1994.

M. Kocvara, M. Stingl, and R. Werner. MOPED User's Guide. Version 1.02. Research Report 262, Institute of Applied Mathematics,University of Erlangen, 2000.

M. Kocvara, M. Zibulevsky, and J.
Zowe. Mechanical design problems with unilateral contact. *M2AN
Mathematical Modelling and Numerical Analysis*, 32:255-282, 1998.

J. Zowe, M. Kocvara, and M. Bendsøe.
Free material optimization via mathematical programming. *Mathematical
Programming, Series B*, 79:445-466, 1997.

Michal Kocvara