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Update 2017

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.

Free Material Optimization

One of the basic problems of structural engineering reads: For a given set of boundary conditions and a given set of loads, find the lightest structure that is able to carry the loads.

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.

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

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G. Allaire and R. Kohn. Optimal design for minimum weight ans compliance in plane stress using extremal microstructures. European J. on Mechanics (A/Solids), 12:839-878, 1993.

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.

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Michal Kocvara
February 5, 2017