# Abstracts

## Boris Belinskiy, University of Tennessee

### Exact Controllability of Some 1D Mechanical Systems With Parameters That Vary in Time

We study the exact controllability problem for some 1D mechanical systems, such as a string with fixed end points, rotating string, and a ring, under stretching tension that varies in time. We are looking for a force (or a couple of forces in case of ring), which drive the state solution to rest. To prove our results we apply the method of moments that has been widely used in control theory of distributed parameter systems since the classical papers of H.O. Fattorini and D.L. Russell. The solution of the problem of moments is based on an auxiliary basis property results. Both method of moments and proof of the basis property are developed for the model with time-dependent parameters.

## Martin Bendsoe, Technical University of Denmark

### Relaxation to Airplanes

Konstantin Lurie has been instrumental in the development of the ideas that lie behind today's computer tools for optimal topology design of structures. This talk will disCuss features of a certain variant of the material distribution method called the `free material method'. This model is ideal for large scale computations and for designing composite parts as required in aeronautical applications.

## Victor Berdichevsky, Wayne State University

### Homogenization Problem and Entropy of Microstructure

It will be argued in this talk that there is an important macroscopic characteristic of random structures, entropy of microstructure. It contains the information about many aspects of the homogenization problem: effective characteristics, size effects and the local field distributions. Computation of entropy of microstructure involves the information on microstructure available from experiments. A variational formula for entropy of microstructure is derived for a number of cases.

## Elena Cherkaev, University of Utah

### Min-max problems of robust optimal design

The talk discusses a problem of robust optimal design of elastic structures when the loading is unknown, and only an integral constraint for the loading is given. The optimal design problem is formulated as minimization of the principal compliance of the domain equal to the maximum of the stored energy over all admissible loadings. The principal compliance is the maximal compliance under the extreme, worst possible applied force. The robust optimal design is a min-max problem for the energy stored in the structure. The minimum is taken over the design parameters, while the maximum of the energy is chosen over the constrained class of forces. It is shown that the problem for the extreme force is reduced to an elasticity problem with mixed nonlinear boundary conditions; the last problem may have multiple stationary solutions. The optimization takes into account the possible multiplicity of extreme loadings and designs the structure that equally withstands all of them. Continuous change of the loading constraint causes bifurcation of the solution of the optimization problem. It is shown that an invariance of the constraints under a symmetry transformation leads to a symmetry of the optimal design. Examples of optimal design are investigated; symmetries and bifurcations of the solutions are discussed. The talk is based on our joint work with Andrej Cherkaev.

## Constantine Dafermos, Brown University

### Hyperbolic Conservation Laws with Involutions and Contingent Entropies

The lecture will discuss the existence and stability of classical solutions, within the framework of admissible weak solutions, for systems of hyperbolic conservation laws endowed with involutions and polyconvex entropies. Examples of systems with this special structure arise in elastodynamics and in electromagnetism.

## Margarita Eglit, Moscow State University

### Dispersion of Elastic Waves in Inhomogeneous Media and Structures

(N.S. Bakhvalov, M.E. Eglit, Lomonosov Moscow State University, Moscow, Russia)

Elastic wave propagation in periodically stratified media as well as in thin layered plates and rods is studied. The ratio d of the length scale of inhomogeneity, or the thickness of the plate and the rod, to the typical wavelength is supposed to be small. The so-called effective equations are derived by the method of two-scale asymptotic expansions [1] over d. The aim is to describe dispersion of waves. That is why the equations up to terms of the second order over d are considered. These terms contain the third and the fourth derivatives of the averaged displacements over time and space coordinates.

It is shown that, in contrast to the situation in homogeneous isotropic materials, for inhomogeneous structures, in general, the terms with the third derivatives do enter the equations [2, 3, 4] with the skew-symmetric matrix of the coefficients. This matrix is equal to zero in some cases, e.g., for waves in media consisting of a repeated system of two homogeneous layers with arbitrary local anisotropy; but it differs from zero, e.g., for a two-layer plate even if the layers are homogeneous and isotropic.

The dependence of the wave velocity on its length is studied analytically and numerically for structures with different types of symmetry. It is shown, in particular, that typically at least one of possible waves displays negative dispersion for every direction of wave propagation, i.e., its velocity decreases as its frequency increases.

The work was supported by Russian Foundation of Basic Research (projects 05-01-00375, 05-01-00511).

#### References

- Bakhvalov N.S., Panasenko G.P., 1984. Homogenization. Averaging Processes in Periodic Media. Mathematical Problems in Mechanics of Composite Materials. Nauka, Moscow.
- Bakhvalov N.S., Eglit M.E., 2000. Effective equations with dispersion for wave propagation in periodic media. Doklady RAN, vol. 370, No 1, 1-4.
- Bakhvalov N.S., Eglit M.E., 2002. Investigation of the effective equations with dispersion for wave propagation in stratified media and thin plates. Doklady RAN, vol. 383, No 6, 742-746.
- Bakhvalov N.S., Bogachev K.Yu., Eglit M.E., 2002. Investigation of the effective equations with dispersion for wave propagation in inhomogeneous thin bars Doklady RAN, vol. 387, No 6, 749-753. No 1, 1-4.

## Brian King, Worcester Polytechnic Institute

The advantages of composite materials have been discovered and applied long ago to good effect. However, a new class of materials, spatio-temporal composites, or dynamic materials provide a promising new chapter to augment the rich history on the subject. By precise control of the temporal properties of the microstructured material, a number of novel and radically different phenomena emerge. This may enable practical applications that were hereforeto unobtainable. Some of the possible applications include dynamic broadband electromagnetic stealth (high-efficiency electromagnetic shielding) as well as creation of left-handed materials (LHMs) with tunable properties. LHMs allow for very compact imaging systems with imaging resolutions well beyond diffraction-limited sizes, which may significantly improve the state-of-the-art in medical imaging and data storage devices. This talk discusses how we may practically implement dynamic materials as well as some of the potential merits and applications of this new class of structures.

## Robert Kohn, Courant Institute, NYU

### A Deterministic-Control-Based Approach to Motion by Curvature

I will present recent work with Sylvia Serfaty. The main focus is motion by curvature in two space dimensions. The level-set formulation of this interface evolution law is a degenerate parabolic equation. We show it can be interpreted as the value function of a deterministic two-person game. More precisely, we give a family of discrete-time, two-person games whose value functions converge in the continuous-time limit to the solution of the motion- by-curvature PDE. This result is unexpected, because the value function of a deterministic control problem is normally a first-order Hamilton-Jacobi equation.

## Gilbert Strang, Massachusetts Institute of Technology

### Dual Problems of Mechanics in *L*^{1} and *L*^{8}

^{1}

^{8}

Leplace's equation comes from minimizing the *L ^{2}* norm of
grad

*u*. We consider the corresponding problems in

*L*and

^{1}*L*over a plane domain. We may minimize the norm of grad

^{8}*u*subject to boundary conditions, or we may minimize the distance between grad

*u*and a given vector field

*v(x,y)*. Of these four problems, some can be solved explicitly (with connections to a continuous max flow-min cut theorem). The problems have equivalent forms, using duality.

Some of our problems have explicit solutions. Others (sometimes even the dual) are unsolved. We also discuss the fundamental (linear) problem of applied mathematics and scientific computing, discrete and continuous.

## Robert Lipton, Louisiana State University

### Stress Transfer Between Macroscopic and Microscopic Length Scales in Random Media

Many structures are hierarchical in nature and are made up of substructures distributed across several length scales. Examples include aircraft wings made from fiber reinforced laminates and naturally occurring structures like bone. From the perspective of failure initiation it is crucial to quantify the load transfer between length scales. The presence of geometrically induced stress or strain singularities at either the structural or substructural scale can have influence across length scales and initiate nonlinear phenomena that result in overall structural failure. In this presentation we examine load transfer between length scales for hierarchical structures when the substructure is known exactly or only in a statistical sense.

New mathematical objects dubbed macrostress modulation functions are presented that facilitate a quantitative description of the load transfer in hierarchical structures.

Several concrete physical examples are provided illustrating how these quantities can be used to quantify the stress and strain distribution inside multi-scale structures. It is then shown how to turn the problem around and use the macrostress modulation functions to design graded microstructures for control of local stress near reentrant corners, bolt holes and other stress risers.

## Isaak Kunin, Professor Emeritus, University of Houston

### Control as an Effective Bridge Between Mathematical Models and Their Realizations

Last Century discoveries in fundamental sciences attracted attention to the following problem. The classical continuum mathematics appears to be inadequate for describing new phenomena such as chaos and complexity, media with microstructure and dislocations, space-time quantization, and so on. Typical approaches that have been suggested for resolving the problem introduce special mathematical structures, e.g. finite Hilbert spaces or cellular automata.

This presentation reviews the results of a different approach carried out by the author in collaboration with specialists in different disciplines including computer science. The main idea is to preserve the universally adopted mathematical concepts and models while using for corrections the corresponding classes of observables and generalized control, which may depend on physical realizations of models as well as include gauge fields, space-time quantization, etc.

The emphasis of the presentation will be on instructive examples and problems that may be of interest to the audience.

New participants in this research are welcome.

## Iliya I. Blekhman, Institute for Problems of Mechanical Engineering, Russian Academy of Sciences and "Mekhanobr-Tekhnika" Corp., Saint Petersburg, Russia

### Design of Vibrational Dynamic Materials as a Problem of Vibrational Rheology

In their joint work of 1999, following the original paper by K. Lurie dated 1997, this author and Professor Lurie introduced the idea of creating dynamic materials. This talk gives an outline of the development this idea has received in connection with the so-called vibrational dynamic materials. Such materials can be defined as having material properties (particularly, elastic and dissipative) that could be substantially affected and controlled by fast mechanical vibration. A theoretical basis for designing such materials is provided by vibrational rheology. The presentation reviews the results obtained by the author and a number of other researchers related to the study and build up of vibrational dynamic materials.

## Uldis Raitums, University of Latvia

### Laminates and Optimal Control Problems for Elliptic Equations

In the first part of the talk we briefly discuss the role of laminate structures in necessary optimality conditions and relaxation procedures for optimal control problems governed by a scalar elliptic equation. In the second part of the talk we consider the possibility to apply laminates to problems governed by linear elliptic systems and non-weakly continuous cost functionals.

## Andrej Cherkaev and Nathan Albin, University of Utah

### Translation-optimal structures of multimaterial conducting mixtures.

The paper considers the G-closure of several conductivities. The problem is to characterize the set of all effective properties tensors of multiphase mixtures composed from given materials taken in prescribed proportions. Konstantin Lurie in 1981 put forward this problem (and coined its name) and obtain in 1982 the first solutions for the two-material composites. The following development of the techniques provided the solutions of many problems in two-phase conducting, elastic, elasto-conducting and other materials. However, the generalization of the technique to multimaterial mixtures progressed more slowly. The set of all effective properties tensor of three-material mixtures is still not known. We present a large class of optimal periodic structures in the case of two-dimensional multimaterial conductivity. These structures are optimal in the sense that they belong to the boundary of the G-closure.

The technique to determine the G-closure begins by establishing a bound—a geometry—independent inequality satisfied by the effective tensors of all microstructures. The bound is sharp if there exists an optimal microstructure whose effective tensor attains the bound: the corresponding inequality becomes an equality It is known that the translation bound, which comes from the polyconvex envelope of the energy, is always sharp for two-material composites. For multimaterial composites, however, the bound is attainable only for a restricted set of volume fractions of the components. Particularly, when the volume fraction of the weakest or strongest material goes to zero, the bound does not tend to the bound for the remaining set of materials (Milton, 1988); therefore, the bound is not sharp for small volume fractions of that material. Additionally, the translation bound is known not to be sharp for extremely anisotropic multiphase composites.

We address the G-closure problem for three conducting materials in two dimensions, though the results are also immediately applicable to the problem of optimal two-dimensional elastic structures of maximal stiffness. The class of optimal structures we introduce improve upon the set of previously known optimal structures. Particularly, the optimal structures of multimaterial composites found by Milton & Kohn (1986) lie inside the set of attainability, and the extremal Gibiansky

The technique to determine the G-closure begins by establishing a bound--a geometry-independent inequality satisfied by the effective tensors of all microstructures. The bound is sharp if there exists an optimal microstructure whose effective tensor attains the bound: the corresponding inequality becomes an equality. It is known that the translation bound, which comes from the polyconvex envelope of the energy, is always sharp for two-material composites. For multimaterial composites, however, the bound is attainable only for a restricted set of volume fractions of the components. Particularly, when the volume fraction of the weakest or strongest material goes to zero, the bound does not tend to the bound for the remaining set of materials (Milton, 1988); therefore, the bound is not sharp for small volume fractions of that material. Additionally, the translation bound is known not to be sharp for extremely anisotropic multiphase composites.

We address the G-closure problem for three conducting materials in two dimensions, though the results are also immediately applicable to the problem of optimal two-dimensional elastic structures of maximal stiffness. The class of optimal structures we introduce improve upon the set of previously known optimal structures. Particularly, the optimal structures of multimaterial composites found by Milton & Kohn (1986) lie inside the set of attainability, and the extremal Gibiansky & Sigmund (1998) structure lies on the boundary of the set.

Our results are based on two new concepts. First, we introduce a regular method for producing optimal structures by examining the local fields. Second, we introduce an improvement on the translation bound itself.

The structures which attain the translation bound are generated by satisfying conditions found by the analysis of the fields in optimal structures. Furthermore, new constraints on the local fields inside any translation-optimal structure are found that supplement the constraint of the translation bound. These additional constraints are satisfied as equalities in the structures that lie on the boundary of the attainable set. We also introduce anisotropic structures that form the boundary of attainability of the translation bound. These structures together with their natural continuation (optimality of which is conjectured but not yet proven) form the inner bound of the multiphase G-closure.

Next, we improve the translation bound by a new concept of localized polyconvex envelope (LPE) for the energy. The LPE differs from the polyconvex envelope by establishing additional inequalities on the supporting fields (the local fields in all optimal microstructures). The new bound degenerates into the translation bound in the range of volume fractions where the translation bound is sharp; in that range, the additional constraints on the fields are satisfied as inequalities. At the boundary of the set of attainability, the new constraints become active and the corresponding effective properties bound becomes tighter. Unlike the translation bound, the new bound satisfies the correct asymptotics when one of the volume fractions vanishes. This bound is attainable, in a special case it coincides with the Nesi bound.

## Suzanne Weekes, Worcester Polytechnic Institute

### On Dynamic Materials

Dynamic materials are formations assembled from materials that are distributed on a microscale in space and in time. This material concept takes into consideration inertial, elastic, electromagnetic and other material properties that affect the dynamic behavior of various mechanical, electrical and environmental systems. When it comes to dynamic applications, one needs temporal variability in the material properties in order to adequately match the changing environment.

A dynamic disturbance on a scale much greater than the scale of a spatio-temporal microstructure will perceive this formation as a new material with its own effective properties. With spatio-temporal variability in the material constituents, one is able to effectively control the dynamic processes by creating effects that are unachievable through purely spatial material design. The control possibilities offered by dynamic materials include, though by no means are restricted to, low frequency waves that can be redirected, transferred into other frequency bands, and so on. Many effects based on parametric resonance such as optical pumping, high energy pulse compression, etc., also represent a great reserve for important applications.

In this talk, we present some results obtained from our ongoing analytical and numerical study of dynamic materials.

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