Optimal shape design of structural elements based on boundary variations results in final designs that are topologically equivalent to the initial choice of design, and general, stable computational schemes for this approach often require some kind of remeshing of the finite element approximation of the analysis problem. This paper presents a methodology for optimal shape design where both these drawbacks can be avoided. The method is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, i~otropic material, with the requirement that the resulting structure can carry the given loads as well as satisfy other design requirements. The computation of effective material properties for the anisotropic material is carried out using the method of homogenization. Computational results are presented and compared with results obtained by boundary variations.

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Generating optimal topologies in structural design using a homogenization method

Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.

Optimal shape design as a material distribution problem

Some of the greatest scientists including Poisson, Faraday, Maxwell, Rayleigh, and Einstein have contributed to the theory of composite materials Mathematically, it is the study of partial differential equations with rapid oscillations in their coefficients Although extensively studied for more than a hundred years, an explosion of ideas in the last five decades (and particularly in the last three decades) has dramatically increased our understanding of the relationship between the properties of the constituent materials, the underlying microstructure of a composite, and the overall effective (electrical, thermal, elastic) moduli which govern the macroscopic behavior This renaissance has been fueled by the technological need for improving our knowledge base of composites, by the advance of the underlying mathematical theory of homogenization, by the discovery of new variational principles, by the recognition of how important the subject is to solving structural optimization problems, and by the realization of the connection with the mathematical problem of quasiconvexification This 2002 book surveys these exciting developments at the frontier of mathematics

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The Theory of Composites

Following an idea of G. Nguetseng, the author defines a notion of “two-scale” convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in $L^2 (\Omega )$ are proven to be relatively compact with respect to this new type of convergence. A corrector-type theorem (i.e., which permits, in some cases, replacing a sequence by its “two-scale” limit, up to a strongly convergent remainder in $L^2 (\Omega )$) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the so-called energy method of Tartar. The power and simplicity of the two-scale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear second-order elliptic equations.

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Homogenization and two-scale convergence

This tutorial/survey paper: (1) provides a concise point of departure for researchers and practitioners alike wishing to assess the current state of the art in the control and monitoring of civil engineering structures; and (2) provides a link between structural control and other fields of control theory, pointing out both differences and similarities, and points out where future research and application efforts are likely to prove fruitful. The paper consists of the following sections: section 1 is an introduction; section 2 deals with passive energy dissipation; section 3 deals with active control; section 4 deals with hybrid and semiactive control systems; section 5 discusses sensors for structural control; section 6 deals with smart material systems; section 7 deals with health monitoring and damage detection; and section 8 deals with research needs. An extensive list of references is provided in the references section.

Structural control: past, present, and future

Using nonparametric identification techniques, equivalent nonlinear memoryless systems are determined for a class of hysteretic oscillators subjected to broadband nonstationary random excitation The oscillators are assumed to have time-invariant bilinear hysteretic properties Numerical techniques are used to determine the time history of the response and excitation of the “exact” oscillator under a single random excitation record This response is subsequently used with a time-domain nonparametric identification technique to determine by means of least-squares methods an equivalent nonlinear oscillator characterized by a doubly-indexed series of Chebyshev polynomials The agreement between the mean-square response of the exact and approximate models is found to be good even when the oscillators are undergoing substantial nonlinear deformations

Response of Hysteretic Oscillators Under Nonstationary Random Excitation

The flow through and around the rotor of a turbomachine exerts a force on the rotor and, hence, rotor shaft and bearing system. In some circumstances this force may lead to excitation of shaft whirl in the direction of impeller rotation. Recent international research of this phenomenon is briefly reviewed; these findings suggest that turbomachines intended to operate well above the first critical speed should take the effect into account.

Impeller-Induced Rotor-Dynamic Forces

Since the attitude instability experience by Explorer 1, many papers have been written on the effects if internal dissipation on the attitude stability of spinning satellites. In the engineering literature, stability analysis is restricted to the variational or linearized perturbational equations, despite the fact that spinning satellites are almost always critical cases in the Liapunov-Poincare stability theory. This is certainly true in the case of dual spin satellites, which have the further complication that the linearized perturbational equations have periodic coefficients. 

The purpose of this note is to treat some problems of attitude stability of spinning satellites in a rigorous manner and to show that, with certain restrictions, the linearized stability analysis correctly predicts the attitude stability of spinning satellites.

Attitude stability of spinning satellites

In this work we examine the free and forced oscillation of a class of two-Degree-of Freedom Undampted Nonlinear Systems. We examine the existence and stability of Similar Normal Modes of the Unsymmetric system and we analyse the subharmonic periodic orbits of the weakly coupled symmetric oscillator. Furthermore, exact solutions for the steady state forced motions of the strongly nonlinear system are derived and specific applications for systems with cubic nonlinearities are given. Finally we give a methodology for examining the Nonsimilar Normal Modes of such systems and we apply the analysis to the case of a weakly coupled system with cubic nonlinearities.

http://authors.library.caltech.edu/26570/1/DYNL89-1A.pdf

Thomas K. Caughey

Papers

Response of Hysteretic Oscillators Under Nonstationary Random Excitation

Impeller-Induced Rotor-Dynamic Forces

Attitude stability of spinning satellites

Some topics in the free and forced oscillations of a class of nonlinear systems