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

A mathematical model is presented which gives a fairly comprehensive explanation of the strange behavior of a certain type of stone, called a Celt or Rattleback, which if placed on a flat horizontal surface and caused to spin in a particular direction about a vertical axis, will quickly stop spinning, oscillate about a horizontal axis, and then start to spin in the opposite direction. In many cases the stone will show spin-reversal for only one direction of initial spin, spinning stably for initial spin in the opposite direction.

A mathematical model of the “Rattleback”

The role played by fluid forces in determining the rotordynamic stability of a centrifugal pump is gaining increasing attention. The present research investigates the contributions to the rotordynamic forces from the discharge-to-suction leakage flows between the front shroud of the rotating impeller and the stationary pump casing. In particular, the dependency of the rotordynamic characteristics of leakage flows on the swirl at the inlet to the leakage path was examined. An inlet guide vane was designed for the experiment so that swirl could be introduced at the leakage flow inlet. The data demonstrate substantial rotordynamic effects and a destabilizing tangential force for small positive whirl ratios; this force decreased with increasing flow rate. The effect of swirl on the rotordynamic forces was found to be destabilizing.

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The Effect of Inlet Swirl on the Rotordynamic Shroud Forces in a Centrifugal Pump

A study is made of the general behavior of a semi-active impact damper. The system consists of an undamped forced torsional oscillator, and a flywheel which can be locked to the oscillator through a clutch. The impact which results during clutch engagement is effective in reducing the vibration amplitude level of the oscillator when it is subjected to bounded excitation

Stability of a Semi-Active Impact Damper: Part I—Global Behavior

Random Excitation of a Loaded Nonlinear String

In recent years, increasing attention has been give to fluid-structure interaction problems in turbomachines. The present research focuses on just one such fluid-structure interaction problem, namely, the role played by fluid forces in determining the rotordynamic stability and characteristics of a centrifugal pump. The emphasis of this study is to investigate the contributions to the rotordynamic forces from the discharge-to-suction leakage flows between the front shroud of the rotating impeller and the stationary pump casing. An experiment was designed to measure the rotordynamic shroud forces due to simulated leakage flows for different parameters such as flow rate, shroud clearance, face-seal clearance and eccentricity. The data demonstrate substantial rotordynamic effects and a destabilizing tangential force for small positive whirl frequency ratios; this force decreased with increasing flow rate. The rotordynamic forces appear to be inversely proportional to the clearance and change significantly with the flow rate. Two sets of data taken at different eccentricities yielded quite similar nondimensional rotordynamic forces indicating that the experiments lie within the linear regime of eccentricity.

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Thomas K. Caughey

Papers

A mathematical model of the “Rattleback”

The Effect of Inlet Swirl on the Rotordynamic Shroud Forces in a Centrifugal Pump

Stability of a Semi-Active Impact Damper: Part I—Global Behavior

Random Excitation of a Loaded Nonlinear String

Experimental Results for the Rotordynamic Characteristics of Leakage Flows in Centrifugal Pumps