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

This paper explores the potential of a time-domain identification procedure to detect structural changes on the basis of noise-polluted measurements. The method of approach requires the use of excitation and acceleration response records, to develop an equivalent multi-degree-of-freedom (MDOF) mathematical model whose order is compatible with the number of sensors used. Application of the identification procedure under discussion yields the optimum value of the elements of equivalent linear system matrices. By performing the identification task before and after potential structural changes (damage) in the physical system have occurred, quantifiable changes in the identified mathematical model can be detected. The usefulness of the identification procedure under discussion for damage detection is demonstrated by means of an example of three-degree-of-freedom (DOF) linear system. This system is used to conduct synthetic experiments to generate noise-polluted “data” sets that are subsequently analyzed to determine the mean, variance, and probability density function corresponding to each element of the identified system matrices. Different versions of the model are investigated in which the location as well as the magnitude of the “damage” is varied. On the basis of this exploratory study, it appears that determining the probability density functions of the identified system matrices may furnish useful indices that can be conveniently extracted during an experimental test, to quantify changes in the characteristics of physical systems.

System identification approach to detection of structural changes

Finite-element methods for nonlinear elastodynamics which conserve energy.

Steady and unsteady diffuser vane pressure measurements have been conducted with a two-dimensional test impeller, in an experimental investigation of rotor-stator interaction within a centrifugal pump having several vaned diffusers, under conditions of different flow coefficients and different radial gaps between the impeller blade trailing edge and the diffuser vane leading edge. The largest pressure fluctuations on the diffuser vanes and the impeller blades were found to be of the same order of magnitude as the total pressure rise across the pump. Increasing the number of diffuser vanes was found to result in a significant decrease of impeller blade pressure fluctuations.

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Experimental Investigation of Rotor-Stator Interaction in a Centrifugal Pump With Several Vaned Diffusers

Nonparametric Identification of Nearly Arbitrary Nonlinear Systems

A self-starting multistage, time-domain procedure is presented for the identification of nonlinear, multi-degree-of-freedom systems undergoing free oscillations or subjected to arbitrary direct force excitations and/or nonuniform support motions. Recursive least-squares parameter estimation methods combined with nonparametric identification techniques are used to represent, with sufficient accuracy, the identified system in a form that allows the convenient prediction of its transient response under excitations that differ from the test signals. The utility of this procedure is demonstrated in a companion paper.

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Thomas K. Caughey

Papers

System identification approach to detection of structural changes

Finite-element methods for nonlinear elastodynamics which conserve energy.

Experimental Investigation of Rotor-Stator Interaction in a Centrifugal Pump With Several Vaned Diffusers

Nonparametric Identification of Nearly Arbitrary Nonlinear Systems

Identification of nonlinear vibrating structures: Part I -- Formulation