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 general procedure is presented for analyzing dynamic response measurements from complex multi-degree-of-freedom nonlinear systems incorporating arbitrary types of nonlinear elements. The analysis procedure develops a reduced-order, nonlinear model whose format is convenient for numerical simulation studies. No information about the system’s mass properties is needed, and only the applied excitations and corresponding response are needed to develop the model whose dimension is compatible with the number of available sensors. The utility of the approach is demonstrated by means of a three-degree-of-freedom system incorporating polynomial-type nonlinear features with hardening as well as softening characteristics.

A General Data-Based Approach for Developing Reduced-Order Models of Nonlinear MDOF Systems

The concept of similar normal mode is used to study the free oscillations of a class of strongly non-linear, two degree of freedom undamped systems. By using the so-called balancing diagrams, the bifurcations of normal modes can be conveniently studied for different forms of stiffness non-linearities. Consequently, the stability of the resulting free motions is analytically examined and the theoretical results are checked by numerical integration of the equations of motion.

Analytical study of similar normal modes and their bifurcations in a class of strongly non-linear systems

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Stochastic Problems in Dynamics

The procedure uses nonlinear auxiliary mass dampers with adjustable motion-limiting stops located at selected positions throughout a given nonlinear system. The degree of the primary structure oscillation near each vibration damper determines the damper's actively-controlled gap size and activation time. By using control energy to adjust the damper parameters instead of directly attenuating the motion of the primary system, a significant improvements is achieved in the total amount of energy expended to accomplish a given level of vibration control. In a related paper, the direct method of Lyapunov is used to establish that the response of the controlled nonlinear primary structure is Lagrange stable

Thomas K. Caughey

Papers

A General Data-Based Approach for Developing Reduced-Order Models of Nonlinear MDOF Systems

Analytical study of similar normal modes and their bifurcations in a class of strongly non-linear systems

Stochastic Problems in Dynamics

Active Parameter Control of Nonlinear Vibrating Structures

Existence, uniqueness, and stability of solutions of a class of nonlinear partial differential equations