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

Physical and numerical experiments show that deterministic noise, or chaos, is ubiquitous. While a good understanding of the onset of chaos has been achieved, using as a mathematical tool the geometric theory of differentiable dynamical systems, moderately excited chaotic systems require new tools, which are provided by the ergodic theory of dynamical systems. This theory has reached a stage where fruitful contact and exchange with physical experiments has become widespread. The present review is an account of the main mathematical ideas and their concrete implementation in analyzing experiments. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information), and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities, as well as their experimental determination, are discussed. The systematic investigation of these quantities provides us for the first time with a reasonable understanding of dynamical systems, excited well beyond the quasiperiodic regimes. This is another step towards understanding highly turbulent fluids.

Ergodic theory of chaos and strange attractors

Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition)

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

A level set method for structural topology optimization

Partial regularity of suitable weak solutions of the navier‐stokes equations

Il existe une constante positive C telle que l'inegalite suivante est vraie pour tout u∈C 0 ∞ (R n ): ||X| G u| L r≤C||x|α|Du|| L p 2 ||x|βu| L q 1−a si et seulement si on a 1/r+γ/n=a(1/p+(α−1)/n)+(1−a)(1/q+β/u), 0≤α−σ si a>0 et α−σ≤1 si a>0 et 1/p+(α−1)/n=1/r+γ/n

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First order interpolation inequalities with weights

A. P. Calderon pose la question: est-il possible de determiner la conductivite thermique d'un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite? On demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

Determining conductivity by boundary measurements

: This paper has attempted a synthesis of three subjects that have developed separately - optimal design, homogenization, and relaxation in the calculus of variations. The underlying theme is that of oscillations. They occur with certain definite patterns in minimizing sequences for the variational problem. The same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized. Each problem is resolved by determining the structure of those oscillations, and above all their macroscopic averages. It is those averages that solve the relaxed problem. They also determine the optimal design and the solution of the homogenized equation. We emphasize also the role of optimization. The homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal, since they correspond to minimizing sequences for the variational problem. The effective properties of these composites attain the optimal bounds for given materials in fixed proportions.

Optimal design and relaxation of variational problems, III

: This reprint studies the blow-up of solutions of a nonlinear heat equation. The main goal is to characterize the asymptotic behavior of u near a singularity, assuming a suitable upper bound on the rate of blowup. Additional keywords: Liouville theorem; and Asymptotic normality.

Robert V. Kohn

Papers

Partial regularity of suitable weak solutions of the navier‐stokes equations

First order interpolation inequalities with weights

Determining conductivity by boundary measurements

Optimal design and relaxation of variational problems, III

Asymptotically self‐similar blow‐up of semilinear heat equations