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

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Asymptotic Analysis of Periodic Structures

A paradigm shift is taking place in medicine from using synthetic implants and tissue grafts to a tissue engineering approach that uses degradable porous material scaffolds integrated with biological cells or molecules to regenerate tissues. This new paradigm requires scaffolds that balance temporary mechanical function with mass transport to aid biological delivery and tissue regeneration. Little is known quantitatively about this balance as early scaffolds were not fabricated with precise porous architecture. Recent advances in both computational topology design (CTD) and solid free-form fabrication (SFF) have made it possible to create scaffolds with controlled architecture. This paper reviews the integration of CTD with SFF to build designer tissue-engineering scaffolds. It also details the mechanical properties and tissue regeneration achieved using designer scaffolds. Finally, future directions are suggested for using designer scaffolds with in vivo experimentation to optimize tissue-engineering treatments, and coupling designer scaffolds with cell printing to create designer material/biofactor hybrids.

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Porous scaffold design for tissue engineering

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

Distributions and Sobolev spaces.- Operators in Banach spaces.- Examples of boundary value problems.- Semigroups and laplace transform.- Homogenization of second order equations.- Homogenization in elasticity and electromagnetism.- Fluid flow in porous media.- Vibration of mixtures of solids and fluids.- Examples of perturbations for elliptic problems.- The Trotter-Kato theorem and related topics.- Spectral perturbation. Case of isolated eigenvalues.- Perturbation of spectral families and applications to selfadjoint eigenvalue problems.- Stiff problems in constant and varialbe domains.- Averaging and two-scale methods.- Generalities and potential method.- Functional methods.- Scattering problems depending on a parameter.

Enrique Sanchez-Palencia

Papers

Non-Homogeneous Media and Vibration Theory

Homogenization Techniques for Composite Media

Comportements local et macroscopique d'un type de milieux physiques heterogenes

On boundary conditions for fluid flow in porous media

Homogenization method for the study of composite media