Structural Optimization and Biological “Designs”

Abstract

The amazing rationality of biological “constructions” excites the interest to modelling them by using the mathematical tools developed in the theory of structural optimization. The structural optimization solves a geometrical problem of the “best” displacements of different materials in a given domain, under certain loadings. Of course, this approach simplifies the real biological problem, because the questions of the mechanism of the building and maintaining of structures are not addressed. The main problem is to guess a functional for the optimization of a living organism. The optimal designs are highly inhomogeneous; their microstructures may be geometrically different, but possess the same effective properties. Therefore the comparing of the various optimal geometries is not trivial. We show, that the variety of optimal geometries shares the same characteristics of the stress tensor in any optimal structure. Namely, special norm of this tensor stay constant within each phase of the optimal mixture.