This research was performed at the School of Mechanical Engineering , Tel Aviv University by Moshe Fuchs, Moshe Paley and graduate students Evgenyi Miroshnik, Eyal Moses.

- Paley M., Fuchs M.B. and Miroshnik, E. (1996),
**The Aboudi Micromechanical Model for Shape Design of Structures**,*The Third International Conference on Computational Structures Technology*, Budapest 21-23 Aug.

The micromechanical approach to topology design of structures consists in defining a domain of porous material, including the loads and supports, and finding an optimal distribution of the densities. The usual method is to mesh the structure and assign a different density and material orientation to every element. Optimal topologies emerge when using the densities and orientations as design variables in a mathematical programming formulation in conjunction with a finite element analysis program. This paper presents the*Aboudi method of cells*, an analytical method to compute the elastic properties of composite material, and applies it to determine the mechanical properties of porous material as a function of the density of material. It is shown that the results are similar to those obtained by special finite element analysis set up to compute the elasticity matrix. The procedure is visualized on simple cantilever design problems and used within the context of a*stress-ratio redesign scheme*. (*Postscript*file, 277K)

- Fuchs M.B. and Moses E. (1997),
**Shape Design of Structures Using a Bi-material Composite with a Stress-Ratio Redesign Algorithm**,*WCSMO-2, The Second World Congress of Structural and Multidisciplinary Optimization*, Zakopane 26-30 May.

The present paper falls in the category of methods for the shape design of structures whithout an initial topology. As usual a uniform domain is meshed into finite elements where every element is made of a composite with a bimaterial periodic microstructure, each with an intrinsic material density and orientation. One phase of the composite is given a very low stiffness and is assumed to represent voids and the other phase is made of the actual material of the structure. Optimal topologies emerge when using the densities and material orientation as design variables in a mathematical programming formulation in conjunction with a finite element structural analysis. Here we apply shape optimization techniques by using commercially available software with little additional algorithms. The numerical techniques are based on (a) a finite element package (Ansys) for meshing and analyzing the initial and intermediate structural domains, (b) the Aboudi method of cells to determine the mechanical properties of a porous orthotropic material as a function of the density of material and (c) a stress-ratio redesign algorithm to perform the actual optimization. Some numerical results are reported. (*Postscript*file, 277K)

- Fuchs M.B. and Moses E. (1999),
**Topological design solutions for structures under transmissible loads**,*5th International Conference on Computational Structures Technology*, Leuven 6-8 September, 2000

A numerical approach to topological design starts with a domain of material to which the external loads and support conditions are applied. The initial domain is homogeneous at the macroscopical scale, a condition which precludes any preferential or intuitive design concepts. The optimization algorithm then proceeds with carving out ineffectual material in order to generate best structural solutions. A similar procedure is followed by an artist when producing a sculpture. As noted by Paul Auster (The invention of Solitude, Faber and Faber, 1988): "the figure is already there in the material; the artist merely hews away at the excess matter until the true form is revealed." The objective function is often the compliance, that is, the flexibility of the structure under the given loads, subject to a constant volume of material constraint. The design variables of the optimization problem are the densities of the elements. It will be noted that in many solutions one can discern a marked difference between the underlying load-bearing structure and the load-transfer part. The main purpose of the present work is to induce the algorithm to produce the 'structural' part of the structure and unclogging it from secondary load-transfer components. This is done by allowing the external loads to move along their line of action. In other words, we have a set of loads to be supported by the structure but we do not specify the exact location of the points of application of the loads along their line of action. The rationale is to push the underlying idea of topological design one step further. Since the point of application of a load influences the topology of the structure let us allow the load to move along its line of actions and let the algorithm also determine optimal load locations. The concept of movable loads is structural optimization is not new. It appears in the definition of Prager structures (Rozvany and Prager, 1999, Comp. Meth. Appl. Mech. Engrg., Vol. 19, 49-58) which are stress-constrained least-weight trusses where the sign of the member stresses must be the same in all elements and the loads are allowed to move along their line of action. This work relates also to a 1999 publication by Hammer and Olhoff (Topology optimization with design dependent loads, 3rd WCSMO, Short Paper Proceedings, Buffalo) since it can be construed as a structure for which the boundary is allowed to move. (*Postscript*file, 186K)