The desired results of variable topology material layout computations are stable and discrete material distributions that optimize the performance of structural systems. To achieve such material layout designs a continuous topology design framework based on hybrid combinations of classical Reuss (compliant) and Voigt (sti ) mixing rules is investigated. To avoid checkerboarding instabilities, the continuous topology optimization formulation is coupled with a novel spatial ÿltering procedure. The issue of obtaining globally optimal discrete layout designs with the proposed formulation is investigated using a continuation method which gradually transitions from the sti Voigt formulation to the compliant Reuss formulation. The very good performance of the proposed methods is demonstrated on four structural topology design optimization problems from the literature.

Discrete ground structure topology optimization design methods have to date received considerable attention in structural engineering. An alternative class of structural topology optimization methods which not yet received much attention in structural engineering, but which have undergone considerable development in the past decade are the so-called continuum formulations. In this work, a continuum structural topology optimization formulation is presented and applied to the concept design optimization of structural bracing systems which are needed to stiffen tall structures against sidesway under lateral wind and seismic type loadings. While demonstrated here in the context of these specific design examples, continuum structural topology optimization methods are believed to hold potential as a design tool for wide range of civil engineering type structures. A variety of continuum topology design formulations, including static compliance minimization and eigenvalue optimization, are explored, and solution parameters are varied to show that a number of design possibilities can be realized as solutions.

This work is directed toward optimizing concept designs of structures featuring inelastic material behaviours by using topology optimization. In the proposed framework, alternative structural designs are described with the aid of spatial distributions of volume fraction design variables throughout a prescribed design domain. Since two or more materials are permitted to simultaneously occupy local regions of the design domain, small-strain integration algorithms for general two-material mixtures of solids are developed for the Voigt (isostrain) and Reuss (isostress) assumptions, and hybrid combinations thereof. Structural topology optimization problems involving non-linear material behaviours are formulated and algorithms for incremental topology design sensitivity analysis (DSA) of energy type functionals are presented. The consistency between the structural topology design formulation and the developed sensitivity analysis algorithms is established on three small structural topology problems separately involving linear elastic materials, elastoplastic materials, and viscoelastic materials. The good performance of the proposed framework is demonstrated by solving two topology optimization problems to maximize the limit strength of elastoplastic structures. It is demonstrated through the second example that structures optimized for maximal strength can be signiÿcantly di erent than those optimized for minimal elastic compliance. ?

It is considered that asymmetrical material layout design solutions are caused by numerical roundo and the convexity characteristics of alternative topology design formulations. Emphasis is placed here not on analyzing potential instabilities that lead to asymmetrical designs, but on a method to stabilize topology design formulations. A novel symmetry reduction method is proposed, implemented and studied. While enforcing symmetry and signi®cantly reducing the size of the optimization problem, the symmetry reduction method is shown to have the added bene®t of greatly simpli®ed design sensitivity analysis of non-simple repeated vibrational eigenvalues which occur in many symmetrical structures. #

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