A b s t r a c tThis paper addresses a novel method of topology and shape optimization, The basic idea is the iterative positioning of new holes (so-called "bubbles") into the present structure of the component. This concept is therefore called the "bubble method". The iterative positioning of new bubbles is carried out by means of different methods, among others by solving a variational problem. The insertion of a new bubble leads to a change of the class of topology. For these different classes of topology, hierarchically structured shape optimizations that determine the optimal shape of the current bubble, as well as the other variable boundaries, are carried out. I n t r o d u c t i o nThe position and arrangement of the structural elements within a component, i.e. the change of its topology, decisively influence the deformation and the stress-field. When developing new structures, it is important to optimally determine the topology. Topology optimization aims at replacing the more intuitive variation of constructions in the design phase by mathematical-mechanical strategies for the sake of greater efficiency.As there are other types of design variables apart from the topology optimization of structures, a classification according to these types seems sensible in order to discuss the recent developments of topology optimization in the scope of structural optimization (Schmit and Mallet 1963). Many structural improvements can be achieved either by an optimal choice of cross-sections or by a best-possible shaping. Apart from optimally choosing advanced materials, a number of research groups have increasingly dealt with topology optimization. The state-of-the-art in this field of research will be described briefly in the following.Michell (1904) developed a design theory for the topology of bar structures that are optimal with regard to weight. The bars in these structures are all perpendicular t o each other and therefore form an optimal arrangement in the sense of maximum tensile and compressive stresses. The development of finite element procedures and of further efficient structural computation methods in the course of the 1960's has facilitated the structural optimization of more complex problems. In this respect, research into topology optimization is yet in its early stages. Important initial steps in this direction were taken by Prager (1974) and Prager and Rozvany (1977), who *Humboldt scholar, visiting the institute of Mechanics and Control Engineering, University of Siegen, 1990Siegen, -1991 solved topology optimization problems by analytical procedures.

A microscopic theory for the dependence on external strain, stress, and shear rate of the transient localization length, elastic modulus, alpha relaxation time, shear viscosity, and other dynamic properties of glassy colloidal suspensions is formulated and numerically applied. The approach is built on entropic barrier hopping as the elementary physical process. The concept of an ideal glass transition plays no role, and dynamical slowing down is a continuous, albeit precipitous, process with increasing colloid volume fraction. The relative roles of mechanically driven motion versus thermally activated barrier hopping and transport have been studied. Various scaling behaviors are found for the relaxation time and shear viscosity in both the controlled stress and shear rate mode of rheological experiments. Apparent power law and/or exponential dependences of the elastic modulus and perturbative and absolute yield stresses on colloid volume fraction are predicted. A nonmonotonic dependence of the absolute yield strain on volume fraction is also found. Qualitative and quantitative comparisons of calculations with experiments on high volume fraction glassy colloidal suspensions show encouraging agreement, and multiple testable predictions are made. The theory is generalizable to treat nonlinear rheological phenomena in other soft glassy complex fluids including depletion gels.

A theoretical analysis of Coulomb systems on lattices in general dimensions is presented. The thermodynamics is developed using Debye-Hückel theory with ionpairing and dipole-ion solvation, specific calculations being performed for 3D lattices.As for continuum electrolytes, low-density results for sc, bcc and fcc lattices indicate the existence of gas-liquid phase separation. The predicted critical densities have values comparable to those of continuum ionic systems, while the critical temperatures are 60-70% higher. However, when the possibility of sublattice ordering as well as Debye screening is taken into account systematically, order-disorder transitions and a tricritical point are found on sc and bcc lattices, and gas-liquid coexistence is suppressed. Our results agree with recent Monte Carlo simulations of lattice electrolytes. 2

A microscopic activated barrier hopping theory of the viscoelasticity of colloidal glasses and gels has been generalized to treat the nonlinear rheological behavior of particle-polymer suspensions. The quiescent cage constraints and depletion bond strength are quantified using the polymer reference interaction site model theory of structure. External deformation (strain or stress) distorts the confining nonequilibrium free energy and reduces the barrier. The theory is specialized to study a limiting mechanical description of yielding and modulus softening in the absence of thermally induced barrier hopping. The yield stress and strain show a rich functional dependence on colloid volume fraction, polymer concentration, and polymer-colloid size asymmetry ratio. The yield stress collapses onto a master curve as a function of the polymer concentration scaled by its ideal mode-coupling gel boundary value, and sufficiently deep in the gel is of an effective power-law form with a universal exponent. A similar functional and scaling dependence of the yield stress on the volume fraction is found, but the apparent power-law exponent is nonuniversal and linearly correlated with the critical gel volume fraction. Stronger gels are generally, but not always, predicted to be more brittle in the strain mode of deformation. The theoretical calculations appear to be in accord with a broad range of observations.

The naive mode coupling-polymer reference interaction site model (MCT-PRISM) theory of gelation and elasticity of suspensions of hard sphere colloids or nanoparticles mixed with nonadsorbing polymers has been extended to treat the emergence of barriers, activated transport, and viscous flow. The barrier makes the dominant contribution to the single particle relaxation time and shear viscosity, and is a rich function of the depletion attraction strength via the polymer concentration, polymer-particle size asymmetry ratio, and particle volume fraction. The dependences of the barrier on these three system parameters can be accurately collapsed onto a single scaling variable, and the resultant master curve is well described by a power law. Nearly universal master curves are also constructed for the hopping or alpha relaxation time for system conditions not too close to the ideal MCT transition. Based on the calculated barrier hopping time, a theory for kinetic gel boundaries is proposed. The form and dependence on system parameters of the kinetic gel lines are qualitatively the same as obtained from prior ideal MCT-PRISM studies. The possible relevance of our results to the phenomenon of gravity-driven gel collapse is studied. The general approach can be extended to treat nonlinear viscoelasticity and rheology of polymer-colloid suspensions and gels.