This paper discusses the homogenization method to determine the effective average elastic constants of linear elasticity of general composite materials by considering their microstructure. After giving a brief theory of the homogenization method, a finite element approximation is introduced with convergence stuay and corresponding error estimate. Applying these, computer programs PREMAT and POSTMAT are developed for preprocessing and postprocessing of material characterization of composite materials. Using these programs, the homogenized elastic constants for macroscopic stress analysis are obtained for typical composite materials to show their capability. Finally, the adaptive finite element method is introduced to improve the accuracy of the finite element approximation.

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Material based models for topology optimization of linear elastic solids with a low volume constraint generate very slender structures composed mainly of bars and beam elements. For this type of structure the value of the buckling critical load becomes one of the most important design criteria and so its control is very important for meaningful practical designs. This paper tries to address this problem, presenting an approach to introduce the possibility of critical load control into the topology optimization model.Using the material based formulation for topology design of structures, the problem of optimal structural reinforcement for a critical load criterion is formulated. The stability problem is conveniently reduced to a linearized eigenvalue problem assuming only material effective properties and macroscopic instability modes. The respective optimality criteria are presented by introducing the Lagrangian associated with the optimization problem. Based on this Lagrangian a first-order method is used as a basis for the numerical update scheme. Two numerical examples to validate the developments are presented and analysed.

This paper deals with the simultaneous optimization of material and structure for minimum compliance. Material properties are represented in the most general form possible for a (locally) linear elastic continuum, namely the unrestricted set of elements of positive semi-definite constitutive tensors and cost measures based on certain invariants of the tensors. Analytical forms are derived for the optimized material properties. These results, which apply in general, indicate that the optimized material is orthotropic with the directions of orthotropy following the directions of principal strains. The analysis for optimization of the material leads to a reduced structural optimization problem, for which the existence of solutions can be shown and for which effective methods for computational solution can be devised.

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