Generating optimal topologies in structural design using a homogenization method

Optimal shape design of structural elements based on boundary variations results in final designs that are topologically equivalent to the initial choice of design, and general, stable computational schemes for this approach often require some kind of remeshing of the finite element approximation of the analysis problem. This paper presents a methodology for optimal shape design where both these drawbacks can be avoided. The method is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, isotropic material, with the requirement that the resulting structure can carry the given loads as well as satisfy other design requirements. The computation of effective material properties for the anisotropic material is carried out using the method of homogenization. Computational results are presented and compared with results obtained by boundary variations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27079/1/0000070.pd

An interpretation for min-max structural design problems including a method for relaxing constraints

Min-max type problems arise in structural design when the objective is to minimize the maximum value of some local measure of system response, e.g. design to minimize the maximum stress or displacement. A method is described whereby the min-max problem is interpreted as a simple min problem. Governing equations for the adjoint structure are derived directly from the Lagrangian for this min problem by using the generalized multiplier rule on the original state equations. Also certain advantages are demonstrated for a modified form of the min-max problem, a form obtained by introducing a relaxation on the local constraints. The analysis is applied for examples of structural design with stress and displacement criteria, and for the design of an elastic foundation to minimize support pressure.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24998/1/0000425.pd

Design of optimal material properties for structures composed of nonlinear material

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/76247/1/AIAA-1994-4367-370.pd

Topology Optimization of Continuum Structures with Stress Constraints

We introduce an extension of topology optimization of continuum structures to deal with local stress criteria. We first consider relevant stress criteria for porous composite materials, initially by studying the stress states of the so-called rank 2 layered materials. Then, an empirical model is proposed for the power law materials (also called SIMP materials). In a second part, solution aspects of topology problems are considered. To deal with the so-called 'singularity' phenomenon of stress constraints in topology design, an $\epsilon$ constraint relaxation of the stress constraints is used. We describe the mathematical programming approach that is used to solve the numerical optimization problems. The proposed strategy is applied to illustrative applications

A mutual energy formulation for optimal structural design

This paper presents a variational formulation for the design of elastic structures where the function to be minimized by the optimal design, i.e. the objective, is expressed in abstract form. The resulting statement of necessary conditions is uniformly applicable for all admissible objectives. Both state and adjoint state variables appear directly in the problem statement, and all objectives and the arguments of constraints are scalars. The adjoint pair of state variables appear in symmetric roles via the expression termed “mutual energy". Application of the generalized formulation is demonstrated by treatment of the following examples: design to minimize the maximum value of displacement or to minimize a global measure of stress, design for generalized compliance, design where self-weight is taken into account, and multicriterion design.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41906/1/158-22-2-95_10220095.pd