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, i~otropic 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.

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Generating optimal topologies in structural design using a homogenization method

Shape optimization in a general setting requires the determination of the optimal spatial material distribution for given loads and boundary conditions. Every point in space is thus a material point or a void and the optimization problem is a discrete variable one. This paper describes various ways of removing this discrete nature of the problem by the introduction of a density function that is a continuous design variable. Domains of high density then define the shape of the mechanical element. For intermediate densities, material parameters given by an artificial material law can be used. Alternatively, the density can arise naturally through the introduction of periodically distributed, microscopic voids, so that effective material parameters for intermediate density values can be computed through homogenization. Several examples in two-dimensional elasticity illustrate that these methods allow a determination of the topology of a mechanical element, as required for a boundary variations shape optimization technique.

Optimal shape design as a material distribution problem

A level set method for structural topology optimization

In topology optimization of structures, materials and mechanisms, parametrization of geometry is often performed by a grey-scale density-like interpolation function. In this paper we analyze and compare the various approaches to this concept in the light of variational bounds on effective properties of composite materials. This allows us to derive simple necessary conditions for the possible realization of grey-scale via composites, leading to a physical interpretation of all feasible designs as well as the optimal design. Thus it is shown that the so-called artificial interpolation model in many circumstances actually falls within the framework of microstructurally based models. Single material and multi-material structural design in elasticity as well as in multi-physics problems is discussed.

/pdf/material-interpolation-schemes-in-topology-optimization-1ps0tr57ei.pdf

Material interpolation schemes in topology optimization

A simple evolutionary procedure for structural optimization

Comment on: On the theory of optimal, constant thickness, fibre-reinforced plates. P. G. Lowe and R. E. Melchers, Int. J. mech. Sci. 14, 311 (1972)

After discussing the aims and significance of topology optimization, layout optimization of grid-like structures is discussed in detail. In the proposed method, the ground structure (structural universe) approach is used, together with continuum-type optimality criteria. Layout optimization may be carried out using (i) analytical methods for exact, often closed-form, continuum-type solutions or (ii) numerical methods for approximate, discretized solutions. Due to the high optimization capability of modern OC methods, the latter provides simultaneous optimization of geometry and topology, with a high degree of accuracy. Finally, a new, extended optimal layout theory for multiload, multipurpose structures is outlined and applications to particular classes of problems given.

Discretized Method for Topology Optimization

This paper concerns a general theory for elastic structures with prescribed deflections and constrained ``taper'' (= rate of spatial change of the cross-sectional area). The proposed theory is applied to minimum weight beams with a linear relationship between the flexural stiffness and the cross-sectional area. This application is then illustrated through an example of a cantilever beam with constrained taper, having a prescribed slope at its free end. As this structure is statically determinate, the ``associated'' displacement field (that is, the Lagrangian for the equilibrium condition) is not employed directly in obtaining the soptimal solution. However, it can be used for checking the validity of the solution through a dual formulation. In this example, the optimal length of the section with constrained taper is given by another Lagrangian. In addition to verifying them through the dual formula, the results are checked by another independent method (differential calculus for an assumed class of soluti...

Optimal Elastic Design with Constrained Taper: Prescribed Deflections

Using extensions of Prager's theories of optimal plastic design, slabs and grillages are optimized within various geometrical constraints ensuring simplicity and practicality. The two classes of problems considered are: (1)Grillages consisting of prismatic beams of preassigned directions and length but variable spacing and balanced prestressed elastic plates having tendons of preassigned directions terminated at the edges only; and (2)slabs with curtailed negative reinforcement of preassigned length. The results given represent rigorous analytical optima as well as identical upper and lower bounds on the limit load. A comprehensive set of solutions is tabulated for rectangular domains with various boundary conditions and the moment volumes for constrained geometry are compared graphically with absolute minimum volumes of unconstrained solutions. Finally, the most economic length of negative reinforcement is calculated for rectangular slabs.

Optimal Slabs and Grillages of Constrained Geometry

Symmetry and non-uniqueness principles play an important role in structural topology optimization. In this paper, they are discussed in the context of analytically derived leastweight trusses and grillages, but these concepts can be extended to other structures and also to numerical, discretized solutions. It is found that in general the optimal topology is unique and for problems with symmetric domain boundaries, loads and supports it is also symmetric. However, in a few exceptional cases the optimal topology is non-unique, and then we have an infinite number of topologies with the same minimum volume or weight, and some of these may be non-symmetric. The deeper reasons for the above findings are explained on the basis of the optimal layout theory of Prager and the author.

Symmetry and non-uniqueness in exact topology optimization of structures

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