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.

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Material interpolation schemes in topology optimization

A simple evolutionary procedure for structural optimization

Structural layout theory is based on two underlying concepts, i.e., the Prager-Shield theory of plastic design and the notion of "structural universe." In this paper, optimal layout theory is extended to allow for the effect of selfweight (dead load). The proposed extension is of practical importance because in the design of long-span structures selfweight is a significant load component and the total weight is strongly dependent on the choice of layout. The application of the modified optimality criteria is illustrated with examples of minimum weight grillages (truss grids) as well as plane frames. The degree of economy achieved is then demonstrated through comparisons with nonoptimal solutions. It is also shown that in the preceding examples both primal and dual formulations give the same minimum weight. This comparison is a convenient check on the optimality of the proposed solutions.

Optimal Layout Theory: Allowance for selfweight

Prager's layout theory: a nonnumeric computer method for generating optimal structural configurations and weight-influence surfaces

In this paper, a systematic exploration of Michell layouts for various combinations of line supports is continued. The solutions discussed include those for reentrant corners, a line support and two free edges, line supports forming any convex polygon, supports consisting of two symmetric curves, and finally, rectangular domains bounded by one line support and three free edges.

Exact Michell layouts for various combinations of line supports-Part II

Optimal beam layouts: Allowance for cost of shear

In optimizing plane flexural systems such as grillages and fiber-reinforced plates of a prescribed depth, the solution can now be obtained relatively easily for any combination of clamped and simply supported boundaries. However, attempts to extend the same theory to systems with free (unsupported) edges failed in the past and it has even been suggested that solutions for such boundary conditions may not have to satisfy certain generally accepted static-kinematic optimality criteria. The reasons for these difficulties are explained herein by considering the optimization of a simple grillage subjected to point loads. It is demonstrated that, when discrete solutions having a prescribed number of beams are considered, then the optimal structural weight can be reduced further by increasing the number of beams specified. The limiting case giving the absolute minimum structural weight appears to consist of an infinite number of beams some of which take on an infinitesimal length. The foregoing layout satisfies the Prager-Shield optimality criterion and is made plausible by establishing very close upper and lower bounds on it. The proposed solution is useful both in furnishing the absolute limits of economy and in providing efficient beam directions for the design of discrete grillages with free edges.

George I. N. Rozvany

Papers

Optimal Layout Theory: Allowance for selfweight

Prager's layout theory: a nonnumeric computer method for generating optimal structural configurations and weight-influence surfaces

Exact Michell layouts for various combinations of line supports-Part II

Optimal beam layouts: Allowance for cost of shear

Optimal Beam Layouts: The Free Edge Paradox