Optimization of Structural Topology, Shape, and Material

The efficient use of materials is important in many different settings. The aerospace industry and the automotive industry, for example, apply sizing and shape optimization to the design of structures and mechanical elements. Shape optimization is also used in the design of electromagnetic, electrochemical and acoustic devices. The subject of non-linear, finite-dimensional optimization for this type of problem is now relatively mature. It has produced a number of successful algorithms that are widely used for structural optimization, including some that have been incorporated in commercial finite element codes. However, these methods are unable to cope with the problem of topology optimization, for either discrete or continuum structures.

In this chapter we present an overview of the basic ingredients of the so-called homogenization method for finding the optimum layout of a linearly elastic structure. In this context the “layout” of the structure includes information on the topology, shape and sizing of the structure and the homogenization method allows for addressing all three problems simultaneously.

In the homogenization method described in chapter 1 the main goal of the optimization is to determine the spatial distribution of material. In order to parameterize material distribution by a continuum density variable and in order to avoid ill-posedness of the problem statement, an extra set of variables were introduced namely the variables defining local microstructure. These variables can be thought of as auxiliary variables, which of course should be chosen optimally as well. However, as we allow the material variables to vary from point to point it seems reasonable to distinguish between the optimization of the spatial distribution of material and the local optimal choice of microstructure. This perspective gives the inspiration for some alternative approaches to solving the homogenization topology problem.

In a number of recent papers on the optimization of structures, the distribution of material as well as the material properties themselves have been considered as design variables [25]. The goal of these studies is to formulate a structural optimization problem in a form that encompasses the design of structural material in a broad sense, while also encompassing the provision of predicting the structural topologies and shapes associated with the optimum distribution of the optimized materials. This goal is accomplished by representing as design variables the material properties in the most general form possible for a (locally) linear elastic continuum namely as the unrestricted set of positive semi-definite constitutive tensors.

Topology optimization of trusses in the form of grid-like continua is a classical subject in structural design. The study of fundamental properties of optimal grid like continua was pioneered by Michell, 1904, but this interesting field has only much later developed into what is now the well-established lay-out theory for frames and flexural systems [3], [8], [16]. The application of numerical methods to discrete truss topology problems and similar structural systems, which is the subject of this chapter, has a shorter history with early contributions in, for example, Dorn, Gomory and Greenberg, 1964 and Fleron, 1964 (see also [26]). The development of computationally efficient methods is not only of great importance for the truss topology problem in itself. It is likewise of interest for solving the reduced problems which arise in the study of simultaneous design of material and structure, as described in chapter 3.

The topology design problems treated so far have all been characterized by being compliance problems for structures with linearly elastic response. The reason for this is the simplifications that are implied for analysis and numerical algorithms. However, we have also seen that these problems in their apparent simplicity are fairly complicated to treat because of the large scale nature of the problems and because of the need to relax the continuum formulations.

The bibliography consists of two parts, bibliographical notes and the list of references at the end of the book.