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Structural optimization. – The interaction between form and mechanics. (English) Zbl 0943.74052

The authors attempt to offer both a review and an insight into some recent ideas concerning lay-out and a choice of topology for optimization. They start with the correct optimal design of cantilever beam given by Galileo in 1638 in publication “Discorsi e demonstranzioni matematiche intorno a due nuove scienze\(\dots\)” Using variational arguments, the authors give a short proof that the Galileo’s solution was correct. They note that only very few design problems can be solved by classical analytical arguments. Techniques commonly used at present time include: a) discretization combined with local approximation technique, such as spline approximations, Bezier interpolation schemes, etc. for determining shape parameters; b) discretization relying primarily on finite model representation to determine gradients with respect to optimization parameters; c) the use of continuous mathematical model with some constraints corresponding to physical limitations (including laws of physics), and other constraints dictated by cost or arbitrary limitations. Generally, analytic problems so formulated are too difficult to solve. Thus, various approximations are considered, and iterative schemes are suggested. Frequently design changes alter the validity of constitutive equations, or of constraints, and an adaptive process has to be initiated. Sensitivity analysis becomes of utmost importance. The reviewer comments that in “real” problems one is rarely interested in attaining an exact optimum. In fact, in many cases it may not be a good idea to do so. W. Prager made a wise remark: “Iterate a few times than just quit.”
The authors continue to look at the Galileo’s problem with the height of the beam regarded as variable, and apply finite element method to compute the beam response. The nodal displacements are obtained by solving a system of linear equations \(Ku=P\), where \(K\) is the stiffness matrix, \(u\) is the vector of generalized displacements, and \(P\) is the vector of generalized loads. In a manner similar to Castigliano’s theorem, the derivatives of equilibrium condition with respect to the design variables produce generalized displacements, which could be associated with fictitious loads. In the opinion of the reviewer, this idea has not been yet sufficiently developed.
Even at this point the authors note that the near-optimal solutions so derived have been tied to assumptions about the topology of the designed shape. Maxwell was first who established fundamental principles for the lay-out design of structures. The beautiful curves of A. G. M. Michell (circa 1900) inspired much research, despite their apparent practical uselessness. The present techniques of changing topology consists of inserting small holes, or “bubbles” of variable shape. Some sophisticated programs use a binary characteristic function with 0-1 optimization of a mesh imposed on available space. Till the technology progressed, such efforts have been expensive in the past, and suffered from non-uniqueness of the final outcome. Regularization (or relaxation) of such schemes looks at such 01-1 porous (at the micro-scale) material, and averages local oscillations of material distribution. R. V. Kohn and G. Strang produced a set of criteria for weak convergence to the solution of originally posed optimization problem.
Here the authors discuss the problems for orthotropic materials, including rank-2 materials (strong and weak material laminates). The authors proceed to give excellent examples of reducing the design process to several adaptive stages. At each stage the results of the previous optimization are used as a starting point in a refined mesh of 0-1 cells. Prager’s predictions still work. The optimal design may produce hard to identify large quantity of porous material, and the layout of homogeneous material following such design is tricky. However, settling for near-optimal solutions, one can generally identify a clear 0-1 shape on a macroscopic scale.
The present overview of topology optimization was badly needed, and the authors should be congratulated for this well-written and expert presentation.

MSC:

74P15 Topological methods for optimization problems in solid mechanics
74E05 Inhomogeneity in solid mechanics
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74S05 Finite element methods applied to problems in solid mechanics
70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
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