In this chapter, we demonstrate that (a) substituting the vector of eigenvalues of a symmetric n x n matrix into a convex permutation symmetric function of n real variables results in a convex function of the matrix, and (b) that if g is a convex function on the real axis, and G is the set of symmetric matrices of a given size with spectrum in the domain of g, then G is a convex set, and when X is a matrix from G, the trace of the matrix g(X), is a convex function of X; here g(X) is the matrix acting at a spectral subspace of X associated with eigenvalue v as multiplication by g(v); both these facts will be heavily used when speaking about cone-convexity is chapter 21.

In this chapter, we (a) outline the subject and the terminology of mathematical and convex programming, (b) introduce the Slater and relaxed Slater conditions and formulate the Convex Theorem on the Alternative -- the basis of Lagrange duality theory in convex programming, (c) introduce the notions of cone-convexity and of the convex programming problem in cone-constrained form, thus extending the standard mathematical programming setup of convex optimization, and (d) formulate and prove the Convex Theorem on the Alternative in cone-constrained form, justifying, as a byproduct, the standard Convex Theorem on the Alternative.

In this chapter, we derive the standard first- and second-order necessary/sufficient conditions for local optimality of a feasible solution to a (possibly nonconvex) mathematical programming problem. We conclude the chapter by illustrating these on the S-Lemma.

In this chapter, we (a) discuss the notion of lower semicontinuity of a function and demonstrate that functions with this property have closed epigraphs, (b) show that the pointwise supremum of a family of lower semicontinous functions is lower discontinuous, (c) demonstrate that a proper lower semiconscious convex function is the pointwise supremum of the affine minorants of the function, (d) introduce the notion of a subgradient and the subdifferential of a convex function at a point and demonstrate existence of subgradients at points from the relative interior of the function’s domain, (e) outline elementary rules of subdifferential calculus, and (f) establish basic properties of the directional derivatives of convex functions and the connection between directional derivatives and subdifferentials.

In this chapter, we extract from the results of Chapter 3 the basic theory of finite systems of linear inequalities - Farkas’ Lemmas, General Theorem on the Alternative, certificates for feasibility/infeasibility of polyhedral sets, and linear programming Duality Theorem.

In this chapter, we (a) introduce the notion of Legendre transformation of a proper convex function, (b) establish basic properties of the Legendre transform, in particular, demonstrate that the transform of a proper lower semicontinuous convex function is itself a proper lower semicontinous convex function and that its Legendre transformation is the original function, (c) demonstrate that the set of minimizers of a proper lower semicontinuous convex function is the subdifferential, taken at the origin, of the function’s Legendre transform, and (d) derive the Young, Holder, and moment inequalities and discuss dual (a.k.a. conjugate) norms.

In contrast to the previous two chapters, which detailed L-system topology optimization approaches that interpret gene-informed rules into a complex set of layout-building instructions, this chapter introduces a grammar-to-layout approach known as the Arrangement L-system (ALS). Here, developmental operations that mimic the processes of cellular division, growth, and movement are directly informed by the genes and then iteratively applied to an iteratively changing topological layout that, once complete, represents an individual. The differences between formulations of the L-system, parameterized L-system, and ALS are discussed; examples of how the cellular division processes are used to develop a topological layout are provided; and extensions to the ALS such as directed search cellular dynamics and cellular division via the two-point topological derivative are detailed. The applicability of the ALS to a variety of structural design problems will be demonstrated, and it will be shown that this approach compares favorably with both conventional topology optimization methods discussed throughout this work as well as the graph-based SPIDRS approach introduced in the previous chapter.

Topology optimization is a powerful tool that, when employed at the preliminary stage of the design process, can determine potential structural configurations that best satisfy specified performance objectives. This chapter explores both the different classifications of topology optimization methodologies and their implementation within the design process, specifically highlighting potential areas where such techniques may fall short. This motivates a discussion on the relevance of a bioinspired approach to topology optimization known as EvoDevo, where topologies developed by interpreting instructions from a Lindenmayer system (L-system) encoding are evolved using a genetic algorithm. Such an approach can lend itself well to multiobjective design problems with a vast design space and for which users have little/no experience or intuition.

To this point, the proposed L-system topology optimization methods have been considered in the context of benchmark structural topology optimization problems, as such problems afford an opportunity for comparison to both other topology optimization methodologies and mathematically proven optimal or ideal solutions. However, the motivation behind the development of these approaches stems from the need for preliminary design method capable of considering complex multiobjective problems involving multiple physics for which the user may not have an intuition. This chapter briefly summarizes several multiphysical problems that have been approached using L-system topology optimization, including fluid transport, heat transfer, electrical, and aeroelastic applications. By no means an exhaustive survey, these examples are intended to provide an overview of potential applications and hopefully provoke opportunities for future efforts.

To address the need for an inherently multiobjective preliminary design tool, this chapter introduces a heuristic alternative to the conventional topology optimization approaches discussed in the previous chapter. Specifically, a parallel rewriting system known as a Lindenmayer system (L-system) is used to encode a limited number of design variables into a string of characters which, when interpreted using a deterministic algorithm, governs the development of a topology. The general formulation of L-systems is provided before discussing how L-system encodings can be interpreted using a graphical method known as turtle graphics. Turtle graphics constructs continuous, straight line segments by tracking the spatial position and orientation of a line-constructing agent, leading to the creation of branched structures that mimic those found in numerous natural systems. The performance of the proposed method is then assessed using simple, well-known topology optimization problems and comparisons to mathematically known optimal or ideal solutions as well as those generated using conventional topology optimization methodologies.