In the vast majority of papers on topology optimization, minimum compliance is used as the objective function. For the practicing engineer, other quantities, such as buckling, natural frequency, tip deflection, strength, inter-story drift, cost and combinations thereof are of interest during the design of a structure. This section reviews current methods available in the literature, specifically for buckling and multi-objective optimization.
4.2.1 Buckling
Typical topology optimization results often contain very slender members; therefore, the work by [122] was one of the first in which a buckling load criterion was considered to address this issue. This was later expanded upon by [121], in which the homogenization method was used for the optimal reinforcement design of portal (single-story) frames under a buckling load. Similarly, the optimal design of plate reinforcements using a non-smooth buckling load criterion was explored in the work by [67]. Later, [143, 142, 181] applied the use of buckling as an optimization criterion for the design of sparse, long-span bridges. Other contributors to the continuum topology optimization problem with buckling as an objective are [211, 163, 29]. Extensions for buckling problems in materials can be found in the paper by [123]. To make design problems more realistic, geometric nonlinearities were incorporated in the stability problem for “perfect” and “imperfect” structures in [91] by directly determining the critical load factor and including it as an inequality constraint.
It should be noted that many numerical issues are encountered in the buckling optimiza- tion problem. One of the main issues associated with topology optimization for buckling is the presence of localized eigenmodes. To eliminate this effect, many techniques artificially re-
move “void” elements from the optimization problem; however, this may produce erroneous solutions since when an element is removed, it cannot re-enter the optimization problem [135]. An alternative methodology to eliminate localized eigenmodes in low density areas for cases in which the problem is not formulated as reinforcement of an existing structure is presented in [135]. Other numerical issues include the case of multiple eigenvalues, which are typically present in symmetric structures and non-smoothness of the eigenvalues [130]. For a review of multiple eigenvalues in structural optimization problems and how to treat them, the reader can refer to the work by [164]. In the implementation proposed here, we include an adapted version of the method suggested by [129] to stabilize the structure and eliminate problems with local effects. Furthermore, in this work, we assume simple (non-repeated) eigenvalues, though differentiability issues associated with repeated eigenvalues might be avoided by reducing the design space in accordance with structural symmetry [96].
4.2.2 Multi-objective optimization
Topology optimization based on a single objective function (i.e. minimum compliance, de- flection) can be quite valuable in solving a variety of engineering problems. However, in the context of buildings, it is more significant to optimize a structure for several engineering criteria together, such as minimum compliance, natural frequency or period, critical buck- ling load, and/or tip deflection. This chapter discusses the development of a multi-objective framework involving these design criteria for topology optimization of civil engineering struc- tures, with particular focus on high-rise buildings.
Previous work on multi-objective optimization can be found in [40, 155, 98]. An attempt towards multi-objective optimization for structures is made in [98], however, this paper uses evolutionary algorithms to optimize for structural compliance, natural frequency and mass subject to constraints on the stress levels. This methodology is not suitable for the gradient- based formulation applied in this research.
In general, multi-objective problems must be solved as a single-objective problem, which can be formulated using a variety of techniques, including the weighted-sum method, weighted min-max method, weighted global criteria, weighted product method, and exponential weighted criterion. [109, 12]. However, one of the major challenges of multi-objective optimization techniques lies in the process of selecting the vector of weights. These weights can be selected by ranking methods (where the importance determines the order of the objective) [202], categorization methods (grouping objectives according to importance), eigenvalue methods (based on comparison between objectives) [157], rating methods (assigning relative impor- tance to objectives), and/or ratio questioning methods (pair-wise comparison of objectives). In this work, the weighted-sum method was chosen to formulate the multi-objective opti- mization problem for its simplicity and the flexibility for the designer to weight the criteria according to the problem at hand. For more details on these methods and their implemen-
tation, the reader can refer to [109, 82, 190].
Since it is very rare for the multi-objective optimization problem to reach the absolute minimum of each objective function simultaneously, we must discuss the notion of Pareto optimal solutions in the context of this work. As given in [109], the optimal solution for a multi-objective optimization problem (using a weighted sum or norm of the vector of objective functions) may not be Pareto optimal (i.e. there may be other points that improve at least one of the objectives). However, the work of [203] showed that if all of the weights are positive, the minimization of the above problem statement is sufficient for Pareto optimality. This concept is described in more detail later in the numerical examples.
Other challenges of multi-objective optimization methods include: (i) an accurate, com- plete set of Pareto solutions many not be given by arbitrarily varying the weights [51], (ii) a priori selection of weights does not imply an acceptable final solution (i.e. one may need to reevaluate the weights throughout the process) [109], and (iii) a weighted sum approach limits the set of solutions to obtain points only on a convex Pareto optimal set [51]. It is important to note that approaches other than the weighted sum may be substituted in the following formulation as deemed appropriate by the designer and the problem under consideration.