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We consider the problem of designing lightweight load-bearing frame structures with additive manufacturability constraints. Specifically, we focus on mathematical programming approaches to finding exact globally optimal solutions, given a pre-specified discrete ground structure and continuous design element dimensions. We take advantage of stiffness matrix decomposition techniques and expand on some of the existing modeling approaches, including exact mixed-integer nonlinear programming and its mixed-integer linear programming restrictions. We propose a (non-convex) quadratic formulation using semi-continuous variables, motivated by recent progress in state-of-the-art quadratic solvers, and demonstrate how some additive-specific restrictions can be incorporated into mathematical optimization. While we show with numerical experiments that the proposed methods significantly reduce the required solution time for finding global optima compared to other formulations, we also observe that even with these new techniques and advanced computational resources, discrete modeling of frame structures remains a tremendously challenging problem.

Traffic crashes and congestion generate high social and economic costs, yet traditional traffic monitoring methods, such as police patrols, fixed cameras, and helicopters, are costly, labor-intensive, and limited in spatial coverage. This paper presents a novel Drone Routing and Scheduling with Flexible Multiple Visits (DRSFMV) framework, an optimization model for planning drone-based highway monitoring under realistic operational constraints, including battery limits, variable monitoring durations, recharging at a depot, and target-specific inter-visit time limits. A mixed-integer nonlinear programming (MINLP) model and a linearized version (MILP) are presented to solve the problem. Due to the NP-hard nature of the underlying problem structure, a heuristic solver, Hexaly, is also used. A case study using real traffic census data from three Southern California counties tests the models across various network sizes and configurations. The MILP solves small and medium instances efficiently, and Hexaly produces high-quality solutions for large-scale networks. Results show clear trade-offs between drone availability and time-slot flexibility, and demonstrate that stricter revisit constraints raise operational cost.

We consider the problem of designing lightweight, load-bearing planar frame structures for additive manufacturing, which can be formulated as a nonlinear, non-convex mathematical programming problems. Even using state-of-the-art commercial solvers, exact methods are only capable of solving small unrealistic instances (with very few variables). In this paper, we develop a heuristic method which is fast and capable of solving the design problem for larger-scale, weight-optimized, planar frame structures for additive manufacturing. The approach explicitly considers manufacturability constraints stemming from the use of additive manufacturing technology and leverages the problem structure imposed by these constraints. The proposed heuristic method is based on iteratively resolving a relaxed master problem on a reduced ground structure. The approach differs from the existing methods in two important aspects. First, we consider planar frame optimization master problem directly (instead of simpler but less relevant truss optimization). Secondly, we employ both element and node addition, which allows us to enforce additive manufacturability constraints without using binary variables (and hence, avoiding the need for computationally expensive integer programming).

Lightweight structures have many applications in different engineering areas, such as automotive, aerospace, or medical industries, among many others. Optimal design of lightweight structures deals with finding the most economical distribution of the material in the design domain. This concept becomes even more important when additive manufacturing (AM) is considered for fabrication of the parts, since it allows for exceptional freedom in the design process. In this dissertation, we consider the design optimization problem for additively manufactured planar frame structures. We specifically consider three different optimization approaches in tackling the problem of finding lightest planar frames which can withstand the external loads. We apply exact optimization methods in Chapter 3, where we propose a novel mixed integer quadratically constrained optimization model for the problem and compare its performance to the existing models from the literature. We then propose a problem-specific heuristic method in Chapter 4, which is capable of solving large-scale problems that couldn’t be handled by exact optimization methods. This heuristic method is a combination of a member-node adding approach and nonlinear optimization, in which the solving process starts from a version of ground structure with a minimal number of elements and then gradually includes elements with the most promising contribution in reducing the stress in the structure. In Chapter 5 we test the ability of metaheuristics, specifically Genetic Algorithm (GA) to solve this mathematical optimization problem. In the proposed hybrid approach, we combine GA with nonlinear optimization. To this end, we designed a new encoding of the candidate solutions together with the GA operators that in addition to the stochastic nature of the GA in solving combinatorial problems, combined with the deterministic exactness of nonlinear optimization provides a novel way to solve the design problem. We conclude the dissertation by providing the main findings and future research in Chapter 6.