An electric motor drive is a power-electronic based system that is used to precisely control the position, speed or torque developed by motor. With the growing complexity of drive systems and the expansion of the use of fast acting power-electronic controllers, computer simulation models are used instead of an explicit mathematical description of a complex system. The aim of this research is to study the use of the simulation based design method for advanced motor drives. The major problem for simulation of a drive system performance is the presence of both fast and slow dynamics in its response that result in relatively long time simulations with a small time step. Moreover, the simulation-based optimal design has a repetitive nature. Therefore, the simulation-based optimal design of a drive system is massively time consuming and requires extensive computing resources. In this research reduced intensity computer models are used to overcome this problem.

Strong gravitational lensing produces multiple distorted images of a background source when it is closely aligned with a mass distribution along the line of sight. The lensed images provide constraints on the parameters of a model of the lens, and the images themselves can be inverted providing a model of the source. Both of these aspects of lensing are extremely valuable, as lensing depends on the total matter distribution, both luminous and dark. Furthermore, lensed sources are commonly located at cosmological distances and are magnified by the lensing effect. This provides a chance to image sources that would be unobservable when viewed with conventional optics. The semilinear method expresses the source modeling step as a least-squares problem for a given set of lens model parameters. The blurring effect due to the point spread function of the instrument used to observe the lensed images is also taken into account. In general, regularization is needed to solve the source deconvolution problem. We use Krylov subspace methods to solve for the pixelated sources. These optimization techniques, such as the Conjugate Gradient method, provide natural regularizing effects from simple truncated iteration. Using these routines, we are able to avoid the explicit construction of the lens and blurring matrices and solve the least squares source optimization problem iteratively. We explore several regularization parameter selection methods commonly used in standard image deconvolution problems, which lead to previously derived expressions for the number of source degrees of freedom. The parameters that describe the lens density distribution are found by global optimization methods including genetic algorithms and particle swarm optimizers. In general, global optimizers are useful in non-linear optimization problems such as lens modeling due to their parameter space mapping capabilities. However, these optimization methods require many function evaluations and iterative approaches to the least squares problem are beneficial due to the speed advantage that they offer. We apply our modeling techniques to a subset of gravitational lens systems from the Sloan Lens ACS (SLACS) survey, and are able to reliably recover the parameters of the lens mass distribution with both analytical and regularized pixelated sources.

The trim-loss, or cutting stock, problem arises whenever material manufactured continuously or in large pieces has to be cut into pieces of sizes ordered by customers. The problem is so to organize the cutting as to minimize the amount of waste (trim-loss) resulting from it. Brown (1971) remarks that no practical solution method has been found for the generalized 2-dimensional trim-loss problem. This thesis discusses the applicability of heuristic search methods as solution techniques for this and other problems. Chapter 2 describes three types of combinatorial search method, state-space search, problem reduction, and branch-and-bound. There is a discussion of the ways in which heuristic information can be incorporated into these methods, and descriptions of the versions of the methods used in the work described in succeeding chapters. In the 1-dimensional trim-loss problem order lengths of some material such as steel bars must be cut from stock lengths held by the supplier. Gilmore and Gomory (1961, 1963) have formulated a mathematical programming solution of this problem, which also arises with the slitting of steel rolls, cutting of metal pipe and slitting of cellophane rolls. Their approach has been developed by Haessler (1971,1975) who is particularly concerned with problems arising in the paper industry. In the 1½-dimensional case the material is manufactured as a continuous sheet of constant width and it is required to minimize the length produced to satisfy orders for rectangular pieces. In the 2-dimensional case the orders are again for rectangular pieces, but here the stock is held as large rectangular sheets. In both cases there may be restrictions as to the way in which the material may be cut; the generalized problem in each case occurs when no such restrictions exist. The 1½-dimensional problem appears to be easier of solution than the 2-dimensional case since in the latter it is necessary not only to determine the relative positions of the required pieces in a cutting pattern, but also to partition the pieces into sets to be cut from separate stock sheets. A solution method for the easier problem might provide some insight into possible methods of solution of the more difficult. In chapter 3, a state-space search method for the solution of generalized 1½-dimensional problems where the number of pieces in the order list is fairly small and the dimensions are small integers is described. This method can be developed to solve 2-dimensional problems in which the order list is fairly small and the size of stock sheets variable but affecting the cost of the material. This development is described in chapter 4. A similarly structured state-space search can be used for finding solutions to optimal network problems. Such searches do not prove the solutions they find to be optimal, so it is of interest also to develop a method for finding solutions to the problems that proves them to be optimal. In chapter 5 the state-space search method is compared with one using branch-and-bound.problems change when large numbers of identical pieces are ordered, so a solution method with a different structure is required. Chapter 6 describes a problem reduction method for generalized 2-dimensional problems in which the order lists are large and the dimensions are small integers. Even when there are restrictions on the way in which the material may be cut, the presence of other constraints may make a mathematical formulation of the 2-dimensional trim-loss problem intractable, so again a heuristic solution method may be desirable. In a problem where there are sequencing constraints on the design of successive cutting patterns, problem reduction is again found to provide a useful solution method. This is described in chapter 7. Some conclusions about the efficacy and potential of the methods used are drawn in chapter 8. The remainder of the present chapter is concerned with setting the work described in this thesis in the context of other work on the same and related problems.

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The main subject of the research in this thesis is the study of conjugate gradient methods for optimization and the development of improved algorithms. After an introductory first chapter, Chapter 2 contains a background of numerical methods for optimization in general and of conjugate gradient-type algorithms in particular. In Chapter 3 we study the convergence properties of conjugate gradient methods and discuss Powell's (1983) counter example that proves that there exist twice continuously differentiable functions with bounded level sets for which the Polak-Ribiere method fails to achieve global convergence whereas the Fletcher-Reeves method is shown to be globally convergent, despite the fact that in numerical computations the Polak-Ribiere method is far more efficient than that of Fletcher-Reeves. Chapters 4 and 5 deal with the development of a number of new hybrid algorithms, three of which are shown to satisfy the descent property at every iteration and achieve global convergence regardless of whether exact or inexact line searches are used. A new restarting procedure for conjugate gradient methods is also given that ensures a descent property to hold and global convergence for any conjugate gradient method using a non negative update. The application of these hybrid algorithms and that of the new restarting procedure to a wide class of well-known test problems is given and discussed in the final Chapter "Discussions and Conclusions". The results obtained, given in the appendices, show that a considerable improvement is achieved by these hybrids and by methods using the new restarting procedure over the existing conjugate gradient methods and also over quasi-Newton methods.

This work focuses on an investigation of multi-modality in typical aerodynamic shape optimization problems and development of optimization algorithms that can find a global optimum. First, a classification of problems based on the degree of multi-modality is introduced. Then, two optimization algorithms are described that can find a global optimum in a computationally efficient manner: a gradient-based multi-start Sobol algorithm, and a hybrid optimization algorithm. Two additional algorithms are considered as well: a gradient-based optimizer and a genetic algorithm. Finally, we consider a set of typical aerodynamic shape optimization problems. In each problem, the primary objectives are to classify the problem according to the degree of multi-modality, and to select the preferred optimization algorithm for the problem. We find that typical two-dimensional airfoil shape optimization problems are unimodal. Three-dimensional shape optimization problems may contain local optima. In these problems, the gradient-based multi-start Sobol algorithm is the most efficient algorithm.

High-performance composite materials exhibit both anisotropic strength and stiffness properties. These anisotropic properties can be used to produce highly-tailored aircraft structures that meet stringent performance requirements, but these properties also present unique challenges for analysis and design. New tools and techniques are developed to address some of these important challenges. A homogenization-based theory for beams is developed to accurately predict the through-thickness stress and strain distribution in thick composite beams. Numerical comparisons demonstrate that the proposed beam theory can be used to obtain highly accurate results in up to three orders of magnitude less computational time than three-dimensional calculations. Due to the large finite-element model requirements for thin composite structures used in aerospace applications, parallel solution methods are explored. A parallel direct Schur factorization method is developed. The parallel scalability of the direct Schur approach is demonstrated for a large finite-element problem with over 5 million unknowns. In order to address manufacturing design requirements, a novel laminate parametrization technique is presented that takes into account the discrete nature of the ply-angle variables, and ply-contiguity constraints. This parametrization technique is demonstrated on a series of structural optimization problems including compliance minimization of a plate, buckling design of a stiffened panel and layup design of a full aircraft wing. The design and analysis of composite structures for aircraft is not a stand-alone problem and cannot be performed without multidisciplinary considerations. A gradient-based aerostructural design optimization framework is presented that partitions the disciplines into distinct process groups. An approximate Newton--Krylov method is shown to be an efficient aerostructural solution algorithm and excellent parallel scalability of the algorithm is demonstrated. An induced drag optimization study is performed to compare the trade-off between wing weight and induced drag for wing tip extensions, raked wing tips and winglets. The results demonstrate that it is possible to achieve a 43% induced drag reduction with no weight penalty, a 28% induced drag reduction with a 10% wing weight reduction, or a 20% wing weight reduction with a 5% induced drag penalty from a baseline wing obtained from a structural mass-minimization problem with fixed aerodynamic loads.

Consideration of the environmental impact of aircraft has become critical in commercial aviation. The continued growth in air traffic has come with increasing concerns and demands to reduce aircraft emissions and this has imposed new constraints on the de- sign and development of future airplane concepts. In this work, an environmental design framework has been developed to design and optimize aircraft for specific environmental metrics. Multidisciplinary design optimization is used to optimize aircraft by simulta- neously considering airframe, engine and mission design. The environmental metrics considered include fuel burn, landing-takeoff NOx and fuel burn per distance flown. Additional concepts such as the design of large aircraft for short ranges are also presented. Multi-objective optimization is also used to illustrate the tradeoffs between the various environmental objective functions.

Many space vehicles have been launched with large flexible components such as booms and solar panels. These large space structures (LSSs) have the potential to make attitude control unstable due to their lightly damped vibration. These vibrations can be controlled using a collection of control moment gyros (CMGs). CMGs consist of a spinning wheel in gimbals and produce a torque when the orientation of the wheel is changed. This study investigates the optimal distribution of these CMGs on LSSs for vibration suppression. The investigation considers a beam and a plate structure with evenly placed CMGs. The optimization allocates the amount of stored angular momentum possessed by these CMGs according to a cost function dependent on how quickly vibration motions are damped and how much control effort is exerted. The optimization results are presented and their effect on the motions of the beam and plate are investigated.

High-performance composite materials exhibit both anisotropic strength and stiffness properties. These anisotropic properties can be used to produce highly-tailored aircraft structures that meet stringent performance requirements, but these properties also present unique challenges for analysis and design. New tools and techniques are developed to address some of these important challenges. A homogenization-based theory for beams is developed to accurately predict the through-thickness stress and strain distribution in thick composite beams. Numerical comparisons demonstrate that the proposed beam theory can be used to obtain highly accurate results in up to three orders of magnitude less computational time than three-dimensional calculations. Due to the large finite-element model requirements for thin composite structures used in aerospace applications, parallel solution methods are explored. A parallel direct Schur factorization method is developed. The parallel scalability of the direct Schur approach is demonstrated for a large finite-element problem with over 5 million unknowns. In order to address manufacturing design requirements, a novel laminate parametrization technique is presented that takes into account the discrete nature of the ply-angle variables, and ply-contiguity constraints. This parametrization technique is demonstrated on a series of structural optimization problems including compliance minimization of a plate, buckling design of a stiffened panel and layup design of a full aircraft wing. The design and analysis of composite structures for aircraft is not a stand-alone problem and cannot be performed without multidisciplinary considerations. A gradient-based aerostructural design optimization framework is presented that partitions the disciplines into distinct process groups. An approximate Newton--Krylov method is shown to be an efficient aerostructural solution algorithm and excellent parallel scalability of the algorithm is demonstrated. An induced drag optimization study is performed to compare the trade-off between wing weight and induced drag for wing tip extensions, raked wing tips and winglets. The results demonstrate that it is possible to achieve a 43% induced drag reduction with no weight penalty, a 28% induced drag reduction with a 10% wing weight reduction, or a 20% wing weight reduction with a 5% induced drag penalty from a baseline wing obtained from a structural mass-minimization problem with fixed aerodynamic loads.