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A significant limitation of the conventional level set method in topology optimization is that it can not create new holes in the design domain. Therefore, the topological derivative approach is proposed to overcome this problem. In this part of the thesis, we investigated the use of the topological derivative in combination with the level set method for topology optimization of solid structures. The topological derivative can indicate the appropriate location to create new holes so that the strong dependency of the optimal topology on the initial design can be alleviated. We also develop an approach to evolve the level set function by replacing the gradient item with a Delta function in the standard Hamilton-Jacobi equation. We find that this handling can create new holes in the solid domain, grow a structure from an empty domain, and improve the convergence rate of the optimization process. The success of our approach is demonstrated by several numerical examples. / Following those methods some numerical implementation issues are discussed, and numerical examples of 2D structural topology optimization problems of minimum compliance design are given and combined with a comparative study where the efficiency, convergence and accuracy of the present methods are highlighted. Finally, conclusions are given. / In the second part of this thesis, we implement another variational level set method, the piecewise constant level set (PCLS) method. This method was first proposed by Lie-Lysaker-Tai in the interface problem field for such tasks as image segmentation and denoising problems. In this approach, by defining a piecewise density function over the whole design domain, the sensitivity of the objective function in respect to the design variable, the level set surface, can be explicitly obtained. Thus, the piecewise density function can be viewed as a bridge establishing the relationship between the implicit level set function and the performance function defined on the design domain. This proposed method retains the advantages of the implicit level set representation, such as the capability of the interface to develop sharp corners, break apart and merge together in a flexible manner. Because the PCLS method is implemented by an implicit iteration differential scheme rather than solving the Hamilton-Jacobi equation, it is not only free of the CFL condition and the reinitialization scheme, but it is also easy to implement. These favorable properties lead to a great timesaving advantage over the conventional level set method. Two other meaningful advantages are the natural nucleation property with which the proposed PCLS method need not incorporate any artificial nucleation scheme and the dependence of the initial design is greatly alleviated. / In the third part of this thesis, we apply a parametric scheme by combining the conventional level set method with radial basis functions (RBFs). This method is introduced because the conventional level set function has no analytical form then the entire design domain must be made discrete in an artificial manner using a rectilinear grid for level set processing - often through a distance transform. The classical level set method for structural topology optimization requires a careful choice of an upwind scheme, extension velocity and a reinitialization algorithm. With the versatile tool, RBF, the original problem can be converted to a parametric optimization problem. Therefore, the costly Hamilton-Jacobi PDE solving procedure can be easily replaced by a standard gradient method or another mature conventional optimization method in the parameter space such as MMA, OC, mathematic programming and so on. / Keywords: structural optimization, level set method, topological derivative, radial basis functions, piecewise constant level set method. (Abstract shortened by UMI.) / The concept of structural optimization has been more and more widely accepted in many engineering fields during the past several decades, because the optimization can result in a much more reasonable and economical structure design with even less material consumption. / Wei Peng. / "June 2007." / Adviser: Yu Michael Wang. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0640. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 166-180). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Boundary variation method and material distribution method are distinct approaches for structural optimization. In the early days, due to the fact that boundary variation methods were generally not able to handle topological changes, it was applied only in shape optimization problems where the topology of initial design is fixed during optimization process. To enable topological changes that are essential to deliver major performance improvements, material distribution method was introduced in the work of Bendsoe and Kikuchi, and thereafter widely adopted in nearly all aspects of topology optimization. Recently a novel boundary variation method for topology optimization was developed based on level set method, in which topological changes is allowed for. In the thesis, we study the level set based boundary variation method and material distribution method for structure optimization problem. / Finally, we studied the semi-Lagrange scheme to solve the Hamilton-Jacobi equation in level set based boundary variation method. In level set method, the free boundary of a structure is optimized via solution of a Hamilton-Jacobi equation. The numerical stability condition in explicit schemes for discrete Hamilton-Jacobi equation severely restricts the time step. To improve the numerical efficiency, we employ a semi-Lagrange scheme to solve Hamilton-Jacobi equation. Therefore, much larger time steps can be obtained and the number of iterations before convergence is greatly reduced. / Firstly, we studied the minimum compliance optimization problem of thermoelastic structures. In this optimization problem, we find that the optimal structures given by the state-of-art material distribution method, SIMP i method, generally have large area of intermediate density values that are not feasible in practical engineering applications because of their poor manufacturability and high costs. Therefore, we apply level set based boundary variation method in the optimization problem. As numerical results show, the optimal structures obtained are well suited to engineering applications. / To sum up, we explore in this thesis the boundary variation method and material distribution method for structure optimization problem. Several meaningful results and conclusions are obtained. / We secondly studied the stress minimization problem. In practical applications the most important requirement on a structure is often the strength of structure which characterizes the resistance to failure. In stress minimization problem, the objective is to minimize the distribution of von Mises stress in a structure. Here, level set method gives a significant convenience for stress optimization, in particular, we need not to incorporate any stress amplification factor of material microstructure which would be an important issue in material distribution method. Moreover, in order to derive more control of maximum stress, we utilize the Kreisselmeier-Steinhauser function to aggregate stresses at each point in a structure into a single global function. / Xia, Qi. / "October 2007." / Adviser: Michael Yu Wang. / Source: Dissertation Abstracts International, Volume: 69-08, Section: B, page: 4993. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 102-111). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Thesis (Ph. D.)--University of Washington, 2007. / Vita. Includes bibliographical references (p. 101-110).

Evolutionary Structural Optimization (ESO) is a relatively new design tool used to improve and optimise the design of structures. In this method, a few elements of an initial design domain of finite elements are iteratively removed. Such a process is carried out repeatedly until an optimum design is achieved, or until a desired given area or volume is reached. In structural design, there is the demand for the development of design tools and methods that includes optimization. This need is the reason behind the development of methods like Evolutionary Structural Optimization (ESO). It is also this demand that this thesis seeks to satisfy. This thesis develops and examines the program named EVO, with the concept of structural optimization in the ESO process. Taking into account the stiffness and stress constraints, EVO allows a realistic and accurate approach to optimising a model in any given environment. Finally, in verifying the ESO algorithm&rsquo / s and EVO program&rsquo / s usefulness to the practical aspect of design, the work presented herein applies the ESO method to case studies. They concern the optimization of 2-D frames, and the optimization of 3-D spatial frames and beams with the prepared program EVO. Comparisons of these optimised models are then made to those that exist in literature.

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Thesis (Ph. D.)--University of Toronto, 2006. / Source: Dissertation Abstracts International, Volume: 67-06, Section: B, page: 3254. Includes bibliographical references.

Thesis (M.S.)--University of Delaware, 2008. / Principal faculty advisor: Robert G. Hunsperger, Dept. of Electrical & Computer Engineering. Includes bibliographical references.

Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1994. / Vita. Abstract. Includes bibliographical references (leaves 111-118). Also available via the Internet.

Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1993. / Vita. Abstract. Includes bibliographical references (leaves 139-151). Also available via the Internet.