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.

Another interesting area we investigate is topology optimization with geometric control. Our initial research interests in topology optimization with geometric width control were motivated by the challenge of how to get more manufacturable compliant mechanism designs for MEMS devices. Considering MEMS fabrication technologies, say LIGA, it is natural that designs with specified feature width are more preferable with respect to those with free-form geometries. We propose a variational approach to this problem. A novel quadratic energy functional is employed to govern the geometric feature width of the design. This geometry describing functional is added to the performance-describing functionals. In this way both the performance and the geometric width of the design are optimized simultaneously. The preliminary results show that this method is capable of generating strip-like (or beam) designs with specified feature width, which is a highly desirable characteristic and uniquely distinguishes the proposed method. / Compliant mechanisms are involved in many applications both in the macro and in the micro world. But for a long time, the design procedure of compliant mechanisms was rather a handicraft than a technology. The conventional way is made on an ad hoc basis which to a large extent depends on the designers intuition, experience and inspiration. The limitations of such a trial-and-error approach are obvious: it is not always guaranteed to work, especially when the design is very complicated or when topology and multi-material problems are taken into account. The practical design and application of compliant mechanisms are in need of a systematic approach to create conceptual design. Here, we take a level-set-based new approach to solve this problem. / Considering the actual requirements on reliability, we also investigate how to get conceptual designs with distributed compliance, which is the core part of this thesis. We find the intrinsic defects in the widely used spring model and prove that it will inevitably cause designs with both large output displacements and low strain energies. We will show low strain energy does not guarantee high stiffness. To evenly distribute the compliance, we propose a new method considering the "characteristic stiffness" at interested points. In this way, the strength (stiffness) at the output port of the system is involved into the objective function and optimized directly. This new method is applied to some benchmark examples of both structure optimization and compliant mechanism optimization to validate its performance. / In our proposed method, the compliant mechanism design problem is recast as an infinite dimensional optimization problem, where the design variable is the geometric shape of the compliant mechanism and the goal is to find a suitable shape in the admissible design space so that the objective functional can reach a minimum. The geometric shape of the compliant mechanism is represented as the zero level set of a one-higher dimensional level set function, and the dynamic variations of the shape are governed by the Hamilton-Jacobi partial differential equation. The application of level set methods endows the optimization process with the particular quality that topological changes of the boundary, such as merging or splitting, can be handled in a natural fashion. By making a connection between the velocity fields in the Hamilton-Jacobi partial differential equation with the shape gradient of the objective functional, we go further to transform the optimization problem into that of finding a steady state solution of the partial differential equation. / Our research follows the route from ease to difficulty, reflecting our understanding of the compliant mechanism design problem at different stages. The first problem addressed in this thesis is how to maintain the structural connectivity during the topology optimization process. De facto hinges are known to be a fairly typical phenomenon in topology optimization of compliant mechanisms; they represent highly localized compliance regions. A most adverse side effect caused by de facto hinges is that they are prone to cause a structurally disconnected design, especially to that with a low volume ratio. To solve this problem, a digital topological connectivity scheme is integrated within the level set model, which ensures connectivity while allows topology changes. This is our first step in the research process. / Chen, Shikui. / "January 2007." / Adviser: Michael Yu Wang. / Source: Dissertation Abstracts International, Volume: 68-08, Section: B, page: 5513. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 149-162). / 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 (M.Sc.). / Written for the School of Computer Science. Title from title page of PDF (viewed 2008/01/15). Includes bibliographical references.

Our method can be naturally applied in the application of manga stylization rendering. We propose a novel colorization technique that propagates color over manga regions exhibiting pattern-continuity as well as intensity-continuity. The proposed method works effectively on colorizing black-and-white manga which contains intensive amount of strokes, hatching, halftoning and screening. Once the user scribbles on the drawing, a local, statistical based pattern feature obtained with Gabor wavelet filters is applied to measure the pattern-continuity. The boundary is then propagated by the level set method that monitors the pattern-continuity. Regions with open boundaries or multiple disjointed regions with similar patterns can be sensibly segmented by a single scribble. With the segmented regions, various colorization techniques can be applied to replace colors, colorize with stroke preservation, or even convert pattern to shading. Based on the observation of the manga features, we then propose a framework to generate manga-style backgrounds from real photographs. It frees manga artists from the tedious and time-consuming background production. To mimic how manga artists produce the tidy background, it composes of two major steps, the line drawing and the screen laying. A line importance model is proposed to simplify and embolden lines in a stylish way. During the screen laying, texture analysis is utilized to automatically match the regions in photographs with screens in the database. Two types of screening mechanisms are proposed for matching shading & texture as well as the high-level structures. / Segmentation has always been a crucial and challenging topic in various computer vision and graphics applications. Information from image features can be incorporated into image segmentation techniques to make the process more accurate and robust. This dissertation focuses on encoding these feature information into the image processing procedures, including image segmentation and manga stylization rendering. / The fundamental part of this dissertation includes the discussion on the construction of the speed function, which is important in applying the curve-evolution based image segmentation. We firstly investigate the curvature term in the speed function, and then show how to transform the image segmentation problem into an interface propagating problem. We propose two formulations to enhance the speed function in level set methods, in order to tackle the segmentation problem of tagged MR images. First, a relaxation factor is introduced, aiming at relaxing the boundary condition when the boundary is unclear or blurry. Second, in order to incorporate human visual sensitive information from the image, a simple and general model is introduced to incorporate shape, texture and color features. By further extending this model, we present a unified approach for segmenting and tracking of the high-resolution color anatomical Chinese Visible Human (CVH) data. The underlying relationship of these two applications relies on the proposed variational framework for the speed function. Our proposed method can be used to segment the first slice of the volume data. Then based on the extracted boundary on the first slice, our method can also be adapted to track the boundary of the homogeneous organs among the subsequent serial images. In addition to the promising segmentation results, the tracking procedure requires only a small amount of user intervention. / Qu Yingge. / "July 2007." / Advisers: Heng Pheng Ann; Wong Tien Tsin. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0433. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 107-120). / 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. / Abstract in English and Chinese. / School code: 1307.

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2004. / Includes bibliographical references (leaves 65-69). Also available in electronic version. Access restricted to campus users.

Thesis (Ph. D.)--University of Iowa, 2007. / Supervisors: Ching-Long Lin, Larry J. Weber. Includes bibliographical references (leaves 153-162).

A finite element (FE) based level set method is proposed for structural topology optimization problems in this thesis. The level set method has become a popular tool for structural topology optimization in recent years because of its ability to describe smooth structure boundaries and handle topological changes. There are commonly two stages in the optimization process: the stress analysis stage and the boundary evolution stage. The first stage is usually performed with the finite element method (FEM) while the second is often realized by solving the level set equation with the finite difference method (FDM). The first motivation for developing the proposed method is the desire to unify the techniques of both stages within a uniform framework. In addition, there are many problems involving irregular design domains in practice, the FEM is more powerful than the FDM in dealing with these problems. This is the second motivation for this study. / Numerical examples are involved in this thesis to illustrate the reliability of the proposed method. Problems on both regular and irregular design domains are considered and different meshes are tested and compared. / Solving the level set equation with the standard Galerkin FEM might produce unstable results because of the hyperbolic characteristic of this equation. Therefore, the streamline diffusion finite element method (SDFEM), a stabilized method, is employed to solve the level set equation. In addition to the advantage of simplicity, this method generates a system of equations with a constant, symmetric, and positive defined coefficient matrix. Furthermore, this matrix can be diagonalized by virtue of the lumping technique in structural dynamics. This makes the cost in solving and storing quite low. It is more important that the lumped coefficient matrix may help to improve the stability under some circumstance. / The accuracy of the finite element based level set method (FELSM) is compared with that of the finite difference based level set method (FDLSM). The FELSM is a first-order accurate algorithm but we prove that its accuracy is enough for structural optimization problems considered in this study. Even higher-order accurate FDLSM schemes are used, the numerical results are still the same as those obtained by FELSM. It is also shown that if the Courant-Friedreichs-Lewy (CFL) number is large, the FELSM is more robust and accurate than FDLSM. / The reinitialization equation is also solved with the SDFEM and an extra diffusion term is added to improve the stability near the boundary. We propose a criterion to select the factor of the diffusion term. Due to numerical errors and the diffusion term, boundary will drift during the process of reinitialization. To constrain the boundary from moving, a Dirichlet boundary condition is enforced. Within the framework of FEM, this enforcement can be conveniently preformed with the Lagrangian multiplier method or the penalty method. / Velocity extension is discussed in this thesis. A natural extension method and a partial differential equation (PDE)-based extension method are introduced. Some related topics, such as the "ersatz" material approach and the recovery of stresses, are discussed as well. / Xing, Xianghua. / Adviser: Michael Yu Wang. / Source: Dissertation Abstracts International, Volume: 71-01, Section: B, page: 0628. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 102-113). / 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 Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese.

Lui, Fung Yee. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 83-92). / Abstracts in English and Chinese. / Acknowledgement --- p.iii / Abbreviation --- p.xii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.2 --- Related Work --- p.6 / Chapter 1.2.1 --- Parametric Optimization Method and Radial Basis Functions --- p.6 / Chapter 1.3 --- Contribution and Organization of the Dissertation --- p.7 / Chapter 2 --- Level Set Method for Structure Shape and Topology Optimization --- p.8 / Chapter 2.1 --- Primary Ideas of Shape and Topology Optimization --- p.8 / Chapter 2.2 --- Level Set models of implicit moving boundaries --- p.11 / Chapter 2.2.1 --- Representation of the Boundary via Level Set Method --- p.11 / Chapter 2.2.2 --- Hamilton-Jacobin Equations --- p.13 / Chapter 2.3 --- Numerical Techniques --- p.13 / Chapter 2.3.1 --- Sign-distance function --- p.14 / Chapter 2.3.2 --- Discrete Computational Scheme --- p.14 / Chapter 2.3.3 --- Level Set Surface Re-initialization --- p.16 / Chapter 2.3.4 --- Velocity Extension --- p.16 / Chapter 3 --- Structure Topology Optimization with Discrete Level Sets --- p.18 / Chapter 3.1 --- A Level Set Method for Structural Shape and Topology Optimization --- p.18 / Chapter 3.1.1 --- Problem Definition --- p.18 / Chapter 3.2 --- Shape Derivative: an Engineering-oriented Deduction --- p.21 / Chapter 3.2.1 --- Sensitivity Analysis --- p.23 / Chapter 3.2.2 --- Optimization Algorithm --- p.28 / Chapter 3.3 --- Limitations of Discrete Level Set Method --- p.30 / Chapter 4 --- RBF based Parametric Level Set Method --- p.32 / Chapter 4.1 --- Introduction --- p.32 / Chapter 4.2 --- Radial Basis Functions Modeling --- p.33 / Chapter 4.2.1 --- Inverse Multiquadric (IMQ) Radial Basis Functions --- p.38 / Chapter 4.3 --- Parameterized Level Set Method in Structure Topology Optimization --- p.39 / Chapter 4.4 --- Parametric Shape and Topology Structure Optimization Method with Radial Basis Functions --- p.42 / Chapter 4.4.1 --- Changing Coefficient Method --- p.43 / Chapter 4.4.2 --- Moving Knot Method --- p.45 / Chapter 4.4.3 --- Combination of Changing Coefficient and Moving Knot method --- p.46 / Chapter 4.5 --- Numerical Implementation --- p.48 / Chapter 4.5.1 --- Sensitivity Calculation --- p.48 / Chapter 4.5.2 --- Optimization Algorithms --- p.49 / Chapter 4.5.3 --- Numerical Examples --- p.52 / Chapter 4.6 --- Summary --- p.65 / Chapter 5 --- Conclusion and Future Work --- p.80 / Chapter 5.1 --- Conclusion --- p.80 / Chapter 5.2 --- Future Work --- p.81 / Bibliography --- p.83

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Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Computer Science." Includes bibliographical references (p. 63-66).