基於計算機仿真的優化設計方法如今已成為產品設計的重要工具之一。其最主要特點包括縮短產品開發週期，降低物理實驗成本，保證產品質量以及利用科學方法推動設計創新等。與此同時，計算機輔助設計，仿真，優化的一體化策略也得到了學術界和工業界的廣泛關注。許多新的研究成果都致力於提高以往算法的效率和適用性。 / 基於水平集的形狀和拓撲優化算法是設計輕量化連續結構體的強有力的工具之一。相比於基於有限單元網格的材分佈算法，前者能夠更清楚地表a達所設計結構的幾何邊界和特徵。這個優勢使得該算法能更好的與計算機輔助幾何設計方法相結合，例如構造立體幾何法 (Constructive Solid Geometry)。另外，最新的研究表明，基於水平集的幾何表達方法能夠很好地與擴展有限元分析(Extended Finite ElementAnalysis) 相結合，實現高效的仿真優化計算。這種結合的主要特點包括統一的數據表達，高精度的結構分析和優化計算，以及優化過程中無需重新劃分有限單元網格等。 / 近年來，儘管水平集結構優化算法得到了廣泛的發展，許多基於該方法的應用也層出不窮，但仍有一些相對實際的問題亟待解決。例如，如何提高水平集優化效率，如何增強該方法的設計能力以及適用性等。本論文致力於研究上述問題并提出了一些實用的新方法。 / 首先，我們結合semi-Lagrangian 數值方法和最優化線搜索算法，提出了一種新的水平集結構優化方法。在求解水平集方程的過程中，semi-Lagrangian 方法允許相對較大的時間步長並且無需受CFL(Courant-Friedrichs-Lewy) 條件的限制。基於這個特點，本文提出的最優化線搜索策略能夠自適應地計算出每一步的最佳時間步長，并充分考慮拓撲優化過程中的實際特徵。實驗表明，本算法能夠有效地減少優化迭代次數，同時降低整體優化計算的時間。另外，我們還提出了一種新的敏度計算方法。其思想與有限維度問題中的共軛梯度法相似。實驗表明該方法能夠替代廣泛運用于水平集優化的最速下降法，得到滿意的優化結果。 / 其次，我們提出了一種在水平集結構優化過程中設計幾何特徵的方法。幾何特徵指模型中包含加工、組裝或者特定功能信息的簡單幾何形狀。在優化設計中加入特徵設計功能有顯著的實際意義。本文中，我們結合水平集方法和構造立體幾何法的優勢，首先在建模時分離出具有特徵的幾何元素體（特徵體）和包含自由邊界的幾何元素體（自由體），然後分別在各自的設計策略下實現同步的優化計算。對於特徵體的設計，我們利用仿射變換驅動幾何形狀的改變并時刻保持關鍵的幾何特徵。其中，仿射變換的速度場通過擬合連續體設計的速度場得到，實際變換則採用粒子水平集方法。另一方面，自由體的形狀和拓撲通過標準的水平集方法進行優化設計。實驗表明，該方法能夠在結構形狀及拓撲優化過程中，保持並設計包含不用實際工程信息的幾何特徵，實現了真正意義上的含有幾何特徵的最優結構設計。本文中，我們將用數個二維和三維的算例來說明該方法的設計潛力和適用性。 / 最後，我們討論并實現了基於自適應水平集方法的三維結構優化算法。該方法在計算過程中結合了顯示和隱式幾何表達的雙重優點。首先，我們用八叉樹網格來表示隱式水平集模型以及其對應的二維流型三角片網格模型。在優化迭代過程中，隱式水平集模型的邊界演化採用semi-Lagrangian 方法。其中，有向距離函數通過直接計算當前顯示模型得到，而非插值。之後，新的三角片網格模型從更新的距離場中提取出來，作為下一步的輸入。這種混合表達和自適應的網格策略不僅實現了窄帶計算，而且能夠很好跟擴展有限元分析方法相結合。此外，我們在計算過程中還提出并加入了一種能夠保持幾何特徵和模型表面拓撲的網格簡化算法以提高計算效率。值得注意的是，這種自適應水平集方法成功地在結構優化過程中植入了幾何模型處理方法。這為進一步發展水平集結構優化提供了一個新的方向。 / In modern product design practice, adopting simulation based optimization has become a standard procedure to reduce experimental cost, shorten development time, assure product quality and promote innovation. Both industries and academics have put great efforts in exploring new approaches to integrate computer aided design (CAD), simulation and optimization processes in an effective and truly applicable way. / For general lightweight structural design of continuum, the level set method is a promising tool for shape and topology optimization. Compared to traditional approaches such as Finite Element (FE) mesh based shape optimization and material based topology optimization, the level set based method excels in its flexibility in handling both shape and topological change as well as the capability in representing a clear structural geometry. The later advantage allows for a intuitive integration of computer aided design and engineering (CAD/CAE), because the level set model can be easily extended to constructive solid geometry, which is a fundamental geometry description of CAD. Meanwhile, recent research progress indicates that coupling level set method with extended finite element (XFEM) analysis for simulation based design possesses tremendous values, such as data compatibility, free of re-meshing and good accuracy. / Although the basic theory of level set based structural optimization has been well established and many applications have been reported in the last decade, the realm is still under investigation for a number of practical issues, such as to improve computational efficiency, optimal search effectiveness, design capability and industrial applicability. This thesis presents some recent research progress and novel techniques towards these common goals. / Firstly, an efficient and numerically stable semi-Lagrangian level set method is proposed for structural optimization with a line search algorithm and a sensitivity modulation scheme. The semi-Lagrange method has an advantage to allow for a large time step without the limitation of Courant- Friedrichs-Lewy (CFL) condition. The line search attempts to adaptively determine an appropriate time step in each iteration of optimization. With consideration of some practical characteristics during topology optimization process, incorporating the line search into semi-Lagrange optimization method can yield fewer design iterations and thus improve the overall computational efficiency. The sensitivity modulation is inspired from the conjugate gradient method in finite-dimensions, and provides an alternative to the standard steepest descent search in level set based optimization. Two benchmark examples are presented to compare the sensitivity modulation and the steepest descent techniques with and without the line search respectively. / Secondly, a generic method to design engineering features for level set based structural optimization is presented. Engineering features are regular and simple shape units containing specific engineering significance for manufacture and assembly consideration. It is practically useful to combine feature design with structural optimization. In this thesis, a Constructive Solid Geometry (CSG) based Level Sets description is proposed to represent a structure based on two basic entities: a level set model containing either a feature shape or a freeform boundary. By treating both entities implicitly and homogeneously, optimal feature design and freeform boundary design are unified under the level set framework. For feature models, a constrained motion of affine transformations is utilized, where the design velocity is obtained through a least square approximation of continuous shape variation. An accurate particle level set updating scheme is employed for the transformation. Meanwhile, freeform models undergo a standard level set updating process using a semi-Lagrange scheme. With this method, various feature characteristics are identified through carefully constructing a CSG model tree with flexible entities and preserved by imposing motion constraints to different stages of the tree. Moreover, because a free shape and topology optimization is enabled over non-feature regions, a truly optimal structural configuration with engineering features can be designed in a convenient way. Several 2D and 3D generative feature design examples are provided to show the applicability of this approach. / Finally, a 3D implementation using adaptive level set method is discussed. This method utilizes both explicit and implicit geometric representations for computation. An octree grid is adopted to accommodate the free structural interface of an implicit level set model and a corresponding 2-manifold triangle mesh model. Within each iteration of optimization, the interface evolves implicitly using a semi-Lagrange level set method, during which the signed distance field is evaluated directly and accurately from the current surface model other than interpolation. After that, another mesh model is extracted from the updated field and serves as the input of subsequent process. This hybrid and adaptive representation scheme not only achieves "narrow band computation", but also facilitates the structural analysis by using a geometry-aware mesh-free approach. Moreover, a feature preserving and topological errorless mesh simplification algorithm is proposed to enhance the computational efficiency. Remarkably, the adaptive level set scheme opens up a gate to incorporate geometric editing into structural optimization in an effective way, which creates a new dimension of opportunity to further develop level set based structural optimization in this direction. A three dimensional benchmark example and possible extensions are presented to demonstrate the capability and potential of this method. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Zhou, Mingdong. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 123-135). / Abstract also in Chinese. / Abstract --- p.i / Acknowledgement --- p.v / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background of Structural Optimization --- p.2 / Chapter 1.2 --- Research Issues and Contributions --- p.7 / Chapter 1.3 --- Content Outline --- p.11 / Chapter 2 --- Structural Optimization with Level Set Method --- p.13 / Chapter 2.1 --- Dynamic Level Set Method --- p.14 / Chapter 2.1.1 --- Implicit Model Description and Hamilton-Jacobi Equation --- p.15 / Chapter 2.1.2 --- Model Update and Re-Initialization --- p.16 / Chapter 2.2 --- Application in Structural Optimization Problem --- p.19 / Chapter 2.2.1 --- Problem Formulation of Linear Elastic Continuum --- p.19 / Chapter 2.2.2 --- Design Sensitivity Analysis --- p.21 / Chapter 2.2.3 --- Optimization Strategy --- p.24 / Chapter 2.3 --- Couple with Extended Finite Element Method --- p.26 / Chapter 2.3.1 --- X-FEM for Structural Analysis --- p.28 / Chapter 2.3.2 --- Numerical Integration --- p.30 / Chapter 2.3.3 --- Imposing Boundary Conditions --- p.31 / Chapter 2.4 --- Summary --- p.33 / Chapter 3 --- A semi-Lagrangian level set method for structural optimization --- p.34 / Chapter 3.1 --- Introduction --- p.35 / Chapter 3.2 --- Semi-Lagrangian Level Set Method --- p.37 / Chapter 3.3 --- A Line Search Algorithm --- p.38 / Chapter 3.4 --- A Sensitivity Modulation Scheme --- p.41 / Chapter 3.5 --- Numerical Examples --- p.43 / Chapter 3.5.1 --- Cantilever beam --- p.44 / Chapter 3.5.2 --- Bridge-type structure --- p.48 / Chapter 3.6 --- Summary --- p.54 / Chapter 4 --- Engineering Feature Design in Structural Optimization --- p.58 / Chapter 4.1 --- Introduction --- p.59 / Chapter 4.2 --- CSG based Level Sets --- p.64 / Chapter 4.3 --- Structural Optimization with CSGLS --- p.67 / Chapter 4.4 --- Constrained Motion with Affine Transformation --- p.71 / Chapter 4.4.1 --- 2D Algorithm --- p.71 / Chapter 4.4.2 --- 3D Algorithm --- p.74 / Chapter 4.5 --- Design Sharp Characteristics --- p.79 / Chapter 4.6 --- Numerical Examples --- p.79 / Chapter 4.6.1 --- Moment of Inertia (MOI) Maximization --- p.79 / Chapter 4.6.2 --- Feature Design in Structural Topology Optimization --- p.81 / Chapter 4.6.3 --- Generative Feature Design --- p.85 / Chapter 4.6.4 --- A 3D Feature Based Optimal Design --- p.92 / Chapter 4.7 --- Summary --- p.93 / Chapter 5 --- Adaptive level set implementation for 3D problems --- p.97 / Chapter 5.1 --- Introduction and Algorithm Overview --- p.98 / Chapter 5.2 --- Hybrid Model Representation and Interface Tracking --- p.100 / Chapter 5.2.1 --- Octree Based Implicit Model --- p.101 / Chapter 5.2.2 --- Triangle Mesh Based Explicit Model --- p.102 / Chapter 5.2.3 --- Interface Tracking --- p.102 / Chapter 5.3 --- Engineering Model Simplification --- p.103 / Chapter 5.3.1 --- Introduction --- p.104 / Chapter 5.3.2 --- Algorithm of Progressive Multi-Pass Simplification --- p.105 / Chapter 5.3.3 --- Numerical Results of Mesh Simplification --- p.109 / Chapter 5.4 --- Structural Analysis --- p.115 / Chapter 5.5 --- Numerical Example of A 3D Optimal Design --- p.116 / Chapter 5.6 --- Summary --- p.116 / Chapter 6 --- Conclusions and Future work --- p.118 / Chapter 6.1 --- Conclusions --- p.118 / Chapter 6.2 --- Future Work --- p.120 / Bibliography --- p.123 / Publications --- p.136

在過去的幾十年，結構優化已經成為了一個能夠啓發工程師們來獲得更加合理和經濟的設計的強大工具。同以前發展的基於材料的結構優化的方法相比，最近水平集方法由於其在邊界表達上面的靈活性以及能夠處理複雜的結構拓撲變化而受到歡迎。結構的邊界被一個隱性的水平集函數追蹤以及它的演化被速度場所驅動，這個速度場是從結合有限元結果的靈敏度分析獲得的。然而，傳統的有限元方法無論在處理介於實體材料和空材料之間的移動邊界上，還是在處理拓撲變化上，都是令人討厭的，因為有限元的網格需要跟結構的邊界保持一致，這樣就會導致耗時的網格重建。隨著擴展有限元方法的出現，傳統有限元方法的型函數被擴展函數所擴展。這樣就使得擴展有限元適合用來表達前面提到的移動邊界，而且通常固定的均勻的網格被採用從而使得網格管理的困難可以被避免。然而，爲了捕獲準確的邊界，更密集的網格是所渴望的，這樣在一定程度上會降低擴展有限元方法的效率，同時有限元分析被認為是最耗時的過程並且在優化過程中的每一步都會進行。在基於水平集的結構優化中，邊界是我們最為關心的地方。更密集的網格最被渴望出現在邊界處，而在遠離邊界的區域只需要更稀疏的網格。因此，爲了改善擴展有限元方法的效率以及縮短優化過程，適當地調整底層的網格是必要的。解決的方案是高分辨率的有限元網格分佈在邊界的附近區域而相對更低密度的網格分佈在遠離邊界的區域，這樣一來，既能縮短計算時間同時又能保證精度。這篇論文的動機就是為在水平集框架下的結構優化開發一個高效又準確的具有自適應網格的擴展有限元方案. / 基於對具有固定網格的傳統的有限元方法的前期研究，二維和三維的h-自適應的擴展有限元方法被調查和發展。通過網格稀疏化的方法從初始的密集的均勻的網格中逐漸去除不需要的有限元單元，多層次自適應的網格被生成來適應但是沒必要完全契合結構的邊界。底層的網格是用四叉樹（二維）或者八叉樹（三維）來描述的。四叉樹或者八叉樹的表達一方面適合用來管理多層次的數據，另外一方面也會使得自適應網格的生成有效率。在本論文中，自適應網格被限制為1-不規則的網格，同時在網格稀疏化的過程中，具有懸掛節點的有限元單元會被生成。這些懸掛的節點會同常規的節點一樣對待，既具有自由度又有相應的型函數。對於每一個單元來說，這些型函數要被修改以滿足單位分解的屬性。對不同類型的單元的積分被研究並且對於那些被邊界劃過的單元以及擁有懸掛節點的單元，特殊的方案應該被採用。因為擴展有限元方法使得有限元方法一般化來處理邊界，這些邊界沒有必要被恰好吻合的網格覆蓋，因此邊界條件有可能在網格內部從而在引進邊界條件的時候會出現困難。一方面，力的邊界條件的引進的困難會少一些，因為它只需要把積分域從單元的邊上修改到單元內部的邊界上。另外一方面，引進位移的邊界條件確實不容易的。在本論文中，尼采方法被用來強加位移邊界條件。爲了驗證邊界條件的引進，以及本論文所提出來的擴展有限元方法的準確性、效率和收斂性，有理論界的二維的例子被用來作為參考標準以及三維的數值算例通過跟ANSYS軟件產生的結果作比較被執行。 / 通過組合水平集方法和本文提出的h-自適應的擴展有限元方法，結構優化的柔度問題被研究。值得注意的是，擴展有限元方法是在自適應網格上實現的而水平集的演化確實在均勻的網格上進行的。在優化的每一步中，隨著結構邊界的傳播，自適應網格會被相應的更新，同時很多成熟的算法可以直接用於均勻網格的水平集的演化。在文獻中常被使用的二維和三維的數值的算例被當做參考標準，特別是兩個實用的應用進一步的驗證了本論文中提出來的擴展有限元方法的可靠性。 / Structural optimization has become a powerful tool to inspire engineers for more reasonable and economical designs during the past decades. Compared to previously developed material based approaches, the level set method for structural optimization is gaining popularity recently due to its exibility in boundary representation and handling complex topological changes of structure. The structural boundary is tracked by an implicit level set function and its evolution is driven by boundary velocity which is derived from sensitivity analysis with the result of finite element analysis. However, conventional Finite Element Method (FEM) is troublesome in handling either moving boundaries between solid material and voids or topological changes, as the finite element meshes need to conform to the boundaries of structure resulting in the time-consuming remeshing process. With the advent of Extended Finite Element Method (X-FEM), shape functions of conventional FEM are extended with enrichment functions, which make X-FEM suitable for representing aforementioned moving boundaries, and usually fixed uniform meshes are employed so that mesh management difficulties can be avoided. However, to capture precision boundaries, denser meshes are desired, which to some extent decreases the efficiency of the XFEM, and meanwhile finite element analysis is regarded as the most time-consuming process and conducted at each iterative step during the optimization. In the level set based structural optimization, boundaries are most concerned where denser meshes are most desired while the regions far away from boundaries only need coarser meshes. Therefore, to improve the efficiency of the X-FEM and shorten the optimization process, it is essential to adjust underlying meshes adequately. The solution scheme is that finite element meshes of higher resolution are distributed in the vicinity of the boundaries while meshes of relatively lower resolution are in the regions far away from the boundaries to significantly decrease the computational time while ensuring the accuracy. The motivation of this dissertation is to develop an efficient and accurate X-FEM scheme with adaptive meshes for structural optimization in the level set framework. / Based on previous studies on conventional X-FEM with fixed uniform meshes, h-Adaptive X-FEM is investigated and developed in both two and three-dimensions. Multilevel adaptive meshes are generated to fit but not necessary to conform to structural boundaries by the method of mesh coarsening to gradually remove unnecessary elements from initial fine uniform meshes. The underlying meshes are depicted by Quadtree(for 2D) or Octree(for 3D) representations which are suitable for managing multilevel data on the one hand and make the generation of adaptive meshes efficient on the other hand. In this thesis, adaptive meshes are restrained to 1-irregular meshes and elements with hanging nodes are produced during the process of mesh coarsening. The hanging nodes are treated as regular nodes with degrees of freedoms (DOFs) and shape functions which are modified to satisfy Partition of Unity(POU) property for each element. Quadrature for different types of elements is studied and special schemes should be adopted for elements crossed by boundaries and elements with hanging nodes where kinks would exist. As the X-FEM generalizes conventional FEM to handle structure whose boundaries are not necessarily covered by conforming meshes, the boundary conditions are possible inside the meshes which presents difficulties while imposing boundary conditions. On the one hand, the imposition of Neumann boundary conditions is not difficult because it only requires a modification of the integral domain from borders of elements to the boundaries insides elements. However, imposing Dirichlet boundary conditions is non-trivial. In this dissertation, Nitsche's method is employed to enforce Dirichlet boundary conditions. In order to verify the imposition of boundary conditions, accuracy, efficiency and convergent rate of the proposed X-FEM, 2D examples with theoretical solution are treated as benchmarks and 3D numerical examples are conducted by the comparison with solutions produced by ANSYS software. / The mean compliance problems of structural optimization are investigated by combining level set method and the proposed h-Adaptive X-FEM. Notably, the X-FEM is achieved on the adaptive meshes while the evolution of level set is conducted on the fine uniform meshes. The adaptive meshes are updated accordingly along with the propagation of structural boundaries at each optimization step, and meanwhile lots of mature algorithms can be used for level set evolution with uniform grids directly. Numerical examples both in 2D and 3D commonly used in literatures are treated as benchmarks especially two practical applications further verify the reliability of the proposed X-FEM. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Tian, Xuefeng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 114-130). / Abstract also in Chinese. / Abstract --- p.i / Acknowledgement --- p.viii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Structural Optimization --- p.1 / Chapter 1.2 --- Finite Element Analysis for Level based Structural Optimization --- p.4 / Chapter 1.3 --- Motivation and Objective --- p.6 / Chapter 1.4 --- Contributions and Organization of the Dissertation --- p.7 / Chapter 2 --- Implementation of h-Adaptive X-FEM --- p.9 / Chapter 2.1 --- X-FEM for Material-void Boundaries --- p.9 / Chapter 2.1.1 --- General Form of X-FEM --- p.9 / Chapter 2.1.2 --- X-FEM for Material-void Boundaries --- p.11 / Chapter 2.2 --- Implicit Representation of Structural Model --- p.12 / Chapter 2.3 --- Implementation of Adaptive Meshes --- p.13 / Chapter 2.4 --- 1-irregular Meshes --- p.15 / Chapter 2.5 --- Criterion of Adaptiveness --- p.16 / Chapter 2.6 --- Quadtree (for 2D) and Octree (for 3D) Representations --- p.17 / Chapter 2.7 --- Quadrature --- p.19 / Chapter 2.7.1 --- Quadrature for Standard Elements without Enrichment --- p.20 / Chapter 2.7.2 --- Quadrature for Elements Crossed by Boundaries --- p.23 / Chapter 2.7.3 --- Quadrature for Elements with Hanging Nodes . --- p.26 / Chapter 2.8 --- Imposition of Boundary Conditions --- p.34 / Chapter 2.8.1 --- Imposition of Neumann Boundary Conditions . --- p.35 / Chapter 2.8.2 --- Imposition of Dirichlet Boundary Conditions --- p.37 / Chapter 2.9 --- Other Numerical Issues --- p.40 / Chapter 2.9.1 --- Small Area/volume Fraction Elements --- p.40 / Chapter 2.9.2 --- Stress Smoothing Strategy --- p.41 / Chapter 3 --- Numerical Validation of h-Adaptive X-FEM --- p.45 / Chapter 3.1 --- A Plate Under Uniaxial Tension --- p.45 / Chapter 3.2 --- A Thick Cylinder Model Under Internal Pressure --- p.51 / Chapter 3.3 --- A Cantilever Beam --- p.57 / Chapter 3.4 --- Infinite Plate with a Circular Hole --- p.61 / Chapter 3.5 --- A Clamp Model (3D) --- p.65 / Chapter 3.6 --- I-Beam (3D) --- p.68 / Chapter 4 --- Level Set Based Structural Optimization Using h-Adaptive X-FEM --- p.71 / Chapter 4.1 --- Structural Optimization Problem --- p.71 / Chapter 4.2 --- Sensitivity Analysis --- p.74 / Chapter 4.3 --- Level Set Evolution --- p.77 / Chapter 4.4 --- Flowchart of Level Set Based Structural Optimization Coupling with h-Adaptive X-FEM --- p.81 / Chapter 5 --- Numerical Examples of Level Set Based Structural Optimization Using h-Adaptive X-FEM --- p.83 / Chapter 5.1 --- A Cantilever Beam (2D) --- p.84 / Chapter 5.2 --- A Michell-Type Structure (2D) --- p.88 / Chapter 5.3 --- A L-Shape Structure (3D) --- p.90 / Chapter 5.4 --- A Michell-type Structure (3D) --- p.95 / Chapter 5.5 --- An Electrical Mast (3D) --- p.99 / Chapter 5.6 --- A Chair Design (3D) --- p.103 / Chapter 6 --- Conclusions and Future Work --- p.109 / Chapter 6.1 --- Conclusions --- p.109 / Chapter 6.2 --- Future Work --- p.111 / Chapter 6.2.1 --- An Easy-to-use FEM Tool --- p.111 / Chapter 6.2.2 --- Large-scale problems and Parallelization --- p.112 / Chapter 6.2.3 --- Higher-order X-FEM --- p.112 / Chapter 6.2.4 --- Other Structural Optimization Problems --- p.112 / Bibliography --- p.114

First of all, the PDE form of the classical level set function phi is parameterized with an analytical form of Radial Basis Function (RBF), which is real-valued and continuously differentiable. Such that the upwind scheme, extension velocity and reinitialization algorithms in solving the discrete Hamilton-Jacobi equation can be waived in the numerical process, the whole framework is transformed into a standard mathematical programming problem in which the linear objective function can be directly optimized by a gradient algorithm - shape sensitivity. The minimization of the mean compliance is studied and presented to demonstrate the advantages of the parametrical method. / Parametrization substantially reduces the complexity of the original discrete PDE level set method. However, the result shows that the high number of RBF knots leads to dense coefficient matrices. Thus, it induces numerical instabilities, slow convergence and less accuracy in the process. Consequently, we then study the distribution of knots density for faster computation. By updating the movement of the knot, the knot moves towards the position where the change is directly determined by the shape sensitivity. In such case, we may use lesser number of knots to describe the properties of the system while the smoothness of the implicit function is satisfied. The sensitivity study is evaluated carefully and discussed in detail. Results show a significant improvement in the computational speed and stability. / The study found significant improvement obtained in the structural optimization with the parametric level set method, both the stability and efficiency were given as the benefits of using the method of the parametrization. / Traditional structural optimization approaches can be referred to as sizing optimization, since their design variables are the proportions of the structure or material. A major restriction in the sizing problem is that the shape and the topology of the structure are fixed a priori. Undoubtedly, changes in shape (e.g., curved boundary) and topology (e.g., holes in a member) could produce more significant improvement in dynamic performance than modifications in size alone. A recent development of shape and topology optimization based on the implicit moving boundaries with the use of the renowned level set method is regarded as one of the most sophisticated methods in handling the change of the structural topology. In this thesis, we study the parametrization of the classical level set method for the structural optimization and the associated computational methodology. / Usually, a large-scale model will lead to bulk coefficient matrices in the RBF optimization and the linear function normally require O (N3) flops and O (N2) memory while processing. It is becoming impractical to solve as N goes over 10,000. In fact, the dense system equation matrix frequently leads to the numerical instabilities and the failure of the optimization. Finally, we introduce the method of Partition of Unity (POU) to deal with this problem. POU is often used in 3D reconstruction of implicit surfaces from scattered point sets. It breaks the global domain into smaller overlapping subdomains such that the implicit functions can be more efficiently interpolated. Meanwhile, the global solution is obtained by blending all the local solutions with a set of weighting functions. The algorithm of POU is presented here, and we analyze and discuss the numerical results accordingly. / Ho, Hon Shan. / Adviser: Michael Y. Wang. / Source: Dissertation Abstracts International, Volume: 73-03, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 106-119). / 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, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

Sonic boom annoyance is an important technical showstopper for commercial supersonic aircraft operations. It has been proposed that aircraft can be shaped to alleviate sonic boom. Choosing the right aircraft shape reflecting the design requirements is a fundamental and most important step that is usually over simplified in the conceptual stages of design by resorting to a qualitative selection of a baseline configuration based on historical designs and designers perspective. Final aircraft designs are attempted by minor shape modifications to this baseline configuration. This procedure may not yield large improvements in the objectives, especially when the baseline is chosen without a rigorous analysis procedure. Traditional analyses and implementations tend to have a complex algorithmic flow, tight coupling between tools used and computational limitations. Some of these shortcomings are overcome in this study and a diverse mix of tools is seamlessly integrated to provide a simple, yet powerful and automatic procedure for sonic boom minimization. A shape optimization procedure for supersonic aircraft design using better geometry generation and improved analysis tools has been successfully demonstrated. The geometry engine provides dynamic reconfiguration and efficient manipulation of various components to yield unstructured watertight geometries. The architecture supports an assimilation of different components and allows configuration changes to be made quickly and efficiently because changes are localized to each component. It also enables an automatic way to combine linear and non-linear analyses tools. It has been shown in this study that varying atmospheric conditions could have a huge impact on the sonic boom annoyance metrics and a quick way of obtaining probability estimates of relevant metrics was demonstrated. The well-accepted theoretical sonic boom minimization equations are generalized to a new form and the relevant equations are derived to yield increased flexibility in aircraft design process. Optimum aircraft shapes are obtained in the conceptual design stages weighing in various conflicting objectives. The unique shape optimization procedure in conjunction with parallel genetic algorithms improves the computational time of the analysis and allows quick exploration of the vast design space. The salient features of the final designs are explained. Future research recommendations are made.