This thesis aims to analyse a finite element method applied to an adjusted Cahn-Hilliard equation that has been used for digital image inpainting applications. We consider both the standard model with a smooth double well potential and an alternative where an obstacle potential has been used. Existence and uniqueness results are derived for both formulations by adapting techniques existing in literature for other problems. For each formulation we then propose approximations, by discretising first in space and then in time, and we derive error bounds between the weak solution of the original formulation and the solution of the discrete approximations in terms of the discretisation parameters. We then propose and implement a practical numerical scheme for both models and investigate their use in applications, alongside some other models from literature. We investigate various real digital image examples and compare the resulting inpaintings for these competing models, considering their suitability for real-world applications.

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

No description available.

A dissertation submitted to the Faculty of Engineering, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering. / In this research a finite element analysis program was developed for the modelling of general compressible Euler flows. An explicit Taylor-Galerkin algorithm was used as the flow solver and was used in conjunction with a flux-corrected transport algorithm in order to obtain high shock resolution without numerical oscillations and overshoots. The solver was applied to two and three dimensional geometries. An axisymmetric extension of the Taylor Galerkin algorithm was also developed. For the two dimensional code, a fully automatic mesh generator was implemented which was able to generate meshes for completely arbitrary geometries, as well as an adaptive refinement algorithm which performs an error analysis on the solution and refines and coarsens the mesh appropriately in order to maintain an optimal mesh resolution. The automatic mesh generator dramatically reduced problem setup time and the adaptive refinement algorithm reduced compllter time by up to 90%" A number of test cases were performed covering a wide range of compressible flows including steady and unsteady flows in air, using the ideal gas model, and shocks in liquids, using the Tait model. Within the limitations of the inviscid and real gas assumptions made, accurate results were obtained, / AC 2018

No description available.

Abstract Grinding introduces unavoidable residual stresses of significant but unknown magnitudes. The effect of residual stresses in surface integrity is related to the nature of the residual stresses which relies purely on the process parameters and the workmaterial properties. It is a well-known fact that the fatigue strength of a ground component is increased by introducing compressive stresses. On the other hand, fatigue cracks may originate at regions of maximum tensile stress and usually at the surface of the material. Moreover, stress corrosion cracking is another consequence of critical surface tensile stress. Added to that, the residual stresses may result in dimension alteration and surface distortion, particularly for thin products such as plates. The beneficial effects of compressive residual stresses have been widely recognized in industry. The wise application of such a principle would bring about improved economical use of parts subjected to fatigue loading and aggressive environmental conditions. Therefore a better understanding of residual stress mechanisms is necessary to increase the dimensional accuracy and improve the surface integrity of ground elements, particularly for parts with high precision and manufactured by automated production lines. Consequently, the development of reliable models for predicting residual stresses is of great value in reducing the amount of measurements and experimental tests of residual stresses. Unfortunately, little effort has been devoted so far to develop appropriate models to take into account grinding conditions, workmaterial properties and boundary conditions. This thesis aims to investigate the residual stress mechanisms induced by grinding in terms of grinding parameters. In order to obtain a full understanding, both the roles of individual factors causing residual stresses (i.e. mechanical, thermal and phase transformation) and their couplings were carefully studied with the aid of the finite element method. The studies include: (1) residual stresses due to thermal grinding conditions, (2) residual stresses due to iso-thermal mechanical grinding conditions, (3) coupling of thermo-mechanical conditions, (4) coupling of thermo-phase transformation, and (5) the full coupling of all the factors. It is found that under sole thermal grinding conditions, the heat flux associated with up-grinding may lead to a higher grinding temperature compared with that of down-grinding. A constant flux introduces the least temperature rise if the total grinding energy is the same. Higher convection heat transfer not only decreases the grinding temperature but also makes the temperature rise occur mainly within a thin surface layer. A similar effect can be achieved by applying higher table speeds. When the grinding temperature is less than the austensing temperature, surface residual stresses are tensile. The heat generated within the grinding zone causes a very non-uniform temperature field in the workpiece. The part of the workmaterial subjected to a higher temperature rise expands more significantly and causes compressive stresses because of the restraint from its surrounding material that expands less. When the surface heat flux moves forward, the material outside the grinding zone contracts under cooling. Since the workmaterial has been plastically deformed during thermal loading, the contraction is restrained and thus a tensile stress field is generated locally. If a workpiece material experiences a critical temperature variation in grinding, phase transformation takes place and a martensite layer appears in the immediate layer underneath the ground surface. It was found that the growth of martensite develops a hardened zone with a higher yield stress that expands with the movement of the heat flux. A tensile surface residual stress is then developed. When the volume growth of material takes place during phase change, compressive residual stresses may also be generated. Under iso-thermal grinding conditions, it was found that plane stress is mainly compressive regardless of the distribution of surface traction and the direction of the tangential grinding force. With up-grinding, the residual stress in the grinding direction is always tensile. However, down-grinding may yield compressive surface residual stresses if the magnitude of the ratio of horizontal to vertical grinding forces is sufficiently large. Moreover, it is noted that discrete surface traction, which is more reasonable in terms of simulating the individual cutting of abrasive grits, would bring about more complex residual stress distribution that is very sensitive to the combined effect of individual cutting grits. If thermal and mechanical grinding conditions are coupled, a state free from residual stresses may be achieved if grinding heat is low and either the convection heat transfer or the table speed is high. However, it is found that the full coupling of the mechanical deformation, the thermal deformation and deformation by phase change results in tensile residual stresses. The effects of cooling and mechanical traction in this case however are minor. In summary, the research of this thesis explored the following: (a) grinding temperature development in terms of a wide range of grinding parameters together with the effect of temperature-dependent material properties, (b) the origin and onset of irreversible deformation due to mechanical loading, thermal loading and phase change under critical grinding conditions, (c) the effects of individual residual stress mechanisms and their partial and full couplings, and (d) the selection of grinding conditions to achieve beneficial residual stresses. Finally, based on the new findings in this research, a more comprehensive methodology is suggested for further study.

2 v. : / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (D.E.)--Dept. of Electrical Engineering, University of Adelaide

Thesis (M. Phil.)--University of Hong Kong, 2006. / Title proper from title frame. Also available in printed format.

Thesis (M.S.)--West Virginia University, 2006. / Title from document title page. Document formatted into pages; contains xv, 147 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 145-147).

Thesis (Ph. D.)--Drexel University, 2006. / Includes abstract and vita. Includes bibliographical references.