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水準集結構拓撲優化方法同傳統的基於材料的拓撲優化方法相比具有明顯的優勢。由於採用了隱式的邊界表達，水準集方法能方便地處理結構形狀和拓撲的變化，且在優化過程中可以保持邊界的光滑。但這種動態結構邊界需要一種有限元分析方法可以適應其動態變化且能夠獲得足夠的計算精度。本文對傳統水準集結構拓撲優化中有限元分析存在的關鍵問題進行研究，同時針對應力約束下的結構拓撲優化，提出了一種新的拓撲優化方法。 / 首先， 擴展有限元法作為一種相對精確和高效的結構分析方法，本文將其引入到水準集結構拓撲優化中。引入擴展有限元法主要是為了處理優化過程中結構邊界上出現的材料的高度不連續情況，從而避免耗時的網格重新劃分。如果結構邊界從有限元單元內部通過，為了獲得足夠的計算精度，該單元內需要採用特殊的數值積分方法。常用的方法是將這個單元內被材料填充的區域劃分成小的子單元去適應單元內部的邊界，然後在各個子單元內採用高斯數值積分來獲得該單元的剛度矩陣。對於二維問題，如果結構邊界從一個單元內部通過，將單元分成幾部分，首先把單元內部的實體區域劃分成子三角形，然後計算出每個子三角形內的高斯積分點，最後單元剛度矩陣通過把所有子三角形的高斯積分點處的數值積分迭加得到。對於三維問題，則是將一個六面體單元分解為幾個四面體，然後在每一個四面體內部通過結構上定義的水準集函數值得到邊界，對於實體的部分劃分為子四面體，在每一個子四面體內計算出高斯積分點，此四面體的單元剛度矩陣為所有子四面體剛度矩陣的迭加，因此，該六面體的單元剛度矩陣為所劃分的四面體單元剛度矩陣的迭加。 / 其次，本文研究了提高擴展有限元法的計算精度和效率的方法。採用擴展有限元法進行結構分析時，如果被結構邊界剖分的有限元單元中實體部分體積比小到一定程度，將會影響到計算精度，本論文給出了處理擴展有限元中這種小單元情形的具體辦法。擴展有限元法作為一種結構分析計算方法，除了必須考慮精度外，效率也是一個重要的指標，尤其對於拓撲優化問題，因拓撲優化問題通常需要多步的反覆運算來獲取最優或局部最優解。為提高擴展有限元法的計算效率，相對於前面的基於剖分單元為子單元進行積分的辦法，本文提出了一種更高效的積分方法，即去除積分單元剖分，通過直接積分來計算被結構邊界剖分的單元的剛度矩陣。這種直接積分的方法不僅能保證結構分析的精度，更能顯著的提高計算效率，這對於水準集結構拓撲優化是非常有意義的。同時高階單元被用來從另一個角度分析擴展有限元法計算精度與效率之間的關係，換言之，可以用高階單元在相對粗的網格上來獲取同低階單元在相對密的網格上相同的分析結果精度，從而提高計算效率。但是這個問題需要找出計算精度在網格密度和單元階次之間的關係。 / 第三，本文以二維和三維結構的柔度最小化問題為例驗證了上述擴展有限元演算法在結構拓撲優化問題中應用的有效性。 / 最後，本文研究了基於應力約束的結構拓撲優化問題，並採用前面提出的擴展有限元法與水準集結合的拓撲優化方法。由於採用擴展有限元法進行結構分析可以獲得較準確的應力計算結果，特別是在結構邊界附近，這對於基於應力的拓撲優化問題有很大的優勢。而且，本文提出了一種形狀等效約束法來有效地控制局部應力約束，數值算例也證明擴展有限元法與形狀等效約束方法相結合對處理應力約束問題是一種非常有效的。同時，本文還提出了一種全新的通過拓撲優化來實現應力隔離結構設計的方法。通過在拓撲優化問題中不同區域施加不同的應力約束來有效地模擬這種應力隔離的問題。最終數值算例證明，該方法可以通過改變力的傳播途徑來達到有效地形成結構的應力隔離。 / Level set method is an elegant approach for structural shape and topology optimization, compared to the conventional material based topology optimization methods. The structural boundary is implicitly represented by a moving level set function. Thus, the shape and topology optimization can be processed simultaneously while maintaining a smooth boundary. The moving structural boundary demands a finite element analysis adaptable to the dynamic boundary changes and meeting required accuracy. In this thesis, the key issues of finite element methods of structural analysis for level set optimization method are investigated and an approach to stress-constrained topology optimization is presented. / Firstly, the extended finite element method (XFEM) is introduced into the level set method structural shape and topology optimization for obtaining a considerably accurate and efficient result of finite element analysis. In fact, the XFEM is employed as a structural analysis method to solve the problems of strong discontinuities between material and void domain during the level set optimization process in order to avoid the time cost remeshing. To achieve a reasonably accurate result of finite element analysis in the element intersected by structural boundary, special numerical integral schemes of XFEM are studied. The partition method is adopted to divide the integral domain into sub-cells, in which Gauss quadrature is utilized to calculate the element stiffness matrix. For two-dimensional (2D) problems, the integral domain is divided into sub-triangles, and the Gauss quadrature points in each sub-triangle are used to evaluate the element stiffness matrix which is the sum of all contributions of these sub-triangles. For three-dimensional (3D) problems, the hexahedral element is decomposed into multiple tetrahedra, and the integral domain in each tetrahedron is divided into sub-tetrahedra for obtaining the Gauss quadrature points. Therefore, the stiffness of each tetrahedron is obtained by summing all contributions of the sub-tetrahedra, which means the hexahedral element stiffness matrix is the accumulation of element stiffness matrixes with all these tetrahedra. / Secondly, the methods for improving the computational accuracy and efficiency of XFEM are studied. First of all, the practical solutions for dealing with the small volume fraction element of the proposed XFEM are provided since this kind of situation may result in the accuracy losing of finite element analysis. Besides computational accuracy of structural analysis, the efficiency is another sufficiently important issue of structural optimization problem. Therefore, a new XFEM integral scheme without quadrature sub-cells is developed for improving the computational efficiency of XFEM compared to the XFEM integral scheme with partition method, which can yield similar accuracy of structural analysis while prominently reducing the computational cost. Numerical experiments indicate that this performance is excellent for level set method shape and topology optimization. Moreover, XFEM with higher order elements are involved to improve the accuracy of structural analysis compared to the corresponding lower order element. Consequently, the computational cost is increased, therefore, the balance of computational cost between FE system scale and the order of element is discussed in this thesis. / Thirdly, the reliability and advantages of the proposed XFEM schemes are illustrated with several 2D and 3D mean compliance minimization examples that are widely employed in the recent literature of structural topology optimization. / Finally, the stress-based topology optimization problems with the proposed XFEM schemes are investigated. Due to the accuracy of structural analysis, XFEM schemes have natural advantages for solving the stress-based topology optimization problems using the level set method. Moreover, the shape equilibrium constraint approach is developed to effectively control the local stress constraint. Some numerical examples are solved to prove the high-performance of the proposed shape equilibrium constraint approach and XFEM schemes in the stress-constrained topology optimization problem. Meanwhile, a new approach of stress isolation design is presented through topology optimization. The stress isolation problem is modeled into a topology optimization problem with multiple stress constraints in different regions. Numerical experiments demonstrate that this approach can change the force transmission paths to successfully realize stress isolation in the structure. / 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. / Li, Li. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 113-123). / Abstract also in Chinese. / Abstract --- p.I / 摘要 --- p.IV / Acknowledgement --- p.VI / Contents --- p.VII / List of Figures --- p.XI / List of Tables --- p.XV / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.1 / Chapter 1.2 --- Related Works --- p.3 / Chapter 1.3 --- XFEM for Structural Optimization --- p.4 / Chapter 1.4 --- Topology Optimization with Stress Constraint --- p.7 / Chapter 1.5 --- Contributions and Organization of the Dissertation --- p.10 / Chapter 2 --- Level Set Method for Structural Optimization --- p.12 / Chapter 2.1 --- Structural Optimization Problem --- p.12 / Chapter 2.2 --- Implicit Level Set Representation --- p.14 / Chapter 2.3 --- Evolution of the Level Set Function --- p.15 / Chapter 2.4 --- Level Set Surface Reinitialization --- p.16 / Chapter 2.5 --- Velocity Extension --- p.17 / Chapter 3 --- Extended Finite Element Method (XFEM) --- p.19 / Chapter 3.1 --- Global Enrichment --- p.19 / Chapter 3.2 --- Local Enrichment --- p.20 / Chapter 3.3 --- Enrichment Function --- p.22 / Chapter 3.3.1 --- Enrichment for Strong Discontinuity --- p.22 / Chapter 3.3.2 --- Enrichment for Weak Discontinuity --- p.23 / Chapter 3.4 --- XFEM used in Structural Optimization --- p.23 / Chapter 4 --- Implementation of XFEM for Structural Optimization --- p.25 / Chapter 4.1 --- 2D XFEM Scheme --- p.26 / Chapter 4.1.1 --- Numerical Integral Scheme in 2D --- p.26 / Chapter 4.1.2 --- Evaluation of the 2D XFEM Scheme --- p.27 / Chapter 4.2 --- 3D XFEM Scheme --- p.30 / Chapter 4.2.1 --- Numerical Integral Scheme in 3D --- p.30 / Chapter 4.2.2 --- Evaluation of the 3D XFEM Scheme --- p.33 / Chapter 5 --- Computational Accuracy and Efficiency Aspects of XFEM --- p.36 / Chapter 5.1 --- XFEM Scheme for Small Volume Fraction Element --- p.38 / Chapter 5.1.1 --- Problem Definition --- p.39 / Chapter 5.1.2 --- Numerical Example --- p.41 / Chapter 5.2 --- Stress Smoothing in XFEM --- p.46 / Chapter 5.3 --- XFEM Integral Scheme without Quadrature Sub-cells --- p.50 / Chapter 5.3.1 --- 2D XFEM Integral Scheme without Quadrature Sub-cells --- p.50 / Chapter 5.3.2 --- 3D XFEM Integral Scheme without Quadrature Sub-cells --- p.53 / Chapter 5.4 --- Higher Order Elements with XFEM Scheme --- p.55 / Chapter 5.4.1 --- Higher Order Elements --- p.55 / Chapter 5.4.2 --- Numerical Example --- p.57 / Chapter 6 --- Minimum Compliance Optimization using XFEM --- p.64 / Chapter 6.1 --- Level Set Formulation of the Optimization Problem --- p.64 / Chapter 6.2 --- Finite Element Analysis with XFEM --- p.65 / Chapter 6.3 --- Shape Sensitivity Analysis --- p.65 / Chapter 6.4 --- Numerical Examples --- p.68 / Chapter 6.4.1 --- A 2D Short Cantilever Beam --- p.68 / Chapter 6.4.2 --- A 3D Short Cantilever Beam --- p.75 / Chapter 6.4.3 --- A Michell-type Structure in 3D --- p.77 / Chapter 7 --- Stress-Constrained Topology Optimization using XFEM --- p.81 / Chapter 7.1 --- Shape Equilibrium Approach to Stress Constraint --- p.81 / Chapter 7.1.1 --- Problem Formulation of Stress-Constrained Topology Optimization --- p.81 / Chapter 7.1.2 --- Shape Equilibrium Constraint Approach --- p.82 / Chapter 7.1.3 --- Material Derivatives of Stress Constraint --- p.83 / Chapter 7.1.4 --- Shape Sensitivity Analysis --- p.85 / Chapter 7.2 --- Finite Element Analysis with XFEM --- p.87 / Chapter 7.3 --- Minimal Weight Design with Stress Constraint --- p.88 / Chapter 7.3.1 --- Problem Definition --- p.88 / Chapter 7.3.2 --- Numerical Example --- p.89 / Chapter 7.4 --- Stress Isolation design --- p.94 / Chapter 7.4.1 --- Problem Definition --- p.94 / Chapter 7.4.2 --- Shape Sensitivity Analysis --- p.95 / Chapter 7.4.3 --- Numerical Examples --- p.97 / Chapter 8 --- Conclusions and Future Works --- p.109 / Chapter 8.1 --- Conclusions --- p.109 / Chapter 8.2 --- Future Works --- p.110 / Chapter 8.2.1 --- Adaptive XFEM --- p.111 / Chapter 8.2.2 --- Extend Shape Equilibrium Constraint Approach to 3D --- p.112 / Chapter 8.2.3 --- Extend the Stress Isolation Design Method into Industrial Applications --- p.112 / Bibliography --- p.113

Wong Chak-fu. / "September 2004." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (p. 451-475). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

In the first part, an efficient and reliable a posteriori error estimate is derived for solving three-dimensional static Maxwell's equations based on the lowest order edge elements of the first family. We propose an adaptive finite element method and establish convergence of the adaptive scheme in energy norm under a restriction on the initial mesh size. Any prescribed error tolerance is thus achieved in a finite number of steps. For discretization based on the lowest order edge elements of the second family, a similar adaptive method is designed which guarantees convergence without any initial mesh size restriction. The proofs rely mainly on error and oscillation reduction estimates as well as the Galerkin orthogonality of the edge element approximation. For time-dependent Maxwell's equations, we deduce an efficient and reliable a posteriori error estimate, upon which an adaptive finite element method is built. / In this thesis, we will address three typical problems with discontinuous coefficients in a general Lipschitz polyhedral domain, which are often encountered in numerical simulation of electromagnetism. / The second part deals with a saddle point problem arising from Maxwell's equations. We present an adaptive finite element method on the basis of the lowest order edge elements of the first family and prove its convergence. The main ingredients of the proof are a novel quasi-orthogonality, which replaces the usual Pythagoras relation, which fails in this case, all error reduction depending on an efficient and reliable a posteriori error estimate and an oscillation reduction. We show that this adaptive scheme is a contraction for the sum of some energy error plus the oscillation. Likewise, the above result is generalized to the discretization by the lowest order edge elements of the second family. / We introduce in the third part an adaptive finite element method for solving the eigenvalue problem of the Maxwell system based on an inverse iterative method. By modifying the exact inverse iteration algorithm involving an inner saddle point solver, we construct an adaptive inverse iteration finite element algorithm, which consists of an inexact inner adaptive procedure for a discrete mixed formulation in place of the original saddle point problem. An efficient and reliable a posteriori error estimate is obtained and the convergence of the inner adaptive method is proved. In addition, the important convergence property of the algorithm is studied, which ensures the errors between true solutions (eigenfunction and eigenvalue) and iterative ones to fall below any given tolerance within a finite number of iterations. / Xu, Yifeng. / "June 2007." / Adviser: Jun Zou. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0357. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 166-175). / 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.

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A ideia de realizar este trabalho surgiu durante do curso de pós-graduação, ministrado pelo Prof. M. Drigas, \"Tópicos especiais sobre máquinas elétricas\", realizado no 2º semestre de 1980 na EPUSP, onde foi observada a necessidade do conhecimento das distribuições de campos magnéticos em dispositivos eletromecânicos com o objetivo de se prever seu desempenho na fase de projeto. Nesta época, já havia sido apresentada a tese do Prof. Janiszewski, o primeiro trabalho, de nosso conhecimento realizado no Brasil nesta área, onde foi desenvolvida a técnica de resolução de problemas de Campos Magnéticos em Regime Estacionário, que, evidentemente, não pode ser aplicada na resolução de problemas onde a variável tempo está envolvida; baseado neste tese, em 1982 o Prof. Luiz Lebensztajn, reproduziu o trabalho do Dr. Janiszewski o qual foi aplicado para verificar a consistência dos resultados práticos na tese de Livre Docência do Prof.. Dr. Aurio Gilberto Falcone. As formulações mais frequentes do Método dos Elementos Finitos (MEF), publicada nos periódicos internacionais, são baseadas no Cálculo Variacional, onde o sistema de equações algébricas não linear resultante, é derivado a partir da obtenção do extremo de uma funcional que em algumas situações não pode ser obtida, limitando assim sua aplicação. Em decorrência deste fato, o primeiro objetivo deste trabalho foi organizar os procedimentos para obtenção do sistema de equações de MEF aplicado à resolução de problemas de campo descritos por equações diferenciais não lineares, sem a necessidade. Algumas contribuições interessantes são encontradas no Capítulo II, referente à formulação do MEF para problemas de campo descrito por operadores diferenciais não auto-adjuntos.No Capítulo III são apresentadas as técnicas de montagem das matrizes, bem como aquelas de introdução das condições de contorno, originárias deste método, que muito embora sejam técnicas de aplicação corriqueiras, ajudarão em muito o pesquisador iniciante nesta área, sem a necessidade de recorrer a outro texto. No Capítulo VI são apresentadas as formulações necessárias para a solução de problemas de campos eletromagnéticos estáticos, para elementos de quatro lados retos (e curvos) assim como a técnica utilizada na obtenção da relutividade em meios não lineares. No Capítulo V são tratados os problemas de campo, onde a variável tempo está envolvida, permitindo assim a resolução de uma série enorme de problemas referentes aos campos de natureza eletromagnética, tais como os fenômenos transitórios e o Regime Permanente Senoidal. Os aspectos computacionais ligados ao trabalho estão expostos no Capítulo VI, onde são apresentadas as rotinas de resolução do sistema de equações resultante adaptadas às particularidades do problema, e as rotinas de integração numérica de problemas descrito por equações diferenciais dependentes do tempo de primeira e segunda ordem. Algumas técnicas apresentadas nestes Capítulos, são aplicadas espe3cificamente para a obtenção da distribuição de campo magnético no Capitulo VII deste trabalho, com o objetivo de analisar o desempenho de um transformador em regime transitório, onde é confirmada a consistência do método. / The idea of making this work came during a graduation course, \" Special topics on electric machines\", lectured by Prof. Dr. M. Drigas during the 2nd semester of 1980 at EPUSP, when the need of knowing the distribution of magnetic fields in electromechanics devices was notices, in order to foresse its performance during design. At that time, the first work about this subject realized made in Brazil was presented in prof. Janiszewski\'s thesis, where a technique was developed to solve Steady-State Magnetic Fields. However, it is clear that when the time variable is considered, this technique cannot be applied. The usual formulations of the Finite Element Method, published in international journals, was based on Variational Calculations, where the resulting non-linear algebraic equations system is derived from the extreme of a functional, which sometimes cannot be obtained, limiting in this way its application. Consequently, the first aim of this work is to organize procedures to obtain the Finite Method equations system, in order solve non-linear differential equations of fields, without the need of a previous functional for the problem. In Chapter II, one will find some interesting contributions referred to the Finite Element Method formulation, in the description of field problems by the use of non self-adjacent differentials operations.Matrix building techniques are presented in Chapter III, as well as the introduction of boundary conditions in this method. In spite of being an ordinary technique, it will help the beginners a lot, eliminating the need of other sources. Chapter IV presents the necessary formulations, which solve static electromagnetic fields for elements of four square (and curved) sides, and the technique used in the determination of non-linear media reluctivity. In Chapter V, the time variable of electromagnetic fields is treated, making possible the solution of problems of this nature, such as transient phenomena and sinusoidal steady-state. Computer aspects of the work are shown in Chapter VI, presenting resolution routines of the equation system fitted to the problem, and numeric integration routines described by first and second order differential equations, which depend on the time. Some techniques showed in those previous Chapters are specifically used in Chapter VII to obtain the magnetic field distribution, which analyses transformer performance during transients. The coherence of the method is also confirmed.

Composite materials are becoming popular in almost all industries. Carbon-fiber and glass-fiber composites are used in aircraft, sports equipment, boats, prosthetics, and wind turbine blades. In all these applications, the composites are subjected to different loads. Loads can take the form of impact or cyclic/fatigue loading, both of which decrease the strength of composites as micro-cracks grow through the composite. Composite laminates are made up of fiber plies (thin layers of fiber) and the fibers are surrounded by a resin like epoxy. It is common for laminates to fail because of delamination growth (plies peeling apart). Small delaminations do not fail a composite, but as delaminations grow, the composite weakens and eventually fails. Composites behave differently than metals do, and failure analysis is more complicated because of the various directions of fibers. Numerical methods (specifically Finite Element Analysis) exist for predicting when failure will occur, but improvements are needed to make these numerical methods more accurate and efficient. The method created, for this thesis, is computationally efficient because it doesn’t require the analyst or computer to adjust the simulation based on where the delamination is (or what kind of shape it is). Energy values are extracted directly from the delamination front and not averaged from nearby locations.

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This thesis provides a contribution towards a general procedure for solving robustly and efficiently limit and shakedown analyses of engineering structures within the static approach which has been chosen for its simplicity of implementation. Throughout the thesis, attempts at improving the robustness and efficiency of the computations are presented. Beginning with efforts to prevent volumetric locking, which is a severe shortcoming of traditional low order h-type displacement elements, the investigation proposes the use of the high order p-version of the finite element method. It is shown theoretically and confirmed numerically that this p-method is not only robust in preventing locking, but also provides very accurate results. However, the use of uniformly distributed high order p-elements may be computationally demanding when the size of the problem becomes large. This difficulty is tackled by two main approaches: use of a p-adaptive procedure at the elastic computation stage and use of approximate piecewise linear yield functions. The p-adaptive scheme produces a non-uniform p-distribution and helps to greatly reduce the number of degrees of freedom needed while still guaranteeing the required level of accuracy. The overall gain is that the sizes of the models are reduced significantly and hence also the computational effort. The adoption of piecewise linear yield surfaces helps to further increase the efficiency at the expense of possibly slightly less accurate, but still very acceptable, results. State-of-the-art linear programming solvers based on the very efficient interior point methodology are used. Significant gains in efficiency are achieved. A heuristic, semi-adaptive scheme to piecewise linearize the yield surfaces is then developed to further reduce the size of the underlying optimization problems. The results show additional gains in efficiency. Finally, major conclusions are summarized, and various aspects suitable for further research are highlighted.

This PhD thesis presents research aims to improving the dynamic modelling of coupled plate structures across a wide frequency range by using analytical, statistical and experimental methods. The analytical waveguide method is used to model the flexural displacement of coupled plate structures which are simply supported along two parallel edges. A method of quickly predicting the average energy level in a plate from details of the waveguide model is described, and used for comparison with SEA models. The Poynting and Impedance methods of predicting the energy flow in coupled plate structures are investigated. Transmission coefficients for coupled plate structures are evaluated using the analytical waveguide method for both semi-infinite and finite coupled plate structures. Finite transmission coefficients have traditionally been more difficult to evaluate due to the presence of a reverberant field, but in this work a novel method of separating the reverberant field using a scattering matrix method is presented. The transmission coefficients for semi-infinite and finite structures are then compared for L-shaped plates. A modal transmission coefficient is also defined and for the cases considered, and is used to develop an alternative method of deriving the transmission coefficient in a finite structure. Frequency averaged transmission coefficients are also considered, and the transmission coefficients derived for finite and semi-infinite structures are found to be very similar after frequency averaging. Statistical Energy Analysis models of coupled plates are evaluated using transmission coefficients derived from waveguide models. The results of the SEA models are compared to those predicted by the analytical waveguide method. A modal transmission coefficient based SEA model is also investigated. In an attempt to validate the numerical work presented in this thesis, experiments have been conducted. Using a wave extraction technique, both the wave amplitudes and plate properties have been evaluated from experimental data, and are subsequently used to experimentally measure the transmission coefficient for two plates coupled at different angles.