Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisor: Clint N. Dawson. Vita. Includes bibliographical references.

This work is concerned with the use of isogeometric analysis based on Non- Uniform Rational B-Splines (NURBS) to develop efficient and robust numerical techniques to deal with the problems of incompressibility in the fields of solid and fluid mechanics. Towards this, two types of formulations, mixed Galerkin and least-squares, are studied. During the first phase of this work, mixed Galerkin formulations, in the context of isogeometric analysis, are presented. Two-field and three-field mixed variational formulations - in both small and large strains - are presented to obtain accurate numerical solutions for the problems modelled with nearly incompressible and elasto-plastic materials. The equivalence of the two mixed formulations, for the considered material models, is derived; and the computational advantages of using two-field formulations are illustrated. Performance of these formulations is assessed by studying several benchmark examples. The ability of the mixed methods, to accurately compute limit loads for problems involving elastoplastic material models; and to deal with volumetric locking, shear locking and severe mesh distortions in finite strains, is illustrated. Later, finite element formulations are developed by combining least-squares and isogeometric analysis in order to extract the best of both. Least-squares finite element methods (LSFEMs) based on the use of governing differential equations directly - without the need to reduce them to equivalent lower-order systems - are developed for compressible and nearly incompressible elasticity in both the small and finite strain regimes; and incompressible Navier-Stokes. The merits of using Gauss-Newton scheme instead of Newton-Raphson method to solve the underlying nonlinear equations are presented. The performance of the proposed LSFEMs is demonstrated with several benchmark examples from the literature. Advantages of using higher-order NURBS in obtaining optimal convergence rates for non-norm-equivalent LSFEMs; and the robustness of LSFEMs, for Navier-Stokes, in obtaining accurate numerical solutions without the need to incorporate any artificial stabilisation techniques, are demonstrated.

In this thesis we develop and analyze a Galerkin-Characteristic method to integrate the potential vorticity equations of a baroclinic ocean. The method proposed is a two stage inductive algorithm. In the first stage the material derivative of the potential vorticity is approximated by combining Galerkin-Characteristic and Particle methods. This yield a computationally efficient algorithm for this stage. Such an algorithm consists of updating the dependent variable at the grid points by cubic spline interpolation at the foot of the characteristic curves of the advective component of the equations. The algorithm is unconditionally stable and conservative for Δt = O(h). The error analysis with respect to L² -norm shows that the algorithm converges with order O(h); however, in the maximum norm it is proved that for sufficiently smooth functions the foot of the characteristic curves are superconvergent points of order O(h⁴ /Δt). The second stage of the algorithm is a projection of the Lagrangian representation of the flow onto the Cartesian space-time Eularian representation coordinated with Crank-Nicholson Finite Elements. The error analysis for this stage with respect to L²-norm shows that the approximation component of the global error is O(h²) for the free-slip boundary condition, and O(h) for the no-slip boundary condition. These estimates represent an improvement with respect to other estimates for the vorticity previously reported in the literature. The evolutionary component of the global error is equal to K(Δt² + h), where K is a constant that depends on the derivatives of the advective quantity along the Characteristic. Since the potential vorticity is a quasi-conservative quantitiy, one can conclude that K is in general small. Numerical experiments illustrate our theoretical results for both stages of the method. / Science, Faculty of / Mathematics, Department of / Graduate

published_or_final_version / Mechanical Engineering / Doctoral / Doctor of Philosophy

Mak, Tsz Fan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 33-36). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

在本文中，我們為了三維空間的馬克士威方程組(Maxwell’s equation)制定和分析了一套新種類的交錯間斷伽遼金(discontinuous Galerkin)方法，同時考慮了時間依賴性和時間諧波的馬克士威方程組。我們用了空間離散上交錯笛卡兒網格，這種方法具有許多良好的性質。首先，我們的方法所得出的數值解保留了電磁能量，並自動符合了高斯定律的離散版本。第二，質量矩陣是對角矩陣，從而時間推進是顯式和非常有效的。第三，我們的方法是高階準確，最佳收斂性在這裏會被嚴格地證明。第四，基於笛卡兒網格，它也很容易被執行，並可視為是典型的Yee’s Scheme的以及四邊形的邊有限元的推廣。最後，超收斂結果也會在這裏被證明。 / 在本文中，我們還提供了幾個數值結果驗證了理論的陳述。我們計算了時間依賴性和時間諧波的馬克士威方程組數值收斂結果。此外，我們計算時間諧波馬克士威方程組特徵值問題的數值特徵值，並與理論特徵值比較結果。最後，完美匹配層(Perfect Matching Layer)吸收邊界的問題也有實行其數值結果。 / We develop and analyze a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell’s equations in this paper. Both time-dependent and time-harmonic Maxwell’s equations are considered. The spatial discretization is based on staggered Cartesian grids which possess many good properties. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Second, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Third, our method is high order accurate and the optimal order of convergence is rigorously proved. Fourth, it is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee’s scheme as well as the quadrilateral edge finite elements. Lastly, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved. / In this paper, we also provide several numerical results to verify the theoretical statements. We compute the numerical convergence order using L2-norm and discrete-norm respectively for both the time-dependent and time-harmonic Maxwell’s equations. Also, we compute the numerical eigenvalues for the time-harmonic eigenvalue problem and compare the result with the theoretical eigenvalues. Lastly, applications to problems in unbounded domains with the use of PML are also presented. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Yu, Tang Fei. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 46-49). / Abstracts also in Chinese. / Chapter 1 --- Introduction and Model Problems --- p.1 / Chapter 2 --- Staggered DG Spaces --- p.4 / Chapter 2.1 --- Review on Gauss-Radau and Gaussisan points --- p.5 / Chapter 2.2 --- Basis functions --- p.6 / Chapter 2.3 --- Finite Elements space --- p.7 / Chapter 3 --- Method derivation --- p.14 / Chapter 3.1 --- Method --- p.14 / Chapter 3.2 --- Time discretization --- p.17 / Chapter 4 --- Energy conservation and Discrete Gauss law --- p.19 / Chapter 4.1 --- Energy conservation --- p.19 / Chapter 4.2 --- Discrete Gauss law --- p.22 / Chapter 5 --- Error analysis --- p.24 / Chapter 6 --- Numerical examples --- p.29 / Chapter 6.1 --- Convergence tests --- p.30 / Chapter 6.2 --- Diffraction by a perfectly conducting object --- p.30 / Chapter 6.3 --- Perfectly matched layers --- p.37 / Chapter 7 --- Time Harmonic Maxwell’s equations --- p.40 / Chapter 7.1 --- Model Problems --- p.40 / Chapter 7.2 --- Numerical examples --- p.40 / Chapter 7.2.1 --- Convergence tests --- p.41 / Chapter 7.2.2 --- Eigenvalues tests --- p.41 / Chapter 8 --- Conclusion --- p.45 / Bibliography --- p.46

The time-dependent elastic wave equation is the foundation of seismology. It is very useful in studying the wave propagation in elastic solids. Simulation of Rayleigh waves, which is governed by the equation, requires high accuracy solutions. Finite difference method have been widely used for the simulation of Rayleigh waves. However, it is not obvious how to effectively impose the free surface boundary condition on a curved surface. On the other hand, discontinuous Galerkin methods are more flexible in handling complex geometries. / In this thesis, we develop a class of discontinuous Galerkin methods for time-dependentelastic wave equation in isotropic homogeneous medium. This method is explicit, locally and globally energy conserving. Also, the L² convergence of this method is optimal and the convergence in energy norm is independent of Poisson's ratio. / Besides, we apply our method to simulate Rayleigh waves on curved free surfaces. We also impose a perfectly matched layer to absorb the outward waves. Numerical examples show that our method can accurately capture features of Rayleigh waves even in a domain with high Poisson's ratio. / 時間依賴型彈性波動方程」是地震學的基礎。這組方程對於波在彈性固體中傳播的研究非常有用。雷利波是由這個方程所控制。模擬雷利波須要有非常準確的解。有限差分法廣泛地應用在雷利波的模擬上，可是如何有效地施加自由表面邊界條件於曲面上的方法並不明顯。另一方面，間斷伽遼金方法能更靈活地處理複雜的幾何形狀。 / 在這篇論文中，我們發展了一類間斷伽遼金方法去求「在均勻各向同性的介質上的時間依賴型彈性波動方程」的解。我們將表明，這種方法是顯式的，局部及全域能量守恆的，而它的收斂是最優的和獨立於泊松比的。 / 除此之外，我們運用這個方法來模擬雷利波在具有起伏的自由表面的半空間模型的傳播。我們會使用完美匹配層去吸收朝外的波動。數值算例反映，即使在高柏松比的介質中，這個方法也可以準確地捕捉雷利波的特徵。 / Lam, Chi Yeung. / Thesis M.Phil. Chinese University of Hong Kong 2014. / Includes bibliographical references (leaves 44-47). / Abstracts also in Chinese. / Title from PDF title page (viewed on 06, October, 2016). / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 214-220). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph.D.)--City University of Hong Kong, 2009. / "Submitted to the Department of Building and Construction in partial fulfillment of the requirements for the degree of Doctor of Philosophy." Includes bibliographical references (leaves [170]-184)

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