In structural analysis and design it is important to consider the effects of uncertainties in loading and material properties in a rational way. Uncertainty in material properties such as heterogeneity in elastic and mass properties can be modeled as a random field. For computational purpose, it is essential to discretize and represent the random field. For a field with known second order statistics, such a representation can be achieved by Karhunen-Lo`eve (KL) expansion. Accordingly, the random field is represented in a truncated series expansion using a few eigenvalues and associated eigenfunctions of the covariance function, and corresponding random coefficients. The eigenvalues and eigenfunctions of the covariance kernel are obtained by solving a second order Fredholm integral equation. A closed-form solution for the integral equation, especially for arbitrary domains, may not always be available. Therefore an approximate solution is sought. While finding an approximate solution, it is important to consider both accuracy of the solution and the cost of computing the solution. This work is focused on exploring a few numerical methods for estimating the solution of this integral equation. Three different methods:(i)using finite element bases(Method1),(ii) mid-point approximation(Method2), and(iii)by the Nystr¨om method(Method3), are implemented and numerically studied. The methods and results are compared in terms of accuracy, computational cost, and difficulty of implementation. In the first method an eigenfunction is first represented in a linear combination of a set of finite element bases. The resulting error in the integral equation is then minimized in the Galerkinsense, which results in a generalized matrix eigenvalue problem. In the second method, the domain is partitioned into a finite number of subdomains. The covariance function is discretized by approximating its value over each subdomain locally, and thereby the integral equation is transformed to a matrix eigenvalue problem. In the third method the Fredholm integral equation is approximated by a quadrature rule, which also results in a matrix eigenvalue problem. The methods and results are compared in terms of accuracy, computational cost, and difficulty of implementation. The first part of the numerical study involves comparing these three methods. This numerical study is first done in one dimensional domain. Then for study in two dimensions a simple rectangular domain(referred toasDomain1)is taken with an uncertain material property modeled as a Gaussian random field. For the chosen covariance model and domain, the analytical solutions are known, which allows verifying the accuracy of the numerical solutions. There by these three numerical methods are studied and are compared for a chosen target accuracy and different correlation lengths of the random field. It was observed that Method 2 and Method 3 are much faster than the Method 1. On the other hand, for Method 2 and 3, additional cost for discretizing the domain into nodes should be considered whereas for a mechanics-related problem, Method 1 can use the available finite element mesh used for solving the mechanics problem. The second part of the work focuses on studying on the effect of the geometry of the model on realizations of the random field. The objective of the study is to see the possibility of generating the random field for a complicated domain from the KL expansion for a simpler domain. For this purpose, two KL decompositions are obtained: one on the Domain1, and another on the same rectangular domain modified with a rectangular hole (referredtoasDomain2) inside it. The random process is generated and realizations are compared. It was observed from the studies that probability density functions at the nodes on both the domains, that is, on Domain 1 and Domain 2, are similar. This observation leads to a possibility that a complicated domain can be replaced by a corresponding simpler domain, thereby reducing the computational cost.

A new method is presented for the numerical computation of the generalized eigen- values of real Hamiltonian or symplectic pencils and matrices. The method is strongly backward stable, i.e., it is numerically backward stable and preserves the structure (i.e., Hamiltonian or symplectic). In the case of a Hamiltonian matrix the method is closely related to the square reduced method of Van Loan, but in contrast to that method which may suffer from a loss of accuracy of order sqrt(epsilon), where epsilon is the machine precision, the new method computes the eigenvalues to full possible accuracy.

In this paper we develop new preconditioned iterative methods for solving monotone nonlinear eigenvalue problems. We investigate the convergence and derive grid-independent error estimates for these methods. Numerical experiments demonstrate the practical effectiveness of the proposed methods for a model problem.

Let F be a field, let G be an undirected graph on n vertices, and let SF(G) be the set of all F-valued symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let MRF(G) be defined as the set of matrices in SF(G) whose rank achieves the minimum of the ranks of matrices in SF(G). We develop techniques involving Z-hat, a process termed nil forcing, and induced subgraphs, that can determine when diagonal entries corresponding to specific vertices of G must be zero or nonzero for all matrices in MRF(G). We call these vertices nil or nonzero vertices, respectively. If a vertex is not a nil or nonzero vertex, we call it a neutral vertex. In addition, we completely classify the vertices of trees in terms of the classifications: nil, nonzero and neutral. Next we give an example of how nil vertices can help solve the inverse inertia problem. Lastly we give results about the inverse eigenvalue problem and solve a more complex variation of the problem (the λ, µ problem) for the path on 4 vertices. We also obtain a general result for the λ, µ problem concerning the number of λ’s and µ’s that can be equal.

In this thesis, we study the relationship between the generalized eigenvalue problem (GEP) $Ax=\lambda Bx$, and systems of differential equations. We examine both the Jordan canonical form and Kronecker's canonical form (KCF). The first part of this work provides an introduction to the fundamentals of generalized eigenvalue problems and methods for solving this problem. We discuss the QZ algorithm, which can be used to determine the generalized eigenvalues and also how it can be implemented on MATLAB with the built in function 'eig'.  One essential facet of this work is the exploration of symmetric matrix pencils, which arise when A and B are both symmetric matrices. Furthermore we discuss discrete velocity models (DVMs) focusing specifically on a 12-velocity model on the plane. The results obtained are then applied to half-space problems for discrete velocity models, with a focus on planar stationary systems.

A investigação dos mecanismos de instabilidade pode ter um papel importante no entendimento do processo laminar para turbulento de um escoamento. Análise de instabilidade de uma camada limite de uma linha de estagnação compressível foi realizada no contexto de teoria linear BiGlobal. O estudo dos mecanismos de instabilidade deste escoamento pode proporcionar uma visão útil no desenho aerodinâmico das asas. Um novo procedimento foi desenvolvido e implementado computacionalmente de maneira sequencial e paralela para o estudo de instabilidade BiGlobal. O mesmo baseia-se em formar a matriz esparsa associada ao problema discretizado por dois métodos: pontos de colocação de Chebyshev-Gauss-Lobatto e diferenças finitas, além das combinações destes métodos. Isto permitiu o uso de bibliotecas computacionais eficientes para resolver o sistema linear associado ao problema de autovalor utilizando o algoritmo de Arnoldi. O desempenho do método numérico e código computacional proposto são analisados do ponto de vista do uso de métodos de ordenação dos elementos da matriz, coeficientes de preenchimento, memória e tempo computacional a fim de determinar a solução mais eficiente para um problema físico geral com técnicas de matrizes esparsas. Um estudo paramétrico da instabilidade da camada limite de uma linha de estagnação foi realizado incluindo o estudo dos efeitos de compressibilidade. O excelente desempenho código computacional permitiu obter as curvas neutras e seus respectivos valores críticos para a faixa de número de Mach 0 \'< OU =\' Ma \'< OU =\' 1. Os resultados confirmam a teoria assintótica apresentada por (THEOFILIS; FEDOROV; COLLIS, 2004) e mostram que o incremento do número de Mach reduz o numero de Reynolds crítico e a faixa instável do número de ondas. / Investigation of linear instability mechanisms is essential for understanding the process of transition from laminar to turbulent flow. An algorithm for the numerical solution of the compressible BiGlobal eigenvalue problem is developed. This algorithm exploits the sparsity of the matrices resulting from the spatial discretization of the enigenvalue problem in order to improve the performance in terms of both memory and CPU time over previous dense algebra solutions. Spectral collocation and finite differences spatial discretization methods are implemented, and a performance study is carried out in order to determine the best practice for the efficient solution of a general physical problem with sparse matrix techniques. A combination of spectral collocation and finite differences can further improve the performance. The code developed is then applied in order to revisit and complete the parametric analyses on global instability of the compressible swept Hiemenz flow initiated in (THEOFILIS; FEDOROV; COLLIS, 2004) and obtain neutral curves of this flow as a function of the Mach number in the 0 \'< OU =\' Ma \'< OU =\' 1 range. The present numerical results fully confirm the asymptotic theory results presented in (THEOFILIS; FEDOROV; COLLIS, 2004). This work presents a complete parametric study of the instability properties of modal three dimensional disturbances in the subsonic range for the flow conguration at hand. Up to the subsonic maximum Mach number value studied, it is found that an increase in this parameter reduces the critical Reynolds number and the range of the unstable spanwise wavenumbers.

No description available.

In this thesis three methods LMGS, TQR and GR are applied to solve an inverseeigenvalue problem. We list the numerical results and compare the accuracy of the computed Jacobi matrix $T$ and the associated orthogonal matrix $Q$, wherethe columns of $Q^T$ are the eigenvectors of $T$. In the application of this inverse eigenvalue problem, the Fourier coefficients of $h(x)=e^x$ relative to the orthonormal polynomials associatedwith $T$ are evaluated, and these values are used to compute the least squarescoefficients of $h$ relative to the Chebyshev polynomials. We list thesenumerical results and compare them as our conclusion.

Dans cette thèse, nous menons à l'aide de la méthode des équations intégrales des analyses de sensitivité de solutions ou de spectres de l'équation de conductivité par rapport aux variations géométriques ou de paramètres de l'équation. En particulier, nous considérons le problème de conductivité dans des milieux à forts contrastes, le problème de perturbation du bord d'une inclusion de conductivité, le problème de valeurs propres du Laplacien dans des domaines perturbés et le problème d'ouverture de gap dans le spectre des cristaux photoniques.

<p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">This thesis consists of two different parts, related to two different fields in mathematical physics: wave propagation and kinetic theory of gases. Various mathematical and computational problems for equations from these areas are treated.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">The first part is devoted to high order finite difference methods for the Helmholtz equation and the wave equation. Compact schemes with high order accuracy are obtained from an investigation of the function derivatives in the truncation error. With the help of the equation itself, it is possible to transfer high order derivatives to lower order or to transfer time derivatives to space derivatives. For the Helmholtz equation, a compact scheme based on this principle is compared to standard schemes and to deferred correction schemes, and the characteristics of the errors for the different methods are demonstrated and discussed. For the wave equation, a finite difference scheme with fourth order accuracy in both space and time is constructed and applied to a problem in discontinuous media.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small;"><span style="font-family: Times New Roman;">The second part addresses some problems related to kinetic equations. A direct simulation Monte-Carlo method is constructed for the Landau-Fokker-Planck equation, and numerical tests are performed to verify the accuracy of the algorithm. A formal derivation of the method from the Boltzmann equation with grazing collisions is performed. The linear and linearized Boltzmann collision operators for the hard sphere molecular model are studied using exact reduction of integral equations to ordinary differential equations. It is demonstrated how the eigenvalues of the operators are found from these equations, and numerical values are computed. A proof of existence of non-zero discrete eigenvalues is given. The ordinary diffential equations are also used for investigation of the Chapman-Enskog distribution function with respect to its asymptotic behavior.</span></span></span></p><p class="MsoNormal" style="margin: 0cm 0cm 0pt;"><span style="mso-ansi-language: EN-US;" lang="EN-US"><span style="font-size: small; font-family: Times New Roman;"> </span></span></p>