Wind power is a renewable energy source that is today the fastest growing solution to reduce CO2 emissions in the electric energy mix. Upwind horizontal axis wind turbine with three blades has been the preferred technical choice for more than two decades. This horizontal axis concept is today widely leading the market. The current PhD thesis will cover an alternative type of wind turbine with straight blades and rotating along the vertical axis. A brief overview of the main differences between the horizontal and vertical axis concept has been made. However the main focus of this thesis is the aerodynamics of the wind turbine blades. Making aerodynamically efficient turbines starts with efficient blades. Making efficient blades requires a good understanding of the physical phenomena and effective simulations tools to model them. The specific aerodynamics for straight bladed vertical axis turbine flow are reviewed together with the standard aerodynamic simulations tools that have been used in the past by blade and rotor designer. A reasonably fast (regarding computer power) and accurate (regarding comparison with experimental results) simulation method was still lacking in the field prior to the current work. This thesis aims at designing such a method. Analytical methods can be used to model complex flow if the geometry is simple. Therefore, a conformal mapping method is derived to transform any set of section into a set of standard circles. Then analytical procedures are generalized to simulate moving multibody sections in the complex vertical flows and forces experienced by the blades. Finally the fast semi analytical aerodynamic algorithm boosted by fast multipole methods to handle high number of vortices is coupled with a simple structural model of the rotor to investigate potential aeroelastic instabilities. Together with these advanced simulation tools, a standard double multiple streamtube model has been developed and used to design several straight bladed rotor ranging from 2 kW to 20 kW. / Felaktigt tryckt som Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 704

The class of computational problems I consider in this thesis share the common trait of requiring consideration of pairs (or higher-order tuples) of data points. I focus on the problem of kernel summation operations ubiquitous in many data mining and scientific algorithms. In machine learning, kernel summations appear in popular kernel methods which can model nonlinear structures in data. Kernel methods include many non-parametric methods such as kernel density estimation, kernel regression, Gaussian process regression, kernel PCA, and kernel support vector machines (SVM). In computational physics, kernel summations occur inside the classical N-body problem for simulating positions of a set of celestial bodies or atoms. This thesis attempts to marry, for the first time, the best relevant techniques in parallel computing, where kernel summations are in low dimensions, with the best general-dimension algorithms from the machine learning literature. We provide a unified, efficient parallel kernel summation framework that can utilize: (1) various types of deterministic and probabilistic approximations that may be suitable for both low and high-dimensional problems with a large number of data points; (2) indexing the data using any multi-dimensional binary tree with both distributed memory (MPI) and shared memory (OpenMP/Intel TBB) parallelism; (3) a dynamic load balancing scheme to adjust work imbalances during the computation. I will first summarize my previous research in serial kernel summation algorithms. This work started from Greengard/Rokhlin's earlier work on fast multipole methods for the purpose of approximating potential sums of many particles. The contributions of this part of this thesis include the followings: (1) reinterpretation of Greengard/Rokhlin's work for the computer science community; (2) the extension of the algorithms to use a larger class of approximation strategies, i.e. probabilistic error bounds via Monte Carlo techniques; (3) the multibody series expansion: the generalization of the theory of fast multipole methods to handle interactions of more than two entities; (4) the first O(N) proof of the batch approximate kernel summation using a notion of intrinsic dimensionality. Then I move onto the problem of parallelization of the kernel summations and tackling the scaling of two other kernel methods, Gaussian process regression (kernel matrix inversion) and kernel PCA (kernel matrix eigendecomposition). The artifact of this thesis has contributed to an open-source machine learning package called MLPACK which has been first demonstrated at the NIPS 2008 and subsequently at the NIPS 2011 Big Learning Workshop. Completing a portion of this thesis involved utilization of high performance computing resource at XSEDE (eXtreme Science and Engineering Discovery Environment) and NERSC (National Energy Research Scientific Computing Center).

This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion. As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems: the need to construct many dense matrices and to evaluate many singular and near-singular integrals. The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested. The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal. In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments.

Dans cette thèse, nous nous intéressons à la modélisation mathématique des hétérogénéités de longueurs caractéristiques beaucoup plus petites que la longueur d'ondes. La thèse consiste en deux parties. La partie théorique est dédiée à l'obtention d'un développement asymptotique raccordé: la solution est décrite à l'aide d'un développement de champ proche au voisinage de l'obstacle et par un développement de champ lointain hors de ce voisinage. Le développement de champ lointain met en jeu des solutions singulières de l'équation des ondes tandis que le champ proche lui est régi par un modèle quasi-statique. Ces deux développements sont alors raccordés dans une zone intermédiaire dite de raccord. Nous obtenons alors des estimations d'erreurs permettant de rendre rigoureux ce développement asymptotique formel. La deuxième partie est numérique. Elle décrit à la fois la méthode de Galerkine discontinue, une méthode de raffinement de maillage espace-temps et propose une discrétisation des modèles asymptotiques obtenues précédemment. Elle est illustrée par un certain nombre de tests numériques. / In this thesis, we focus our attention on the modeling of heterogeneities which are smaller than the wavelength. The document is decomposed into two parts : a theoretical one and a numerical one. In the first part, we derive a matched asymptotic expansion composed of a far-field expansion and a near-field expansion. The terms of the far-field expansion are singular solutions of the wave equation whereas the terms of the near-field expansion satisfy quasistatic problems. These expansions are matched in an intermediate region. We justify mathematically this theory by proving error estimates. In the second part, we describe the Discontinuous Galerkin method, a local time stepping method and the implementation of the matched asymptotic method. Numerical simulations illustrate these results.

High-frequency wave phenomena is observed in many physical settings, most notably in acoustics, electromagnetics, and elasticity. In all of these fields, numerical simulation and modeling of the forward propagation problem is important to the design and analysis of many systems; a few examples which rely on these computations are the development of metamaterial technologies and geophysical prospecting for natural resources. There are two modes of modeling the forward problem: the frequency domain and the time domain. As the title states, this work is concerned with the former regime. The difficulties of solving the high-frequency wave propagation problem accurately lies in the large number of degrees of freedom required. Conventional wisdom in the computational electromagnetics commmunity suggests that about 10 degrees of freedom per wavelength be used in each coordinate direction to resolve each oscillation. If K is the width of the domain in wavelengths, the number of unknowns N grows at least by O(K^2) for surface discretizations and O(K^3) for volume discretizations in 3D. The memory requirements and asymptotic complexity estimates of direct algorithms such as the multifrontal method are too costly for such problems. Thus, iterative solvers must be used. In this dissertation, I will present fast algorithms which, in conjunction with GMRES, allow the solution of the forward problem in O(N) or O(N log N) time. / text

This work addresses the question of the efficient numerical solution of time-domain boundary integral equations with retarded potentials arising in the problems of acoustic and electromagnetic scattering. The convolutional form of the time-domain boundary operators allows to discretize them with the help of Runge-Kutta convolution quadrature. This method combines Laplace-transform and time-stepping approaches and requires the explicit form of the fundamental solution only in the Laplace domain to be known. Recent numerical and analytical studies revealed excellent properties of Runge-Kutta convolution quadrature, e.g. high convergence order, stability, low dissipation and dispersion. As a model problem, we consider the wave scattering in three dimensions. The convolution quadrature discretization of the indirect formulation for the three-dimensional wave equation leads to the lower triangular Toeplitz system of equations. Each entry of this system is a boundary integral operator with a kernel defined by convolution quadrature. In this work we develop an efficient method of almost linear complexity for the solution of this system based on the existing recursive algorithm. The latter requires the construction of many discretizations of the Helmholtz boundary single layer operator for a wide range of complex wavenumbers. This leads to two main problems: the need to construct many dense matrices and to evaluate many singular and near-singular integrals. The first problem is overcome by the use of data-sparse techniques, namely, the high-frequency fast multipole method (HF FMM) and H-matrices. The applicability of both techniques for the discretization of the Helmholtz boundary single-layer operators with complex wavenumbers is analyzed. It is shown that the presence of decay can favorably affect the length of the fast multipole expansions and thus reduce the matrix-vector multiplication times. The performance of H-matrices and the HF FMM is compared for a range of complex wavenumbers, and the strategy to choose between two techniques is suggested. The second problem, namely, the assembly of many singular and nearly-singular integrals, is solved by the use of the Huygens principle. In this work we prove that kernels of the boundary integral operators $w_n^h(d)$ ($h$ is the time step and $t_n=nh$ is the time) exhibit exponential decay outside of the neighborhood of $d=nh$ (this is the consequence of the Huygens principle). The size of the support of these kernels for fixed $h$ increases with $n$ as $n^a,a<1$, where $a$ depends on the order of the Runge-Kutta method and is (typically) smaller for Runge-Kutta methods of higher order. Numerical experiments demonstrate that theoretically predicted values of $a$ are quite close to optimal. In the work it is shown how this property can be used in the recursive algorithm to construct only a few matrices with the near-field, while for the rest of the matrices the far-field only is assembled. The resulting method allows to solve the three-dimensional wave scattering problem with asymptotically almost linear complexity. The efficiency of the approach is confirmed by extensive numerical experiments.