Radial basis functions (RBFs) are probably best known for their applications to scattered data problems. Until the 1990s, RBF theory only involved functions that were scalar-valued. Matrix-valued RBFs were subsequently introduced by Narcowich and Ward in 1994, when they constructed divergence-free vector-valued functions that interpolate data at scattered points. In 2002, Lowitzsch gave the first error estimates for divergence-free interpolants. However, these estimates are only valid when the target function resides in the native space of the RBF. In this paper we develop Sobolev-type error estimates for cases where the target function is less smooth than functions in the native space. In the process of doing this, we give an alternate characterization of the native space, derive improved stability estimates for the interpolation matrix, and give divergence-free interpolation and approximation results for band-limited functions. Furthermore, we introduce a new class of matrix-valued RBFs that can be used to produce curl-free interpolants.

This thesis presents a preconditioned solenoidal basis method to solve the algebraic system arising from the linearization and discretization of primitive variable formulations of Navier-Stokes equations for incompressible fluid flows. The system is restricted to a discrete divergence-free space which is constructed from the incompressibility constraint. This research work extends an earlier work on the solenoidal basis method for two-dimensional flows and three-dimensional flows that involved the construction of the solenoidal basis P using circulating flows or vortices on a uniform mesh. A localized algebraic scheme for constructing P is detailed using mixed finite elements on an unstructured mesh. A preconditioner which is motivated by the analysis of the reduced system is also presented. Benchmark simulations are conducted to analyze the performance of the proposed approach.

Background. Fluid simulation is an area of ongoing research. In recent years, simulators have become more realistic and stable, partly by employing the condition of having divergence-free velocity fields. A divergence-free velocity field is a strict constraint that requires a high level of correctness in a simulation. Another recent development is in the subject of performance optimization, where asynchronous time integration is used. Asynchronous time integration means integrating different parts of a fluid with varying time step sizes. Doing so leads to overall larger time step sizes, which improves performance. This thesis combines the divergence-free velocity field condition with asynchronous time stepping in a particle-based simulator. Objectives. This thesis aims to achieve a performance speedup by implementing asynchronous time integration into an existing particle-based simulator that assures the velocity field is divergence-free. Methods. With an open source simulator employing a divergence-free velocity field as a starting point, asynchronous time integration is implemented. This is achieved by dividing the fluid into three regions, each with their own time step sizes. Introducing asynchronous time integration means significantly lowering the stability of a simulation. This is countered by implementing additional steps to increase stability. Results. Roughly a 40\% speedup is achieved in two out of three scenes, with similar visual results as the original synchronous simulation. In the third scene, there is no performance speedup as the performance is similar to that of the original simulation. The two first scenes could be sped up further with more aggressive settings for asynchronous time integration. This is however not possible due to stability issues, which are also the cause for the third scene not resulting in any speedup. Conclusions. Asynchronous simulation is shown to be a valid option even alongside a divergence solver. However, occasional unrealistic behavior resembling explosions among the particles do occur. Besides from being undesirable behavior, these explosions also decrease performance and prevent more aggressive performance settings from being used. Analysis of their cause, attempted solutions and potential future solutions are provided in the discussion chapter.

Approximation and interpolation employing radial basis functions has found important applications since the early 1980's in areas such as signal processing, medical imaging, as well as neural networks. Several applications demand that certain physical properties be fulfilled, such as a function being divergence free. No such class of radial basis functions that reflects these physical properties was known until 1994, when Narcowich and Ward introduced a family of matrix-valued radial basis functions that are divergence free. They also obtained error bounds and stability estimates for interpolation by means of these functions. These divergence-free functions are very smooth, and have unbounded support. In this thesis we introduce a new class of matrix-valued radial basis functions that are divergence free as well as compactly supported. This leads to the possibility of applying fast solvers for inverting interpolation matrices, as these matrices are not only symmetric and positive definite, but also sparse because of this compact support. We develop error bounds and stability estimates which hold for a broad class of functions. We conclude with applications to the numerical solution of the Navier-Stokes equation for certain incompressible fluid flows.

4D flow MRI is an imaging technique able to provide relevant information on patients’ cardiac health condition both from a visual and a quantitative point of view. Its applicability is however limited by uncertainty in the data due to the presence of noise. A new class of filters, called divergence free filters, was recently proposed. They incorporate physical knowledge into the filtering of 4D flow data. One way to implement divergence filters is via wavelet transform. The filtering process using the Divergence Free Wavelet Transform can be carried out in a completely automated fashion and was shown to hold promising results. The focus of this thesis was thus put towards assessing the effect produced by these filters on a large cohort of patients. Time-resolved segmentations were incorporated into the filtering process as this was thought to enhance divergence reduction. They were also used to investigate the filtering in every region of the thoracic cardiovascular system. The assessment of the filters was carried out both from a visual and a quantitative perspective. In-house tools were used to compute clinically used parameters on the data before and after the filtering to investigate the introduced change. The results showed that the used method was able to reduce divergence like noise while preserving all the relevant information contained in the original data, in all the regions of the heart. Flow quantifications were essentially unchanged by the filtering suggesting that the method can be safely applied on 4D flow data.

abstract: Divergence-free vector field interpolants properties are explored on uniform and scattered nodes, and also their application to fluid flow problems. These interpolants may be applied to physical problems that require the approximant to have zero divergence, such as the velocity field in the incompressible Navier-Stokes equations and the magnetic and electric fields in the Maxwell's equations. In addition, the methods studied here are meshfree, and are suitable for problems defined on complex domains, where mesh generation is computationally expensive or inaccurate, or for problems where the data is only available at scattered locations. The contributions of this work include a detailed comparison between standard and divergence-free radial basis approximations, a study of the Lebesgue constants for divergence-free approximations and their dependence on node placement, and an investigation of the flat limit of divergence-free interpolants. Finally, numerical solvers for the incompressible Navier-Stokes equations in primitive variables are implemented using discretizations based on traditional and divergence-free kernels. The numerical results are compared to reference solutions obtained with a spectral method. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2016

Ce travail traite de la modélisation des plasmas de fusion qui est ici abordée à l'aide d'un modèle Euler bi-températures et du modèle de la magnétohydrodynamique (MHD) idéale et résistive. Ces modèles sont tout d'abord établis à partir des équations de la MHD bi-fluide et nous montrons qu'ils correspondent à des régimes asymptotiques différents pour des plasmas faiblement ou fortement magnétisés. Nous décrivons ensuite les méthodes de volumes finis pour des maillages structurés et non-structurés qui ont été utilisées pour approcher les solutions de ces modèles. Pour les trois modèles mathématiques étudiés dans cette thèse, les méthodes numériques reposent sur des schémas de relaxation. Afin d'appliquer ces méthodes aux problèmes de fusion par confinement magnétique, nous décrivons comment modifier les méthodes de volumes finis pour les appliquer à des problèmes formulés en coordonnées cylindriques tout en gardant une formulation conservative forte des équations. Enfin nous étudions une stratégie pour maintenir la contrainte de divergence nulle du champ magnétique qui apparait dans les modèles MHD. Une série de cas tests numériques pour les trois modèles est présentée pour différentes géométries afin de valider les méthodes numériques proposées. / This work deals with the modeling of fusion plasma which is discussed by using a bi-temperature Euler model and the ideal and resistive magnetohydrodynamic (MHD) ones. First, these models are established from the bi-fluid MHD equations and we show that they correspond to different asymptotic regimes for lowly or highly magnetized plasma. Next, we describe the finite volume methods for structured and non-structured meshes which have been used to approximate the solution of these models. For the three mathematical models studied in this thesis, the numerical methods are based on relaxation schemes. In order to apply those methods to magnetic confinement fusion problems, we explain how to modify the finite volume methods to apply it to problem given in cylindrical coordinates while keeping a strong conservative formulation. Finally, a strategy dealing with the divergence-free constraint of the magnetic field is studied. A set of numerical tests for the three models is presented for different geometries to validate the proposed numerical methods.

No description available.