8 research outputs found

    Multigrid methods for nonlinear second order partial differential operators

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    This thesis is concerned with the efficient numerical solution of nonlinear partial differential equations (PDEs) of elliptic and parabolic type. Such PDEs arise frequently in models used to describe many physical phenomena, from the diffusion of a toxin in soil to the flow of viscous fluids. The main focus of this research is to better understand the implementation and performance of nonlinear multigrid methods for the solution of elliptic and parabolic PDEs, following their discretisation. For the most part finite element discretisations are considered, but other techniques are also discussed. Following discretisation of a PDE the two most frequently used nonlinear multigrid methods are Newton-Multigrid and the Full Approximation Scheme (FAS). These are both very efficient algorithms, and have the advantage that when they are applied to practical problems, their execution times scale linearly with the size of the problem being solved. Even though this has yet to be proved in theory for most problems, these methods have been widely adopted in practice in order to solve highly complex nonlinear (systems of) PDEs. Many research groups use either Newton-MG or FAS without much consideration as to which should be preferred, since both algorithms perform satisfactorily. In this thesis we address the question as to which method is likely to be more computationally efficient in practice. As part of this investigation the implementation of the algorithms is considered in a framework which allows the direct comparison of the computational effort of the two iterations. As well as this, the convergence properties of the methods are considered, applied to a variety of model problems. Extensive results are presented in the comparison, which are explained by available theory whenever possible. The strength and range of results presented allows us to confidently conclude that for a practical problem, discretised using a finite element discretisation, an improved efficiency and stability of a Newton-MG iteration, compared to an FAS iteration, is likely to be observed. The relative advantage of a Newton-MG method is likely to be larger the more complex the problem being solved becomes

    Mixed-Cell Methods for Diffusion Problems in Multiphase Systems.

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    We simulate diffusion in multimaterial systems with a cell-centered Eulerian mesh in two dimensions. A system with immiscible fluids contains sharp interfaces. An Eulerian mesh is fixed in space and does not move with the material. Therefore, cells with an interface contain multiple fluids; these are known as mixed cells. The treatment of mixed cells can vary in computational cost and accuracy. In some cases, the primary source of inaccuracy can be attributed to approximations made in modeling the mixed cells. This thesis focuses on the treatment of mixed cells based on the diffusion approximation of the transport equation. We introduce five subgrid, mixed-cell models. Two models have a single temperature for each cell, while the other three allow a separate temperature for each phase. The single-temperature models are implemented using the Support-Operators Method, which is derived herein. The first single-temperature model utilizes an effective tensor diffusivity that distinguishes diffusion tangent and normal to the interface. The second single-temperature model specifies a unique diffusivity in each corner of a mixed cell, which is effectively a mesh refinement of the mixed cell. The three multi-temperature models have increasingly accurate levels of approximation of the flux: (i) flux is calculated between cell-centers for each phase, (ii) flux is calculated between the centroid of each phase, and (iii) flux normal to an interface is calculated between centroids of each phase. The physical interpretations of these models are: (i) each phase occupies the entire cell, (ii) oblique flux is continuous, (iii) only normal flux is continuous. The standard approximation, using the harmonic mean of the diffusivities present in a mixed cell as an effective diffusivity, is also tested for comparison. We also derive two time-dependent analytical solutions for diffusion in a two-phase system, in both one and two dimensions. With the standard model as a reference point, the accuracy of the new models is quantified, and the convergence rates of the error are determined between pairs of spatial resolutions for the two problems with analytical solutions. Simulations of multiphysics and multimaterial phenomenon may benefit from increased mixed-cell fidelity achieved in this dissertation.PHDApplied PhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/107150/1/leftynm_1.pd

    Measuring and Modeling Fluid Dynamic Processes using Digital Image Sequence Analysis

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    In this thesis novel motion models have been developed and incorporated into an extended parameter estimation framework that allows to accurately estimate the parameters and regularize them if needed. The performance of this framework has been increased to real time and implemented on inexpensive graphics hardware. Confidence and situation measures have been designed to discard inaccurate estimates. A phase field approach was developed to estimate piecewise smooth motion while detecting object boundaries at the same time. These algorithmic improvements have been successfully applied to three areas of fluid dynamics: air-sea interaction, microfluidics and plant physiology. At the ocean surface, the fluxes of heat and momentum have been measured with thermographic techniques, both spatially and temporally highly resolved. These measurement techniques present milestones for research in air-sea interaction, where point measurements and particle based laboratory measurements represent the state-of-the art. Calculations were done with two models, both making complement assumptions. Still, results derived from both models agree remarkably well. Measurements were conducted in laboratory settings as well as in the field. Microfluidic flow was measured with a new approach to molecular tagging velocimetry that explicitly models Taylor dispersion. This has lead to an increase in accuracy and applicability. Inaccuracies and problems of previous approaches due to Taylor dispersion were successfully evaded. Ground truth test measurements have been conducted, proving the accuracy of this novel technique. For the first time, flow velocities were measured in the xylem of plant leaves with active thermography. This represents a technique for measuring these flows on extended leaf areas on free standing plants, minimizing the impact caused by the measurement. Ground truth measurements on perfused leafs were performed. Measurements were also conducted on free standing plants in a climatic chamber, to measure xylem flows and relate flow velocities to environmental parameter. With a cuvette, environmental factors were varied locally. These measurements underlined the sensitivity of the new approach. A linear relationship in between flow rates and xylem diameter was found
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