High Order Finite Element Discretization of the Compressible Euler and Navier-Stokes Equations

We present a high order accurate streamline-upwind/Petrov-Galerkin (SUPG) algorithm for the solution of the compressible Euler and Navier-Stokes equations. The flow equations are written in terms of entropy variables which result in symmetric flux Jacobian matrices and a dimensionally consistent Finite Element discretization. We show that solutions derived from quadratic element approximation are of superior quality next to their linear element counterparts. We demonstrate this through numerical solutions of both classical test cases as well as examples more practical in nature

Foundations of space-time finite element methods: polytopes, interpolation, and integration

The main purpose of this article is to facilitate the implementation of space-time finite element methods in four-dimensional space. In order to develop a finite element method in this setting, it is necessary to create a numerical foundation, or equivalently a numerical infrastructure. This foundation should include a collection of suitable elements (usually hypercubes, simplices, or closely related polytopes), numerical interpolation procedures (usually orthonormal polynomial bases), and numerical integration procedures (usually quadrature rules). It is well known that each of these areas has yet to be fully explored, and in the present article, we attempt to directly address this issue. We begin by developing a concrete, sequential procedure for constructing generic four-dimensional elements (4-polytopes). Thereafter, we review the key numerical properties of several canonical elements: the tesseract, tetrahedral prism, and pentatope. Here, we provide explicit expressions for orthonormal polynomial bases on these elements. Next, we construct fully symmetric quadrature rules with positive weights that are capable of exactly integrating high-degree polynomials, e.g. up to degree 17 on the tesseract. Finally, the quadrature rules are successfully tested using a set of canonical numerical experiments on polynomial and transcendental functions.Comment: 34 pages, 18 figure

Volume Tracking: A new method for quantitative assessment and visualization of intracardiac blood flow from three-dimensional, time-resolved, three-component magnetic resonance velocity mapping

<p>Abstract</p> <p>Background</p> <p>Functional and morphological changes of the heart influence blood flow patterns. Therefore, flow patterns may carry diagnostic and prognostic information. Three-dimensional, time-resolved, three-directional phase contrast cardiovascular magnetic resonance (4D PC-CMR) can image flow patterns with unique detail, and using new flow visualization methods may lead to new insights. The aim of this study is to present and validate a novel visualization method with a quantitative potential for blood flow from 4D PC-CMR, called Volume Tracking, and investigate if Volume Tracking complements particle tracing, the most common visualization method used today.</p> <p>Methods</p> <p>Eight healthy volunteers and one patient with a large apical left ventricular aneurysm underwent 4D PC-CMR flow imaging of the whole heart. Volume Tracking and particle tracing visualizations were compared visually side-by-side in a visualization software package. To validate Volume Tracking, the number of particle traces that agreed with the Volume Tracking visualizations was counted and expressed as a percentage of total released particles in mid-diastole and end-diastole respectively. Two independent observers described blood flow patterns in the left ventricle using Volume Tracking visualizations.</p> <p>Results</p> <p>Volume Tracking was feasible in all eight healthy volunteers and in the patient. Visually, Volume Tracking and particle tracing are complementary methods, showing different aspects of the flow. When validated against particle tracing, on average 90.5% and 87.8% of the particles agreed with the Volume Tracking surface in mid-diastole and end-diastole respectively. Inflow patterns in the left ventricle varied between the subjects, with excellent agreement between observers. The left ventricular inflow pattern in the patient differed from the healthy subjects.</p> <p>Conclusion</p> <p>Volume Tracking is a new visualization method for blood flow measured by 4D PC-CMR. Volume Tracking complements and provides incremental information compared to particle tracing that may lead to a better understanding of blood flow and may improve diagnosis and prognosis of cardiovascular diseases.</p

Iron and bismuth bound human serum transferrin reveals a partially-opened conformation in the N-lobe

Human serum transferrin (hTF) binds Fe(III) tightly but reversibly, and delivers it to cells via a receptor-mediated endocytosis process. The metal-binding and release result in significant conformational changes of the protein. Here, we report the crystal structures of diferric-hTF (Fe N Fe C-hTF) and bismuth-bound hTF (Bi N Fe C-hTF) at 2.8 and 2.4 Å resolutions respectively. Notably, the N-lobes of both structures exhibit unique 'partially-opened' conformations between those of the apo-hTF and holo-hTF. Fe(III) and Bi(III) in the N-lobe coordinate to, besides anions, only two (Tyr95 and Tyr188) and one (Tyr188) tyrosine residues, respectively, in contrast to four residues in the holo-hTF. The C-lobe of both structures are fully closed with iron coordinating to four residues and a carbonate. The structures of hTF observed here represent key conformers captured in the dynamic nature of the transferrin family proteins and provide a structural basis for understanding the mechanism of metal uptake and release in transferrin families. © 2012 Macmillan Publishers Limited. All rights reserved.published_or_final_versio

The Importance of Eigenvectors for Local Preconditioners of the Euler Equations

The design of local preconditioners to accelerate the convergence to a steady state for the compressible Euler equations has so far been solely based on eigenvalue analysis. However, numerical evidence exists that the eigenvector structure also has an influence on the performance of preconditioners, and should therefore be included in the design process. In this paper, we present the mathematical framework for the eigenvector analysis of local preconditioners for the multi-dimensional Euler equations. The non-normality of the preconditioned system is crucial in determining the potential for transient amplification of perturbations. Several existing local preconditioners are shown to possess a highly nonnormal structure for low Mach numbers. This non-normality leads to significant robustness problems at stagnation points. A modification to these preconditioners which eliminates the non-normality is suggested, and numerical results are presented showing the marked improvement in robustne..

Comparisons of Experimental and Numerical Results for Axisymmetric Vortex Breakdown in Pipes

Experimental data for the bubble mode of breakdown in pipe flows is compared with calculations of the axisymmetric form of the Navier-Stokes equations. The location of breakdown in the calculations is found to be extremely sensitive to the inlet conditions; however, generally good agreement between the experimental and numerical breakdown locations is obtained. Detailed comparisons of the velocity distributions show that the bubble form of breakdown is well approximated by the axisymmetric Navier-Stokes equations for the flow upstream of the middle of the recirculation region. In the bubble tail, larger discrepancies exist between the experiment and calculation suggesting that the bubble form of breakdown is unstable to non-axisymmetric disturbances in the tail. Also, high Reynolds number bubble structures are found to exhibit periodic vortex ring shedding from the tail of the breakdown bubble. 1 Introduction Vortex breakdown has been a widely researched fluid phenomenon since its dis..

A Robust Multigrid Algorithm for the Euler Equations with Local Preconditioning and Semi-coarsening

A semi-coarsened multigrid algorithm with a point block Jacobi, multi-stage smoother for second order upwind discretizations of the two-dimensional Euler equations is presented which produces convergence rates independent of grid size for moderate subsonic Mach numbers. By modification of this base algorithm to include local preconditioning for low Mach number flows, the convergence becomes largely independent of grid size and Mach number over a range of flow conditions from nearly incompressible to transonic flows including internal and external flows. A local limiting technique is introduced to increase the robustness of preconditioning in the presence of stagnation points. Computational timings are made showing that the semi-coarsening algorithm requires O(N) time to lower the fine grid residual six orders of magnitude where N is the number of cells. By comparison, the same algorithm applied to a full-coarsening approach requires O(N 3=2 ) time, and, in nearly all cases, the semi-..