151 research outputs found
A highly parallel multigrid-like method for the solution of the Euler equations
We consider a highly parallel multigrid-like method for the solution of the two dimensional steady Euler equations. The new method, introduced as filtering multigrid, is similar to a standard multigrid scheme in that convergence on the finest grid is accelerated by iterations on coarser grids. In the filtering method, however, additional fine grid subproblems are processed concurrently with coarse grid computations to further accelerate convergence. These additional problems are obtained by splitting the residual into a smooth and an oscillatory component. The smooth component is then used to form a coarse grid problem (similar to standard multigrid) while the oscillatory component is used for a fine grid subproblem. The primary advantage in the filtering approach is that fewer iterations are required and that most of the additional work per iteration can be performed in parallel with the standard coarse grid computations. We generalize the filtering algorithm to a version suitable for nonlinear problems. We emphasize that this generalization is conceptually straight-forward and relatively easy to implement. In particular, no explicit linearization (e.g., formation of Jacobians) needs to be performed (similar to the FAS multigrid approach). We illustrate the nonlinear version by applying it to the Euler equations, and presenting numerical results. Finally, a performance evaluation is made based on execution time models and convergence information obtained from numerical experiments
An algebraic multigrid method for mixed discretizations of the Navier-Stokes equations
Algebraic multigrid (AMG) preconditioners are considered for discretized
systems of partial differential equations (PDEs) where unknowns associated with
different physical quantities are not necessarily co-located at mesh points.
Specifically, we investigate a mixed finite element discretization of
the incompressible Navier-Stokes equations where the number of velocity nodes
is much greater than the number of pressure nodes. Consequently, some velocity
degrees-of-freedom (dofs) are defined at spatial locations where there are no
corresponding pressure dofs. Thus, AMG approaches leveraging this co-located
structure are not applicable. This paper instead proposes an automatic AMG
coarsening that mimics certain pressure/velocity dof relationships of the
discretization. The main idea is to first automatically define coarse
pressures in a somewhat standard AMG fashion and then to carefully (but
automatically) choose coarse velocity unknowns so that the spatial location
relationship between pressure and velocity dofs resembles that on the finest
grid. To define coefficients within the inter-grid transfers, an energy
minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific
coarsening schemes and grid transfer sparsity patterns, and so it is applicable
to the proposed coarsening. Numerical results highlighting solver performance
are given on Stokes and incompressible Navier-Stokes problems.Comment: Submitted to a journa
Analysis of a parallel multigrid algorithm
The parallel multigrid algorithm of Frederickson and McBryan (1987) is considered. This algorithm uses multiple coarse-grid problems (instead of one problem) in the hope of accelerating convergence and is found to have a close relationship to traditional multigrid methods. Specifically, the parallel coarse-grid correction operator is identical to a traditional multigrid coarse-grid correction operator, except that the mixing of high and low frequencies caused by aliasing error is removed. Appropriate relaxation operators can be chosen to take advantage of this property. Comparisons between the standard multigrid and the new method are made
Variables Impacting Southern Illinois Airport Activity Between The Years 2000 And 2010
Southern Illinois Airport is a publicly used and operated airport that forecasts its airport activity for airport planning purposes. This research uses linear regression analysis to identify independent variables impacting based aircraft, local civilian operations and itinerant general aviation aircraft operations at Southern Illinois Airport between 2000 and 2010. Regression analysis is a Federal Aviation Administration approved method in determining relationships between airport activity factors and other variables, but is typically used in large scale airport system planning and not at general aviation airports such as Southern Illinois Airport. The results appear promising for future use in airport planning as the test did identify significant relationships between Southern Illinois Airport activity and independent variables
Compression and Reduced Representation Techniques for Patch-Based Relaxation
Patch-based relaxation refers to a family of methods for solving linear
systems which partitions the matrix into smaller pieces often corresponding to
groups of adjacent degrees of freedom residing within patches of the
computational domain. The two most common families of patch-based methods are
block-Jacobi and Schwarz methods, where the former typically corresponds to
non-overlapping domains and the later implies some overlap. We focus on cases
where each patch consists of the degrees of freedom within a finite element
method mesh cell. Patch methods often capture complex local physics much more
effectively than simpler point-smoothers such as Jacobi; however, forming,
inverting, and applying each patch can be prohibitively expensive in terms of
both storage and computation time. To this end, we propose several approaches
for performing analysis on these patches and constructing a reduced
representation. The compression techniques rely on either matrix norm
comparisons or unsupervised learning via a clustering approach. We illustrate
how it is frequently possible to retain/factor less than 5% of all patches and
still develop a method that converges with the same number of iterations or
slightly more than when all patches are stored/factored.Comment: 16 pages, 5 figure
USING TARGETED, SMALL-GROUP INSTRUCTION AND CULTURALLY RESPONSIVE TEACHING TO IMPROVE READING ENGAGEMENT AND CONFIDENCE
The purpose of this study was to investigate what happens to reading engagement and student confidence when targeted instruction and culturally responsive teaching is used in a middle school literacy classroom. Individual instruction in the areas of reading comprehension and fluency, as well as book clubs and literature circles, were used over a three-month period in an effort to motivate striving and reluctant readers and build confidence in their reading skills. Eleven sixth grade students participated in the study, and both qualitative and quantitative research methods were used. Data was collected using a teacher\u27s research journal, audio recordings, surveys, interviews, and student work samples. Patterns of reading volume, stamina, and student responses to targeted instruction were analyzed using triangulation methods and coding in order to determine common themes. Based on the findings of the study, students tended to be more motivated to read texts that included topics in which they had prior knowledge or otherwise interested them. Reading engagement also tended to increase surrounding social interactions with peers about shared texts. Increased reading engagement, as well as evident progress with fluency and reading comprehension skills, tended to cause student confidence in reading to increase as well. This study was conducted during the Covid-19 pandemic. Therefore, further research is needed during a more typical school year
Performance of a parallel code for the Euler equations on hypercube computers
The performance of hypercubes were evaluated on a computational fluid dynamics problem and the parallel environment issues were considered that must be addressed, such as algorithm changes, implementation choices, programming effort, and programming environment. The evaluation focuses on a widely used fluid dynamics code, FLO52, which solves the two dimensional steady Euler equations describing flow around the airfoil. The code development experience is described, including interacting with the operating system, utilizing the message-passing communication system, and code modifications necessary to increase parallel efficiency. Results from two hypercube parallel computers (a 16-node iPSC/2, and a 512-node NCUBE/ten) are discussed and compared. In addition, a mathematical model of the execution time was developed as a function of several machine and algorithm parameters. This model accurately predicts the actual run times obtained and is used to explore the performance of the code in interesting but yet physically realizable regions of the parameter space. Based on this model, predictions about future hypercubes are made
A cognitive framework for analyzing and describing introductory students' use and understanding of mathematics in physics
Many introductory, algebra-based physics students perform poorly on mathematical problem solving tasks in physics. There are at least two possible, distinct reasons for this poor performance: (1) students simply lack the mathematical skills needed to solve problems in physics, or (2) students do not know how to apply the mathematical skills they have to particular problem situations in physics. While many students do lack the requisite mathematical skills, a major finding from this work is that the majority of students possess the requisite mathematical skills, yet fail to use or interpret them in the context of physics.
In this thesis I propose a theoretical framework to analyze and describe students' mathematical thinking in physics. In particular, I attempt to answer two questions. What are the cognitive tools involved in formal mathematical thinking in physics? And, why do students make the kinds of mistakes they do when using mathematics in physics?
According to the proposed theoretical framework there are three major theoretical constructs: mathematical resources, which are the knowledge elements that are activated in mathematical thinking and problem solving; epistemic games, which are patterns of activities that use particular kinds of knowledge to create new knowledge or solve a problem; and frames, which are structures of expectations that determine how individuals interpret situations or events.
The empirical basis for this study comes from videotaped sessions of college students solving homework problems. The students are enrolled in an algebra-based introductory physics course. The videotapes were transcribed and analyzed using the aforementioned theoretical framework.
Two important results from this work are: (1) the construction of a theoretical framework that offers researchers a vocabulary (ontological classification of cognitive structures) and grammar (relationship between the cognitive structures) for understanding the nature and origin of mathematical use in the context physics, and (2) a detailed understanding, in terms of the proposed theoretical framework, of the errors that students make when using mathematics in the context of physics
- …