44 research outputs found
Monolithic Multigrid for Magnetohydrodynamics
The magnetohydrodynamics (MHD) equations model a wide range of plasma physics
applications and are characterized by a nonlinear system of partial
differential equations that strongly couples a charged fluid with the evolution
of electromagnetic fields. After discretization and linearization, the
resulting system of equations is generally difficult to solve due to the
coupling between variables, and the heterogeneous coefficients induced by the
linearization process. In this paper, we investigate multigrid preconditioners
for this system based on specialized relaxation schemes that properly address
the system structure and coupling. Three extensions of Vanka relaxation are
proposed and applied to problems with up to 170 million degrees of freedom and
fluid and magnetic Reynolds numbers up to 400 for stationary problems and up to
20,000 for time-dependent problems
Beyond deficit-based models of learners' cognition: Interpreting engineering students' difficulties with sense-making in terms of fine-grained epistemological and conceptual dynamics
Researchers have argued against deficit-based explanations of students'
troubles with mathematical sense-making, pointing instead to factors such as
epistemology: students' beliefs about knowledge and learning can hinder them
from activating and integrating productive knowledge they have. In this case
study of an engineering major solving problems (about content from his
introductory physics course) during a clinical interview, we show that "Jim"
has all the mathematical and conceptual knowledge he would need to solve a
hydrostatic pressure problem that we posed to him. But he reaches and sticks
with an incorrect answer that violates common sense. We argue that his lack of
mathematical sense-making-specifically, translating and reconciling between
mathematical and everyday/common-sense reasoning-stems in part from his
epistemological views, i.e., his views about the nature of knowledge and
learning. He regards mathematical equations as much more trustworthy than
everyday reasoning, and he does not view mathematical equations as expressing
meaning that tractably connects to common sense. For these reasons, he does not
view reconciling between common sense and mathematical formalism as either
necessary or plausible to accomplish. We, however, avoid a potential "deficit
trap"-substituting an epistemological deficit for a concepts/skills deficit-by
incorporating multiple, context-dependent epistemological stances into Jim's
cognitive dynamics. We argue that Jim's epistemological stance contains
productive seeds that instructors could build upon to support Jim's
mathematical sense-making: He does see common-sense as connected to formalism
(though not always tractably so) and in some circumstances this connection is
both salient and valued.Comment: Submitted to the Journal of Engineering Educatio
An events based algorithm for distributing concurrent tasks on multi-core architectures
In this paper, a programming model is presented which enables scalable parallel performance on multi-core shared memory architectures. The model has been developed for application to a wide range of numerical simulation problems. Such problems involve time stepping or iteration algorithms where synchronization of multiple threads of execution is required. It is shown that traditional approaches to parallelism including message passing and scatter-gather can be improved upon in terms of speed-up and memory management. Using spatial decomposition to create orthogonal computational tasks, a new task management algorithm called H-Dispatch is developed. This algorithm makes efficient use of memory resources by limiting the need for garbage collection and takes optimal advantage of multiple cores by employing a “hungry” pull strategy. The technique is demonstrated on a simple finite difference solver and results are compared to traditional MPI and scatter-gather approaches. The H-Dispatch approach achieves near linear speed-up with results for efficiency of 85% on a 24-core machine. It is noted that the H-Dispatch algorithm is quite general and can be applied to a wide class of computational tasks on heterogeneous architectures involving multi-core and GPGPU hardware.Schlumberger-Doll Research CenterSaudi Aramc
MPAS-Albany Land Ice (MALI): a variable-resolution ice sheet model for Earth system modeling using Voronoi grids
We introduce MPAS-Albany Land Ice (MALI) v6.0, a new variable-resolution land ice model that uses unstructured Voronoi grids on a plane or
sphere. MALI is built using the Model for Prediction Across Scales (MPAS)
framework for developing variable-resolution Earth system model components
and the Albany multi-physics code base for the solution of coupled systems of
partial differential equations, which itself makes use of Trilinos solver
libraries. MALI includes a three-dimensional first-order momentum balance
solver (Blatter–Pattyn) by linking to the Albany-LI ice sheet velocity
solver and an explicit shallow ice velocity solver. The evolution of ice
geometry and tracers is handled through an explicit first-order horizontal
advection scheme with vertical remapping. The evolution of ice temperature is
treated using operator splitting of vertical diffusion and horizontal
advection and can be configured to use either a temperature or enthalpy
formulation. MALI includes a mass-conserving subglacial hydrology model that
supports distributed and/or channelized drainage and can optionally be
coupled to ice dynamics. Options for calving include eigencalving, which
assumes that the calving rate is proportional to extensional strain rates. MALI is
evaluated against commonly used exact solutions and community benchmark
experiments and shows the expected accuracy. Results for the MISMIP3d
benchmark experiments with MALI's Blatter–Pattyn solver fall between
published results from Stokes and L1L2 models as expected. We use the model
to simulate a semi-realistic Antarctic ice sheet problem following the
initMIP protocol and using 2 km resolution in marine ice sheet regions. MALI
is the glacier component of the Energy Exascale Earth System Model (E3SM)
version 1, and we describe current and planned coupling to other E3SM
components.</p
Research on Teaching and Learning Mathematics at the Tertiary Level:State-of-the-art and Looking Ahead
This topical survey focuses on research in tertiary mathematics education, a field that has experienced considerable growth over the last 10 years. Drawing on the most recent journal publication as well as the latest advances from recent high quality conference proceedings, our review culls out the following five emergent areas of interest: mathematics teaching at the tertiary level; the role of mathematics in other disciplines; textbooks, assessment and students’ studying practices; transition to the tertiary level; and theoretical-methodological advances. We conclude the survey with a discussion of some potential ways forward for future research in this new and rapidly developing domain of inquiry
Sand2006--2256
The application of the finite element method to nonlinear solid mechanics problems results in the neccessity to repeatedly solve a large nonlinear set of equations. In this paper we limit ourself to problems arising in constrained solid mechanics problems. It is common to apply some variant of Newton's method or a Newton-- Krylov method to such problems. Often, an analytic Jacobian matrix is formed and used in the above mentioned methods. However, if no analytic Jacobian is given, Newton methods might not be the method of choice. Here, we focus on a variational nonlinear multigrid approach that adopts the smoothed aggregation algebraic multigrid method to generate a hierachy of coarse grids in a purely algebraic manner. We use preconditioned nonlinear conjugent gradient methods and/or quasi--Newton methods as nonlinear smoothers on fine and coarse grids. In addition we discuss the possibility to augment this basic algorithm with an automatically generated Jacobian by applying a block colored finite differencing scheme. After outlining the fundamental algorithms we give some examples and provide documentation for the parallel implementation of the described method within the Trilinos framework