99,124 research outputs found
Natural discretizations for the divergence, gradient, and curl on logically rectangular grids
AbstractThis is the first in series of papers creating a discrete analog of vector analysis on logically rectangular, nonorthogonal, nonsmooth grids. We introduce notations for 2-D logically rectangular grids, describe both cell-valued and nodal discretizations for scalar functions, and construct the natural discretizations of vector fields, using the vector components normal and tangential to the cell boundaries. We then define natural discrete analogs of the divergence, gradient, and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates, such as length of elements of coordinate lines, areas of elements of coordinate surfaces, and elementary volumes.We introduce the discrete volume integral of scalar functions, the discrete surface integral, and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects. These theorems include the following: the discrete analog of relationship div A→ = 0 if and only if A→ = curl B→; curl A→ = 0 if and only if A→ = grad ϕ; if A→ = grad ϕ, then the line integral does not depend on path; and if the line integral of a vector function is equal to zero for any closed path, then this vector is the gradient of a scalar function.Last, we define the discrete operators DIV, GRAD, and CURL in terms of primitive differencing operators (based on forward and backward differences) and primitive metric operators (related to multiplications of discrete functions by length of edges, areas of surfaces, and volumes of 3-D cells). These formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints
Multi-field approach in mechanics of structural solids
We overview the basic concepts, models, and methods related to the
multi-field continuum theory of solids with complex structures. The multi-field
theory is formulated for structural solids by introducing a macrocell
consisting of several primitive cells and, accordingly, by increasing the
number of vector fields describing the response of the body to external
factors. Using this approach, we obtain several continuum models and explore
their essential properties by comparison with the original structural models.
Static and dynamical problems as well as the stability problems for structural
solids are considered. We demonstrate that the multi-field approach gives a way
to obtain families of models that generalize classical ones and are valid not
only for long-, but also for short-wavelength deformations of the structural
solid. Some examples of application of the multi-field theory and directions
for its further development are also discussed.Comment: 25 pages, 18 figure
Feature Lines for Illustrating Medical Surface Models: Mathematical Background and Survey
This paper provides a tutorial and survey for a specific kind of illustrative
visualization technique: feature lines. We examine different feature line
methods. For this, we provide the differential geometry behind these concepts
and adapt this mathematical field to the discrete differential geometry. All
discrete differential geometry terms are explained for triangulated surface
meshes. These utilities serve as basis for the feature line methods. We provide
the reader with all knowledge to re-implement every feature line method.
Furthermore, we summarize the methods and suggest a guideline for which kind of
surface which feature line algorithm is best suited. Our work is motivated by,
but not restricted to, medical and biological surface models.Comment: 33 page
Multi-field continuum theory for medium with microscopic rotations
We derive the multi-field, micropolar-type continuum theory for the
two-dimensional model of crystal having finite-size particles. Continuum
theories are usually valid for waves with wavelength much larger than the size
of primitive cell of crystal. By comparison of the dispersion relations, it is
demonstrated that in contrast to the single-field continuum theory constructed
in our previous paper the multi-field generalization is valid not only for long
but also for short waves. We show that the multi-field model can be used to
describe spatially localized short- and long wavelength distortions. Short-wave
external fields of forces and torques can be also naturally taken into account
by the multi-field continuum theory.Comment: 14 pages, 4 figures, submitted to International Journal of Solids and
Structure
Nonaxisymmetric, multi-region relaxed magnetohydrodynamic equilibrium solutions
We describe a magnetohydrodynamic (MHD) constrained energy functional for
equilibrium calculations that combines the topological constraints of ideal MHD
with elements of Taylor relaxation.
Extremizing states allow for partially chaotic magnetic fields and
non-trivial pressure profiles supported by a discrete set of ideal interfaces
with irrational rotational transforms.
Numerical solutions are computed using the Stepped Pressure Equilibrium Code,
SPEC, and benchmarks and convergence calculations are presented.Comment: Submitted to Plasma Physics and Controlled Fusion for publication
with a cluster of papers associated with workshop: Stability and Nonlinear
Dynamics of Plasmas, October 31, 2009 Atlanta, GA on occasion of 65th
birthday of R.L. Dewar. V2 is revised for referee
The Topology ToolKit
This system paper presents the Topology ToolKit (TTK), a software platform
designed for topological data analysis in scientific visualization. TTK
provides a unified, generic, efficient, and robust implementation of key
algorithms for the topological analysis of scalar data, including: critical
points, integral lines, persistence diagrams, persistence curves, merge trees,
contour trees, Morse-Smale complexes, fiber surfaces, continuous scatterplots,
Jacobi sets, Reeb spaces, and more. TTK is easily accessible to end users due
to a tight integration with ParaView. It is also easily accessible to
developers through a variety of bindings (Python, VTK/C++) for fast prototyping
or through direct, dependence-free, C++, to ease integration into pre-existing
complex systems. While developing TTK, we faced several algorithmic and
software engineering challenges, which we document in this paper. In
particular, we present an algorithm for the construction of a discrete gradient
that complies to the critical points extracted in the piecewise-linear setting.
This algorithm guarantees a combinatorial consistency across the topological
abstractions supported by TTK, and importantly, a unified implementation of
topological data simplification for multi-scale exploration and analysis. We
also present a cached triangulation data structure, that supports time
efficient and generic traversals, which self-adjusts its memory usage on demand
for input simplicial meshes and which implicitly emulates a triangulation for
regular grids with no memory overhead. Finally, we describe an original
software architecture, which guarantees memory efficient and direct accesses to
TTK features, while still allowing for researchers powerful and easy bindings
and extensions. TTK is open source (BSD license) and its code, online
documentation and video tutorials are available on TTK's website
Combinatorial Gradient Fields for 2D Images with Empirically Convergent Separatrices
This paper proposes an efficient probabilistic method that computes
combinatorial gradient fields for two dimensional image data. In contrast to
existing algorithms, this approach yields a geometric Morse-Smale complex that
converges almost surely to its continuous counterpart when the image resolution
is increased. This approach is motivated using basic ideas from probability
theory and builds upon an algorithm from discrete Morse theory with a strong
mathematical foundation. While a formal proof is only hinted at, we do provide
a thorough numerical evaluation of our method and compare it to established
algorithms.Comment: 17 pages, 7 figure
Optimal topological simplification of discrete functions on surfaces
We solve the problem of minimizing the number of critical points among all
functions on a surface within a prescribed distance {\delta} from a given input
function. The result is achieved by establishing a connection between discrete
Morse theory and persistent homology. Our method completely removes homological
noise with persistence less than 2{\delta}, constructively proving the
tightness of a lower bound on the number of critical points given by the
stability theorem of persistent homology in dimension two for any input
function. We also show that an optimal solution can be computed in linear time
after persistence pairs have been computed.Comment: 27 pages, 8 figure
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