5,132 research outputs found
An Overview of Variational Integrators
The purpose of this paper is to survey some recent advances in variational
integrators for both finite dimensional mechanical systems as well as continuum
mechanics. These advances include the general development of discrete
mechanics, applications to dissipative systems, collisions, spacetime integration algorithms,
AVIâs (Asynchronous Variational Integrators), as well as reduction for
discrete mechanical systems. To keep the article within the set limits, we will only
treat each topic briefly and will not attempt to develop any particular topic in
any depth. We hope, nonetheless, that this paper serves as a useful guide to the
literature as well as to future directions and open problems in the subject
Point vortices on the sphere: a case with opposite vorticities
We study systems formed of 2N point vortices on a sphere with N vortices of
strength +1 and N vortices of strength -1. In this case, the Hamiltonian is
conserved by the symmetry which exchanges the positive vortices with the
negative vortices. We prove the existence of some fixed and relative
equilibria, and then study their stability with the ``Energy Momentum Method''.
Most of the results obtained are nonlinear stability results. To end, some
bifurcations are described.Comment: 35 pages, 9 figure
Hamiltonian approach to hybrid plasma models
The Hamiltonian structures of several hybrid kinetic-fluid models are
identified explicitly, upon considering collisionless Vlasov dynamics for the
hot particles interacting with a bulk fluid. After presenting different
pressure-coupling schemes for an ordinary fluid interacting with a hot gas, the
paper extends the treatment to account for a fluid plasma interacting with an
energetic ion species. Both current-coupling and pressure-coupling MHD schemes
are treated extensively. In particular, pressure-coupling schemes are shown to
require a transport-like term in the Vlasov kinetic equation, in order for the
Hamiltonian structure to be preserved. The last part of the paper is devoted to
studying the more general case of an energetic ion species interacting with a
neutralizing electron background (hybrid Hall-MHD). Circulation laws and
Casimir functionals are presented explicitly in each case.Comment: 27 pages, no figures. To appear in J. Phys.
Geometric, Variational Integrators for Computer Animation
We present a general-purpose numerical scheme for time integration of Lagrangian dynamical systemsâan important
computational tool at the core of most physics-based animation techniques. Several features make this
particular time integrator highly desirable for computer animation: it numerically preserves important invariants,
such as linear and angular momenta; the symplectic nature of the integrator also guarantees a correct energy
behavior, even when dissipation and external forces are added; holonomic constraints can also be enforced quite
simply; finally, our simple methodology allows for the design of high-order accurate schemes if needed. Two key
properties set the method apart from earlier approaches. First, the nonlinear equations that must be solved during
an update step are replaced by a minimization of a novel functional, speeding up time stepping by more than a
factor of two in practice. Second, the formulation introduces additional variables that provide key flexibility in the
implementation of the method. These properties are achieved using a discrete form of a general variational principle
called the Pontryagin-Hamilton principle, expressing time integration in a geometric manner. We demonstrate
the applicability of our integrators to the simulation of non-linear elasticity with implementation details
Discrete Lie Advection of Differential Forms
In this paper, we present a numerical technique for performing Lie advection
of arbitrary differential forms. Leveraging advances in high-resolution finite
volume methods for scalar hyperbolic conservation laws, we first discretize the
interior product (also called contraction) through integrals over Eulerian
approximations of extrusions. This, along with Cartan's homotopy formula and a
discrete exterior derivative, can then be used to derive a discrete Lie
derivative. The usefulness of this operator is demonstrated through the
numerical advection of scalar fields and 1-forms on regular grids.Comment: Accepted version; to be published in J. FoC
Nonaffine Correlations in Random Elastic Media
Materials characterized by spatially homogeneous elastic moduli undergo
affine distortions when subjected to external stress at their boundaries, i.e.,
their displacements \uv (\xv) from a uniform reference state grow linearly
with position \xv, and their strains are spatially constant. Many materials,
including all macroscopically isotropic amorphous ones, have elastic moduli
that vary randomly with position, and they necessarily undergo nonaffine
distortions in response to external stress. We study general aspects of
nonaffine response and correlation using analytic calculations and numerical
simulations. We define nonaffine displacements \uv' (\xv) as the difference
between \uv (\xv) and affine displacements, and we investigate the
nonaffinity correlation function
and related functions. We introduce four model random systems with random
elastic moduli induced by locally random spring constants, by random
coordination number, by random stress, or by any combination of these. We show
analytically and numerically that scales as A |\xv|^{-(d-2)}
where the amplitude is proportional to the variance of local elastic moduli
regardless of the origin of their randomness. We show that the driving force
for nonaffine displacements is a spatial derivative of the random elastic
constant tensor times the constant affine strain. Random stress by itself does
not drive nonaffine response, though the randomness in elastic moduli it may
generate does. We study models with both short and long-range correlations in
random elastic moduli.Comment: 22 Pages, 18 figures, RevTeX
Mapping of serotype-specific, immunodominant epitopes in the NS-4 region of hepatitis C virus (HCV):use of type-specific peptides to serologically differentiate infections with HCV types 1, 2, and 3
The effect of sequence variability between different types of hepatitis C virus (HCV) on the antigenicity of the NS-4 protein was investigated by epitope mapping and by enzyme-linked immunosorbent assay with branched oligopeptides. Epitope mapping of the region between amino acid residues 1679 and 1768 in the HCV polyprotein revealed two major antigenic regions (1961 to 1708 and 1710 to 1728) that were recognized by antibody elicited upon natural infection of HCV. The antigenic regions were highly variable between variants of HCV, with only 50 to 60% amino acid sequence similarity between types 1, 2, and 3. Although limited serological cross-reactivity between HCV types was detected between peptides, particularly in the first antigenic region of NS-4, type-specific reactivity formed the principal component of the natural humoral immune response to NS-4. Type-specific antibody to particular HCV types was detected in 89% of the samples from anti-HCV-positive blood donors and correlated almost exactly with genotypic analysis of HCV sequences amplified from the samples by polymerase chain reaction. Whereas almost all blood donors appeared to be infected with a single virus type (97%), a higher proportion of samples (40%) from hemophiliacs infected from transfusion of non-heat-inactivated clotting factor contained antibody to two or even all three HCV types, providing evidence that long-term exposure may lead to multiple infection with different variants of HCV
Un-reduction
This paper provides a full geometric development of a new technique called
un-reduction, for dealing with dynamics and optimal control problems posed on
spaces that are unwieldy for numerical implementation. The technique, which was
originally concieved for an application to image dynamics, uses Lagrangian
reduction by symmetry in reverse. A deeper understanding of un-reduction leads
to new developments in image matching which serve to illustrate the
mathematical power of the technique.Comment: 25 pages, revised versio
On the use of projectors for Hamiltonian systems and their relationship with Dirac brackets
The role of projectors associated with Poisson brackets of constrained
Hamiltonian systems is analyzed. Projectors act in two instances in a bracket:
in the explicit dependence on the variables and in the computation of the
functional derivatives. The role of these projectors is investigated by using
Dirac's theory of constrained Hamiltonian systems. Results are illustrated by
three examples taken from plasma physics: magnetohydrodynamics, the
Vlasov-Maxwell system, and the linear two-species Vlasov system with
quasineutrality
Multisymplectic Geometry and Multisymplectic Preissman Scheme for the KP Equation
The multisymplectic structure of the KP equation is obtained directly from
the variational principal. Using the covariant De Donder-Weyl Hamilton function
theories, we reformulate the KP equation to the multisymplectic form which
proposed by Bridges. From the multisymplectic equation, we can derive a
multisymplectic numerical scheme of the KP equation which can be simplified to
multisymplectic forty-five points scheme.Comment: 17 papges, 8 figure
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