5,825 research outputs found
Isoparametric and Dupin Hypersurfaces
A hypersurface in a real space-form , or is
isoparametric if it has constant principal curvatures. For and
, the classification of isoparametric hypersurfaces is complete and
relatively simple, but as Elie Cartan showed in a series of four papers in
1938-1940, the subject is much deeper and more complex for hypersurfaces in the
sphere . A hypersurface in a real space-form is proper Dupin if
the number of distinct principal curvatures is constant on , and
each principal curvature function is constant along each leaf of its
corresponding principal foliation. This is an important generalization of the
isoparametric property that has its roots in nineteenth century differential
geometry and has been studied effectively in the context of Lie sphere
geometry. This paper is a survey of the known results in these fields with
emphasis on results that have been obtained in more recent years and discussion
of important open problems in the field.Comment: This is a contribution to the Special Issue "Elie Cartan and
Differential Geometry", published in SIGMA (Symmetry, Integrability and
Geometry: Methods and Applications) at http://www.emis.de/journals/SIGM
Geometric rigidity of constant heat flow
Let be a compact Riemannian manifold with smooth boundary and let
be the solution of the heat equation on , having constant unit
initial data and Dirichlet boundary conditions ( on the
boundary, at all times). If at every time the normal derivative of is
a constant function on the boundary, we say that has the {\it constant
flow property}. This gives rise to an overdetermined parabolic problem, and our
aim is to classify the manifolds having this property. In fact, if the metric
is analytic, we prove that has the constant flow property if and only
if it is an {\it isoparametric tube}, that is, it is a solid tube of constant
radius around a closed, smooth, minimal submanifold, with the additional
property that all equidistants to the boundary (parallel hypersurfaces) are
smooth and have constant mean curvature. Hence, the constant flow property can
be viewed as an analytic counterpart to the isoparametric property. Finally, we
relate the constant flow property with other overdetermined problems, in
particular, the well-known Serrin problem on the mean-exit time function, and
discuss a counterexample involving minimal free boundary immersions into
Euclidean balls.Comment: Replaces the earlier version arXiv: 1709.03447. To appear in Calculus
of Variations and PD
Analysis of a high order Trace Finite Element Method for PDEs on level set surfaces
We present a new high order finite element method for the discretization of
partial differential equations on stationary smooth surfaces which are
implicitly described as the zero level of a level set function. The
discretization is based on a trace finite element technique. The higher
discretization accuracy is obtained by using an isoparametric mapping of the
volume mesh, based on the level set function, as introduced in [C. Lehrenfeld,
\emph{High order unfitted finite element methods on level set domains using
isoparametric mappings}, Comp. Meth. Appl. Mech. Engrg. 2016]. The resulting
trace finite element method is easy to implement. We present an error analysis
of this method and derive optimal order -norm error bounds. A
second topic of this paper is a unified analysis of several stabilization
methods for trace finite element methods. Only a stabilization method which is
based on adding an anisotropic diffusion in the volume mesh is able to control
the condition number of the stiffness matrix also for the case of higher order
discretizations. Results of numerical experiments are included which confirm
the theoretical findings on optimal order discretization errors and uniformly
bounded condition numbers.Comment: 28 pages, 5 figures, 1 tabl
Trace Finite Element Methods for PDEs on Surfaces
In this paper we consider a class of unfitted finite element methods for
discretization of partial differential equations on surfaces. In this class of
methods known as the Trace Finite Element Method (TraceFEM), restrictions or
traces of background surface-independent finite element functions are used to
approximate the solution of a PDE on a surface. We treat equations on steady
and time-dependent (evolving) surfaces. Higher order TraceFEM is explained in
detail. We review the error analysis and algebraic properties of the method.
The paper navigates through the known variants of the TraceFEM and the
literature on the subject
Adaptive beamforming of steerable array monopole antenna for WLAN application
The modern communication systems are using smart portable devices that operate on WLAN frequency of 2.45 GHz. One of the serious limitations of handled devices is difficult to achieve a direct connection between the transmitter and receiver. Therefore, a smart steerable patter array antenna is highly recommended for new generation communication. Using low-cost steerable passive monopole array antenna can achieve a beam steering and high gain. Loading an additional reactance to the passive elements of the array are changed the mutual coupling between the arrays, which leads to steering the pattern to the desired direction. However, this needs fast process accurate optimised parameters. In this study, four passives one active monopole array antenna is proposed and simulated by using CST Microwave Studio software. The adaptive beamforming is proposed by using downhill simplex algorithm. The results show that the optimum reactance values are suggested after 0.074 second with 94 iterations to achieve a direction of arrival of 180° and 0°. The simulated radiation is successfully steered to the direction of 180 ° by adding the suggested reactance into the passive elements. Furthermore, the antenna gain is improved by 1.3 dBi that achieved a value of 5.3 dBi.The envelope-cross-correlation (ECC) shows magnitudes less than 0.5 among the elements. This algorithm successfully is provided with the optimum reactance values. The proposed approach can be considered a fast and significant candidate for new generation of smart communication WLAN applications
Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes
We present a new family of high order accurate fully discrete one-step
Discontinuous Galerkin (DG) finite element schemes on moving unstructured
meshes for the solution of nonlinear hyperbolic PDE in multiple space
dimensions, which may also include parabolic terms in order to model
dissipative transport processes. High order piecewise polynomials are adopted
to represent the discrete solution at each time level and within each spatial
control volume of the computational grid, while high order of accuracy in time
is achieved by the ADER approach. In our algorithm the spatial mesh
configuration can be defined in two different ways: either by an isoparametric
approach that generates curved control volumes, or by a piecewise linear
decomposition of each spatial control volume into simplex sub-elements. Our
numerical method belongs to the category of direct
Arbitrary-Lagrangian-Eulerian (ALE) schemes, where a space-time conservation
formulation of the governing PDE system is considered and which already takes
into account the new grid geometry directly during the computation of the
numerical fluxes. Our new Lagrangian-type DG scheme adopts the novel a
posteriori sub-cell finite volume limiter method, in which the validity of the
candidate solution produced in each cell by an unlimited ADER-DG scheme is
verified against a set of physical and numerical detection criteria. Those
cells which do not satisfy all of the above criteria are flagged as troubled
cells and are recomputed with a second order TVD finite volume scheme. The
numerical convergence rates of the new ALE ADER-DG schemes are studied up to
fourth order in space and time and several test problems are simulated.
Finally, an application inspired by Inertial Confinement Fusion (ICF) type
flows is considered by solving the Euler equations and the PDE of viscous and
resistive magnetohydrodynamics (VRMHD).Comment: 39 pages, 21 figure
The quarter-point quadratic isoparametric element as a singular element for crack problems
The quadratic isoparametric elements which embody the inverse square root singularity are used for calculating the stress intensity factors at tips of cracks. The strain singularity at a point or an edge is obtained in a simple manner by placing the mid-side nodes at quarter points in the vicinity of the crack tip or an edge. These elements are implemented in NASTRAN as dummy elements. The method eliminates the use of special crack tip elements and in addition, these elements satisfy the constant strain and rigid body modes required for convergence
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