79,330 research outputs found
Volume of minimal hypersurfaces in manifolds with nonnegative Ricci curvature
We establish a min-max estimate on the volume width of a closed Riemannian
manifold with nonnegative Ricci curvature. More precisely, we show that every
closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse
function whose level set volume is bounded in terms of the volume of the
manifold. As a consequence of this sweep-out estimate, there exists an
embedded, closed (possibly singular) minimal hypersurface whose volume is
bounded in terms of the volume of the manifold.Comment: 17 pages, 1 figur
Variational Theory and Domain Decomposition for Nonlocal Problems
In this article we present the first results on domain decomposition methods
for nonlocal operators. We present a nonlocal variational formulation for these
operators and establish the well-posedness of associated boundary value
problems, proving a nonlocal Poincar\'{e} inequality. To determine the
conditioning of the discretized operator, we prove a spectral equivalence which
leads to a mesh size independent upper bound for the condition number of the
stiffness matrix. We then introduce a nonlocal two-domain variational
formulation utilizing nonlocal transmission conditions, and prove equivalence
with the single-domain formulation. A nonlocal Schur complement is introduced.
We establish condition number bounds for the nonlocal stiffness and Schur
complement matrices. Supporting numerical experiments demonstrating the
conditioning of the nonlocal one- and two-domain problems are presented.Comment: Updated the technical part. In press in Applied Mathematics and
Computatio
Studies of Thermally Unstable Accretion Disks around Black Holes with Adaptive Pseudo-Spectral Domain Decomposition Method I. Limit-Cycle Behavior in the Case of Moderate Viscosity
We present a numerical method for spatially 1.5-dimensional and
time-dependent studies of accretion disks around black holes, that is
originated from a combination of the standard pseudo-spectral method and the
adaptive domain decomposition method existing in the literature, but with a
number of improvements in both the numerical and physical senses. In
particular, we introduce a new treatment for the connection at the interfaces
of decomposed subdomains, construct an adaptive function for the mapping
between the Chebyshev-Gauss-Lobatto collocation points and the physical
collocation points in each subdomain, and modify the over-simplified
1-dimensional basic equations of accretion flows to account for the effects of
viscous stresses in both the azimuthal and radial directions. Our method is
verified by reproducing the best results obtained previously by Szuszkiewicz &
Miller on the limit-cycle behavior of thermally unstable accretion disks with
moderate viscosity. A new finding is that, according to our computations, the
Bernoulli function of the matter in such disks is always and everywhere
negative, so that outflows are unlikely to originate from these disks. We are
encouraged to study the more difficult case of thermally unstable accretion
disks with strong viscosity, and wish to report our results in a subsequent
paper.Comment: 29 pages, 8 figures, accepted by Ap
Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization
In this paper, we propose a general framework for constructing IGA-suitable
planar B-spline parameterizations from given complex CAD boundaries consisting
of a set of B-spline curves. Instead of forming the computational domain by a
simple boundary, planar domains with high genus and more complex boundary
curves are considered. Firstly, some pre-processing operations including
B\'ezier extraction and subdivision are performed on each boundary curve in
order to generate a high-quality planar parameterization; then a robust planar
domain partition framework is proposed to construct high-quality patch-meshing
results with few singularities from the discrete boundary formed by connecting
the end points of the resulting boundary segments. After the topology
information generation of quadrilateral decomposition, the optimal placement of
interior B\'ezier curves corresponding to the interior edges of the
quadrangulation is constructed by a global optimization method to achieve a
patch-partition with high quality. Finally, after the imposition of
C1=G1-continuity constraints on the interface of neighboring B\'ezier patches
with respect to each quad in the quadrangulation, the high-quality B\'ezier
patch parameterization is obtained by a C1-constrained local optimization
method to achieve uniform and orthogonal iso-parametric structures while
keeping the continuity conditions between patches. The efficiency and
robustness of the proposed method are demonstrated by several examples which
are compared to results obtained by the skeleton-based parameterization
approach
On a decomposition of regular domains into John domains with uniform constants
We derive a decomposition result for regular, two-dimensional domains into
John domains with uniform constants. We prove that for every simply connected
domain with -boundary there is a corresponding
partition with such
that each component is a John domain with a John constant only depending on
. The result implies that many inequalities in Sobolev spaces such as
Poincar\'e's or Korn's inequality hold on the partition of for uniform
constants, which are independent of
Computation of bounds for anchor problems in limit analysis and decomposition techniques
Numerical techniques for the computation of strict bounds in limit analyses
have been developed for more than thirty years. The efficiency of these techniques
have been substantially improved in the last ten years, and have been successfully
applied to academic problems, foundations and excavations. We here extend
the theoretical background to problems with anchors, interface conditions, and
joints. Those extensions are relevant for the analysis of retaining and anchored walls,
which we study in this work. The analysis of three-dimensional domains remains
as yet very scarce. From the computational standpoint, the memory requirements
and CPU time are exceedingly prohibitive when mesh adaptivity is employed. For
this reason, we also present here the application of decomposition techniques to
the optimisation problem of limit analysis. We discuss the performance of different
methodologies adopted in the literature for general optimisation problems, such as
primal and dual decomposition, and suggest some strategies that are suitable for the
parallelisation of large three-dimensional problems. The propo sed decomposition
techniques are tested against representative problems.Peer ReviewedPreprin
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