150 research outputs found
Planar splines on a triangulation with a single totally interior edge
We derive an explicit formula, valid for all integers , for the
dimension of the vector space of piecewise polynomial functions
continuously differentiable to order and whose constituents have degree at
most , where is a planar triangulation that has a single totally
interior edge. This extends previous results of Toh\v{a}neanu, Min\'{a}\v{c},
and Sorokina. Our result is a natural successor of Schumaker's 1979 dimension
formula for splines on a planar vertex star. Indeed, there has not been a
dimension formula in this level of generality (valid for all integers and any vertex coordinates) since Schumaker's result. We derive our results
using commutative algebra.Comment: 20 pages, 3 figure
Effective statistical physics of Anosov systems
We present evidence indicating that Anosov systems can be endowed with a
unique physically reasonable effective temperature. Results for the two
paradigmatic Anosov systems (i.e., the cat map and the geodesic flow on a
surface of constant negative curvature) are used to justify a proposal for
extending Ruelle's thermodynamical formalism into a comprehensive theory of
statistical physics for nonequilibrium steady states satisfying the
Gallavotti-Cohen chaotic hypothesis.Comment: 38 pages, 17 figures. Substantially more details in sections 4 and 6;
new and revised figures also added. Typos and minor errors (esp. in section
6) corrected along with minor notational changes. MATLAB code for
calculations in section 16 also included as inline comment in TeX source now.
The thrust of the paper is unaffecte
On the Stretch Factor of Polygonal Chains
Let be a polygonal chain. The stretch factor of
is the ratio between the total length of and the distance of its
endpoints, . For a parameter , we call a -chain if , for
every triple , . The stretch factor is a global
property: it measures how close is to a straight line, and it involves all
the vertices of ; being a -chain, on the other hand, is a
fingerprint-property: it only depends on subsets of vertices of the
chain.
We investigate how the -chain property influences the stretch factor in
the plane: (i) we show that for every , there is a noncrossing
-chain that has stretch factor , for
sufficiently large constant ; (ii) on the other hand, the
stretch factor of a -chain is , for every
constant , regardless of whether is crossing or noncrossing; and
(iii) we give a randomized algorithm that can determine, for a polygonal chain
in with vertices, the minimum for which is
a -chain in expected time and
space.Comment: 16 pages, 11 figure
New Techniques for the Modeling, Processing and Visualization of Surfaces and Volumes
With the advent of powerful 3D acquisition technology, there is a growing demand
for the modeling, processing, and visualization of surfaces and volumes. The
proposed methods must be efficient and robust, and they must be able to extract the essential structure of the data and to easily and quickly convey the most significant information to a human observer. Independent of the specific nature of the data, the following fundamental problems can be identified: shape reconstruction from discrete samples, data analysis, and data compression.
This thesis presents several novel solutions to these problems for surfaces
(Part I) and volumes (Part II). For surfaces, we adopt the well-known triangle
mesh representation and develop new algorithms for discrete curvature estimation,detection of feature lines, and line-art rendering (Chapter 3), for connectivity encoding (Chapter 4), and for topology preserving compression of 2D vector fields (Chapter 5). For volumes, that are often given as discrete samples, we base our approach for reconstruction and visualization on the use of new trivariate spline spaces on a certain tetrahedral partition. We study the properties of the new spline spaces (Chapter 7) and present efficient algorithms for reconstruction and visualization by iso-surface rendering for both, regularly (Chapter 8) and irregularly (Chapter 9) distributed data samples
Mathematical foundations of adaptive isogeometric analysis
This paper reviews the state of the art and discusses recent developments in
the field of adaptive isogeometric analysis, with special focus on the
mathematical theory. This includes an overview of available spline technologies
for the local resolution of possible singularities as well as the
state-of-the-art formulation of convergence and quasi-optimality of adaptive
algorithms for both the finite element method (FEM) and the boundary element
method (BEM) in the frame of isogeometric analysis (IGA)
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by Erd˝os
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version
A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics
We demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss–Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established
Higher order H^1 and H(¿^) FEM techniques with EM applications
Ph.DDOCTOR OF PHILOSOPH
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