8,655 research outputs found
An Optimal-Dimensionality Sampling for Spin- Functions on the Sphere
For the representation of spin- band-limited functions on the sphere, we
propose a sampling scheme with optimal number of samples equal to the number of
degrees of freedom of the function in harmonic space. In comparison to the
existing sampling designs, which require samples for the
representation of spin- functions band-limited at , the proposed scheme
requires samples for the accurate computation of the spin-
spherical harmonic transform~(-SHT). For the proposed sampling scheme, we
also develop a method to compute the -SHT. We place the samples in our
design scheme such that the matrices involved in the computation of -SHT are
well-conditioned. We also present a multi-pass -SHT to improve the accuracy
of the transform. We also show the proposed sampling design exhibits superior
geometrical properties compared to existing equiangular and Gauss-Legendre
sampling schemes, and enables accurate computation of the -SHT corroborated
through numerical experiments.Comment: 5 pages, 2 figure
Locally optimal Delaunay-refinement and optimisation-based mesh generation
The field of mesh generation concerns the development of efficient algorithmic techniques to construct high-quality tessellations of complex geometrical objects. In this thesis, I investigate the problem of unstructured simplicial mesh generation for problems in two- and three-dimensional spaces, in which meshes consist of collections of triangular and tetrahedral elements. I focus on the development of efficient algorithms and computer programs to produce high-quality meshes for planar, surface and volumetric objects of arbitrary complexity. I develop and implement a number of new algorithms for mesh construction based on the Frontal-Delaunay paradigm - a hybridisation of conventional Delaunay-refinement and advancing-front techniques. I show that the proposed algorithms are a significant improvement on existing approaches, typically outperforming the Delaunay-refinement technique in terms of both element shape- and size-quality, while offering significantly improved theoretical robustness compared to advancing-front techniques. I verify experimentally that the proposed methods achieve the same element shape- and size-guarantees that are typically associated with conventional Delaunay-refinement techniques. In addition to mesh construction, methods for mesh improvement are also investigated. I develop and implement a family of techniques designed to improve the element shape quality of existing simplicial meshes, using a combination of optimisation-based vertex smoothing, local topological transformation and vertex insertion techniques. These operations are interleaved according to a new priority-based schedule, and I show that the resulting algorithms are competitive with existing state-of-the-art approaches in terms of mesh quality, while offering significant improvements in computational efficiency. Optimised C++ implementations for the proposed mesh generation and mesh optimisation algorithms are provided in the JIGSAW and JITTERBUG software libraries
Microstructure-based modeling of elastic functionally graded materials: One dimensional case
Functionally graded materials (FGMs) are two-phase composites with
continuously changing microstructure adapted to performance requirements.
Traditionally, the overall behavior of FGMs has been determined using local
averaging techniques or a given smooth variation of material properties.
Although these models are computationally efficient, their validity and
accuracy remain questionable, since a link with the underlying microstructure
(including its randomness) is not clear. In this paper, we propose a modeling
strategy for the linear elastic analysis of FGMs systematically based on a
realistic microstructural model. The overall response of FGMs is addressed in
the framework of stochastic Hashin-Shtrikman variational principles. To allow
for the analysis of finite bodies, recently introduced discretization schemes
based on the Finite Element Method and the Boundary Element Method are employed
to obtain statistics of local fields. Representative numerical examples are
presented to compare the performance and accuracy of both schemes. To gain
insight into similarities and differences between these methods and to minimize
technicalities, the analysis is performed in the one-dimensional setting.Comment: 33 pages, 14 figure
Automatic alignment for three-dimensional tomographic reconstruction
In tomographic reconstruction, the goal is to reconstruct an unknown object
from a collection of line integrals. Given a complete sampling of such line
integrals for various angles and directions, explicit inverse formulas exist to
reconstruct the object. Given noisy and incomplete measurements, the inverse
problem is typically solved through a regularized least-squares approach. A
challenge for both approaches is that in practice the exact directions and
offsets of the x-rays are only known approximately due to, e.g. calibration
errors. Such errors lead to artifacts in the reconstructed image. In the case
of sufficient sampling and geometrically simple misalignment, the measurements
can be corrected by exploiting so-called consistency conditions. In other
cases, such conditions may not apply and we have to solve an additional inverse
problem to retrieve the angles and shifts. In this paper we propose a general
algorithmic framework for retrieving these parameters in conjunction with an
algebraic reconstruction technique. The proposed approach is illustrated by
numerical examples for both simulated data and an electron tomography dataset
Equilibrium and kinetic properties of knotted ring polymers: a computational approach
We provide hereafter a summary of the Thesis organization. Chapter 1 contains a short
introduction to the mathematical theory of knots. Starting from the mathematical
definition of knotting, we introduce the fundamental concepts and knot properties used
throughout this Thesis.
In chapter 2 we tackle the problem of measuring the degree of localization of a knot.
This is in general a very challenging task, involving the assignment of a topological state
to open arcs of the ring. To assign a topological state to an open arc, one must first close it
into a ring whose topological state can be assessed using the tools introduced in chapter 1.
Consequently, the resulting topological state may depend on the specific closure scheme
that is followed. To reduce this ambiguity we introduce a novel closure scheme, the
minimally-interfering closure. We prove the robustness of the minimally-interfering
closure by comparing its results against several standard closure schemes.
We further show that the identified knotted portion depends also on the search
algorithm adopted to find it. The knot search algorithms adopted in literature can be
divided in two general categories: bottom-up searches and top-down searches. We show
that bottom-up and knot-down searches give in general different results for the length of
a knot, the difference increasing with increasing length of the polymer rings. We suggest
that this systematic difference can explain the discrepancies between previous numerical
results on the scaling behaviour of the knot length with increasing length of polymer
rings in good solvent.
In chapter 3 we investigate the mutual entanglement between multiple prime knots
tied on the same ring. Knots like these, which can be decomposed into simpler ones, are
called composite knots and dominate the knot spectrum of sufficiently long polymers [131].
Since prime knots are expected to localize to point-like decorations for asymptotically
large chain lengths, it is expected that composite knots should factorize into separate
prime components [101, 82, 43, 11]. Therefore the asymptotic properties of composite
knots should merely depend on the number of prime components (factor knots) by which
they are formed [101, 82, 43, 11] and the properties of the single prime components should be largely independent from the presence of other knots on the ring. We show that this
factorization into separate prime components is only partial for composite knots which
are dominant in an equilibrium population of Freely Jointed Rings. As a consequence
the properties of those prime knots which are found as separate along the chain depend
on the number of knots tied on it. We further show that these results can be explained
using a transparent one-dimensional model in which prime knots are substituted with
paraknots.
Chapters 4, 5 and 6 are dedicated to investigate the interplay between topological
entanglement and geometrical entanglement produced either by surrounding rings in a
dense solution or spherical confinement.
In chapter 4 we investigate the equilibrium and kinetic properties of solutions of
model ring polymers, modulating the interplay of inter- and intra-chain entanglement by
varying both solution density (from infinite dilution up to 40% volume occupancy) and
ring topology (by considering unknotted and trefoil-knotted chains). The equilibrium
metric properties of rings with either topology are found to be only weakly affected
by the increase of solution density. Even at the highest density, the average ring size,
shape anisotropy and length of the knotted region differ at most by 40% from those
of isolated rings. Conversely, kinetics are strongly affected by the degree of inter-chain
entanglement: for both unknots and trefoils the characteristic times of ring size relaxation,
reorientation and diffusion change by one order of magnitude across the considered range
of concentrations. Yet, significant topology-dependent differences in kinetics are observed
only for very dilute solutions (much below the ring overlap threshold). For knotted rings,
the slowest kinetic process is found to correspond to the diffusion of the knotted region
along the ring backbone.
In chapter 5 we study the interplay of geometrical and topological entanglement in
semiflexible knotted polymer rings under spherical confinement. We first characterize
how the top-down knot length lk depends on the ring contour length, Lc and the radius of
the confining sphere, Rc. In the no- and strong-confinement cases we observe weak knot
localization and complete knot delocalization, respectively. We show that the complex
interplay of lk, Lc and Rc that seamlessly bridges these two limits can be encompassed
by a simple scaling argument based on deflection theory. We then move to study the
behaviour of the bottom-up knot length lsk under the same conditions and observe
that it follows a qualitatively different behaviour from lk, decreasing upon increasing
confinement. The behaviour of lsk is rationalized using the same argument based on deflection theory. The qualitative difference between the two knot lengths highlights a
multiscale character of the entanglement emerging upon increasing confinement.
Finally, in chapter 6 we adopt a complementary approach, using topological analysis
(the properties of the knot spectrum) to infer the physical properties of packaged
bacteriophage genome. With their m long dsDNA genome packaged inside capsids
whose diameter are in the 50 80 nm range, bacteriophages bring the highest level of
compactification and arguably the simplest example of genome organization in living
organisms [31, 40]. Cryo-em studies showed that DNA in bacteriophages epsilon-15 and
phi-29 is neatly ordered in concentric shells close to the capsid wall, while an increasing
level of disorder was measured when moving away from the capsid internal surface. On
the other hand the detected spectrum of knots formed by DNA that is circularised
inside the P4 viral capsid showed that DNA tends to be knotted with high probability,
with a knot spectrum characterized by complex knots and biased towards torus knots
and against achiral ones. Existing coarse-grain DNA models, while being capable of
reproducing the salient physical aspects of free, unconstrained DNA, are not able to
reproduce the experimentally observed features of packaged viral DNA. We show, using
stochastic simulation techniques, that both the shell ordering and the knot spectrum
can be reproduced quantitatively if one accounts for the preference of contacting DNA
strands to juxtapose at a small twist angle, as in cholesteric liquid crystals
Complexity measurement and characterization of 360-degree content
The appropriate characterization of the test material, used for subjective evaluation tests and for benchmarking image and video processing algorithms and quality metrics, can be crucial in order to perform comparative studies that provide useful insights. This paper focuses on the characterisation of 360-degree images. We discuss why it is important to take into account the geometry of the signal and the interactive nature of 360-degree content navigation, for a perceptual characterization of these signals. Particularly, we show that the computation of classical indicators of spatial complexity, commonly used for 2D images, might lead to different conclusions depending on the geometrical domain use
Singular Higher Order Divergence-Conforming Bases of Additive Kind and Moments Method Applications to 3D Sharp-Wedge Structures
We present new subsectional, singular divergence conforming vector bases that incorporate the edge conditions for conducting wedges. The bases are of additive kind because obtained by incrementing the regular polynomial vector bases with other subsectional basis sets that model the singular behavior of the unknown vector field in the wedge neighborhood. Singular bases of this kind, complete to arbitrarily high order, are described in a unified and consistent manner for curved quadrilateral and triangular elements. The higher order basis functions are obtained as the product of lowest order functions and Silvester-Lagrange interpolatory polynomials with specially arranged arrays of interpolation points. The completeness properties are discussed and these bases are proved to be fully compatible with the standard, high-order regular vector bases used in adjacent elements. Our singular bases guarantee normal continuity along the edges of the elements allowing for the discontinuity of tangential components, adequate modelling of the divergence, and removal of spurious solutions. These singular high-order bases provide more accurate and efficient numerical solutions of surface integral problems. Several test-case problems are considered in the paper, thereby obtaining highly accurate numerical results for the current and charge density induced on 3D sharp-wedge structures. The results are compared with other solutions when available and confirm the faster convergence of these bases on wedge problem
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