8,655 research outputs found

    An Optimal-Dimensionality Sampling for Spin-ss Functions on the Sphere

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    For the representation of spin-ss 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 2L2{\sim}2L^2 samples for the representation of spin-ss functions band-limited at LL, the proposed scheme requires No=L2s2N_o=L^2-s^2 samples for the accurate computation of the spin-ss spherical harmonic transform~(ss-SHT). For the proposed sampling scheme, we also develop a method to compute the ss-SHT. We place the samples in our design scheme such that the matrices involved in the computation of ss-SHT are well-conditioned. We also present a multi-pass ss-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 ss-SHT corroborated through numerical experiments.Comment: 5 pages, 2 figure

    Locally optimal Delaunay-refinement and optimisation-based mesh generation

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    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

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    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

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    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

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    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

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    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

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    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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