Regge Finite Elements with Applications in Solid Mechanics and Relativity

University of Minnesota Ph.D. dissertation. May 2018. Major: Mathematics. Advisor: Douglas Arnold. 1 computer file (PDF); ix, 183 pages.This thesis proposes a new family of finite elements, called generalized Regge finite elements, for discretizing symmetric matrix-valued functions and symmetric 2-tensor fields. We demonstrate its effectiveness for applications in computational geometry, mathematical physics, and solid mechanics. Generalized Regge finite elements are inspired by Tullio Regge’s pioneering work on discretizing Einstein’s theory of general relativity. We analyze why current discretization schemes based on Regge’s original ideas fail and point out new directions which combine Regge’s geometric insight with the successful framework of finite element analysis. In particular, we derive well-posed linear model problems from general relativity and propose discretizations based on generalized Regge finite elements. While the first part of the thesis generalizes Regge’s initial proposal and enlarges its scope to many other applications outside relativity, the second part of this thesis represents the initial steps towards a stable structure-preserving discretization of the Einstein’s field equation

Numerical analysis of transport in dynamical systems

Transport processes play an important role in many natural phenomena. Prominent examples are the chaotic advection of fluid particles in geophysical flows or the transport of asteroids and comets in the solar system. Similar transport mechanisms are also at work in chemical physics explaining for example the transition between different conformations of molecules or the kinematics of chemical reactions. Therefore, in the numerical analysis of such dynamical systems one is interested in the identification of those regions in phase space that are involved in the transport process. In this context, invariant manifolds of hyperbolic objects play a crucial role as these structures are known to form natural barriers to transport ...thesi

Taking shape: The data science of elastic shape analysis with practical applications

This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London.A mathematical curve can represent many different objects, both physical and abstract, from the outline curve of an artefact in an image to the weight of growing animal to the set of frequencies used in a sound. Regardless of these variations, the curves can almost always vary non-linearly. One way to study shapes and their potential variations is elastic shape analysis, a rich theory of which has developed over the past twenty years. However, methods of elastic shape analysis are seldom utilized in practical applications on real-world data, especially outside of the mathematical shape analysis community. Our aim in this thesis is to explore some practical applications of elastic shape analysis. To do this, we work with various types of shape data, the majority of which are based on image datasets. As our focus is on two-dimensional curves, it is important to be able to robustly extract contours from images, before we can apply elastic shape analysis tools. In order to analyse the shapes in a dataset, we turn to methods of machine learning, to investigate the applications of elastic shape analysis in classification. In this thesis, we introduce an anthology of projects, in order to emphasise and under- stand the potential of elastic shape analysis in practical applications. There are four main projects in this thesis: (i) Classification of objects using outlines and the comparisons between methods of elastic shape analysis, geometric morphometrics, and human experts, with a focus on ancient Greek vases, (ii) Mussel species identification and a demonstra- tion that shape may not be enough in some applications, (iii) A novel tool to monitor the development of k ̄ak ̄ap ̄o chicks, and (iv) Classifying individual kiwi based on acoustic data from their calls. By combining tools from computer vision and machine learning with methods of elastic shape analysis, we introduce a practical framework for the application of elastic shape analysis, through a data science lens

Proceedings of the Second International Workshop on Mathematical Foundations of Computational Anatomy (MFCA'08) - Geometrical and Statistical Methods for Modelling Biological Shape Variability

International audienceThe goal of computational anatomy is to analyze and to statistically model the anatomy of organs in different subjects. Computational anatomic methods are generally based on the extraction of anatomical features or manifolds which are then statistically analyzed, often through a non-linear registration. There are nowadays a growing number of methods that can faithfully deal with the underlying biomechanical behavior of intra-subject deformations. However, it is more difficult to relate the anatomies of different subjects. In the absence of any justified physical model, diffeomorphisms provide a general mathematical framework that enforce topological consistency. Working with such infinite dimensional space raises some deep computational and mathematical problems, in particular for doing statistics. Likewise, modeling the variability of surfaces leads to rely on shape spaces that are much more complex than for curves. To cope with these, different methodological and computational frameworks have been proposed (e.g. smooth left-invariant metrics, focus on well-behaved subspaces of diffeomorphisms, modeling surfaces using courants, etc.) The goal of the Mathematical Foundations of Computational Anatomy (MFCA) workshop is to foster the interactions between the mathematical community around shapes and the MICCAI community around computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop aims at being a forum for the exchange of the theoretical ideas and a source of inspiration for new methodological developments in computational anatomy. A special emphasis is put on theoretical developments, applications and results being welcomed as illustrations. Following the very successful first edition of this workshop in 2006 (see http://www.inria.fr/sophia/asclepios/events/MFCA06/), the second edition was held in New-York on September 6, in conjunction with MICCAI 2008. Contributions were solicited in Riemannian and group theoretical methods, Geometric measurements of the anatomy, Advanced statistics on deformations and shapes, Metrics for computational anatomy, Statistics of surfaces. 34 submissions were received, among which 9 were accepted to MICCAI and had to be withdrawn from the workshop. Each of the remaining 25 paper was reviewed by three members of the program committee. To guaranty a high level program, 16 papers only were selected

Detecting and analyzing coherent structures in two-dimensional dynamical systems

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2013.Some pages landscape orientation. Cataloged from PDF version of thesis.Includes bibliographical references (pages 211-218).The identification of coherent structures enhances the understanding of transport by complex flows such as those found at the ocean surface. The rapidly developing approach of Lagrangian Coherent Structures (LCSs) is based on the identification of codimension-1 structures that are locally the strongest repelling material surfaces in forwards or backwards-time over a given time window. Current theory and methods regarding LCSs are surveyed, and we present a modified algorithm for their detection, highlighting the pros and cons of the modified approach. One beneficial aspect of the modified approach is that it is possible to classify and advect LCSs through the time window. We apply the improved detection scheme as well as the classification to a high quality, validated simulation of ocean surface flow near the Ningaloo reef along the coast of Western Australia. This region is home to the longest fringing reef in Australia, a diverse marine environment, and a growing offshore drilling industry, and understanding the surface flows will enable better informed decisions to be made in this environmentally delicate domain. In addition to applying the LCS techniques, for the first time we account for the impact of surface winds on the LCS field by creating a hybrid current-wind velocity field. While the LCS approach is based on rigorous dynamical systems theory, its reliance on an accurate velocity field restricts potential ocean applications to simulations or regions with surface velocity measurements via systems like HF radar stations. An untapped resource is the data collected from float trajectories. With the goal of eventual application to these data sets, we develop a coherent structure detection algorithm utilizing sparse trajectory data. This new approach is based on the application of tools from braid theory, which produce a simplified perspective of the mixing of two-dimensional systems that enables rapid analysis. As a first application, our braid-based approach is applied to a periodically stirred system.by Michael R. Allshouse.Ph. D

Surface-guided computing to analyze subcellular morphology and membrane-associated signals in 3D

Signal transduction and cell function are governed by the spatiotemporal organization of membrane-associated molecules. Despite significant advances in visualizing molecular distributions by 3D light microscopy, cell biologists still have limited quantitative understanding of the processes implicated in the regulation of molecular signals at the whole cell scale. In particular, complex and transient cell surface morphologies challenge the complete sampling of cell geometry, membrane-associated molecular concentration and activity and the computing of meaningful parameters such as the cofluctuation between morphology and signals. Here, we introduce u-Unwrap3D, a framework to remap arbitrarily complex 3D cell surfaces and membrane-associated signals into equivalent lower dimensional representations. The mappings are bidirectional, allowing the application of image processing operations in the data representation best suited for the task and to subsequently present the results in any of the other representations, including the original 3D cell surface. Leveraging this surface-guided computing paradigm, we track segmented surface motifs in 2D to quantify the recruitment of Septin polymers by blebbing events; we quantify actin enrichment in peripheral ruffles; and we measure the speed of ruffle movement along topographically complex cell surfaces. Thus, u-Unwrap3D provides access to spatiotemporal analyses of cell biological parameters on unconstrained 3D surface geometries and signals.Comment: 49 pages, 10 figure

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition